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Dissertations in Forestry and Natural Sciences

ISBN 978-952-61-2441-4ISSN 1798-5668


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SUBHAJIT BEJ

LOCAL FIELD CONTROLLED LINEAR AND KERR NONLINEAR OPTICAL PROPERTIES OF

PERIODIC SUBWAVELENGTH STRUCTURES


In this book, local-field controlled linear and Kerr nonlinear optical properties of subwavelength periodic nanostructures

and nanocomposites are studied. Efficient numerical techniques and novel analytical models have been developed to aid in these

studies. In addition, prospect for achieving low energy optical bistability with a silicon nitride

guided mode resonance filter is examined numerically followed by an experimental

demonstration of all-optical modulation using such a structure.

SUBHAJIT BEJ

PUBLICATIONS OF THE UNIVERSITY OF EASTERN FINLANDDISSERTATIONS IN FORESTRY AND NATURAL SCIENCES

N:o 262

Subhajit Bej

LOCAL FIELD CONTROLLED LINEAR

AND KERR NONLINEAR OPTICAL

PROPERTIES OF PERIODIC

SUBWAVELENGTH STRUCTURES

ACADEMIC DISSERTATION

To be presented by the permission of the Faculty of Science and Forestry for publicexamination in the Auditorium F100 in Futura Building at the University of EasternFinland, Joensuu, on March 23rd, 2017, at 12 o’clock.

University of Eastern FinlandDepartment of Physics and Mathematics

Joensuu 2017

Grano OyJyväskylä, 2017

Editors: Pertti Pasanen, Pekka Toivanen,Jukka Tuomela, Matti Vornanen

Distribution:University of Eastern Finland Library / Sales of publications

[email protected]

http://www.uef.fi/kirjasto

ISBN: 978-952-61-2441-4 (print)ISSNL: 1798-5668ISSN: 1798-5668

ISBN: 978-952-61-2442-1 (pdf)ISSNL: 1798-5668ISSN: 1798-5676

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Author’s address: University of Eastern FinlandDepartment of Physics and MathematicsP.O. Box 111FI-80101 JOENSUUFINLANDemail: [email protected]

Supervisors: Professor Jari Turunen, D. Sc.University of Eastern FinlandDepartment of Physics and MathematicsP.O. Box 111FI-80101 JOENSUUFINLANDemail: [email protected]

Professor Yuri P. Svirko, Ph.D.University of Eastern FinlandDepartment of Physics and MathematicsP.O. Box 111FI-80101 JOENSUUFINLANDemail: [email protected]

Docent Jani Tervo, Ph.D.University of Eastern FinlandDepartment of Physics and MathematicsP.O. Box 111FI-80101 JOENSUUFINLANDemail: [email protected]

Reviewers: Associate Professor Hiroyuki Ichikawa, Ph.D.Ehime UniversityDepartment of Electrical and Electronic Engineering790-8577 MatsuyamaJAPANemail: [email protected]

Dr. Alexey Yulin, Ph.D.ITMO UniversityThe Metamaterials LaboratoryBirjevaja line V.O.,14 St. Petersburg199034 RUSSIAemail: [email protected]

Opponent: Dr.-Ing. Bernd KleemannStaff ScientistCorporate Research and TechnologyCarl Zeiss AGD-73446 OberkochenGERMANYemail: [email protected]

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Subhajit BejLocal field controlled linear and Kerr nonlinear optical properties of periodic sub-wavelength structuresJoensuu: University of Eastern Finland, 2017Publications of the University of Eastern FinlandDissertations in Forestry and Natural Sciences

ABSTRACT

Optical properties of structured media are controlled by the local electric and mag-netic fields to a great extent. Nanostructuring allows one to customize the macro-scopic linear and nonlinear optical properties of these artificial media by engineer-ing the local fields precisely. In this thesis, local electric field controlled linear andKerr nonlinear optical properties of subwavelength periodic nanostructures are in-vestigated theoretically, numerically, and experimentally. Both metal-dielectric andall-dielectric structures are examined.

In the subwavelength regime, when the feature sizes approach the wavelength oflight, approximate theoretical models may produce inaccurate results. A numericaltechnique based on the Fourier-Expansion Eigenmode Method has been developedto model optical Kerr effect in periodic structures. This numerical model serves asan efficient and accurate tool for the design and analysis of low power all-opticaldevices which rely on local field enhanced optical Kerr nonlinearity.

Light propagation in a form birefringent medium with optical Kerr medium isconsidered and an analytical model is developed which accurately describes non-linear light-matter interactions.

Possibility to achieve all-optical modulation and optical bistability with waveg-uide grating structures fabricated from silicon nitride thin films grown on top offused silica substrates are analyzed both theoretically and experimentally. Thesilicon nitride films are grown by plasma enhanced chemical vapour deposition(PECVD) and the waveguide grating structures are fabricated by electron beamlithography and reactive ion etching techniques. Experiments are carried out usinga single-walled carbon nanotube (SWCNT) modelocked ultrafast fiber laser with anamplifier and a tunable unit.

A full wave numerical approach based on the Fourier Modal Method is intro-duced to model nanocomposite optical materials. This approach can be used tomodel arbitrary particle geometries and their random arrangements, finite wave-length effects, multipolar effects, and percolation. Furthermore, the proposed methodcan be used to engineer large effective optical nonlinearities at the nanoscale, whichis pivotal for designing novel low power nonlinear photonic devices.

Universal Decimal Classification: 535.1, 535.3, 535.4, 535.41, 535.42, 535.421, 535.8,535.92, 535.13, 535.18, 535.181, 535.317.2, 537.226.2, 537.226.5, 537.8OCIS codes: 050.0050, 050.1960, 050.2065, 050.2555, 050.6624, 050.1755, 050.5745,190.0190, 190.1450, 190.3270, 190.4360, 190.7110, 160.1245, 160.3918, 160.4330, 230.1150Keywords: optics; micro-optics; nanophotonics; nonlinear optics; nonlinear metamaterials;resonant nanophotonics; computational physics; computational electromagnetism; rigorousgrating theory; diffraction gratings; Kerr effect; form birefringence; resonance waveguidegrating; guided mode resonance; optical filters; optical bistability, optical switching; all-

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optical device; all-optical modulation; microfabrication; nanofabrication; fiber laser; electronbeam lithography; reactive ion etching; nanostructured materials; localized surface plasmonresonance; nanocomposites

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ACKNOWLEDGEMENTS

This thesis is a summary of nearly four years of work as an early stage researcherin the Department of Physics and Mathematics at University of Eastern Finland,Joensuu. Four years is quite a long time span and it is phenomenal how fast theworld changed during this tenure- scientists successfully cloned human stem cells,first observation of gravitational waves were made, a historic agreement targetingclimate change was signed in Paris, Google’s artificial intelligence AlphaGo defeatedworld’s top GO Player and many more to mention. In the past two years, the De-partment of Physics and Mathematics at UEF also encountered major changes. Likeeveryone else in our unit, these had either direct or indirect impact on my researchwork and turned my PhD project into a thrilling roller-coaster journey. Neverthe-less, I enjoyed every part of this safari and it taught me how to stay positive andfocus on individual research during unfavourable situations. However, this journeywould be incomplete without constant encouragement and support of my academicsupervisors, my parents, my beloved wife, and close friends.

First of all, thanks to the ’Big Bang’ for creating the universe and the ’Evolution’for creating human beings.

I would like to express my deepest gratitude to Prof. Jari Turunen for his uniquestyle guidance and persistent support during these years. I am thankful to my aca-demic co-supervisor Prof. Yuri Svirko for his valuable instructions and suggestionsfrom time to time. I want to extend my thanks to Dr. Jani Tervo with whom I startedto work back in 2011. Without Jani’s active supervision and hands-on training dur-ing the early years of my doctoral studies, it would be impossible to complete thisthesis.

I am obliged to the past Head of Department Prof. Pasi Vahimaa, Prof. SeppoHonkanen, and the present Head of Department Prof. Timo Jääskeläinen for provid-ing me the opportunity to work in the encouraging atmosphere of our department.

My sincere thanks to Dr. Janne Laukkanen who trained me to work indepen-dently in our lithography laboratory. Dr. Toni Saastamoinen, Dr. Matthieu Roussey,Dr. Viatcheslav Vanyukov, Dr. Tomi Kaplas, Dr. Hemmo Tuovinen, Prof. TeroSetälä, and Prof. Markku Kuittinen deserve special thanks for being able to helpme and spend their valuable time whenever I needed. I also want to thank Dr.Pertti Pääkkönen, Tommi Itkonen, and Timo Vahimaa for their kind support. Forassistance at the administrative level, Dr. Noora Heikkilä, Ms. Katri Mustonen, Ms.Hannele Karppinen, and Ms. Marita Ratilainen deserve my sincerest gratitude.

I am indebted to Prof. Zhipei Sun for giving me an opportunity to visit and workin his optics laboratory at Micronova, Otaniemi, Espoo. I want to acknowledge allother collaborators especially Diao Li from Northwest University (China), and JorgeFrancés from University of Alicante (Spain) for their efforts. Special thanks to Dr.Srikanth Sugavanam from Aston University (UK) for fruitful discussions during theplanning stage of some experimental works.

During the time at UEF, I was fortunate to meet other talented young researchersas well. Thanks to my previous office mates Henri Partanen, and Markus Häyrinenfor the time we spent discussing academic and sometimes more refreshing non-academic stuffs.

I really appreciate the labor made by the pre-examiners Associate Professor Hi-royuki Ichikawa and Dr. Alexey Yulin after accepting the request to review mythesis. Their suggestions and constructive comments helped in increasing the read-ability of this thesis.

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Life is incomplete without friends. Thanks to my good friends Swarup, Avik,Ayan, Sourav, Shovan, Apurbo, Nilabha, Hasanur for the light moments we sharedand memories we created together. Special thanks to my friend, mentor, and drink-ing (coffee) buddy Dr. Md. Sahidullah.

My heartfelt thanks to my beloved parents Mr. Siddheswar Bej and Mrs. TistaBej for their kindness, encouragement, love and enormous support throughout mylife. Thanks to my lovely sister Shampa, my brother-in-law Sanjib, and my cutenephew Songlap for their unconditional love. Finally, I want to thank my soulmate,my beautiful wife Lahari for completely understanding a maniac like me, acceptingme as I am and keeping faith in me even when I had started to loose faith in myself!

To humanity and global peace.

Joensuu, February 26, 2017

Subhajit Bej

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TABLE OF CONTENTS

1 Introduction 11.1 Background ..................................................................................... 11.2 Motivation ....................................................................................... 41.3 Thesis outline .................................................................................. 4

2 Fundamentals of electromagnetic theory 72.1 Complex representation of Electromagnetic field quantities and Fourier

decomposition. ................................................................................. 72.2 Macroscopic Maxwell’s equations and their empirical basis ................ 82.3 Material constitutive equations ......................................................... 92.4 Electromagnetic boundary conditions ................................................ 112.5 Wave equation ................................................................................. 112.6 TE-TM decomposition ..................................................................... 132.7 Electromagnetic energy quantities .................................................... 142.8 Electromagnetic plane wave ............................................................. 142.9 Polarization of an EM plane wave..................................................... 152.10 Plane wave at planar boundary ......................................................... 162.11 General field and angular spectrum representation............................. 212.12 Theory of evanescent waves ............................................................. 222.13 Monochromatic plane wave in anisotropic medium ............................ 232.14 Electromagnetic theory of metals...................................................... 25

2.14.1 Drude model ......................................................................... 262.14.2 Interband transitions model ................................................... 27

2.15 Plasmons ......................................................................................... 282.15.1 Volume plasmons................................................................... 292.15.2 Surface plasmon polaritons .................................................... 292.15.3 Particle plasmons................................................................... 32

2.16 Field encountering stack of thin films ............................................... 342.17 Recursive S-matrix algorithm ............................................................ 362.18 Local field........................................................................................ 382.19 Summary ......................................................................................... 41

3 Rigorous analysis of diffraction gratings 433.1 Working principle of a grating .......................................................... 433.2 Pseuodoperiodicity and grating equations ......................................... 453.3 Diffraction efficiencies ...................................................................... 473.4 Overview of the existing numerical modeling methods ....................... 483.5 Fourier Modal method for diffraction gratings ................................... 48

3.5.1 Fourier factorization rules ...................................................... 493.6 FMM for linear gratings with plane wave illumination ....................... 50

3.6.1 Formulation of the eigenvalue problem ................................... 513.6.2 Solution of electromagnetic boundary conditions .................... 553.6.3 Solution for multilayered gratings ........................................... 56

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3.6.4 Field inside the gratings ......................................................... 583.7 FMM for anisotropic crossed gratings ............................................... 593.8 Staircase approximation ................................................................... 673.9 Summary ......................................................................................... 67

4 Light propagation in Periodic media with optical Kerr nonlinearity 694.1 Light propagation in isotropic third order nonlinear materials ............ 694.2 Theory of the Optical Kerr effect in isotropic media .......................... 704.3 Modeling light-induced anisotropy with the linear FMM .................... 734.4 Symmetries in light-induced anisotropy ............................................. 764.5 Numerical examples ......................................................................... 81

4.5.1 One dimensional metallic gratings with grooves filled with

χ(3) media............................................................................. 824.5.2 1-D binary grating with TiO2 as the Kerr nonlinear material ... 824.5.3 Crossed gratings with the pillars made with Si3N4 .................. 834.5.4 Si3N4 resonance waveguide-grating ........................................ 85

4.6 Summary ......................................................................................... 88

5 Theory of form birefringence in Kerr-type media 895.1 Propagation of light in crystals ......................................................... 895.2 Birefringence of a uniaxial crystal ..................................................... 915.3 Theory of form birefringence ............................................................ 935.4 Form birefringence in Kerr media: analytical formulation................... 965.5 Numerical examples ......................................................................... 1005.6 Summary ......................................................................................... 102

6 All-optical modulation and optical bistability with a Silicon Nitridewaveguide grating 1056.1 Theory of optical bistability - the Fabry-Perot resonator approach ..... 1066.2 Working principle of a waveguide grating .......................................... 1086.3 Silicon Nitride vs. crystalline Silicon as a nonlinear material .............. 1116.4 Fabrication of the waveguide grating structures ................................ 113

6.4.1 Thin film deposition .............................................................. 1136.4.2 Electron beam lithography ..................................................... 1146.4.3 Resist technology. .................................................................. 1166.4.4 Reactive ion etching .............................................................. 116

6.5 Numerical simulation results ............................................................. 1196.6 Experimental results ......................................................................... 1266.7 Summary ......................................................................................... 129

7 Modeling nanocomposite optical materials with FMM 1317.1 Nanocomposite optical materials ...................................................... 1317.2 Quasi-static approximation and its validity ........................................ 1327.3 Common composite geometries ........................................................ 133

7.3.1 Maxwell Garnett geometry . .................................................... 1337.3.2 Bruggeman geometry. ............................................................ 1347.3.3 Layered composite geometry . ................................................. 134

7.4 Optical properties of nanocomposites containing metal nanoparticles. 1347.5 Rigorous modeling- methodology . ..................................................... 1377.6 Numerical examples ......................................................................... 137

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7.6.1 Porous silicon nanostructures ................................................. 1387.6.2 Silver nanospheres on glass substrate ..................................... 1407.6.3 Silver nanorods embedded in a Kerr nonlinear host ................. 143

7.7 Summary ......................................................................................... 145

8 Summary, conclusions and scope of future work 1478.1 Summary with conclusions ............................................................... 1478.2 Scope of future work ........................................................................ 148

BIBLIOGRAPHY 151

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1 Introduction

This chapter includes a general background, the motivation behind this work, andan outline of this thesis.

1.1 BACKGROUND

Nanostructured materials with exciting optical properties can be readily found innature. An example is opal, which contains silica nano spheres arranged in a regularlattice, showing different colours when seen from different angles. Other examplesinclude the butterfly wings which are iridescent due to interference of light in tree-like nanostructures [1], and a special class of beetles with chiral nanostructuresforming its exoskeleton which make it appear either green or black depending onthe handedness of the circularly polarized illuminating light [2]. In the past, strongefforts have been made in mimicking these naturally available nanostructures toyield unprecedented optical properties. These man-made nanostructures are widelyknown as metamaterials [3–5] i.e. materials with properties beyond the conventionalmaterials.

Though the theory of making optical metamaterials have been understood onlyrecently, the first man-made metamaterials were constructed much before, withoutknowing the underlying mechanisms. A particularly interesting example is the Ly-curgus Cup which appears green when seen in reflected light and red in transmittedlight. It was understood later that the reason behind this phenomenon can be at-tributed to plasmon resonances associated with the gold and silver nanoparticlescontained in glass which forms the cup [6].

Understanding the connection between the change of bulk material propertiesand structuring became possible after the construction of a diffraction grating in1785 by D. Rittenhouse with 50 hairs positioned by the threads of two screws [7]and later in 1821 by Joseph Von Fraunhofer who used almost the same techniquefor constructing his wire diffraction grating [8]. Subsequent works include the the-oretical study of a periodic stack of dielectric layers by Rayleigh [9] who realizedthat a particular wavelength can be completely reflected using such a structure, andan experiment carried out by Jagadis C. Bose with mm-wave [10]. Bose found thateven a simple book can have linearly polarizing properties which can be enhancedby introducing metal foils in between the pages. While Bose’s experiment can beattributed to the first systematic study of the electromagnetic properties of a com-posite medium, Rayleigh’s work is thought to be the first step towards the develop-ment of a new class of materials widely known as photonic crystals, where the term’photonic crystals’ was first introduced in 1989 by Eli Yablonovitch [11]. Photoniccrystals form an important class of metamaterials having stop bands which resem-ble the band gap of semiconductor materials. Depending on their lattice structures(which results from artificial structuring) these artificial media can inhibit sponta-neous emission [12] or strongly confine photons [13]. Some of the very first ap-plications of these include development of energy efficient LEDs [14] and photoniccrystal fibers [15].

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The building block of a metamaterial is often termed as a meta-molecule whichis different from a material molecule in the sense that it displays unconventionaloptical properties. Many of these meta-molecules are actually nano-size opticalresonators or resonant nanoantennas which can couple localized electromagneticfields as well as freely propagating radiation [16, 17]. Metal nanoparticles [18] sus-taining surface plasmon-polariton and localized surface plasmon modes as well asdielectric structures with high refractive indices [19, 20] can be employed as suchresonators. The most well known example of an optical resonator is the split-ringresonator which was introduced by Hardy and Whitehead in 1981 [21]. In 1999, itwas realized by John Pendry et al. [22] that these split ring resonators can be usedto create artificial media with negative magnetic response. Within a year, DavidSmith et al. demonstrated a split ring and wire metamaterial with simultaneouslynegative permittivity and negative permeability [23]. These artificial media witheffective negative refractive index invoked the primary research interests in the fieldof metamaterials and it is believed that these media can be useful in realizing opticalcloaking [5, 24–26], and super-resolution imaging [27].

Planar metamaterials are more appealing due to ease of their fabrication. Ex-tensive research in the field of planar chiral metamaterials resulted in developmentof wave plates of essentially zero thicknesses [28, 29], tunable polarization rotators,and circular polarizers [30, 31]. Asymmetric structuring on subwavelength scalecan yield strong resonance which can trap energy on the surface of a planar meta-material and can cause a transparent metamaterial state which is identical to elec-tromagnetically induced transparency [32–36]. Furthermore, planar metamaterialswith optical gain media may enable lasing [37, 38]. Lastly, metasurfaces comprisingof arrays of nanoantennas with subwavelength separation between them and withspatially varying geometric properties can be employed to shape optical wavefrontsvery precisely within a distance much smaller than the wavelength of light. It isbelieved that these metasurfaces will complement conventional imaging lenses innear future [39].

Ever since the experimental demonstration of optical second harmonic gener-ation by Franken et al. in 1961 [40], the field of nonlinear optics is continuouslyblooming. Nonlinear optical processes are relatively weak in nature and strong ef-forts have been made to enhance these either by introducing new materials or byenhancing light-matter interactions with the aid of nanostructuring. Increased effec-tive nonlinear optical responses can be realized either through plasmonic resonancesin metallic nanostructures or geometrical resonances in all-dielectric subwavelengthstructures.

Coupling of light to surface plasmons can produce strong local electromagneticfields [16,41,42] which boost the nonlinear processes. To exemplify, surface plasmonmodes at structured metal surface enhance the inherently weak Raman scattering upto several orders of magnitude and may permit even single-molecule detection [43].Plasmonic structures sustaining localized surface plasmons (LSP) or surface plas-mon polaritons (SPP) can be used to achieve optically tunable optical propertieswhich rely on the mechanisms of optical Kerr nonlinearity. These SPP or LSP reso-nances enhance the effective refractive index change of either the metal particles orthe surrounding media by strong local field confinement and consequently lowerthe light intensity required for observing nonlinearity induced changes. Exam-ples include metal nanoparticles [44, 45], metal doped bulk media [46–48], nanorodand nanosphere assemblies covered with nonlinear media [49, 50] etc. Due to LSPenhanced local electric field, it has been possible also to realize photon tunnel-

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ing through a nanometer size pinhole in a metal film covered with a nonlinearpolymer [51, 52]. Furthermore, enhanced nonlinear effects in plasmonic waveg-uides, such as metal-insulator-metal, V-groove, nano slot, or dielectric loaded waveg-uides [53–57], have been employed to efficiently modulate and switch the SPP sig-nals by all-optical means. SPP or LSP based additional field enhancement effectsin these plasmonic waveguides help in achieving better figure of merit (the ratio ofmodulation performance to size) as compared to the photonic waveguides [58].

Alternatively, nonlinear effects in semiconductors or dielectric materials can beenhanced using waveguide resonances. Some of the most common examples in-clude silicon based slot waveguides where the slots are filled with nonlinear poly-mer materials [59], silicon and silicon nitride ring resonators [60–62], slow-lightphotonic crystal waveguides [63–66], and waveguide gratings [67–69]. Applica-tions include electro-optic and all-optical modulation, all-optical wavelength con-version, all-optical multiplexing and demultiplexing, and supercontinuum genera-tion [70, 71].

In recent years, the field of metamaterials has merged together with the fieldof nonlinear optics to form a new class of artificial materials known as Nonlin-ear metamaterials [72]. These new unconventional materials have huge potential innext-generation optical networks as one can tailor and tune their nonlinear opticalproperties. The essence of constructing a nonlinear metamaterial lies in achievingthe desired nonlinear response by constructing a meta-molecule with specific linearand nonlinear optical responses and arranging these meta-molecules in a specifiedmanner to yield their collective effects. Employing nonlinear metamaterials, it ispossible to enhance a particular nonlinear effect without changing its nature. Oneway to achieve this is to increase the local field confinement in the region containingthe nonlinear material. In 1999, Pendry et al. [73] first suggested that introduction ofa nonlinear material into the gaps of spit ring resonators (SRR) might be beneficial toenhance the nonlinear optical response through local electric field effects. Few yearslater, it was shown [74] that such a trick produces nonlinearity in magnetic responserather than the enhancement of electric nonlinearity. Alternatively, nonlinear pro-cess enhancement can be achieved through mutual interactions between specificallyordered nanostructures. For example, second-harmonic generation due to surfacenonlinearity from an array of metal nanoparticles can be enhanced via better phasematching conditions which can be accomplished by symmetry breaking with specialshaped structures and/or unit cells composed of these structures [75–80]. However,surface defects arising from fabrication errors caused the experimental demonstra-tions of these effects challenging for quite a long time.

Another important class of nonlinear metamaterials employ self-action effects,where the enhanced nonlinear properties also affect the linear properties of themetamaterials. This may result in all-optical tuning, all-optical switching, bista-bility, mutistability, or modulation instability. Several theoretical and experimentalworks on self-tunability of SRRs have been reported, where various nonlinear inser-tions into SRRs were introduced [81–84]. Intensity dependent nonlinear resistancesof these SRRs can tune a metamaterial slab between transmission, reflection, andabsorption states. Optical bistability in a resonant nanostructure occurs as a resultof the resonance peak shift with increasing power where at higher power levelsthe peak becomes asymmetric. Most of the works related to optical bistability inmetamaterials to date are either theoretical or numerical [85–88].

Besides, introducing new metamaterial geometries, there has always been questfor new materials which shows improved optical properties. One way to control the

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material properties is by means of molecular engineering i.e. to intermix two or morematerials at the molecular level and hence form nanocomposite optical materials. Inmany cases, these nanocomposites can display properties superior to those of theirconstituents. Device level applications of these composite media are increasing innumber day by day [89].

Strong theoretical models and numerical techniques are indispensable to de-scribe optical properties of micro and nanostructures in the subwavelength regime.In many cases, effective medium theories are applied to evaluate the macroscopicbulk properties of these structures. However, these theories can be accurate only inthe quasi-static regime i.e. when the feature sizes are much smaller than the wave-length of incoming light. Hence, if the smallest feature size approaches the wave-length, one must employ full-rigorous theories which solves James C. Maxwell’sequations for electrodynamics [90, 91] either in the space-time/space-frequency do-main or in the spatial frequency domain. In the nonlinear domain, full-wave nu-merical simulations are challenging as they require huge computing resources.

However, advanced numerical techniques need to be assisted by improved fab-rication methodology and better experimentation techniques.

1.2 MOTIVATION

Modern nanophotonics aims at manipulating and controlling the optical propertiesof bulk materials either by nanostructuring or by intermixing two or more homo-geneous media at the nanoscale. The primary goal is to develop ultra-compact andultra-fast optical devices for fully functional photonic circuits that can be integratedon chip with electronics [53, 92–94]. Since its development, the area of nanopho-tonics found a variety of applications in different spheres including optical datastorage [95, 96], super-resolution-imaging [97], bio and gas sensing [98–101], photo-voltaics [102] and high performance optical computing [103].

Many areas of nanophotonics employ subwavelength resonant nanostructures.Strong light-matter interactions can take place inside these structures due to en-hanced local field [104] achieved either through the excitation of trapped electro-magnetic modes or by localized resonances arising from the structure geometry [18].These strong interactions may result in unusual macroscopic linear optical prop-erties. Strongly confined local fields are crucial also for the intensity dependentnonlinear optical processes, where the nonlinear response scales to the localizedfield intensity, with a variety of applications ranging from frequency-conversion,wave mixing, Raman scattering to all-optical switching. Besides the field amplitude,structuring on the subwavelength scale allows also to control the phase and thepolarization properties of light and found useful in realization of compact polariz-ers [105], and wave plates [106]. However, to achieve precise control over these localfields in optical nanostructures, one needs excellent harmony between numericalmodeling, fabrication and experimentation techniques. This invoked the motivationfor writing this thesis.

1.3 THESIS OUTLINE

This doctoral thesis is divided altogether in eight chapters. After this general in-troduction, Chapter 2 contains the fundamentals of electromagnetism. This chaptercovers from Macroscopic Maxwell’s equations to the concept of local fields in linear

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and nonlinear optics. The subsequent chapters are based on the theories laid out inChapter 2.

In Chapter 3, we first include a general introduction to the periodically perturbedmedia, working principle of a diffraction grating, and an overview of the existingfull-wave numerical simulation techniques for diffraction gratings. However, in thesubsequent sections, we emphasize the Fourier-Expansion Eigenmode Method orsimply the Fourier Modal Method (FMM) for modeling diffraction gratings. Imple-mentations of the FMM for linear and multilayered two-dimensional (2D) gratingsand lastly for anisotropic crossed gratings (3D) are included.

In Chapter 4, we present an efficient and accurate numerical method to modeloptical Kerr effect (OKE) in periodic structures. This FMM based technique accu-rately estimates local field enhanced optical Kerr nonlinearity in a periodic structureand can be employed for design and analysis of low power all-optical devices.

In Chapter 5, we extend the classical theory of form birefringence to optical Kerrnonlinear media. We develop an analytical model which can describe nonlinearlight-matter interactions in such a medium and verify its accuracy by comparing theresults obtained by this model and the full rigorous FMM based technique devel-oped in Chapter 4.

In Chapter 6, we investigate theoretically, numerically, and experimentally thepossibility to achieve all-optical modulation and optical bistability with a waveguidegrating structure fabricated from silicon nitride thin film grown on top of fused sil-ica substrate. FMM based numerical simulation results showing optical bistability,fabrication methodology, and experimental results demonstrating all-optical modu-lation of transmitted signal are included.

In Chapter 7, we present a FMM based full wave numerical approach which canbe applied to accurately model percolation, arbitrary particle geometry and parti-cle clustering inside a nanocomposite optical medium. The examples presented inChapter 7 include porous silicon nanocomposites, and metal-dielectric nanocom-posites. Numerical experiments demonstrate the effect of local field on the linearand nonlinear optical properties of these nanocomposite optical media.

Finally, we make the conclusions in Chapter 8 along with an outlook for futureresearch directions.

The modeling methodologies introduced in Chapter 4 and Chapter 7, the theoret-ical model presented in Chapter 5 and some of the theoretical as well as numericalresults included in Chapter 4, Chapter 5, and Chapter 7 are either published orsubmitted for publication in peer-reviewed scientific journal articles:

• S. Bej, J. Tervo, Y. Svirko, and J. Turunen, “Modeling the optical Kerr effect inperiodic structures by the linear Fourier modal method," J. Opt. Soc. Am. B31, 2371–2378 (2014).

• S. Bej, J. Tervo, Y. Svirko, and J. Turunen, “Form birefringence in Kerr media:analytical formulation and rigorous theory," Opt. Lett. 40, 2913–2916 (2015) .

• S. Bej, T. Saastamoinen, Y. Svirko, and J. Turunen, “Optical properties ofnanocomposites from grating theory viewpoint," (Submitted) (2017).

Several journal articles related to the subjects covered in Chapter 6 and Chapter 7are under preparation. Some of the results included in this thesis are also publishedin the following conference proceedings:

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• S. Bej, J. Tervo, Y. Svirko, and J. Turunen, “Fourier modal method for crossedgratings with Kerr-type nonlinearity,” Proc. SPIE 9131, 161–169 (2014).

• S. Bej, J. Tervo, Y. Svirko, and J. Turunen, “All-optical control of form birefrin-gence,” in 2015 European Conference on Lasers and Electro-Optics - EuropeanQuantum Electronics Conference, (Optical Society of America, 2015), paperCD_P_16.

• S. Bej, J. Tervo, J. Francés, Y. P. Svirko, and J. Turunen, “Analysis of all-opticallytunable functionalities in subwavelength periodic structures by the Fouriermodal method,” Proc. SPIE 9889(06), 1–9 (2016).

• S. Bej, J. Laukkanen, J. Tervo, Y. P. Svirko, and J. Turunen, “Optical bistabilityin a Silicon Nitride waveguide grating,” Proc. SPIE 9894(0C), 1–10 (2016).

• J. Francés, S. Bleda, S. Bej, J. Tervo, V. Navarro-Fuster, S. Fenoll, F. J. Martínez-Guardiolaa, and C. Neipp, “Efficient split field FDTD analysis of third-ordernonlinear materials in 2D periodic media,” Proc. SPIE 9889(08), 1–8 (2016).

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2 Fundamentals of electromagnetic theory

The main subject of this thesis is the study of local field controlled linear and Kerrnonlinear optical properties of subwavelength nanostructures and nanocomposites.In the beginning, it is necessary to understand the basic principles of electromag-netism. This chapter deals with the basics of the electromagnetic theory for free-space optics. The more advanced topics will be covered in the following chapters.

2.1 COMPLEX REPRESENTATION OF ELECTROMAGNETIC FIELD

QUANTITIES AND FOURIER DECOMPOSITION

After James Clerk Maxwell realized that light is of electromagnetic origin, it wasessential to develop an accurate mathematical model that can describe the propaga-tion of light and also fits well with the available experimental results. In physics,all the field quantities that can be measured are real valued functions of space andtime. Nevertheless, in linear optics, to simplify the underlying mathematics we canemploy complex representation of the electromagnetic field quantities. Using thiscomplex notation, a monochromatic time harmonic stationary field can be describedas

Are(r, t) = ℜA(r) exp (−iωt) , (2.1)

where A(r) is the complex amplitude of the real valued function Are(r, t) and can beassociated with any measurable field quantity. r, t, and ω denote the position vector,time instant, and the angular frequency of the time harmonic field respectively.However, to describe the behavior of natural polychromatic light rigorously, we cannot use the above mentioned approach. In such a scenario, we can define a uniquecomplex counterpart of the real valued field Are(r, t) by introducing the concept oftemporal Fourier integral [107],

Are(r, t) =∫ ∞

−∞Are(r, ω) exp (−iωt)dω. (2.2)

where we assume that Are(r, t) is square integrable with respect to time i.e.∫ ∞

−∞|Are(r, t)|2 dt < ∞. (2.3)

Clearly,

Are(r, ω) =1

2π

∫ ∞

−∞Are(r, t) exp (iωt)dt, (2.4)

where Are(r, ω) represents a space-frequency domain spectral amplitude of the realvalued field Are(r, t). Equations (2.2) and (2.4) show that any space-time domainvector field can be expressed as a superposition of time harmonic fields with spectralcomplex-amplitudes Are(r, ω). Also, since Are(r, t) is real, its Fourier componentsAre(r, ω) satisfy the condition

Are(r,−ω) = A∗re(r, ω) (2.5)

7

where the ∗ symbol is used to represent complex conjugate. The above mentionedcondition unveils that the negative frequency components do not carry additionalinformation which is not already contained in the positive ones. Hence, withoutloss of generality we can define a new space-frequency domain function

A(r, ω) =

0, if ω < 02Are(r, ω) if ω ≥ 0,

(2.6)

where the Fourier conjugate i.e. the complex-valued space-time domain function isdefined as

A(r, t) =∫ ∞

−∞A(r, ω) exp (−iωt)dω. (2.7)

Clearly, from Eq. (2.7), the positive part of the spectrum differs from that of its realcounterpart only by a multiplicative constant and the Fourier spectrum of the newlydefined function is written as

A(r, ω) =1

2π

∫ ∞

−∞A(r, t) exp (iωt)dt, (2.8)

and can be attributed to any measurable physical quantity. Complex representationfor scalar fields can be introduced in a similar fashion.

2.2 MACROSCOPIC MAXWELL’S EQUATIONS AND THEIR EMPIRI-

CAL BASIS

J. C. Maxwell’s theory of electricity and magnetism [90, 91] gives the fundamentallaws of electromagnetism. With the aid of these laws, which are based upon fourequations, electromagnetic field quantities can be treated classically. In principle,electromagnetic fields in any media (which do not contain any abrupt boundary)and in any form can be solved with these equations which are widely known asMaxwell’s equations. In space-time domain, Maxwell’s equations can be representedas a set of four partial differential equations written as

∇× H(r, t) = J(r, t) +∂

∂tD(r, t), (2.9)

∇× E(r, t) = − ∂

∂tB(r, t) , (2.10)

∇ · D(r, t) = ρ(r, t) , (2.11)

∇ · B(r, t) = 0 , (2.12)

where E(r, t), H(r, t), D(r, t), B(r, t), J(r, t), and ρ(r, t) are the electric field, themagnetic field, the electric displacement, the magnetic induction, the electric currentdensity and the electric charge density, respectively. These four equations connectall the measurable electromagnetic field quantities. Each of these equations repre-sents generalization of certain experimental results. Equation (2.9) is an extensionof Ampere’s law, Eq. (2.10) is Faraday’s law of electromagnetic induction in differ-ential form, Eq. (2.11) is Gauss’s law which can be derived from Coulomb’s law,and Eq. (2.12) signifies the fact that magnetic monopoles do not exist. Clearly, asMaxwell’s equations are mathematical expressions of certain experimental results,these can not be proved; however the applicability of these in any condition can bevalidated.

8

Now without any loss of generality, if we make an assumption that all themeasurable electromagnetic field quantities are time-harmonic i.e. of the form ofEq. (2.1), we can derive a new set of Maxwell’s equations in the space-frequencydomain

∇× H(r, ω) = J(r, ω)− iωD(r, ω) , (2.13)

∇× E(r, ω) = iωB(r, ω) , (2.14)

∇ · D(r, ω) = ρ(r, ω) , (2.15)

∇ · B(r, ω) = 0 . (2.16)

Equations (2.13)-(2.16) are useful because often in optics it is more convenient tooperate in space-frequency domain than in space-time domains.

2.3 MATERIAL CONSTITUTIVE EQUATIONS

Both the space-time and the space-frequency domain Maxwell’s equations containmore than four unknown field quantities. Hence, to solve these, we need to intro-duce additional equations which connect these quantities. By introducing two newspace-time domain field quantities known as the electric polarization P(r, t) and themagnetization M(r, t), we may write these additional equations in the form

D(r, t) = ε0E(r, t) + P(r, t), (2.17)

H(r, t) =1

µ0B(r, t)− M(r, t), (2.18)

where ε0 is the electric permittivity of vacuum and µ0 is the magnetic permeabil-ity of vacuum. Equations (2.17) and (2.18) connect the space-time domain electricdisplacement to the electric field and the space-time domain magnetic field to mag-netic induction, respectively. Both the electric polarization and the magnetizationare nonlinear functions of the electric and the magnetic fields. However, at opticalfrequencies, magnetization is typically very small even if the field strength is large.Hence it can often be neglected. However, if intense laser illumination is used,electric polarization must generally be written in the form of a series expansion

P(r, t) = χ(1)(r)E(r, t) + χ(2)(r)E2(r, t) + χ(3)(r)E3(r, t) + . . . (2.19)

where the position-dependent constants χ(j) are susceptibilities of different orders.In the domain of linear optics, it is sufficient to retain only the first term in theright-hand-side of Eq. (2.19). The second and third terms give rise to nonlineareffects of second and third order, respectively. Usually at moderate field intensities,these higher order terms are negligible and the space-time domain relation betweenthe electric polarization and the electric field in this context of linear optics can bewritten as

P(r, t) =ε0

2π

∫ ∞

0χ(r, t′)E(r, t − t′) dt′, (2.20)

where χ(r, t′) is the position dependent real-valued dielectric susceptibility tensor.In an isotropic medium, the susceptibility assumes a scalar value (which still de-pends on r) and can be written as

χ(r, t′) = χ(r, t′)I, (2.21)

9

where I is the identity matrix. The space-time domain electric current density, andthe electric displacement are also connected to the space-time domain electric fieldby equations analogous to Eq. (2.20)

J(r, t) =1

2π

∫ ∞

0σ(r, t′)E(r, t − t′) dt′, (2.22)

D(r, t) =ε0

2π

∫ ∞

0ε(r, t′)E(r, t − t′) dt′, (2.23)

and as we saw before, for isotropic media σ(r, t′) and ε(r, t′) reduce to scalar con-ductivity σ(r, t′) and scalar permittivity ε(r, t′) respectively. Together with the space-time domain Maxwell’s equations (2.9)–(2.12), equations (2.20), (2.22), and (2.23)provide the desired relations between the electromagnetic field quantities in thespace-time domain. We can derive similar expressions in the space-frequency do-main for non-magnetic media by taking the Fourier transforms of equations (2.20),(2.22), (2.23), (2.18) and using the convolution theorem [108]

P(r, ω) = ε0χ(r, ω)E(r, ω), (2.24)

J(r, ω) = σ(r, ω)E(r, ω), (2.25)

D(r, ω) = ε0ε(r, ω)E(r, ω), (2.26)

B(r, ω) = µ0H(r, ω), (2.27)

where ε(r, ω) is defined as the relative permittivity tensor. Clearly, the space-frequency domain equations are easier to handle as these avoid the complicacy ofperforming integrations.

Equations (2.24)–(2.27) are referred as the space-frequency domain material con-stitutive equations. In space-time domain the electric current density and the electricpolarization are related by

J(r, t) =∂

∂tP(r, t). (2.28)

Using Eqs. (2.17), (2.25), (2.26), (2.28) and applying the Fourier transform, we canderive the relationship between the relative complex permittivity (written as ε(r, ω))and the electric conductivity in the form

ε(r, ω) = ε(r, ω) +i

ε0ωσ(r, ω), (2.29)

which includes the effects of both the free electrons and the bound electrons. For anisotropic linear medium the complex refractive index can be defined as

n(ω) = n(ω) + iκ(ω) =√

ε(r, ω), (2.30)

where both n(ω) and κ(ω) are real functions, n(ω) determines the phase velocityof the propagating electromagnetic wave, and κ(ω) determines the strength of itsdamping or attenuation.

Using the complex relative permittivity as defined in Eq. (2.29) and making useof the space-frequency domain constitutive relations Eqs. (2.24)–(2.27), Eq. (2.13) canbe written in the form

∇× H(r, ω) = −iωε0ε(r, ω)E(r, ω). (2.31)

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Furthermore taking divergence of Eq. (2.31) from the left and employing the vectoridentity ∇ · ∇× A = 0, we get

∇ · [ε(r, ω)E(r, ω)] = 0 . (2.32)

Now we may rewrite the space-frequency domain Maxwell’s equations in the fol-lowing form,

∇× H(r, ω) = −iωε0ε(r, ω)E(r, ω) , (2.33)

∇× E(r, ω) = iωB(r, ω) , (2.34)

∇ · [ε(r, ω)E(r, ω)] = 0 , (2.35)

∇ · B(r, ω) = 0 . (2.36)

Much of our discussions below will be based on the set of equations (2.33)-(2.36).

2.4 ELECTROMAGNETIC BOUNDARY CONDITIONS

The validity of Maxwell’s equations in differential form at position r is based on theassumption that the medium in the immediate vicinity of r is continuous. Clearly,the above mentioned criteria for continuity is no longer valid at the boundary be-tween two media. However, Maxwell’s equations in integral form still remain valid.As throughout this thesis we will deal with Maxwell’s equations only in differen-tial form, we need to find additional conditions that describe the field properties atan abrupt boundary between two media. These conditions are known as electro-magnetic boundary conditions [109–113]. Electromagnetic boundary conditions areessential to describe many interesting optical phenomena that occur at boundariesbetween two media such as reflection, refraction, and scattering.

Considering a sharp boundary between two media denoted by 1 and 2, andintroducing u12 as the unit surface normal vector at position r which points frommedium 1 to medium 2, we can write the space-frequency domain electromagneticboundary conditions in the following form

u12(r) · [ε2(r, ω)E2(r, ω)− ε1(r, ω)E1(r, ω)] = 0 , (2.37)

u12(r) · [B2(r, ω)− B1(r, ω)] = 0 , (2.38)

u12(r)× [E2(r, ω)− E1(r, ω)] = 0 , (2.39)

u12(r)× [B2(r, ω)− B1(r, ω)] =1

µ0JS(r, ω) , (2.40)

where JS(r, ω) is the surface current density at position r. It is worth mentioningthat at optical frequencies the surface current density is usually zero. However, incase of an infinitely conducting material it must be set to non-zero value. Equa-tions (2.37)–(2.40) signifies that across the discontinuities between two dielectricsor finitely conducting materials, all the magnetic field components, the tangentialelectric field components, and the normal components of the electric displacementremain continuous.

2.5 WAVE EQUATION

One of the most important consequences of Maxwell’s equations is the equationfor the electromagnetic wave propagation. For a homogeneous, isotropic, source-free linear optical medium, the complex relative permittivity tensor ε(r, ω) reduces

11

to scalar complex permittivity ε(r, ω). Hence, we can apply the curl operator toEq. (2.34) and use Eqs. (2.33) and (2.27) to obtain

∇× [∇× E(r, ω)] = ω2ε0 ε(ω)E(r, ω). (2.41)

Now we use the vector identity ∇× (∇× A) ≡ ∇(∇ · A)−∇2 A and use Eq. (2.35)

to get the wave equation for the electric field (also known as the Helmholtz waveequation for the electric field) in the space-frequency domain

∇2E(r, ω) + k2

0 ε(ω)E(r, ω) = 0, (2.42)

where, λ0 is the vacuum wavelength, c is the speed of light in vacuum, and k0 =2π/λ0 = ω0/c is the free-space wave number. Similarly, proceeding with Eq. (2.33)we can derive the Helmholtz wave equation for the magnetic field

∇2H(r, ω) + k2

0 ε(ω)H(r, ω) = 0. (2.43)

In the above derivations we have assumed the medium to be isotropic, linear andsource-free.

Let’s now try to derive the wave equation for nonlinear media. We begin withthe space-time domain Maxwell’s equations (2.9)–(2.12). As we are interested in thesolution in regions of space which contain no free charges and free currents, wemay put ρ = 0 and J = 0 in the space-time domain Maxwell’s equations. Also, weassume non-magnetic media hence M in Eq. (2.18) becomes zero and we can rewriteEq. (2.18) in the following form

B = µ0 H. (2.44)

We now proceed to derive the wave-equation as in the linear case. Taking curlof Eq. (2.10), interchanging the order of the space-domain and the time-domainderivatives and using Eqs. (2.9), (2.44), and (2.17) we get the most general form ofthe wave-equation in nonlinear optics

∇×∇× E +1c2

∂2

∂t2 E = − 1ε0c2

∂2P

∂t2 , (2.45)

where on the right hand side of Eq. (2.45) we have replaced µ0 by 1/ε0c2. Under cer-tain conditions the above written generalized form of the nonlinear wave equationcan be simplified by using the vector identity ∇× (∇× E) ≡ ∇(∇ · E)−∇

2E andputting ∇ · E = 0 as in the case of isotropic linear media. For a transverse, infiniteplane wave and for pulsed light under slowly varying amplitude approximation∇ · E vanishes and we can write the nonlinear wave equation in the following form

∇2E − 1

c2∂2

∂t2 E =1

ε0c2∂2P

∂t2 . (2.46)

Now, we can split P and D into linear and nonlinear counterparts such that

P = PL + PNL, (2.47)

D = DL + DNL, (2.48)

where DL(r, t) = ε0E(r, t) +PL(r, t) = ε0E(r, t) +χ(1)(r)E(r, t). Hence the nonlinearwave equation (2.46) can be rewritten as

∇2E − 1

ε0c2∂2

∂t2 DL =1

ε0c2∂2PNL

∂t2 . (2.49)

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For an isotropic, lossless, dispersionless medium εL reduces to a scalar quantity εL

and Eq. (2.49) reduces to

∇2E − εL

ε0c2∂2E

∂t2 =1

ε0c2∂2PNL

∂t2 . (2.50)

Clearly, Eq. (2.50) has the form of an inhomogeneous driven wave equation wherethe nonlinearity of the medium acts as a source term. In the absence of this sourceterm Eq. (2.50) reduces to the linear wave-equation (for electric field) for isotropicmedium.

2.6 TE-TM DECOMPOSITION

Let us assume that the field quantities in Maxwell’s equations (2.33)-(2.36) are y-invariant. Also, we assume that the permittivity distribution is y-invariant. Henceall the partial derivatives with respect to y vanish in Maxwell’s equations. Now, ifwe further assume that the incident field is propagating in the xz-plane, we can splitthe Maxwell’s equations (2.33)–(2.36) into two sets of equations

Hx(x, z) =i

k0

√

ε0

µ0

∂

∂zEy(x, z), (2.51)

Hz(x, z) = − ik0

√

ε0

µ0

∂

∂xEy(x, z), (2.52)

∂

∂zHx(x, z)− ∂

∂xHz(x, z) = −ik0 ε(x, z)

√

ε0

µ0Ey(x, z), (2.53)

and

Ex(x, z) = − ik0 ε(x, z)

√

ε0

µ0

∂

∂zHy(x, z), (2.54)

Ez(x, z) =i

k0ε(x, z)

√

ε0

µ0

∂

∂xHy(x, z), (2.55)

∂

∂zEy(x, z)− ∂

∂xEz(x, z) = ik0

√

ε0

µ0Hy(x, z). (2.56)

Clearly, the first set includes only the y-component of the electric field (the compo-nent of the electric field normal to the plane of incidence) and the x, z componentsof the magnetic field. Hence, this set corresponds to transverse electric or TE po-larization. Analogously, the second set is termed as transverse magnetic or TMpolarization. Substituting Eqs. (2.51) and (2.52) into Eq. (2.53) we can obtain a singlepartial differential equation for the TE polarized set

∂2

∂x2 Ey(x, z) +∂2

∂z2 Ey(x, z) + k20 ε(x, z)Ey(x, z) = 0. (2.57)

Similarly proceeding with the TM polarized set we can derive the single partialdifferential equation for the TM polarized set

∂

∂x

[

1ε(x, z)

∂

∂xHy(x, z)

]

+∂

∂z

[

1ε(x, z)

∂

∂zHy(x, z)

]

+k20 ε(x, z)Hy(x, z) = 0. (2.58)

We shall discuss in detail the polarization properties of an electromagnetic wave insection 2.9.

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2.7 ELECTROMAGNETIC ENERGY QUANTITIES

Though we introduced complex notations for the electromagnetic field quantities,measurable field quantities are always real valued. We connect the complex quanti-ties with the real measurable field quantities by defining the concept of the energyof the electromagnetic field. The electric energy density we(r, t), and the magneticenergy density wm(r, t) are defined by the following relations [114]

we(r, t) =12

Ere(r, t) · Dre(r, t), (2.59)

wm(r, t) =12

Hre(r, t) · Bre(r, t). (2.60)

The total instantaneous energy density of the electromagnetic field is defined as thesum of these two quantities. As the frequencies in the optical part of the spectrum isvery high ( 1015s−1) and the response times of the available photo-detectors are sev-eral orders of magnitude larger than the fluctuations of the field quantities, we canonly measure the time-averaged signal [115], where we define the time-averaging ofa function f (t) by

〈 f (t)〉 = limT→∞

12T

∫ T

−Tf (t)dt. (2.61)

The time-averaged electromagnetic energy densities which can be measured by theavailable photo-detectors are given by

〈we(r, t)〉 = 14

ε0ε(r)|E(r)|2, (2.62)

〈wm(r, t)〉 = 14

µ0|H(r)|2. (2.63)

For plane wave illumination we define the direction of the energy flow by the direc-tion of the Poynting vector S(r, t) = Ere(r, t)× Hre(r, t). As usual, we can define thetime-averaged Poynting vector by

〈S(r, t)〉 = 12ℜ [E(r)× H∗(r)] (2.64)

whose magnitude quantifies the intensity of the field. The superscript ∗ denotesthe complex conjugate. The time-averaged Poynting vector can be used to measurethe field intensity also for the non-plane wave illumination. Nevertheless, we musttake extra care in defining the direction of the energy flow in such scenario as thereexist cases where the z-component of the time-averaged Poynting vector might takenegative values in some small but finite regions [116].

2.8 ELECTROMAGNETIC PLANE WAVE

The simplest solution of Maxwell’s equations is the plane wave solution i.e. a fieldwith planar surface of constant phase. In the space-frequency domain, the planewave solutions for the electric field and the magnetic field take the following forms

E(r, ω) = E0(ω) exp (ik · r), (2.65)

H(r, ω) = H0(ω) exp (ik · r), (2.66)

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where E0(ω) and H0(ω) are the vectorial complex electric and magnetic field am-plitudes respectively, and k is the wave vector which defines the normal direction tothe plane of constant phase and also the propagation direction of the plane wave. Incartesian coordinate system k is defined as k = kx x + kyy + kzz with |k| = k = k0n.

2.9 POLARIZATION OF AN EM PLANE WAVE

Like other vector fields, the electromagnetic wave has certain directional properties.The directional information of an electromagnetic wave is characterized by its po-larization [117]. For the most simple case of a plane wave propagating along z-direction, we can solve Eq. (2.42) i.e. the Helmholtz equation for the electric field inthe space-frequency domain and express the electric field as,

E(r, ω) = E0s s exp [iφs(ω)] + E0p p exp[

iφp(ω)]

exp(ikz), (2.67)

where s and p are unit vectors along two mutually orthogonal directions, E0s (al-ternatively E0y) and E0p (alternatively E0x) are the amplitudes of the electric fieldcomponents along s (or y) and p (or x) respectively. φs (alternatively φy) and φp (al-ternatively φx) are the phases of the electric field components vibrating along s- andp- directions respectively. The term polarization in this context is used to describethe relations between the amplitudes and the phases of the electric field compo-nents in two mutually orthogonal directions. If, δ = φp − φs, which is defined asthe relative phase, is a multiple of π or one of the field components is zero, we havelinearly polarized light. In the most general case for non zero field componentsand arbitrary values of δ, we have elliptically polarized light. In a special situation,when E0s = E0p, and δ = ±π/2 ± 2mπ, where m is an integer, we have circularlypolarized light. As the polarized light propagates in an optical medium, the tip ofthe electric field vector traces out a specific characteristic form. In the most generalcase, this trace is an ellipse as shown in Fig. 2.1.

A convenient way of representing polarized light was invented by an Americanphysicist R. Clark Jones in 1941 [118]. The technique prescribed by Jones is conciseand can easily be applied to coherent beams. If Ex and Ey are scalar components ofthe electric field, Jones’ matrix can be written as,

E(t) =

[

Ex(t)Ey(t)

]

. (2.68)

Preserving the phase information of the electric field components we can writeEq. (2.68) in the following form

E(t) =

[

E0x(t) exp (iφx)E0y(t) exp

(

iφy)

]

. (2.69)

Now, we can express the horizontal (TM) and the vertical (TE) polarized light usingJones’ formalism as

EH =

[

E0x(t) exp (iφx)0

]

, (2.70)

EV =

[

0E0y(t) exp

(

iφy)

]

. (2.71)

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Letting E0x = E0y and φx = φy we get,

E = E0x exp (iφx)

[

11

]

. (2.72)

This represents +45 polarized light. However, it is customary to normalize theJones’ vector demanding |Ex|2 + |Ey|2 = 1. After normalization we get the followingform of Jones’ vector for +45 polarized light

E45 =1√2

[

11

]

. (2.73)

Similarly, the normalized Jones vectors for H-polarized, V-polarized, left-circularlypolarized (LCP) and right-circularly polarized (RCP) light can be written as

EH =

[

10

]

, EV =

[

01

]

, ELCP =1√2

[

1i

]

, ERCP =1√2

[

1−i

]

. (2.74)

Here we must mention that describing the polarization properties for a generalelectromagnetic field is more demanding task and is out of scope for this thesis.

2.10 PLANE WAVE AT PLANAR BOUNDARY

Let’s now consider the simplest optical boundary-value problem of a plane waveincident obliquely on a planar interface as shown in Fig. 2.2. The boundary or theinterface is defined by the plane z = 0. The material to the left i.e. in the half spacez < 0 has real valued permittivity ε1 whereas the permittivity of the medium to the

s (y)

p (x)

Es

Ep

δ = 0δ = π/2

δ = π

y′

x′

Figure 2.1: Polarization ellipse i.e. the trace of the tip of the electric field vectorE(r) as a function of time. The propagation direction of the electromagnetic planewave is along the normal to the plane of this paper and points towards the reader.The traces for δ = 0 and δ = π are linearly polarized.

16

right of the boundary (z > 0) ε2 might be complex. We assume that the incidentwave is partly reflected and partly transmitted at the boundary. θin is the angle ofincidence, where the subscript ’in’ stands for incidence. Similarly, the quantitieswith subscripts ’ref’ and ’tra’ denote the reflected and the transmitted field quan-tities respectively. Furthermore, we assume that the reflected and the transmittedfields are also plane waves. Now, omitting the explicit frequency dependence of thefield quantities at the boundary for the sake of brevity we may write the plane wavecomponents at the boundary in the following form

Ein(r) = A exp [i(kin,xx + kin,zz)] , (2.75)

Bin(r) = B0,in exp [i(kin,xx + kin,zz)] , (2.76)

Eref(r) = R exp [i(kref · r)] , (2.77)

Bref(r) = B0,ref exp [i(kref · r)] , (2.78)

Etra(r) = T exp [i(ktra · r)] , (2.79)

Btra(r) = B0,tra exp [i(ktra · r)] . (2.80)

Hence, the total electric field just before the boundary and just after the boundarycan be written as

E(x, y, 0−) = A exp [i(kin,xx)] + R exp [i(kx,refx)] , (2.81)

E(x, y, 0+) = T exp [i(kx,trax)] . (2.82)

kref

θref

θin

kin

ε1 = ℜ [ε1] ε2

ktra

θtra

x

z

Figure 2.2: Geometry of a plane wave incident obliquely on a planar interfacebetween two media.

17

Putting equations (2.81) and (2.82) into the electromagnetic boundary conditions(2.37) and (2.39) and using the uniqueness of the Fourier transform we get

kx,in = kx,ref = kx,tra = kx, (2.83)

ky,ref = ky,tra = 0, (2.84)

Ax + Rx = Tx, (2.85)

Ay + Ry = Ty, (2.86)

ε1 Az + ε1Rz = ε2Tz. (2.87)

The relations between the components of the magnetic induction and the electricfield components are obtained by inserting Eqs. (2.75)–(2.79) into Eq. (2.14) and canbe written as

B0,x,p = − 1ω

kz,pWy, B0,y,p =1ω(kz,pWx − kx,pWz), B0,z,p =

1ω

kx,pWy, (2.88)

where W can be A, R or T, and p may denote the incident (in), reflected (ref) or thetransmitted (tra) field. Now we can apply Eq. (2.88) into the two other electromag-netic boundary conditions i.e. Eqs. (2.38) and (2.40) to get

−ω JS,0,x + kz,in Ay + kz,refRy = kz,traTy, (2.89)

ω JS,0,y + kz,inAx − kx,inAz + kz,refRx − kx,refRz = kz,traTx − kx,traTz, (2.90)

kx,inAy + kx,refRy = kx,traTy, (2.91)

where JS,0,x and JS,0,y are the components of the complex amplitude vector of thesurface current density. Eq. (2.83) and the dispersion relations for the wave vectorsgive

k2x + k2

z,in = k20 ε1, (2.92)

k2x + k2

z,ref = k20 ε1, (2.93)

k2x + k2

z,tra = k20 ε2, (2.94)

Consequently kz,ref = ±kz,in. By examining the propagation directions in Fig. 2.2,we may conclude that kz,ref < 0, kz,in > 0, and kz,tra > 0. Hence,

kz,ref = −kz,in, (2.95)

k2z,tra − k2

z,in = k20(ε2 − ε1). (2.96)

Also, by examining the definitions of angles in Fig. 2.2 and using Eqs. (2.83), (2.95),and (2.96), we find that

θref = −θin, (2.97)

n1 sin θin = n2 sin θtra. (2.98)

Equations (2.97) and (2.98) are the well known law of reflection and the Snell’s law,respectively which connect the propagation angles of the reflected θref (or −θ1) andtransmitted waves θtra (or θ2) to the input angle θin (or θ1). We must note that if n2is not real, then necessarily ℑθtra 6= 0. Now we may use Eqs. (2.95) and (2.83) to

18

rewrite the boundary value conditions i.e. Eqs. (2.85)–(2.91) in the following form

Ax + Rx = Tx, (2.99)

Ay + Ry = Ty, (2.100)

ε1 Az + ε1Rz = ε2Tz, (2.101)

−ω JS,0,x + kz,in(Ay − Ry) = kz,traTy, (2.102)

ω JS,0,y + kz,in(Ax − Rx)− kx,in(Az + Rz) = kz,traTx − kx,traTz, (2.103)

Ay + Ry = Ty. (2.104)

Clearly, Eqs. (2.100), (2.102), (2.104) belong to the TE polarized set and Eqs. (2.99),(2.101), (2.103) belong to the TM polarized set.

The wave vectors for the TE and the TM polarized cases are illustrated in Figs. 2.3and 2.4 respectively. We apply Maxwell’s divergence equation to Eqs. (2.101) and

krefB0,ref,TE

RTE

ATE

kin

B0,in,TE

ε1 = ℜ [ε1] ε2

TTEktra

B0,tra,TE

θ1

θ2

x

z

Figure 2.3: Direction of the field vectors for TE polarized light.

(2.103) to get

kz,traε1(Ax − Rx) = kz,in ε2Tx, (2.105)

kz,inkz,traω JS,0,y + kz,traε1(Ax − Rx) = kz,in ε2Tx. (2.106)

Hence, we conclude JS,0,y = 0 i.e. in TM polarization there is no surface current. ForTE polarization we may assume JS,0,x = 0 as long as external currents are absent.Using Eqs. (2.99) and (2.105) we can derive the following equations which are the

19

transmission and the reflection equations for the TM polarized field

Tx =2kz,traε1

kz,traε1 + kz,inε2Ax =

2n1 cos θ2

n1 cos θ2 + n2 cos θ1Ax , (2.107)

Rx =kz,traε1 − kz,inε2

kz,traε1 + kz,inε2Ax =

n1 cos θ2 − n2 cos θ1

n1 cos θ2 + n2 cos θ1Ax . (2.108)

Inspecting Fig. 2.4 we find out that ATM = sec θ1 Ax, TTM = sec θ2Tx, and RTM =sec θ1Rx . Hence, the complex reflection and transmission amplitude coefficientswhich are also known as Fresnel’s transmission and reflection coefficients are writ-ten as

tTM =TTM

ATM=

2n1 cos θ1

n1 cos θ2 + n2 cos θ1, (2.109)

rTM =RTM

ATM=


n1 cos θ2 + n2 cos θ1. (2.110)

Starting from Eqs. (2.100) and (2.102) and proceeding in a similar way we can derivethe Fresnel’s coefficients for the TE polarized set

tTE =TTE

ATE=

2n1 cos θ1

n1 cos θ1 + n2 cos θ2, (2.111)

rTE =RTE

ATE=


n1 cos θ1 + n2 cos θ2. (2.112)

The transmitted and the reflected efficiencies i.e. the normalized transmitted andreflected energy are defined by noting that the direction of energy flow is along z-

kref

B0,ref,TM

RTM

ATM kin

B0,in,TM

ε1 = ℜ [ε1] ε2

TTM

ktraB0,tra,TM

x

zθ1

θ2

Figure 2.4: Direction of the field vectors for TM polarized light.

20

direction and hence considering the z- component of the spectral Poynting vectorwhich is defined as

Pz =ε0ckz

2k0|E0|2, (2.113)

we can derive the expressions for the transmitted and the reflected efficiencies as

RTM =

∣

∣

∣

∣

Pref,z,TM

Pin,z,TM

∣

∣

∣

∣

= |rTM|2 , (2.114)

RTE = |rTE|2 , (2.115)

TTM =kz,tra

kz,in

∣

∣

∣

∣

TTM

ATM

∣

∣

∣

∣

2

=n2 cos θ2

n1 cos θ1|tTM|2 , (2.116)

TTE =kz,tra

kz,in

∣

∣

∣

∣

TTE

ATE

∣

∣

∣

∣

2

=n2 cos θ2

n1 cos θ1|tTM|2 . (2.117)

The transmitted efficiency is meaningful only if n2 is real. Otherwise we have ab-sorption and the field rapidly decays upon propagation. However, the reflectedefficiency still remains meaningful as we assumed n1 to be real.

2.11 GENERAL FIELD AND ANGULAR SPECTRUM REPRESENTA-

TION

We already saw that the electric and the magnetic field vectors of a plane wave satis-fies, in a homogeneous medium, the Helmholtz equations (2.42) and (2.43). We nowproceed to show that the general solutions of these equations can be representedas superpositions of plane waves. Let us now introduce a Fourier transform pairbetween the space-frequency domain and the wave vector-frequency domain (alsoknown as the spatial-frequency domain) electric field vectors

E(r, ω) =1

4π2

∫∫ ∞

−∞E(kx, ky, z, ω) exp

[

i(kxx + kyy)]

dkx dky, (2.118)

E(kx, ky, z, ω) =∫∫ ∞

−∞E(r, ω) exp

[

−i(kxx + kyy)]

dx dy. (2.119)

If we now insert Eq. (2.118) into the Helmholtz equation for the electric field viz.Eq. (2.42), we get

∂2

∂z2 E(kx, ky, z, ω) + k2z E(kx, ky, z, ω) = 0, (2.120)

where

Kz =

[k2 − k2x − k2

y]1/2 , if k2

x + k2y ≤ k2 ,

i[k2x + k2

y − k2]1/2 , if k2x + k2

y > k2 .(2.121)

A general solution of Eq. (2.120) may be written in the following form

E(kx, ky, z, ω) = U+(kx, ky, ω) exp [ikz(z − z0)] + U−(kx, ky, ω)

× exp [−ikz(z − z0)] , (2.122)

where z = z0 is the reference plane. Depending on the conditions written inEq. (2.121) the solutions of Eq. (2.120) can be divided into two groups. The group

21

having real values of kz represents homogeneous waves. Whereas the group contain-ing imaginary values of kz represents evanescent waves. The ’±’ signs in the sub-scripts represent waves propagating along +z and along −z respectively. Thoughthe evanescent waves rapidly decay exponentially with the propagating distance insubwavelength plasmonic systems, as we will see later, they have significant roles.

Using the known value of the field at z = z0 and assuming that there is no sourcein the positive half space we can write the Fourier transform pair in Eqs.(2.118) and(2.119) in the following reformulated form

E(x, y, z, ω) =∫∫ ∞

−∞U+(kx, ky, ω) exp

[

i(kxx + kyy + kz(z − z0))]

×dx dy, (2.123)

U+(kx, ky, ω) =1

4π2

∫∫ ∞

−∞E(x, y, z0, ω) exp

[

−i(kxx + kyy)]

×dkx dky. (2.124)

The above written Fourier transform pair is known as the angular spectrum repre-sentation of the field. With this representation any general field can be written asa superposition of plane waves. To evaluate the angular spectrum at an arbitraryplane at z > z0, we just need to multiply U+(kx, ky, ω) by a constant exponentialfactor.

2.12 THEORY OF EVANESCENT WAVES

As we mentioned in the earlier section, evanescent waves are extremely importantin subwavelength nanostructures and in the context of plasmonics. To understandtheir behavior let us assume the same planar interface as in Fig. 2.2. By examiningthe definitions of the angles and employing Eqs. (2.83), (2.95), and (2.95) we arriveat the well known Snell’s law

n1 sin θin = n2 sin θtra, (2.125)

which connects the propagation angle of the transmitted wave to the angle of inci-dence. Now we use Eqs.(2.83) and (2.121) to obtain

[kz,tra]2 = ε1k2

0

[

ε2

ε1− sin2 θ1

]

. (2.126)

If we assume that the permittivity of the incident medium is higher than that of thetransmitted medium i.e. ε1 > ε2, we get

θ1 > θc = sin−1 n2

n1, (2.127)

where θc is the critical angle of incidence i.e. the angle of incidence beyond whichlight wave passing through a optically denser medium to the surface of a less densemedium is no longer refracted but totally reflected. Clearly from Eq. (2.126), kz,trabecomes imaginary which represents an evanescent field propagating parallel to theinterface but decaying exponentially along z in medium 2.

22

2.13 MONOCHROMATIC PLANE WAVE IN ANISOTROPIC MEDIUM

So far we have assumed the propagation medium to be isotropic. Let us now in-vestigate the situation when light propagates in an electrically anisotropic mediumi.e. in crystals with only electrical anisotropy. The other assumptions regarding themedium properties still remain valid i.e. we assume homogeneous, non-conducting,and magnetically isotropic medium. In such a medium, the electric displacement Dand the electric field E no longer remains collinear.

The simplest relation between D and E which can account electrical anisotropycan be written in component form as

Dj = ∑k

ε jkEk, (2.128)

where j stands for the cartesian indices (1, 2, 3), and k stands for each of 1, 2 and 3 inturn in the summation on the right hand side of Eq. (2.128). The expressions for theelectric and the magnetic energy densities, and the Poynting vector i.e. Eqs. (2.62),(2.63), and (2.64) still retain their validity and can be written in the following form

we = (1/2)ε0 ∑jk

Ejε jkEk, (2.129a)

wm = (1/2)µ0µr H2, (2.129b)

〈S〉 = (1/2)ℜ [E × H∗] . (2.129c)

Here we have kept the relative magnetic permeability µr (which is a constant) topreserve some symmetry in the formulae and also to include weakly magnetic crys-tals. However, at optical frequencies we can always replace µr by 1 in SI system ofunits. Also, to write Eq. (2.129), we have omitted the explicit position (r) and timedependence of the field quantities.

Let’s now check the consistency of Eq. (2.129) with the energy conservation prin-ciple. For this we multiply space-time domain Maxwell’s equation 1 i.e. Eq. (2.9)with E and Maxwell’s equation 2 i.e. Eq. (2.10) with H and use the vector identityE · (∇× H)− H · (∇× E) = −∇ · (E × H) to write the following equation

12 ∑

jk

Ejε jk∂

∂tEk +

12

µ0∂

∂t

(

µr H2)

= −12∇ · (E × H) , (2.130)

where we have divided both sides of Eq. (2.130) by 2. Clearly, the second term onthe left hand side of Eq. (2.130) represents the rate of change of magnetic energy

per unit volume but the first term i.e. 12 ∑jk Ejε jk

∂∂t Ek = 1

4 ∑jk ε jk

(

Ej∂∂t Ek + Ek

∂∂t Ej

)

represents the rate of change of the electric energy density only if

∑jk

ε jk

(

Ej∂

∂tEk − Ek

∂

∂tEj

)

= 0. (2.131)

However, the term on the right hand side of Eq. (2.130) equals to −∇ · S. Hence,for the validity of the energy theorem, which states that the decrease in the totalelectromagnetic energy per unit time in a certain volume is equal to the net outwardflux per unit time, in differential form viz.

−∇ · S = (∂/∂t) (we + wm), (2.132)

23

we must haveε jk = εkj. (2.133)

In words, the permittivity tensor must be symmetric with only 6 independent ele-ments. Employing condition (2.133), Eq. (2.129a) can be written as

ε11E21 + ε22E2

2 + ε33E23 + 2ε12E1E2 + 2ε13E1E3 + 2ε12E1E2 = 2we. (2.134)

The right-hand side of Eq. (2.134) is a positive constant as we represents electricenergy density. Hence, Eq. (2.134) represents an ellipsoid. Any ellipsoid can bewritten in a coordinate system with its axes parallel to the principal axes of theellipsoid. Assuming that the principal axes of the ellipsoid are along x,y, and z,where x, y, z are also cartesian indices (the coordinate systems (x, y, z) and (1, 2, 3))are rotated with respect to each other), we can write Eq. (2.134) in the followingsimple form

D2x

ǫx+

D2y

ǫy+

D2z

ǫz= 2we, (2.135)

withDx = ǫxEx, Dy = ǫyEy, Dz = ǫzEz. (2.136)

Here ǫx, ǫy, and ǫz are the principal relative permittivities. Equations (2.135) and(2.137) indicate that the electric displacement vector D and the electric field vectorE are in general not parallel to each other unless E directs along one of the princi-pal axes of the crystal or all the principal relative permittivities are equal i.e. theellipsoid degenerates into a sphere. For a plane wave propagating in an anisotropicmedium, the direction of the electromagnetic energy flow is in general differentfrom the direction of the wave normal. Hence in addition to the wave-normal veloc-ity or the phase velocity vp = c/n, n being the position dependent refractive index,we can introduce the concept of ray (or energy) velocity vr = S/w, where S is theamplitude of the Poynting vector and w is the total electromagnetic energy density.Clearly from the definition, the ray velocity is equal to the energy that crosses anarea perpendicular to the flow direction in unit time divided by the energy per unitvolume. Assuming time-harmonic field of the form exp iω [(n/c) (r · u)− t], wecan write Maxwell’s equations (2.9) and (2.10) in the following forms (in regionswithout currents)

(n/c)u × H = −D, (n/c)u × E = µH, (2.137)

where, we have used the relation B = µ0µr H = µH. Let us now eliminate H fromEqs. (2.137) to get the following

D = − n2

µc2 u × (u × E) =n2

µc2 [E − u(u · E)] = (n2/µc2)E⊥. (2.138)

E⊥ is the component of the electric field in a direction perpendicular to the wavenormal and lies in a plane containing E and u. Clearly from Eq. (2.137), the magneticfield H (also the magnetic induction B) vibrates at right angles to E, D, and u.Furthermore, D is orthogonal to u, hence H and D are transversal to the wavepropagation direction. However in an anisotropic medium, E is not transversal tothe wave normal as before. We denote the angle between D and E by α. Let us nowdefine a new unit vector s which defines the direction of the energy flow. From the

24

definition of the Poynting’s vector, both E and H are perpendicular to s. Hence, theangle between u and s is also α. Finally, we may conclude that in an anisotropicmedium, the vectors D, H, and u on one hand, and the vectors E, H, and s on theother hand form orthogonal vector triplets with the common vector H as shown inFig. 2.5. The theorem of equal electric and magnetic energy densities still remain

B

H

s = S|S|

αα

u E

D

E⊥

Figure 2.5: Direction of the field quantities, the wave normal, and the energy flowin an electrically anisotropic medium.

valid and by use of Eqs. (2.129) and (2.137) we can write the total electromagneticenergy density in the following form

w =n

cS · u. (2.139)

It now follows from the definitions of the phase velocity and the ray velocity andEq. (2.139) that

vp = vrs · u = vr cos α, (2.140)

which translates that the phase velocity in an anisotropic medium is the projectionof the ray velocity along the direction of the wave normal.

2.14 ELECTROMAGNETIC THEORY OF METALS

While studying the electromagnetic properties of metals, in many cases it is as-sumed for simplicity that the material media are perfectly conductive. In perfectlyconducting media, the flow of electrons is completely free and it is assumed thatthe ”free electrons” can respond to the incident field infinitely quickly without be-ing scattered. Also for an ideal metal, the electric field is ’zero’ everywhere insidethe material as in the limit ε → −∞, the electrons respond perfectly to the appliedelectric field and thus cancels the external field completely. This also gives zeroelectrical resistance of ideal metals. Although the assumptions made above lead toanalytical solutions of the modes of the optical field, these are inaccurate at opticalfrequencies. The reason is that the free electrons inside a metal have finite mass and

25

they suffer scattering with phonons (lattice vibrations), lattice defects, surface of thematerials as well as with other electrons which impose limit on the response time ofthe metals to the applied external field. In this section, we shall introduce the Drudemodel and the interband transitions model for describing complex permittivities ofmetals. These models are of great importance to understand the interaction betweenmetals and external electromagnetic fields at the atomic level. In many time-domainsolvers of Maxwell’s equations such as in FDTD [119], these can be included directly.

2.14.1 Drude model

In the Drude model (also known as the plasma model), the optical properties ofmetals are portrayed by considering the optical response of a free electron gas inan electric field. It is assumed that the gas of free electrons moves against a fixedbackground of positive ion core [103]. In the plasma model, the lattice potential andelectron-electron interactions are ignored. However, it is assumed that some featuresof band structure are connected to the effective mass (me) of a single electron. Letus denote the free electron density by Ne. Also let us assume that the electronoscillations triggered by the external field are attenuated via collisions (scattering)occurring with characteristic frequency γ = 1/τ, where τ is the relaxation timeof the free electron gas. The Drude model assumes that the valence electrons areidentical to the free electrons. To derive the complex relative permittivity of thematerial using this model, we start from the equation of motion of a free electronin the free electron gas which is subject to the influence of an externally appliedelectric field E(r, ω) [120]:

me∂2

∂t2 r(t) + meγ∂

∂tr(t) = qeE(r) exp(−iωt), (2.141)

where qe denotes the electric charge of a single electron, r(t) is the time-dependentdisplacement of the electron with respect to the ion core, and γ is equivalent to thedamping parameter. Let us attempt a solution of Eq. (2.141) of the form r(t) =r0 exp(−iωt). Substituting this in Eq. (2.141), we get the following expression of theelectron displacement

r(t) =qe

me(ω2 + iωγ)E(r, ω). (2.142)

The electric polarization of a single electron is defined as p(t) = qer(t). Henceassuming that all the electrons in the free electron gas are displaced by an equalamount under the influence of the external field, we get the following expressionfor the total electric polarization [121]:

P(t, ω) = Ne p(t, ω) = Neqer(t) = − Neq2e

me(ω2 + iωγ)E(r, ω). (2.143)

Let us now recall the material constitutive relations i.e. Eqs. (2.17) and (2.24). Withhelp of these and Eq. (2.143), we get the following expression for the complex per-mittivity [122]:

εD(ω) = 1 −ω2

p

ω2 + iγω= 1 −

ω2pτ2

ω2τ2 + iωτ, (2.144)

where ωp is termed as the volume plasma frequency and is given by

ωp =

√

Neqe

ε0me. (2.145)

26

Equation (2.144) is known as the Drude model of the optical response of metals. Thereal and the imaginary part of the complex relative permittivity ε(ω) = ε1(ω) +iε2(ω) can be calculated from

ε1(ω) = 1 −ω2

pτ2

1 + ω2τ2 , (2.146a)

ε2(ω) =ω2

pτ

ω(1 + ω2τ2). (2.146b)

Usually the free electron density (Ne) in metals lies in the range 1028 − 1029 m−3.This leads to the plasma frequency in the ultraviolet region.

Drude model can be well used to describe optical properties of alkali metals inthe visible frequencies but for noble metals this model gives inaccurate results dueto the interband transitions.

2.14.2 Interband transitions model

The Drude model underestimates the imaginary parts of the complex permittivitiesof many noble metals in the visible region. This is because the high frequency (hencehigh energy) photons excite the lower band valence electrons into the conductionband [122]. Such transitions can not be described by the Drude model. In theinterband transitions model, such type of transitions can be taken into account byintroducing oscillations of the bound electrons. In this model, the motion of thebound electrons can be described by the following equation [123]:

me∂2

∂t2 r(t) + meγ∂

∂tr(t) + mω2

0r(t) = qeE(r) exp(−iωt), (2.147)

where ω0 is defined as the resonance frequency of the bound electrons. Consideringa trial solution as in the previous section, we can obtain the following expression forthe complex relative permittivity

εD(ω) = 1 −ω2

p

(ω20 − ω2) + iγω

, (2.148)

where the plasma frequency is defined by

ωp =

√

Neq2e

ε0me, (2.149)

with Ne denoting the density of the bound electrons.In reality, the external field can penetrate the metal surface. Atomic structure of

metals demand that the current density is the highest on the surface and it graduallydecreases to zero when we go deeper inside the metal. Hence, when an externalelectromagnetic field penetrates a metal surface, it gradually decreases to zero. Thedistance at which the electromagnetic field amplitude decreases to 1/e of its value onthe metal surface, is called the skindepth of the metal and is given by the followingrelation

δ =c

κω=

√

2σ0ωµ0

, (2.150)

where κ is defined in Eq. (2.30), and σ0 is the dc-conductivity of the metal as before.From Eq. (2.150), we see that for better conductors, the skin depth is smaller.

27

2.15 PLASMONS

By plasma, we specify a medium consisting of positively and negatively chargedparticles. The free electrons and ion cores inside a metal constitute a plasma withzero net charge. Plasmons are collective oscillations of the free (conduction) elec-trons inside a plasma. There are three types of plasmons each with their own char-acteristic nature. These are outlined below.

ω

k

ωBP

ωSPP

ωLSP

ω = ck0

Figure 2.6: Dispersion diagram of bulk plasmons (black dashed line), SPP(dash-dotted line) and LSP(dotted line) at metal-vacuum interface as compared to thedispersion of light in vacuum (solid black line).

+ +- - +

+

-

- -

-

-

--

-

--

-

-

++

+ +

+

++

+

+

+

+ - - -++

(a) (b) (c)

BP SPP LSP

Figure 2.7: Schematic of (a) bulk plasmon, (b) a surface plasmon polariton at ametal-dielectric interface, and (c) a localized surface plasmon at the interface of aspherical metal nanoparticle and surrounding dielectric.

28

2.15.1 Volume plasmons

In metals, local deviations in free electron density cause restoring forces from thefixed ionic cores which lead to simple harmonic motion of the conduction electrons.The quanta of these oscillations are termed as bulk plasmons. The characteristicfrequencies of bulk plasmons (ωBP) are given by Eq. (2.145). For most of the metals,bulk plasma frequencies are higher than the frequencies of the visible light. Whenlight of lower frequency (than the bulk plasmon frequency) is incident on the metal,the electric field of the incident light causes the conduction electrons to move withrespect to the positively charged ionic cores in such a way that an internal electricfield is generated which cancels out the incident electric field and creates a reflectedwave. However, for frequencies of the incident light much higher than the plasmafrequency i.e. in the limit ω ≪ ωBP, the electrons cannot keep up with light oscil-lations. Hence, we can make the approximation ωτ ≫ 1 and neglect the dampingterm iωτ in Eq. (2.144) to get

ε(ω) = 1 −ω2

p

ω2 . (2.151)

The dispersion relation of the electromagnetic fields can be determined from therelation k2 = |k|2 = εω2/c2:

ω(k) =√

ω2p + (kc)2. (2.152)

Clearly from the dispersion curve in Fig. 2.6, for light waves with frequencies belowthe bulk plasma frequency (ω < ωBP), there is no propagation. The solid blackline represents the dispersion of light in vacuum where ω = ck0, the dashed linedepicts the dispersion of light propagation inside the metal. For ω = ωBP, we getfrom Eq. (2.152), ε(ωBP) = 0. One can show [103] that in such a scenario collectivelongitudinal excitation mode (E ‖ k) is formed with a purely depolarizing field (E =(−1/ε0)P). Since the bulk plasmons are longitudinal waves they cannot couple totransversal electro-magnetic fields and also cannot be excited from direct irradiation.The bulk plasmons are illustrated in Fig. 2.7(a).

2.15.2 Surface plasmon polaritons

There is another kind of plasmon which has coupled longitudinal and transversecomponents along the surface of a metal. Since this type of plasmon has coupledlongitudinal and transverse parts (depicted in Fig. 2.7(b)), it is called surface plas-mon polariton (SPP) [124]. In 1902, Wood observed that if a reflection grating isilluminated with polychromatic light, narrow dark and bright bands appear in thereflection spectrum, which are widely known as Wood’s anomalies. After almostthree decades, these anomalies were explained by Ugo Fano [125–127]. Fano con-cluded that these narrow bands are caused by the surface waves or the surfaceplasmons polaritons, which can be excited on the surface of metallic gratings underspecial conditions.

Let’s now proceed to derive the dispersion relation for a SPP by assuming asmooth planar interface separating two half-spaces with scalar permittivities ε1 andε2 as shown in Fig. 2.2. This is the simplest geometry supporting SPPs. The electricand the magnetic field components in both the half spaces can be decomposed intotwo independent sets as described in section 2.6. Assuming a TE polarized incident

29

plane wave at the interface (located at z = 0), the only non-vanishing electric fieldcomponent in both half spaces can be written as

E(j)y (x, z) = E

(j)y,0 exp

i[

kxx + k(j)z z]

, (2.153)

where the superscript j = 1, 2 denotes the left and the right half-spaces respectively.Using Eqs. (2.51) and (2.52), we can find expressions for the non-vanishing magneticfield components:

H(j)x (x, z) = − k

(j)z

k0

√

ε0

µ0E(j)y,0 exp

i[

kxx + k(j)z z]

, (2.154a)

H(j)z (x, z) =

kx

k0

√

ε0

µ0E(j)y,0 exp

i[

kxx + k(j)z z]

. (2.154b)

As the electromagnetic boundary conditions Eq. (2.39) and (2.40) demand that thetangential field components i.e. Ey and Hx must be continuous across the interface,we must have

k(1)z = k

(2)z . (2.155)

From the definition of the wave vector we see that Eq. (2.155) is satisfied only ifε(1) = ε(2). Hence, we are forced to conclude that a TE polarized wave can not exciteSPPs at the interface between two different media.

If we now assume TM polarized incident field, the only non-vanishing magneticfield component in both half-spaces can be expressed by

H(j)y (x, z) = H

(j)y,0 exp

i[

kxx + k(j)z z]

. (2.156)

By use of Eqs. (2.54) and (2.55) we can find the expressions for the non-vanishingelectric field components

E(j)x (x, z) =

k(j)z

k0 ε(j)

√

µ0

ε0H

(j)y,0 exp

i[

kxx + k(j)z z]

, (2.157a)

E(j)z (x, z) = − kx

k0 ε(j)

√

µ0

ε0H

(j)y,0 exp

i[

kxx + k(j)z z]

. (2.157b)

The electromagnetic boundary conditions i.e. Eqs. (2.38)–(2.40) demand that

ε(1)E(1)z = ε(2)E

(2)z , (2.158a)

E(1)x = E

(2)x , (2.158b)

H(1)y = H

(2)y , (2.158c)

respectively. As the tangential field components Ex and Hy must be continuousacross the interface we get using Eq. (2.157a),

ε(1)

k(1)z

=ε(2)

k(2)z

. (2.159)

30

If we now substitute for [k(1)z ]2 = ε(j)k2

0 − k2x in Eq. (2.159), we obtain the so-called

dispersion relation of the propagating surface plasmons

kx = kSPP = k0

√

ε(1)ε(2)

ε(1) + ε(2). (2.160)

The normal component of the surface plasmon wave vector is given by

k(j)z = k0

√

[ε(j)]2

ε(1) + ε(2). (2.161)

As we are seeking for a wave solution that propagates along the interface and ex-ponentially decays into both half-spaces, we must have real valued tangential wave

vector component kx = kSP. Also, the normal wave vector components k(j)z s must

be imaginary in both half-spaces. The first condition demands that both the numer-ator and the denominator in the right hand side of Eq. (2.160) is either positive ornegative. The second condition demands that the denominator on the right handside of Eq. (2.161) is negative. These requirements are fulfilled at the interface of ametal and a dielectric, where the absolute value of the real part of the permittivityof the metal is larger than the permittivity of the dielectric. For real metals, belowthe plasma frequency ωp, the permittivities are imaginary which leads to complexvalues of kSP. The real and the imaginary parts of kSP determine the surface plas-mon wavelength and the attenuation coefficient (1/e decay length) of the surfaceplasmon wave along the interface respectively, i.e.

λSP =2π

ℜkSPP, (2.162a)

andδSP =

1ℑkSPP

. (2.162b)

In a similar fashion we can define the exponential decay length of the surface plas-mon field perpendicular to the interface i.e. in the two half-spaces by the followingrelation

δ(j) =1

ℑ∣

∣

∣k(j)z

∣

∣

∣

. (2.163)

This 1/e decay length inside the metals is equal to the skin-depth of the metals.If we now assume that the dielectric is vacuum, in the limit of large k and as-

suming a Drude model for the permittivity of the metal we obtain from Eq. (2.160),

ωSPP = ωBP/√

2. (2.164)

The surface plasmon dispersion relation and the definition of the decay length inEq. (2.162b) infer that for the real metals (for example gold, silver, copper, aluminumetc.) there is a momentum (hk, h being the Planck’s constant) mismatch betweenfree-space photons and surface plasmons in the same medium and in the wave-length region where the surface plasmons are expected to propagate. This meansthat the SPPs will not be excited by incident light under normal circumstances. Thisis graphically illustrated in Fig. 2.6 where we see that the SPP line has a higher mo-mentum (larger k) than the free-space light line (solid black line) for all frequencies.

31

Also, we observe that the dispersion curve of light in vacuum (solid black line) isclose to the SPP dispersion curve (black dash-dotted line) at metal-vacuum interfaceonly for shorter frequencies of the incident light. Hence in the lower frequency /higher wavelength regime, the SPP is light-like and the transverse components ofthe SPP field will dominate. Conversely, in lower wavelength regime, the SPP dis-persion is far from the dispersion of light in vacuum and the SPP is plasmon-likewhere the longitudinal components dominate.

In practical applications to excite SPP, the momenta of the incident wave vectorcomponent along the interface and the SPP wave vector need to be matched. Someof the common tricks which are applied to compensate for the momentum mismatchare listed below

• Prism Coupling: The metal film in this case is illuminated through a dielectricprism at an angle of incidence greater than the angle of total internal reflection.The wave vector of incident light is increased in the optically dense medium.At a certain angle of incidence θin, the component of the light wave vector(inside the prism) parallel to the interface matches with the SPP wavevector atmetal-air interface and light is coupled to the SPP mode as shown in Fig. 2.8(a).

• Grating Coupling: A grating with grating vector parallel to the interface is in-troduced in this case as shown in Fig. 2.8(b). Momentum conservation can besatisfied by the mth (m = 1, 2, 3, . . . ) evanescent diffraction order by

|kx + mK| = ω

cℜ√

ε(1) ε(2)

ε(1) + ε(2), (2.165)

where kx = k sin θ is the component of the incident wave vector parallel to theinterface, θ is the angle of incidence and K = 2π/|d| with d representing thegrating vector which directs parallel to the interface.

(a) (b)

θin

θin

d

Figure 2.8: Geometry of (a) prism coupling, and (b) grating coupling for excitingSPPs at a metal-dielectric interface.

2.15.3 Particle plasmons

There exists a third kind of plasma oscillation which is known as particle plasmon.This arises from the scattering problem of a small subwavelength metallic nanopar-ticle under influence of an oscillating external electromagnetic field. Electrons are

32

pulled back to the particle surface by a restoring force which arise from the curva-ture of the particle. At resonance, the fields inside the particle and in its immediatevicinity are greatly enhanced, sometimes their magnitudes can be even 100 ordershigher than that of the exciting field. As the restoring force in this case is the surfacecharge (instead of the ionic cores), the particle plasmon is a transverse standing waveas shown in Fig. 2.7(c). Due to its standing wave nature particle plasmons are alsotermed as localized surface plasmons (LSPs). There are no momentum matchingconditions as the particle plasmons have a net zero momentum (standing waves).From Fig. 2.6, we observe that the light line (solid line) will be able to cross thedotted horizontal line (which represents the dispersion for a particle plasmon) forany value of k.

When the particle is extremely small compared to the wavelength of the incidentfield, the phase of the electric field does not change considerably over the wholenanoparticle. Hence all the conduction electrons in the particle experience the samedriving force, causing them to act coherently. These coherently shifted electronsunder the influence of the fluctuating external electromagnetic field, resembles anoscillator together with the restoring field. The behaviour of this oscillator is deter-mined by the effective electron mass, surface charge density, and the geometry ofthe particle. Most physical effects associated with LSPs can be explained with thehelp of this simple model. Using the damped driven harmonic oscillator model wecan write the following expression

d2

dt2 x(t) = −ω20x(t)− 2γ

d

dtx(t) + E0 exp (iωt) (2.166)

describing the oscillations of the conduction electrons. x(t) is the displacement ofthe conduction electrons under influence of the external alternating electric fieldE0 exp (iωt), the restoring force is the surface polarization charge, which can beimagined as a spring of constant k pulling the electrons back to equilibrium, thedamping term γ may arise due to the decay of the plasmon mode into radiationmode (results in emission of a photon with the same frequency as the incident field),phonon scattering, scattering by a lattice defect, surface scattering or absorption bya surrounding medium. The solution of Eq. (2.166) is given by

x(t) =E0

2meω0

1√

(ω0 − ω)2 + γ2, (2.167)

where ω0 =√

ks/m is the resonance frequency which depends on the surface chargeand hence on the geometry and the size of the particle. The LSP resonance is ob-served at ω = ω0. Since the resonance frequency is proportional to the springconstant ks, a weaker restoring force leads to a lower frequency of oscillation caus-ing redshift of the resonance. Usually, the LSP resonances are observed in the visibleto near-infrared regime.

The geometry of the nanoparticle strongly affects the microscopic polarizability(α) inside the particle which in turn affects the surface charge and hence the reso-nance frequency. For a small spherical particle, assuming a uniform static electricfield throughout its volume, if we have nonabsorbing surrounding medium, themicroscopic polarizability takes the following form [103]

α = 4πε0r3 εm − εd

εm + 2εd, (2.168)

33

where r is the radius of the spherical metal particle with relative permittivity εm,εd is the relative permittivity of the surrounding dielectric medium. Clearly, forεm = −2εd, the denominator on the right hand side of Eq. (2.168) is zero and wehave LSP resonance. Assuming Drude model for the permittivity of the metal andtaking the dielectric to be vacuum as before, we get

ωLSP = ωBP/√

3, (2.169)

which is illustrated in Fig. 2.6.

2.16 FIELD ENCOUNTERING STACK OF THIN FILMS

We next examine a case of special importance in a broad range of applications inrelation to optics. This is the situation when a large number of thin homogeneouslayers are separated by plane interfaces parallel to each other as shown in Fig. 2.9.The positions of the interfaces are given by z(j), j = 0, 1, . . . , J − 1, J, where J is thetotal number of layers. The permittivities of the layers are defined as

ε(z) = ε(j), z(j−1)< z < zj. (2.170)

Here we define region 2 as the layer between z(1) and z(2). Let us now assumeε(0) and ε( J+1) to be real valued i.e. we have real permittivities for the incidentand the output regions. Also, we assume that the incident field is y invariant andpropagates from region 0 towards the positive z-axis. Using the angular spectrumrepresentation, the input electric field can be expressed in the form

E(0+)(x, z) =∫ ∞

−∞U(0+)(kx) exp

[

ikxx + ik(j)z (z − z(0))

]

dkx, (2.171)

U(0+)(kx) =1

2π

∫ ∞

−∞E(0+)(x, z(0)) exp [−ikxx] dx. (2.172)

It can be proved easily that the y-invariance of the incident field leads to y-invariancealso in all other regions. Hence, using the angular spectrum representation theelectric field in the j-th layer can be written as

E(j,+)(x, z) =∫ ∞

−∞U(j,+)(kx) exp

[


]

dkx, (2.173)

U(j,+)(kx) =1

2π

∫ ∞

−∞E(j,+)(x, z(j)) exp [−ikxx] dx, (2.174)

E(j,−)(x, z) =∫ ∞

−∞U(j,−)(kx) exp

[


]

dkx, (2.175)

U(j,−)(kx) =1

2π

∫ ∞

−∞E(j,−)(x, z(j)) exp [−ikxx] dx, (2.176)

for j = 1, 2, . . . , J − 1, J. Here we compute the angular spectrum of the propagatingfield at z(j) for the j-th layer. Now we can decompose Maxwell’s equations in j-th

34

z(1) z(2) z(3) z( J−2) z( J−1) z( J)z(0)

ε(1) ε(2) ε(3) ε( J−1) ε( J) ε( J+1)ε(0)

0

x

z

Figure 2.9: Schematic of light propagation in stack of thin films.

layer into two sets (TE/TM) in a similar way as described in section 2.6

∂

∂zE(j,±)y (x, z) = −iωB

(j,±)x (x, z), (2.177)

∂

∂xE(j,±)y (x, z) = iωB

(j,±)z (x, z), (2.178)

iωµ0ε0 ε(j)E(j,±)y (x, z) =

[

∂

∂xB(j,±)z (x, z)− ∂

∂zB(j,±)x (x, z)

]

, (2.179)

∂

∂zB(j,±)y (x, z) = iωµ0ε0 ε(j)E

(j,±)x (x, z), (2.180)

∂

∂xB(j,±)y (x, z) = −iωE

(j,±)z (x, z), (2.181)

iωB(j,±)y (x, z) =

[

∂

∂zE(j,±)x (x, z)− ∂

∂xE(j,±)z (x, z)

]

, (2.182)

Eqs. (2.177)–(2.179) contain only the y component of the electric field which is per-pendicular o the xz plane i.e. the plane of propagation. Hence these form the TEpolarized set. Similarly Eqs. (2.180)–(2.182) form the TM polarized set. Followingthe electromagnetic boundary conditions we can now demand the continuity of thefield quantities at the boundary z = z(j). For the TE polarized set, continuity of theelectric field gives

E(j,+)y

[

x, z(j)]

+ E(j,−)y

[

x, z(j)]

= E(j+1,+)y

[

x, z(j)]

+ E(j+1,−)y

[

x, z(j)]

, (2.183)

35

where h(j) = z(j)− z(j−1) is the thickness of the j-th layer. Uniqueness of the Fourierrepresentation gives

U(j,+)y + U

(j,−)y = U

(j+1,+)y f

(j+1)− + U

(j+1,−)y f

(j+1)+ , (2.184)

where f(j+1)± = exp

[

±ik(j)z h(j)

]

and we dropped the explicit kx dependence of Uy

for the sake of brevity. As z derivatives of the electric fields are also continuousacross the boundary, we get the following relation

k(j)z U

(j,+)y − k

(j)z U

(j,−)y = k

(j+1)z U

(j+1,+)y f

(j+1)− − k

(j+1)z U

(j+1,−)y f

(j+1)+ . (2.185)

Eqs. (2.184) and (2.185) can be conveniently written in matrix form as[

1 1

k(j)z −k

(j)z

] [

U(j,+)y

U(j,−)y

]

= (2.186)

[

1 1

k(j+1)z −k

(j+1)z

] [

f(j+1)− 0

0 f(j+1)+

] [

U(j+1,+)y

U(j+1,−)y

]

. (2.187)

In the case of TM polarization the electromagnetic boundary conditions demand thecontinuity of the y components of the magnetic induction. Proceeding similarly asin the TE polarized case we arrive at the following matrix equation

[

1 1

k(j)z /ε(j) −k

(j)z /ε(j)

] [

V(j,+)y

V(j,−)y

]

= (2.188)

[

1 1

k(j+1)z /ε(j+1) −k

(j+1)z /ε(j+1)

] [

f(j+1)− 0

0 f(j+1)+

] [

V(j+1,+)y

V(j+1,−)y

]

, (2.189)

where V(j)y denotes the angular spectrum of the magnetic induction. Here, we must

emphasize that though the y components of the magnetic induction are continuousacross the boundary their z-derivatives are discontinuous. However, the z-derivative

of V(j)y divided by the complex permittivity i.e. ε(j)s is continuous across the bound-

ary. Owing to the similarity of the two matrix equations (2.185) and (2.186), we maywrite a single matrix equation for both the TE and the TM polarized fields as

[

1 1g(j) −g(j)

] [

G(j,+)

G(j,−)

]

= (2.190)

[

1 1g(j+1) −g(j+1)

]

[

f(j+1)− 0

0 f(j+1)+

]

[

G(j+1,+)

G(j+1,−)

]

, (2.191)

where g(j) = k(j)z , G(j,±) = U

(j,±)y for TE polarization, and g(j) = k

(j)z /ε(j), G(j,±) =

V(j,±)y for TM polarization. Eq. (2.190) connects the angular spectra in regions j and

j + 1.

2.17 RECURSIVE S-MATRIX ALGORITHM

Several standard matrix algorithms exist for solving the boundary value equationsassociated with the stack of thin films problem discussed in the previous section.

36

These matrix algorithms are required to connect the amplitudes of the input andthe output field quantities. However, all of these algorithms face a common dif-ficulty which is associated with the exponential functions of the spatial variablesin the direction perpendicular to the grating plane, to be more specific with the

f(j)± s. Especially if the complex permittivity of the j-th layer is complex valued,

the imaginary part of k(j)z becomes positive and f

(j)− becomes unstable. If there is

a small numerical error in computing the argument of f(j)− , it grows exponentially

with the growing layer thickness. The well known T-matrix algorithm suffers fromthis type of numerical instability. One way to overcome this is to make sure thatall the angular-spectrum components in each layer propagate only along their real,physical propagation direction, thus eliminating the cumulative exponential errorin each step. This is done using the recursive S-matrix (also known as scatteringmatrix) algorithm [128–130]. In S-matrix algorithm we take the angular spectrum

S(j)

G(j+1,+) G(j+1,−)

G(j,−) G(j,+)

Figure 2.10: Block diagram of S-matrix for two adjacent layers.

components G(j+1,−) and G(j,+) as inputs to the system and try to find the S-matrixto derive the components G(j+1,+) and G(j,−) using propagation along the correctdirection only. Mathematically, this can be expressed as,

[

G(j+1,+)

G(j,−)

]

= S(j)(j+1)[

G(j,+)

G(j+1,−)

]

. (2.192)

Block diagram of S-matrix for two consecutive layers is shown in Fig. 2.10. We canconstruct the S-matrix either for a part or for the whole system. A single S-matrixcontains all of the system’s scattering properties. The elements of the S-matrix canbe deduced from Eq. (2.190). In this context we must mention that the connectionbetween different layer-S-matrices are not straightforward. Lifeng Li [128–130] useda recursive approach to solve the S-matrix problem in an efficient and stable way. Inthis approach, one starts from the sub-system S-matrix S(j+1)( J+1) and employs theboundary value equations to find expression for the sub-system matrix S(j)( J+1) in

37

terms of the elements of S(j+1)( J+1). The desired equation is of the following form

[

G( J+1,+)

G(j,−)

]

= S(j)( J+1)[

G(j,+)

G( J+1,−)

]

= (2.193)

[

S(j)( J+1)11 G(j,+) + S

(j)( J+1)12 G( J+1,−)

S(j)( J+1)21 G(j,+) + S

(j)( J+1)22 G( J+1,−)

]

, (2.194)

where the elements S(j)( J+1)pq can be expressed in terms of the elements S

(j+1)( J+1)pq .

After some lengthy though straightforward calculations and applying the boundary-value condition (2.190) we can evaluate the coefficients of the S-matrix in Eq. (2.193)

S(j)( J+1)21 = Z

(j)( J+1)21 + Z

(j)( J+1)22 g(j), (2.195)

S(j)( J+1)22 =

[

Z(j)( J+1)22 g(j+1)− Z

(j)( J+1)21

]

f(j+1)+ S

(j+1)( J+1)22 , (2.196)

S(j)( J+1)11 = S

(j+1)( J+1)11 f

(j+1)+

[

Z(j)( J+1)11 + Z

(j)( J+1)12 g(j)

]

, (2.197)

S(j)( J+1)12 = S

(j+1)( J+1)11 f

(j+1)+

[

Z(j)( J+1)12 g(j+1) − Z

(j)( J+1)11

]

× f(j+1)+ S

(j+1)( J+1)22 + S

(j+1)( J+1)12 , (2.198)

where g(j) is defined in section 2.16 and the elements of the matrix Z(j)( J+1) aredefined by the equation

Z(j)( J+1) =

[

1 + f(j+1)+ S

(j+1)( J+1)21 f

(j+1)+ −1

g(j+1)[

1 − f(j+1)+ S

(j+1)( J+1)21 f

(j+1)+

]

g(j)

]−1

=

[

Z(j)( J+1)11 Z

(j)( J+1)12

Z(j)( J+1)21 Z

(j)( J+1)22

]

. (2.199)

Eqs. (2.195)–(2.198) are the desired S-matrix recursion formulae for the film stackproblem discussed in the previous section. An S-matrix which is not operating overa boundary between two media cannot cause any difference in the field. Hence, weget the starting point of the recursion as S( J+1)( J+1) = I, where I is the identitymatrix of appropriate size. After we derive the system S-matrix i.e. S(0)( J+1), wecan use Eq. (2.193) to solve the unknown transmitted and the reflected fields.

2.18 LOCAL FIELD

All the field quantities introduced in the previous sections are the average macro-scopic quantities. Here we introduce the concept of local field i.e. the field whichdrives atomic transitions in a material. Local field is generally different from theaverage field inside the material medium. For a medium with low atomic numberdensity the difference between the local field and the macroscopic ensemble averagefield is not significant. However, the difference increases for materials with higheratomic number density ≥ 1015 cm−3 [131]. In such a scenario, the influence of lo-cal field effects must be taken into consideration using proper mathematical modelswhich relate the local fields to the macroscopic field quantities. Using these mod-els we can investigate the effects of the local fields on the optical properties of the

38

materials. However, the choice of the model strongly depends on the material. Fora homogeneous medium, the macroscopic electric field and the local field can berelated by the following relation

E = LEloc, (2.200)

where L is known as the local field correction factor. Different theoretical modelspredict different expressions for this local field correction factor L. The most usedmodel is perhaps the Lorentz model. In the simplest version of this model, themedium is treated as a cubic lattice of point dipoles of the same kind. To evaluatethe local field acting on a particular dipole inside the medium, we need to introducethe concept of an imaginary spherical cavity surrounding the dipole of interest.The radius of the imaginary cavity is assumed to be much larger than the inter-dipole distance though much smaller than the wavelength of the incoming light. Todeduce the expression of L, we take into consideration the exact contributions tothe local field from the dipoles situated inside the imaginary spherical cavity exceptthe dipole of study whereas, the effects of all the other dipoles are expressed interms of the average macroscopic polarization P. This approach gives the followingexpression [132]

Eloc = E +4π

3P. (2.201)

Now assuming the medium to be lossless, linear, and dispersionless we can derivethe well known Clausius-Mossotti relation between the macroscopic average dielec-tric permittivity and the microscopic polarizability α, which is defined as the relativetendency of a dipole to be distorted from its normal shape by an external electricfield and/or by a nearby dipole. Microscopic polarizability relates the microscopicdipole moment (p) induced in a atom of the medium to the local field acting on theatom by the following relation

p = αEloc, (2.202)

where the macroscopic polarization and the microscopic dipole moment are relatedby

P = N p, (2.203)

N being the atomic number density. After some simple and straightforward calcu-lations we arrive at the well-known Clausius-Mossotti (or Lorentz-Lorenz) relation

ε − 1ε + 2

=4π

3Nα. (2.204)

Also, the relation between the macroscopic average electric field and the local fieldreads as

Eloc =ε + 2

3E. (2.205)

Comparing Eqs. (2.200) and (2.205) we get the expression for the local field correc-tion factor

LLor =ε + 2

3. (2.206)

The above written expression for the local field correction factor remains valid forhomogeneous media i.e. when all the particles are of the same type.

Onsager introduced a different macroscopic model [133] for relating the localfield to the macroscopic average electric field. He treated the atom/molecule under

39

study being enclosed in a a real tiny cavity. According to his model, the field actingon the molecule is divided into the cavity field (the field in the center of the realcavity when the molecule is absent) and the reaction field (the cavity field correctionfactor due to the presence of the molecular dipole at the center of the real cavity).Onsager model gives the following relation between Eloc and E

Eloc =3ε

2ε + 1E +

2(ε − 1)(2ε + 1)r3 p, (2.207)

where r is the radius of the cavity. The first term on the right hand side of Eq. (2.207)is the expression for the cavity field and the second term expresses the reaction field.Most of the experimental works can be well described either by the Lorentz modelor by the Onsager model. Both of these models can describe the optical propertiesof a guest-host system. In Lorentz model, the polarizability of the guest atom mustbe the same as that of the host molecules. On the other hand, the Onsager modelis more suitable when the polarizability of the guest molecule/atom is significantlydifferent from that of the host molecules/atoms.

So far, our discussions were restricted to the local fields in the context of linearoptics. However, in the nonlinear optical regime, local field effects are more pro-nounced. To demonstrate this, let us first assume a homogeneous, centrosymmetricmedium which can be described by the Lorentz model of the local field. Also, weassume only third-order nonlinear optical interactions at a single frequency. Hence,the macroscopic medium susceptibility in terms of the macroscopic electric field canbe written as [134]

χ(1) + 3χ(3)|E|2 + . . . . (2.208)

Let us now denote the microscopic counterparts i.e. the linear polarizability and thethird-order hyperpolarizability by γ(1) and γ(3) respectively. We proceed to deriverelations between γ(1) and χ(1), and γ(3) and χ(3) respectively. Considering slowlyvarying amplitudes (i.e. we assume that the amplitudes change negligibly overdistance scales equal to the wavelength of light) of the macroscopic electric field andthe macroscopic polarization, we can write the following expression for the Lorentzlocal field

Eloc = E +4π

3

[

P(1) + P(3) + . . .]

, (2.209)

where P(1) and P(3) are the linear and the third-order nonlinear macroscopic po-larizations respectively which are related to their microscopic counterparts by thefollowing relations

P(1) = Nγ(1)Eloc, (2.210a)

P(3) = Nγ(3)|Eloc|2Eloc. (2.210b)

Combining Eqs. (2.210a) and (2.209) we get,

P(1) =ε − 14π

[

E +4π

3P(3) + . . .

]

. (2.211)

The macroscopic electric displacement vector D by definition (in the Gaussian unit)has the following expression

D = E + 4πP(1) + 4πP(3) + . . . . (2.212)

40

If we now substitute Eq. (2.211) into Eq. (2.212), we obtain

D = εE + 4πPNL, (2.213)

wherePNL = LLor

(

P(3) + . . .)

(2.214)

is termed as the macroscopic nonlinear source polarization [135]. LLor is the localfield correction factor as defined earlier. The total polarization P can be written inthe following form using Eq. (2.210a)

P = χ(1)E + PNL. (2.215)

Furthermore, we can substitute Eq. (2.209) into Eq. (2.210b) and ignore the termsproportional to higher than the third power of the electric field to yield

P(3) = 3Nγ(3)|LLor|2LLor|E|2E. (2.216)

Finally, we substitute Eq. (2.216) into Eq. (2.214) and then substitute Eq. (2.214) intoEq. (2.215) to write the macroscopic polarization in the following form

P = χ(1)E + 3Nγ(3)|LLor|2LLor|E|2E + . . . . (2.217)

We already know that the macroscopic polarization can be written in terms of powerseries expansion of the macroscopic average electric field E i.e. in the following form

P = χE = χ(1)E + 3χ(3)|E|2E + . . . . (2.218)

Hence, comparing Eqs. (2.217) and (2.218) we obtain

χ(1) = Nγ(1)LLor, (2.219a)

χ(3) = Nγ(3)|LLor|2LLor. (2.219b)

Equations (2.219) show that the third-order nonlinear susceptibility scales as thethird power of the local-field correction factor whereas the linear optical susceptibil-ity scales as the first power of the local-field correction factor. Hence, the influenceof the local field on the nonlinear optical properties of a material can be significant.

In this context, we mention that significant control over the local field inside amedium can be achieved by nanostructuring and also by intermixing more than onehomogenous media to form a composite medium. These topics will be discussed indetail in the subsequent chapters.

2.19 SUMMARY

In this chapter we have laid out the theoretical foundation for the rest of the the-sis. The concept of polarization discussed in section 2.9 plays an important partin optics and will be used throughout the following chapters. The notion of localfield introduced in section 2.18 will be helpful for the chapters dealing with the lo-cal field effects on the linear and the nonlinear optical properties of nanostructuredmaterials.

41

3 Rigorous analysis of diffraction gratings

Periodic systems portray important aspects in science and technology. Moreover,desire for order in human society gave birth to the domain of diffraction gratingswhich refer to structures with periodically modulated optical properties [136–138].Regular perturbations affect the propagation of light through these structures andhence novel optical properties can be realized as compared to their bulk constituents.In optics, periodic modulations are seen as changes in the medium permittivity (alsochanges of permeability occurs but throughout this thesis we shall restrict ourselvesto non-magnetic materials). Since their first discovery in the 16th century by D. Rit-tenhouse [139], the diffraction gratings have emerged as useful optical devices andare most commonly used in monochromators, spectrometers, wavelength divisionmultiplexing devices, optical pulse compressing devices, and in many other opticalinstruments.

With the advancement of the domain of gratings, it was needed to develop effi-cient and accurate numerical tools for rigorous theoretical modeling of these. Thereexists a wide variety of grating modeling methods due not only to historical rea-sons, but mainly to the absence of a universal approach that could efficiently solveall diffraction problems. Though some of the methods cover broader domain ofproblems, more specialized ones are usually more efficient. The main subject of thischapter is the Fourier-Expansion Eigenmode Method or simply the Fourier ModalMethod (FMM) for modeling diffraction gratings. Here we shall discuss in detailthe implementations of the FMM for linear and multilayered gratings and lastly foranisotropic crossed gratings. Before going into the details of the FMM, we shallbriefly discuss the working principle of a grating, and some useful terms such aspseudoperiodic fields, grating equations, and the diffraction efficiencies.

3.1 WORKING PRINCIPLE OF A GRATING

In general, diffraction gratings can be broadly divided into two categories:

• Volume gratings: the grating structure is index-modulated, i.e, the permittiv-ity is varying in one or two lateral directions.

• Surface relief gratings: certain spatially varying profile is fabricated into amaterial having constant complex permittivity.

However, there also exist gratings, which combine these two types.In section 2.10 we have studied the diffraction of a plane wave from a plane

interface. This results in a single reflected and a single transmitted wave. Nowwe continue to investigate the situation when we replace the plane interface witha periodically modulated interface as shown in Fig. 3.1. Periodic modulation ofthe interface changes the impulsion (wavevector surface component) of the incidentwave along the surface i.e. kin,S by adding or subtracting an integer number ofgrating impulses (grating vectors) D. This can be expressed by the following relation

km,S = kin,S + mD, (3.1)

43

where D = 2(π/d)d is the grating vector which directs along the unit vector d.Now, if we assume that the periodicity is along x, the interface lies in the xy planeand the incident wave lies in the xz plane as depicted in the geometry of Fig. 3.1(i.e. non-conical mounting), we get the so-called grating equation in reflection

sin θm = sin θin + mλin

d, (3.2)

where m = 0, 1, 2, . . . . Clearly, all the diffraction orders are contained in the xzplane. In case the opto-geometric properties (refractive index and thickness) of theinterface are modulated two-dimensionally we can define two grating vectors D1and D2 along d1 and d2 respectively (i.e. along the directions of periodicity) asshown in Fig. 3.2. These create two sets of diffraction orders together with theirspatial combinations. The grating equation for the 2-D periodic case is written inthe following form

kmn,S = kin,S + mD1 + nD2, (3.3)

with D1 = 2(π/dx)d1 and D2 = 2(π/dy)d2.It is worth to mention here that depending on the ratio between the grating

period d and the wavelength of the incident wave λin, diffraction gratings can bedivided into three sub-groups which are paraxial (d ≫ λin) domain, resonance(d ≈ λin) domain, and quasi-static (d ≪ λin) domain gratings [136]. In the paraxialdomain, the response of the grating becomes almost polarization independent andthe grating acts as a thin transparency [140]. In the resonance domain, the gratingresponse is strongly polarization dependent [141, 142]. Finally, in the quasi-staticdomain the grating acts as a homogeneous anisotropic thin film with effective re-fractive index [143]. Both the quasi-static and the resonance domain gratings will bediscussed in detail in Chapters 4 and 5 respectively.

x

y

z

d

d

θin

Ein

kin

kref,0

kref,1

kref,2

ktra,0ktra,1

ktra,2

Figure 3.1: Lamellar grating under non-conical mounting. TM polarized light isassumed to be obliquely incident on the grating.

44

3.2 PSEUODOPERIODICITY AND GRATING EQUATIONS

Let us now consider a more general interaction geometry which is known as conicalmounting. The geometry is shown in Fig. 3.3. The plane wave with propagationvector k forms an angle θin with z axis as before but it also forms an angle φ withx axis. φ is known as the conical angle. Also the unit electric polarization vectoru forms an angle ψ with the plane of propagation. Considering the geometry of

x

y

z

dx

dy

d2

d1

Figure 3.2: Two dimensionally periodic grating geometry.

x

y

z

φ

u k

θ

Figure 3.3: Wave propagation geometry in case of conical illumination.

45

Fig. 3.3, k can be written in the following form

k = kx x + kyy + kzz = k0n(sin θ cos φx + sin θ sin φy + cos θz), (3.4)

where k0 is the free-space wave number and n is the refractive index of the medium.Now we consider a laterally periodic structure with periods dx and dy along x andy respectively as shown in Fig. 3.2. Also, we assume the periodic region to be multi-layered and bounded between the planes at z = z(0) and z = z( J+1). Complexpermittivities of the half-spaces at z < z(0) and z > z( J+1) i.e. the permittivitiesof the superstrate and the substrate are ε(0) and ε( J+1) respectively. Furthermorewe assume that the complex permittivity is z-invariant inside each layer. Now, theincident unit amplitude plane wave can be expressed as

Ein(r) = u exp[

i(kx,0x + ky,0y + kz,0,0z)]

. (3.5)

Following the conditions of periodicity, the permittivity inside the jth layer (j =1, . . . , J) can be written in the following form

ε(j)(x, y) = ε(j)(x + dx, y + dy). (3.6)

It then follows from the Floque-Bloch theorem [120] that the field inside the gratingregion as well as in the substrate and the superstrate are pseudoperiodic i.e. laterallyperiodic except for a phase factor which depends on the incident wave vector andthe transverse position. Mathematically this can be written as

E(x + dx, y + dy, z) = E(x, y, z) exp[

i(kx,0dx + ky,0dy)]

. (3.7)

If we combine the pseudoperiodic field conditions in Eq. (3.7) and the angularspectrum representation in Eqs. (2.123)–(2.124) we find that the wave vector compo-nents kx and ky can only have discrete values i.e.

kx,m = kx,0 + m2π/dx, (3.8)

ky,n = ky,0 + n2π/dy. (3.9)

These equations are identical to Eq. (3.3) in component form and hold for both thereflected and the transmitted diffraction orders. The z component of the wave vectorfor (m, n)th diffraction order is given by

k(j)z,m,n = k0n(j) cos θ

(j)m,n, (3.10)

where j = 0 for the incident and the reflected orders, while j = J + 1 for the trans-mitted orders. We can now combine Eq. (3.10) and (2.121) to deduce the three-dimensional grating equation

[

n(j) sin θ(j)n,m

]2=[

n(0) sin θ(0) cos φ(0) + mλ0/dx

]2

[

n(0) sin θ(0) sin φ(0) + nλ0/dy

]2. (3.11)

The conical angles φm,n for the reflected and the transmitted orders are deducedfrom the following relation

tanφm,n =ky,n

kx,m=

n(0) sin θ(0) sin φ(0) + nλ0/dy

n(0) sin θ(0) cos φ(0) + mλ0/dx. (3.12)

46

For non-conical incidence φ(0) = 0, if we assume 1-D periodic configuration i.e.dy → ∞, Eq. (3.11) reduces to the conventional 1-D grating equation in transmission

n(j) sin θ(j)m = n(0) sin θ(0) + mλ0/d, (3.13)

where dx has been replaced by d. The 1-D periodic y invariant grating geometry fornon-conical illumination is illustrated in Fig. 3.4. Also, as the angular spectrum isdiscrete which is easy to realize by combining Eqs. (3.8), (3.9) and (2.123), the angularspectrum representations in the homogeneous regions at z < z(0) and z > z( J+1)

reduce to Rayleigh expansions [144]. Hence we arrive at the following expressionsfor the reflected and the transmitted diffracted electric fields i.e. the expressions forthe fields in the homogeneous regions

E(0,−)(r) = ∑m

∑n

U(0,−)m,n

× exp(

i

kx,mx + ky,ny − k(0)z,m,n

[

z − z(0)])

, (3.14)

E( J+1,+)(r) = ∑m

∑n

U( J+1,+)m,n

× exp(

i

kx,mx + ky,ny + k( J+1)z,m,n

[

z − z( J+1)])

, (3.15)

where U(0,−)m,n and U

( J+1,+)m,n are the complex amplitudes of the reflected and the trans-

mitted orders respectively.

3.3 DIFFRACTION EFFICIENCIES

Among all the experimentally measurable quantities in diffractive optics, diffractionefficiency of a grating is one of the main quantities of interest. The diffractionefficiency of the (m, n)th diffraction order which is written as ηm,n is defined as theratio of the time average of the z-component of the Poynting vector 〈Sz,m,n(r, t)〉of the (m, n)th diffraction order and the time average of the z-component of thePoynting vector 〈Sz,in(r, t)〉 of the incident wave. Recalling Eq. (2.64) of section 2.7we get

〈Sz(r, t)〉 = n

2

√

ε0

µ0|Ein|2 cos θ =

12n

√

µ0

ε0|Hin|2 cos θ. (3.16)

Using the definition of the diffraction efficiencies in Eq. (3.16), we can write theefficiencies of the diffraction orders in the following form

η(j)m,n =

〈S(j)z,m,n(r, t)〉

〈Sz,in(r, t)〉 = ℜ

k(j)z,m,n

kz,0,0

∣

∣

∣U

(j)m,n

∣

∣

∣

2

|Uin|2, (3.17)

where U(j)m,n and Uin are the complex amplitudes of the (m, n)th diffraction order and

the incident field respectively. In Eq. (3.17) j = 0 designate the reflected orders andj = J + 1 the transmitted orders. We can now assume the incident field to be of unitamplitude to simplify Eq. (3.17) in the following form

η(j)m,n =

〈Sz,m,n(r, t)〉〈Sz,in(r, t)〉 = ℜ

k(j)z,m,n

kz,0,0

∣

∣

∣U

(j)m,n

∣

∣

∣

2. (3.18)

47

In case of 1-D periodic y-invariant gratings we can define the diffraction efficien-cies for TE and TM polarized light separately by decomposing Maxwell’s equationsinto two sets as described in section 2.6. In non-conical illumination the diffractionefficiencies for TE and TM polarized light can be expressed mathematically in thefollowing forms respectively:

η(j)m = ℜ

k(j)z,m

kz,0

∣

∣

∣U

(j)m

∣

∣

∣

2

|Uin|2, (3.19)

η(j)m = ℜ

n(j)k(j)z,m

n(0)kz,0

∣

∣

∣U

(j)m

∣

∣

∣

2

|Uin|2. (3.20)

3.4 OVERVIEW OF THE EXISTING NUMERICAL MODELING METH-

ODS

In all the standard full-wave numerical simulation techniques for modeling diffrac-tion gratings, the entire three-dimensional space containing the grating is dividedinto three distinct regions. The transparent semi-infinite half-space or the half spacecontaining the incident field located at the top i.e. at z < z(0) (the superstrate), theperiodically modulated region (which might include sub-regions) or the region con-taining the grating, and the bottom region at z > z( J+1) which might be opaqueto the incident field (the substrate). At first, electromagnetic fields in each regionis evaluated separately and expressed mathematically. These mathematical expres-sions contain unknown coefficients. Next, electromagnetic boundary conditions areapplied to the total fields at the interfaces of two subsequent regions or sub-regionsto match the electromagnetic fields at the boundaries and evaluate the unknowncoefficients by solving a set of linear algebraic equations. Values of these coeffi-cients are used to derive quantities of interest for instance the diffraction efficiencies,phases and polarization states of the diffracted fields, near and far field distributionsof the diffracted waves etc.

The three steps described in the above paragraph are shared by many rigorousnumerical methods for modeling diffraction gratings. Depending on the contentsof the first step (i.e. the way we solve the electromagnetic field quantities in eachregion), we can classify the available methods into two groups which are space-domain methods such as the finite-difference time-domain method (FDTD) [145] orthe finite element method [146] and spatial-frequency-domain methods, includingthe differential method [147, 148] and the Fourier modal method (FMM) [149–151].Some of the other widely used numerical methods include the integral method [152]and the coordinate transformation method [153].

3.5 FOURIER MODAL METHOD FOR DIFFRACTION GRATINGS

In a modal method the total electromagnetic fields in each region are constructed bymeans of modes. Modes can be defined as distinct, self-sustainable pattern of mo-tion satisfying the governing law of physics and the internal boundary conditions.Each of these grating modes satisfy Maxwell’s equations and the related internalboundary conditions including the pseudo-periodicity conditions. The final solu-tion of the total fields is obtained by superposing all the modes and applying the

48

external electromagnetic boundary conditions between different regions and the ra-diation conditions at infinity.

In the Fourier Modal Method (FMM), electromagnetic field quantities inside thegrating region are mathematically expressed in terms of Floquet-Fourier series andthe medium permittivities and permeabilities by Fourier series. Whereas, fields inthe superstrate and the substrate are written in terms of Rayleigh expansions whichwe have already seen in Eqs. (3.14)–(3.15). MaxwellâAZs equations are first solvedseparately in each z-independent layer, and then the external boundary conditionsare applied at the interfaces of two subsequent layers as discussed in the previousparagraph to evaluate the unknown coefficients. This is efficiently done by therecursive S-matrix propagation algorithm as discussed in section 2.17. In the FMM,a rectangle (in general a parallelogram) is used as a building block for an arbitrarygrating profile inside a z-independent layer.

3.5.1 Fourier factorization rules

We have already mentioned in the previous section that in the FMM, complex per-mittivities in the modulated region are expressed in terms of Fourier series and thefield quantities in terms of Floquet-Fourier series. It was realized that to obtain bet-ter numerical convergence especially for the metallic gratings, Fourier coefficients inthe Fourier series expansions of the complex permittivities and the field quantitiesshould be treated in a special way. This realization led to the establishment of theFourier factorization rules by Lifeng Li [154]. Introduction of these rules not onlyhelped to improve the convergence of the FMM [155,156] but also enabled a series ofother developments in rigorous grating theory. One should notice that the complexpermittivity is discontinuous along the direction of periodicity (x direction for a 1-Dlamellar grating) and Li’s Fourier Factorization rules are needed to correctly handlethe Fourier coefficients at the boundaries of discontinuity. In section 2.4, we foundthat in electromagnetic boundary conditions often we talk about the continuity ofproduct of two functions at the interface. For example in Eq. (2.37) we have thecontinuity relation for the product of the complex permittivity and the electric field.Correct Fourier factorization of the product of two functions should be carried outin such cases to avoid numerical errors.

Let us now consider three periodic functions of x, f1(x), f2(x), and f3(x). Theperiod is d for each of these functions. Also, we assume the following relationbetween these functions

f3(x) = f1(x) f2(x). (3.21)

As the functions are periodic we can represent these in the form of Fourier seriesexpansion

fi(x) =∞

∑n=−∞

fi,n exp(i2πnx/d), (3.22)

for i = 1, 2, 3. The Fourier coefficients i.e. fi,n are given by

fi,n =1d

∫ d

0fi(x) exp(−i2πnx/d) dx. (3.23)

The Fourier coefficients of f3(x) can be obtained from the Fourier coefficients off1(x) and f2(x) by using the Laurent’s rule:

f3,n =∞

∑m=−∞

f1,n−m f2,m. (3.24)

49

Hence, the Fourier factorization of f3(x) can be expressed by

f3(x) =∞

∑n=−∞

f3,n exp(i2πnx/d) =

∞

∑n=−∞

∞

∑m=−∞

f1,n−m f2,m exp(i2πnx/d). (3.25)

Equation (3.25) contains an infinite number of Fourier coefficients of the functions fi

but to solve the diffraction problem with a computer with limited resources we canonly retain a certain number of Fourier coefficients in the above equation. However,if we retain only up to Mth order Fourier coefficients i.e. m = −M → M in Eq. (3.24),depending on the kind of discontinuities the functions f1(x) and f2(x) possess, theLaurent’s rule might produce numerical errors when constructing f3(x) from theFourier coefficients of f1(x) and f2(x). Li examined three possible cases and madethree clear conclusions which are known as Li’s Fourier factorization rules. Theserules are listed below:

1. If f1(x) and f2(x) are piecewise smooth, bounded periodic functions and ifthey do not have concurrent jump discontinuities, the product (type 1 product)can be factorized by the truncated form of Laurent’s rule i.e.

f(M)3,n =

M

∑m=−M

J f1Kn−m f2,m, (3.26)

where J f1Kn−m is the symmetrically truncated Toeplitz matrix generated by theFourier coefficients of f1. The (m, n)th element of the Toeplitz matrix is f1,n−m.

2. If f1(x) and f2(x) are piecewise smooth, bounded periodic functions andif they have only concurrent pair-wise complementary jump discontinuities,there product (type 2 product) cannot be factorized by the Laurent’s rule. Insuch a scenario we must apply the inverse rule i.e.

f(M)3,n =

M

∑m=−M

J1/ f1K−1n−m f2,m. (3.27)

3. If f1(x) and f2(x) are piecewise smooth, bounded periodic functions and ifthey have concurrent but not pair-wise complementary jump discontinuities,neither the Laurent’s rule nor the inverse rule holds.

3.6 FMM FOR LINEAR GRATINGS WITH PLANE WAVE ILLUMINA-

TION

Let us now go back to the geometry in Fig. 3.4 i.e. we consider a linear y- invariantmultilayered grating under non-conical illumination. The rigorous diffraction the-ory for such a y- invariant but single layered structure under non-conical mountingwas first introduced by Knop [157]. Later the theory was extended for multilayeredgratings [149, 155, 156, 158–166], gratings under conical illumination [163, 167], andfinally to gratings with anisotropic materials [168, 169].

50

For the y- invariant geometry in Fig. 3.4, the complex permittivity in the j-thlayer can be expressed in the following form using Fourier series expansion

ε(j)(x) =∞

∑p=−∞

ε(j)p exp(i2πx/d), (3.28)

where ε(j)p is the pth Fourier coefficient in the expansion of ε(j)(x) and is given by

ε(j)p = (1/d)

∫ d

0ε(j)(x) exp(−i2πpx/d) dx. (3.29)

Putting p = 0 in Eq. (3.29), we can obtain the average value of the complex permit-tivity in the j-th layer.

3.6.1 Formulation of the eigenvalue problem

In subsection 2.6 we have already seen that if the structure is y- invariant, we candecompose Maxwell’s equations into TE and TM polarized sets. Now as the fields inthe regions z < z(0) and z > z( J+1) can be expressed in terms of Rayleigh expansions,for an incident field with complex amplitude Uin we have the following expressions

x

z

z(0) z(1) z(j−1) z(j) z( J−1) z( J)

ε(1)(x) ε(j)(x) ε( J)(x)

ε( J+1)

A(0,−)1

A(0,−)0

Ain

A(0,−)−1

0

A( J+1,+)1

A( J+1,+)0

A( J+1,+)−1

A( J+1,+)2

ε(0)

d

Figure 3.4: Diffraction geometry from a multilayered y-invariant grating.

51

for the incident, the reflected and the transmitted fields

Ey,in(x, z) = Uin exp

i(

kx,0x + kz,0

[

z − z(0)])

, (3.30)

E(0,−)y (x, z) =

∞

∑m=−∞

U(0,−)m exp

(

i

kx,mx − k(0)z,m

[

z − z(0)])

, (3.31)

E( J+1,+)y (x, z) =

∞

∑m=−∞

U( J+1,+)m exp

(

i

kx,mx + k( J+1)z,m

[

z − z( J+1)])

, (3.32)

where kx,m is given by Eq. (3.8) and k(j)z,m in Eqs. (3.31) and (3.32) are given by

k(j)z,m =

√

k20 ε(j) − k2

x,m (3.33)

and U(0,−)m , U

( J+1,+)m are the unknown amplitudes of the reflected and the transmit-

ted orders respectively.Now we proceed to find the solution of the field inside the grating region. The

electric field inside the j-th layer (z(j−1)< z < z(j)) of the modulated region can be

expressed in terms of pseudo-Fourier series

E(j)y (x, z) =

∞

∑m=−∞

A(j)y,m(z) exp (ikx,mx) , (3.34)

where A(j)y,m is the amplitude of the m-th space-harmonic field which is given by

A(j)y,m = (1/d)

∫ d

0E(j)y (x, z) exp (−ikx,mx) dx. (3.35)

If we substitute the expression of the complex permittivity from Eq. (3.28) and theexpression of the electric field from Eq. (3.34) into the Helmholtz equation for theelectric field for the y-invariant structure, i.e. Eq. (2.57) we get

−∞

∑m=−∞

k2x,mA

(j)y,m(z) exp (ikx,mx) +

∞

∑m=−∞

∂2

∂z2 A(j)y,m(z) exp (ikx,mx)

+k20

∞

∑m=−∞

∞

∑p=−∞

εp exp (i2πpx/d) A(j)y,m(z) exp (ikx,mx) = 0. (3.36)

Let us now multiply Eq. (3.36) by the term (1/d) exp (−ikx,q), where q is an integerand integrate it over the grating period i.e. from x = 0 to x = d to get the followingequation

−k2x,qA

(j)y,m(z) +

∂2

∂z2 A(j)y,m(z) + k2

0

∞

∑m=−∞

εq−m A(j)y,m(z) = 0. (3.37)

Taking a close look at the last product term of Eq. (3.37), we may conclude that Lau-rent’s rule i.e. Eq. (3.26) can be applied to correctly estimate the Fourier coefficientsof the product even if truncated summations are used. The reason is the following.In TE polarization, Ey is continuous across the discontinuities along x-direction and

hence the product of εq−m and A(j)y,m(z) is of type 1.

52

Solution for the differential equation (3.37) has a well known form

A(j)y,m(z) = A

(j)m exp

[

iγ(j)z]

, (3.38)

where γ is yet undefined constant. We can insert Eq. (3.38) into Eq. (3.37) to yield

∞

∑m=−∞

ε(j)q−mA

(j)m − k2

x,qA(j)q = A

(j)q

[

γ(j)/k0

]2. (3.39)

The above equation can be conveniently written in the following matrix form

[

Jε(j)Kq−m − Lx

]

A(j) = A(j)[

Λ(j)/k0

]2, (3.40)

where the elements of the matrix Lx are Lx,q,m = k2x,qδq,m with δq,m denoting the

Kronecker delta. A(j) is the eigenvector matrix with its columns containing the

Fourier coefficients of the electric field amplitude in the j-th layer i.e. A(j)m,ps. The

matrix Λ(j) is a diagonal matrix which contains the respective eigenvalues i.e. γ

(j)p s.

After we solve the eigenvalue problem in Eq. (3.40), we can write a single solutionof the electric field in the j-th layer. This solution is of the form

E(j)y,p(x, z) = exp

[

±iγ(j)p z] ∞

∑m=−∞

A(j)m,p exp(ikx,mx). (3.41)

These solutions remain invariant along z- direction and hence they closely resem-ble the so-called waveguide modes. However, the difference between these gratingmodes and the waveguide modes lies in the fact that the grating modes do not re-

main confined to any specific part of the grating period. γ(j)p s in view of Eq. (3.38)

are the propagation constants along z direction. Analogously to the plane wave

solutions of Maxwell’s equations, we may have two solutions for γ(j)p for each eigen-

value. These two different solutions correspond to the field modes propagatingeither along positive z or along negative z directions. To ensure numerical stabilitywhile solving the diffraction problem, it is absolutely necessary to choose the sign

of γ(j)p carefully. The rules for choosing the correct signs are as listed below

• If γ(j)p is imaginary, we choose its sign such that ℑ

γ(j)p

> 0.

• If γ(j)p is real, we choose its sign such that ℜ

γ(j)p

> 0.

Finally we can write the general solution of the electric field inside the j-th layer bysumming the modes propagating along +z and −z directions. The general solutiontakes the following form

E(j)y (x, z) =

∞

∑p=1

(

a(j)p exp

iγ(j)p

[

z − z(j−1)]

+b(j)p exp

−iγ(j)p

[

z − z(j)]

)

∞

∑m=−∞

A(j)m,p exp(ikx,mx), (3.42)

53

where a(j)p and b

(j)p s are yet unsolved complex electric field amplitudes of the forward

propagating and the backward propagating grating modes in the j-th layer. We willsee in the next section that these complex amplitudes can be determined from theelectromagnetic boundary conditions.

In a similar way, we can proceed to write the eigenvalue equation for the TMpolarized set. Let us first write the expressions of the magnetic field outside thegrating region i.e. at z < z(0) and z > z( J+1). The incident, the reflected and thetransmitted fields can be written as

Hy,in(x, z) = Cin exp

i(kx,0x + kz,0

[

z − z(0)]

)

, (3.43)

H(0,−)y (x, z) =

∞

∑m=−∞

C(0,−)m exp

(

i

kx,mx − k(0)z,m

[

z − z(0)])

, (3.44)

H( J+1,+)y (x, z) =

∞

∑m=−∞

C( J+1,+)m exp

(

i

kx,mx + k( J+1)z,m

[

z − z( J+1)])

, (3.45)

where C(0,−)m , C

( J+1,+)m are the unknown amplitudes of the reflected and the trans-

mitted magnetic fields respectively. To solve the field inside the grating region,we must pay special attention to the continuity of the field components across theboundaries of discontinuity. We can now rearrange the Helmholtz equation for themagnetic field i.e. Eq. (2.58) such that it only contains products of type 1 or type 2.Additionally assuming the complex permittivity inside the j-th layer of the gratingregion to be z-independent we can write Eq. (2.58) in the following form

ε(j)(x)

k20H

(j)y (x, z) +

∂

∂x

[

1ε(j)(x)

∂

∂xH

(j)y (x, z)

]

= − ∂2

∂z2 H(j)y (x, z). (3.46)

In TM polarization, H(j)y (x, z) and its z derivatives are continuous across the inter-

faces of discontinuity along x direction. Obviously ε(j)(x) is discontinuous acrossthe boundaries along x. The expression inside the curly bracket on the left handside of Eq. (3.46) is also discontinuous as it contains the term ε(j)(x). However, thediscontinuity jumps of the expression inside the curly bracket and that of ε(j)(x) areconcurrent and pairwise complementary. Hence their product is of type 2 and weneed to apply the inverse rule for correct Fourier factorization and to accurately esti-mate the Fourier coefficients of the product even if truncated summations are used.It is easy to check that the product inside the square brackets on the left hand side ofEq. (3.46) is also of type 2. After some lengthy though straightforward calculationswe arrive at the eigenvalue equation for TM polarized light [155]

Jζ(j)K−1q−m

[

k20 I − LxJε(j)K−1

q−mLx

]

C(j) = C(j)[

Λ(j)]2

, (3.47)

where ζ(x) = 1/ε(x). The elements of the matrix Lx are defined as before. C(j)

is the eigenvector matrix with its columns containing the Fourier coefficients of the

magnetic field amplitude in the j-th layer i.e. C(j)m,ps. The matrix Λ

(j) is a diagonal

matrix which contains the respective eigenvalues i.e. γ(j)p s like in TE polarization. As

in TE case, we can write the general solution of the magnetic field in the following

54

form

H(j)y (x, z) =

∞

∑p=1

(

a(j)1,p exp

iγ(j)p

[

z − z(j−1)]

+b(j)1,p exp

−iγ(j)p

[

z − z(j)]

)

∞

∑m=−∞

C(j)m,p exp(ikx,mx), (3.48)

where a(j)1,p and b

(j)1,ps are yet unsolved complex magnetic field amplitudes of the

forward propagating and the backward propagating grating modes in the j-th layer.

3.6.2 Solution of electromagnetic boundary conditions

Let us first consider a linear y- invariant binary grating i.e. a grating having onlyone layer. Also, let’s assume that the grating region is located between 0 < z < h.The continuity of the electric field across the boundaries of discontinuity along xfor TE polarized light input gives the following expressions at the boundaries z = 0and z = h respectively

Uinδm,0 + U(0,−)m =

∞

∑p=1

[

ap + bp exp(

iγph)]

Am,p, (3.49)

U( J+1,+)m =

∞

∑p=1

[

ap exp(

iγph)

+ bp]

Am,p. (3.50)

From Eq. (2.40) of section 2.4, we see that the x- component of the magnetic fieldover the boundaries at z = 0 and z = h is continuous. Hence we can use Eq. (2.51)to obtain

k(0)z,m

[

Uinδm,0 − U(0,−)m

]

=∞

∑p=1

[

ap − bp exp(

iγph)]

Am,p, (3.51)

k( J+1)z,m U

( J+1,+)m =

∞

∑p=1

γp

[

ap exp(

iγph)

+ bp

]

Am,p. (3.52)

Now we have four sets of equations with unknown coefficients. Solving U(0,−)m and

U( J+1,+)m from Eqs. (3.49) and (3.50), substituting their expressions in Eqs. (3.51) and

(3.52), and rearranging the terms we derive two sets of equations which contain theunknown modal amplitudes in the grating region. These two sets of equations canbe written as

∞

∑p=1

[

k(0)z,m + γp

]

Am,pap +∞

∑p=1

[

k(0)z,m − γp

]

exp(

iγph)

Am,pbp

= 2k(0)z,mUinδm,0, (3.53)

∞

∑p=1

[

k( J+1)z,m − γp

]

exp(

iγph)

Am,pap +∞

∑p=1

[

k( J+1)z,m + γp

]

Am,pbp = 0. (3.54)

Eqs. (3.53) and (3.54) can be conveniently cast in matrix form as[

M1 M2M3 M4

] [

[ap][bp]

]

=

[[

2k(0)z,mUinδm,0

]

0

]

, (3.55)

55

with

M1,m,p =[

k(0)z,m + γp

]

Am,p, (3.56)

M2,m,p =[

k(0)z,m − γp

]

exp(

iγph)

Am,p, (3.57)

M3,m,p =[

k( J+1)z,m − γp

]

exp(

iγph)

Am,p, (3.58)

M4,m,p =[

k( J+1)z,m + γp

]

Am,p. (3.59)

The unknown coefficients ap can be solved from the following matrix equation forforward propagating modal amplitudes where we have replaced the coefficients bp

as a function of ap

[

ap]

=[

M1 − M2M−14 M3

]−1 [2k

(0)z,mUinδm,0

]

. (3.60)

Once we evaluate the coefficients ap, the coefficients bp i.e. the backward propagat-ing amplitudes can be evaluated from the relation

[

bp]

= −M−14 M3

[

ap]

. (3.61)

We can now put the known values of ap and bp in Eqs. (3.49) and (3.50) to evaluatethe reflected and the transmitted electric field amplitudes. Diffraction efficiencies

can be calculated using the known values of U(0,−)m and U

( J+1,+)m and using Eq. (3.19).

Proceeding similarly we can evaluate the complex amplitudes of the magnetic fieldinside and outside the grating region.

3.6.3 Solution for multilayered gratings

If the grating region is multilayered we use the S-matrix approach as described insection 2.17 to solve the boundary-value problem. To simplify the derivation of the

S-matrix in case of multilayered gratings, let’s now introduce the quantities c(j)p such

thatc(j)p = a

(j)p exp

[

iγ(j)p h(j)

]

, (3.62)

which represent the upward propagating modes at the boundary at z = z(j). Thesymbol h(j) denotes the thickness of the j-th layer bounded between the boundariesat z = z(j) and z = z(j−1). According to the electromagnetic boundary conditions,the tangential components of the field are continuous across the boundary at z =

z(j−1). Hence, we may conveniently write the boundary-value problem in matrixform

[

A(j−1) A(j−1)

D(j−1) −D(j−1)

] [

c(j−1)

b(j−1)

]

=

[

A(j) A(j)

D(j) −D(j)

] [

F(j)− c(j)

F(j)+ b(j)

]

, (3.63)

where D(j) = A(j)Λ

(j). The matrix F(j)± contains the elements exp

[

±γ(j)p h(j)

]

δm,p.The above written matrix representation for the field inside the grating region isalso valid in the upper and the lower half-spaces. Hence these remain valid alsoat the boundaries at z = z(0) and z = z( J). However, we must be careful while

56

defining the matrices in the homogeneous regions (the upper and the lower half-

spaces). Clearly, F( J+1)± = I as the boundary at z = z( J+1) does not exist. Also in

the homogeneous regions, γ(j) in the matrix Λ(j) corresponds to the z- component

of the wave vector. c(0) corresponds to the complex amplitudes of the incidentelectric field Uin in Eq. (3.30). For unit amplitude incident field, c(0) = 1 and all theother elements equal to zero. Additionally, b(0) and c( J+1) represent the complexamplitudes of the reflected and the transmitted fields respectively. Furthermore, weassume that the incident field exists only in region 0 and hence b( J+1) = 0.

As in the thin film stack problem, we need to find the S-matrix S( J+1)(0) whichconnects the input and the output field quantities by the following relation

[

c( J+1)

b(0)

]

= S( J+1)(0)

[

c(0)

b( J+1)

]

=

[

S( J+1)(0)11 S

( J+1)(0)12

S( J+1)(0)21 S

( J+1)(0)22

] [

c(0)

b( J+1)

]

, (3.64)

where the superscripts (J + 1) (0) signify that the total S-matrix must be con-structed layer by layer starting from region J + 1 and proceeding towards region 0.Starting from the sub-system S-matrix S(j+1)( J+1) and employing the boundary-value equations we can find an expression for the sub-system matrix S(j)( J+1) interms of the elements of S(j+1)( J+1) in a similar way as discussed in section 2.17.Also, we can find similar expressions i.e. Eqs. (2.193)–(2.198) for the elements of

the matrix S. We note that in the homogeneous regions, the elements S( J+1)(0)12

and S( J+1)(0)22 vanish as b( J+1) = 0. We can solve for the remaining S-matrix ele-

ments S( J+1)(0)11 and S

( J+1)(0)21 using Eqs. (3.63)–(3.64). After some lengthy though

straightforward calculations, we arrive at

S( J+1)(j−1)11 = S

( J+1)(j)11 F

(j)+

×[

Z( J+1)(j−1)11 A(j−1) + Z

( J+1)(j−1)12 D(j−1)

]

, (3.65)

S( J+1)(j−1)21 = Z

( J+1)(j−1)21 A(j−1) − Z

( J+1)(j−1)22 D(j−1), (3.66)

where the elements of the matrix Z( J+1)(j−1) are defined as

Z( J+1)(j−1) =

[

Z( J+1)(j−1)11 Z

( J+1)(j−1)12

Z( J+1)(j−1)21 Z

( J+1)(j−1)22

]

=

[

A(j)[F(j)+ S

( J+1)(j)21 F

(j)+ + I] −A(j−1)

D(j)[I − F(j)+ S

( J+1)(j)21 F

(j)+ ] D(j−1)

]−1

. (3.67)

Finally, the reflected and the transmitted field in the homogeneous regions aresolved using Eq. (3.64) and we get the following expressions for the reflected andthe transmitted field complex amplitudes respectively

b(0) = c(0)S( J+1)(0)21 = U0,−, (3.68)

c( J+1) = c(0)S( J+1)(0)11 = U J+1,+. (3.69)

Once we solve the amplitudes of the electric field, we can estimate the diffractionefficiencies using Eq. (3.19). In this context we must mention some additional prop-erties of the S-matrix. We already know that an S-matrix which is not operating over

57

an interface does not produce any change in the field. Hence we only have the diag-onal elements and these elements are equal to identity matrix I. All the off-diagonalelements vanish for a S-matrix which is not operating over an boundary.

3.6.4 Field inside the gratings

It appears from Eq. (3.64) that the field inside the grating layers can be easily solved.

However, this is not true. The inversion of the S-matrix element S( J+1)(j)11 is unstable

and might lead to large numerical errors. Hence, we need to introduce a new S-matrix for the field calculations inside the grating layers. Let’s start from the originalfield representation in Eq. (3.48) inside the j-th grating layer. We may write theboundary value problem at z = z(j) in matrix form

[

A(j) A(j)

D(j) −D(j)

] [

F(j)+ a(j)

b(j)

]

=

[

A(j+1) A(j+1)

D(j+1) −D(j+1)

] [

a(j+1)

F(j+1)+ b(j+1)

]

. (3.70)

To avoid numerical errors, this time we start constructing the S-matrix from the 0-thlayer and proceed layer by layer to reach the region J + 1. As the field representationsas well as the way of the matrix construction are different, we define a new S-matrixW (0)(j). This W matrix connects the field in region 0 and the field inside the j-thlayer of the grating. The matrix relation reads as

[

a(j)

b(0)

]

= W (0)(j)

[

a(0)

b(j)

]

=

[

W(0)(j)11 W

(0)(j)12

W(0)(j)21 W

(0)(j)22

] [

a(0)

b(j)

]

. (3.71)

As before we start from the sub-system S-matrix W (0)(j) and employing theboundary-value equations proceed to find expression for the sub-system matrixW (0)(j+1) in terms of the elements of the S-matrix W (0)(j). We assume that thereis no incident field in the region J + 1, to obtain the following expression for theamplitudes of the backward propagating field. Hence, using Eqs. (3.62) and (3.64)we get

b(j) = S( J+1)(j)21 F

(j)+ a(j), (3.72)

The forward propagating field amplitudes can be deduced similarly using Eqs. (3.71)and (3.72)

a(j) =[

I −W(0)(j)12 S

( J+1)(j)21 F

(j)+

]−1W

(0)(j)11 a(0). (3.73)

The matrices W(0)(j)11 and W

(0)(j)12 can be derived using Eqs. (3.70) and (3.71).

Again after some lengthy though straightforward calculations, we arrive at

W(0)(j+1)11 = −

[

Y(0)(j+1)11 A(j) + Y

(0)(j+1)12 D(j)

]

F(j)+ W

(0)(j)11 , (3.74)

W(0)(j+1)12 =

[

Y(0)(j+1)11 A(j+1) − Y

(0)(j+1)12 D(j+1)

]

F(j+1)+ , (3.75)

where the elements of the matrix Y (0)(j+1) are defined as

Y (0)(j+1) =

[

Y(0)(j+1)11 Y

(0)(j+1)12

Y(0)(j+1)21 Y

(0)(j+1)22

]

=

[

−A(j+1) A(j)[F(j)+ W

(0)(j)12 + I]

−D(j+1) D(j)[F(j)+ W

(0)(j)12 − I]

]−1

. (3.76)

58

From Eqs. (3.72) and (3.73) we can see that the only non-vanishing element of the

S-matrix S( J+1)(j) is S( J+1)(j)21 . Hence without solving the elements S

( J+1)(j)11 , we

can solve the complex amplitudes of the transmitted field from Eq. (3.75).

3.7 FMM FOR ANISOTROPIC CROSSED GRATINGS

In this section, we shall include the formulation for the most general FMM foranisotropic media where the permittivities and the permeabilities are tensors. Alsoto keep the most general form of the FMM, we assume a slanted coordinate systemas shown in Fig. 3.5. None of the coordinate axes are perpendicular to each other.This formulations is based on Lifeng Li’s article published in 2003 i.e. Ref. [170]. Letus first introduce the covariant and the contravariant basis vectors [150] ui and uj

such that ui · uj = δij, where δij is the Kronecker delta symbol.In the slanted coordinate system of Fig. 3.5 these covariant and contravariant

x = x1, x1Φ x2

ζ

y = x2

x3Θ

z = x3

Figure 3.5: Slanted coordinate system.

tensors can be written in the form

u1 = x, (3.77a)

u2 = x sin ζ + y cos ζ, (3.77b)

u3 = x sin Θ cos Φ + y sin Θ sin Φ + z cos Θ, (3.77c)

u1 = x − y tan ζ + z tan Θ(sin Φ tan ζ − cos Φ), (3.78a)

u2 = y sec ζ − z tan Θ sin Φ sec ζ, (3.78b)

u3 = z sec Θ, (3.78c)

59

x1 = x − y tan ζ + z tan Θ(sin Φ tan ζ − cos Φ), (3.79a)

x2 = y sec ζ − z tan Θ sin Φ sec ζ, (3.79b)

x3 = z sec Θ. (3.79c)

The contravariant elements of the permittivity tensor in terms of the permittivitytensor elements ¯ετχ in the Cartesian system take the form

ερσ =∂xρ

∂xτ

∂xσ

∂xχ¯ετχ, (3.80)

where x1 = x, x2 = y, and x3 = z are the Cartesian coordinates and we have usedEinstein’s summation notation, i.e. we have used summation with respect to a pairof identical covariant and contravariant indices. Proceeding analogously for therelative permeability tensor we find

µρσ =∂xρ

∂xτ

∂xσ

∂xχ¯µτχ. (3.81)

Here we emphasize the fact that though the relative permeability tensor µ = I atoptical frequencies, for non-orthogonal coordinate systems we also have non-zerooff-diagonal elements. Consequently, permeability tensor is no more an identitytensor. Also, we point out that mathematically there is no difference between ma-terial anisotropy and anisotropy arising from the coordinate system. To keep thederivations more general, we assume physically anisotropic media together withthe slanted coordinate system. For isotropic media ¯ετχ = εδτ,χ, ¯µτχ = µδτ,χ andEqs. (3.80) and (3.81) reduce to

ερσ = εgρσ, (3.82a)

µρσ = µgρσ, (3.82b)

where

gρσ =∂xρ

∂xτ

∂xσ

∂xχ(3.83)

is the covariant metric tensor of the Cartesian system. Denoting its (gρσ) reciprocalby g, we get from Eq. (3.78c), g = cos2 ζ cos2 Θ. We now proceed to derive theEigenvalue equations. As before, we assume the optical properties of the materialto be invariant along x3 direction in a single layer j. In the following expressions wedrop the index j as well as the position dependence for the sake of brevity.

At first, we express space-frequency domain Maxwell’s equations i.e. Eqs. (2.13)–(2.16) in covariant forms

κρστ ∂

∂xσEτ = ik0

√gµρσNσ, (3.84a)

κρστ ∂

∂xσNτ = −ik0

√gερσEσ, (3.84b)

∂

∂xτετσEσ = 0, (3.84c)

∂

∂xτµτσNσ = 0, (3.84d)

where κρστ is the Levi-Civita symbol which is defined as

κ123 = κ231 = κ312 = 1, κ321 = κ213 = κ132 = −1,

κρστ = 0 for all other combinations, (3.85)

60

and Nσ =√

µ0/ε0Hσ. As the fields inside the modulated regions are pseudoperi-odic, we can express them in terms of Pseudo-Fourier series as we did in case of y-invariant gratings

Eσ(x1, x2, x3) = ∑p

∑q

Eσpq(x3) exp(

ik1,px1 + ik2,qx2)

, (3.86a)

Nσ(x1, x2, x3) = ∑p

∑q

Nσpq(x3) exp(

ik1,px1 + ik2,qx2)

, (3.86b)

where Eσpq(x3) and Nσpq(x3) are the Fourier coefficients of the electric and the mag-netic fields respectively. Clearly, these coefficients are x3 dependent. From elemen-tary tensor theory we know that the covariant components of a vector a(x1, x2, x3)i.e. av, where v is either ρ or σ, is tangential to the coordinate surface xτ = xτ

0at a fixed point (x1

0, x20, x3

0), whereas the contravariant component aτ is the normalcomponent at the same position. Hence recalling Eq. (2.26) of section 2.3, the con-travariant components Dv of the electric displacement can be written as

D1 = ε11E1 + ε12E2 + ε13E3, (3.87a)

D2 = ε21E1 + ε22E2 + ε23E3, (3.87b)

D3 = ε31E1 + ε32E2 + ε33E3. (3.87c)

Clearly, the contravariant component D1 of the electric displacement is continuousalong x1 direction. Additionally, the covariant electric field components E2 and E3are continuous along x1.

We now write Eqs. (3.87a)–(3.87c) in the form that these includes only type 1 andtype 2 products (we introduced these two types of products while describing Li’sFourier factorization rules in section 3.5.1) along x1 direction.

D1 = ε11[

E1 +

(

ε12

ε11

)

E2 +

(

ε12

ε11

)

E3

]

, (3.88a)

D2 =

(

ε21

ε11

)

D1 +

(

ε22 − ε21ε12

ε11

)

E2 +

(

ε23 − ε21 ε13

ε11

)

E3, (3.88b)

D3 =

(

ε31

ε11

)

D1 +

(

ε32 − ε31ε12

ε11

)

E2 +

(

ε33 − ε31 ε13

ε11

)

E3. (3.88c)

We can conveniently cast Eqs. (3.88a)–(3.88c) in the following block matrix form

D1

D2

D3

= Q

E1E2E3

, (3.89)

where [169]

Q =

⌈ξ11⌉−1 ⌈ξ11⌉−1⌈ε12ξ11⌉⌈ε21ξ11⌉⌈ξ11⌉−1 ⌈ε21ξ11⌉⌈ξ11⌉−1⌈ε12ξ11⌉+ ⌈ε22 − ε21 ε12ξ11⌉⌈ε31ξ11⌉⌈ξ11⌉−1 ⌈ε31ξ11⌉⌈ξ11⌉−1⌈ε12ξ11⌉+ ⌈ε32 − ε31 ε12ξ11⌉

⌈ξ11⌉−1⌈ε13ξ11⌉⌈ε21ξ11⌉⌈ξ11⌉−1⌈ε13ξ11⌉+ ⌈ε23 − ε21 ε13ξ11⌉⌈ε31ξ11⌉⌈ξ11⌉−1⌈ε13ξ11⌉+ ⌈ε33 − ε31 ε13ξ11⌉

. (3.90)

61

⌈·⌉ denotes a Toeplitz matrix formed by the Fourier coefficients in x1 direction, andξ ijs are defined as ξ ij = 1/εij.

To simplify the notations, let us now introduce the following operator nota-tion. We consider an arbitrary matrix O with its elements given by Oρσ, where(ρ, σ)=(1, 2, 3). Now we define the operator l±τ in such a way that when it operateson O we get P = l±τ (O) where

Pρσ =

(Oττ)−1, if ρ = τ, σ = τ

(Oττ)−1Oτσ, if ρ = τ, σ 6= τ

Oρτ(Oττ)−1, if ρ 6= τ, σ = τ

Oρσ ±Oρτ(Oττ)−1Oτσ, otherwise.

(3.91)

In the above definitions we have assumed that the inverse of the matrix Oττ existsand Oρσ are either scalars or square matrices. From the above definitions it is easyto check that l+τ l−τ (O) = l−τ l+τ (O) = O. Furthermore, we define the operator Fτ

such that when it operates on an arbitrary matrix (block or ordinary) with elementsPρσ(x1, x2, x3), the resulting matrix R = Fτ(P) is a block matrix with its elementsRρσ being Toeplitz matrices generated by the Fourier coefficients of Pρσ(x1, x2, x3)with respect to xτ. Clearly the operator Fτ changes the dimensions of the operandmatrix but the operators l±τ retain the dimensions of the operand matrix. Equation(3.90) can be written in a compact form using the operator notations introducedabove

Q = L1(ε), (3.92)

whereLτ = l+τ Fτ l−τ . (3.93)

Now, we go back to Eq. (3.88b) and rewrite this in the following form

D2 = Q21E1 + Q22E2 + Q23E3. (3.94)

The field components D2, E1, and E3 are continuous along x2 direction. In theappendix of Ref [170], Li has shown that the Fourier coefficients of the componentsD2, E1, and E3 calculated along x1 direction are also continuous along x2 direction.Thus, we may conclude that the vectors D2, E1, and E3 are also continuous along x2

direction. Multiplying the left hand side of Eq. (3.94) with (Q22)−1 from left we get

E2 = (Q22)−1D2 −[

(Q22)−1Q21]

E1 −[

(Q22)−1Q23]

E3. (3.95)

Eliminating the discontinuous E2 from Eqs. (3.88a) and (3.88c) we can rewrite thesein the following form

D1 =[

Q12(Q22)−1]

D2 +[

Q11 − Q12(Q22)−1Q21]

E1

+[

Q13 − Q12(Q22)−1Q23]

E3, (3.96)

D3 =[

Q32(Q22)−1]

D2 +[

Q31 − Q32(Q22)−1Q21]

E1

+[

Q33 − Q32(Q22)−1Q23]

E3, (3.97)

Examining Eqs. (3.95)–(3.97) we see that all the products are of type 1 because theelements of the vectors are continuous along x2 direction. Hence, we can applyLaurent’s rule for the Fourier factorization.

62

First we calculate the Fourier coefficients of all the vectors appearing in Eqs. (3.95)–(3.97). From Eq. (3.95) we can now solve the expression of Dτ as

Dρmn = ∑

p,qν

ρσmn,pqEσ,pq, (3.98)

whereν = L2L1(ε). (3.99)

In the notation νρσmn,pq, the first index of each pair (m and p) correspond to the Fourier

coefficients along x1 direction while second index of each pair (n and q) correspondto the Fourier coefficients along x2 direction.

Using Eqs. (3.84b), (3.89), and (3.98) we get

ky,nN3,mn + i∂

∂x3 N2,mn = −k∗0 ∑p,q

ν1σmn,pqEσ,pq, (3.100a)

−kx,mN3,mn − i∂

∂x3 N1,mn = −k∗0 ∑p,q

ν2σmn,pqEσ,pq, (3.100b)

kx,nN2,mn − ky,nN1,mn = −k∗0 ∑p,q

ν2σmn,pqEσ,pq, (3.100c)

where k∗0 is defined as k∗0 = k0√

g. Proceeding analogously, we can derive therelations between the magnetic induction and the magnetic field which gives

Tρmn = ∑

p,qβ

ρσmn,pqNσ,pq, (3.101)

where Tσ =√

µ0/ε0Bσ andβ = L2L1(µ). (3.102)

Proceeding as before we now obtain

ky,nE3,mn + i∂

∂x3 E2,mn = −k∗0 ∑p,q

β1σmn,pqNσ,pq, (3.103a)

−kx,mE3,mn − i∂

∂x3 E1,mn = −k∗0 ∑p,q

β2σmn,pqNσ,pq, (3.103b)

kx,nE2,mn − ky,nE1,mn = −k∗0 ∑p,q

β2σmn,pqNσ,pq. (3.103c)

From Eqs. (3.100a)–(3.100c) and Eqs. (3.103a)–(3.103c) we can now eliminate E3,mn

and N3,mn respectively to obtain first-order differential equations along x3 direction.As in case of a y- invariant linear grating we can express the differential equationsas an eigenvalue problem which can be written as

MG = γG. (3.104)

where γ corresponds to the propagation constants in the x3 dependent term -

exp[

iγ

x3 − x3,(j)]

,

G =

E1E2N1

N2

(3.105)

63

and

M =

−µ23Y − X ǫ31 −µ23X − X ǫ32

µ13Y − Y ǫ31 −µ13X − Y ǫ32

−k∗0 ǫ21 − (1/k∗0)X µ33Y −k∗0 ǫ22 + (1/k∗0)X µ33X

k∗0 ǫ11 − (1/k∗0)Y µ33Y k∗0ǫ12 + (1/k∗0)Y µ33X

k∗0 µ21 + (1/k∗0)X ǫ33Y k∗0 µ22 − (1/k∗0)X ǫ33X

−k∗0 µ11 + (1/k∗0)Y ǫ33Y −k∗0 µ12 − (1/k∗0)Y ǫ33X

−ǫ23Y − X µ31 ǫ23X − X µ32

ǫ13Y − Y µ31 −ǫ13X − Y µ32

. (3.106)

where (X)mn,pq = kx,mδmpδnq, (Y)mn,pq = ky,nδmpδnq and

ǫ = l−3 (ν), µ = l−3 (β). (3.107)

Clearly from Eqs. (3.104) and (3.106), the total number of eigenvalues depend onthe truncation order i.e. on the total number of diffraction orders along x1 andx2 directions respectively. Let us denote the truncation orders along x1 and x2 byP1 and P2 respectively. Hence, the size of the matrix M becomes 4P1P2 × 4P1P2.Also, the number of the grating modes inside the structure as well as the numberof eigenvalues is 4P1P2 × 4P1P2. Each eigenvalue gives the propagation constant ofa mode propagating either along +x3 direction or along −x3 direction inside thej-th layer. To keep the S-matrix recursion stable, we divide the eigenvalues into twosets following the rules introduced in section 3.6.1. The first set corresponds to themodes propagating along +x3 direction whereas the second set corresponds to themodes propagating along −x3 direction (we denote these modes by the superscripts+/−). It is worth to mention that it is better to have an equal number of elements inthe two sets to ensure improved numerical stability. Each mode has certain spatialstructure governed by its corresponding eigenvector. The expressions of the electricand the magnetic fields inside the j-th grating layer can be written in the followingforms

E(j)σ (x1, x2, x3) = ∑

m∑n

∑q

(

u(j)q E

(j,+)σ,mn,q exp

iγ(j,+)q

[

x3 − x3,(j)]

+d(j)q E

(j,−)σ,mn,q exp

iγ(j,−)q

[

x3 − x3,(j)])

exp i(

k1mx1 + k2nx2)

, (3.108)

N(j)σ (x1, x2, x3) = ∑

m∑n

∑q

(

u(j)q N

(j,+)σ,mn,q exp

iγ(j,+)q

[

x3 − x3,(j)]

+d(j)q N

(j,−)σ,mn,q exp

iγ(j,−)q

[

x3 − x3,(j)])

exp i(

k1mx1 + k2nx2)

, (3.109)

where the indices σ = (1, 2, 3), u(j)q and d

(j)q are the forward and the backward

propagating yet unknown modal amplitudes. As before, these modal amplitudescan be solved using S-matrix algorithm.

After splitting the modes equally into two sets, we have 2P1P2 number of modesin each set and for each element E

j,±σ,mn,q or N

j,±σ,mn,q we have P1P2 number of Fourier

coefficients. Hence all the eigenvectors can be arranged into four 2P1P2 × 2P1P2

64

matrices Rj, where j = 1, 2, 3, 4, such that

[

R1(j)]

pq=

[

E(j,+)1,q

]

p, p = 1, 2, . . . P1P2,

[

E(j,+)2,q

]

p, p = P1P2 + 1, P1P2 + 2, . . . 2P1P2,

, (3.110)

[

R2(j)]

pq=

[

E(j,−)1,q

]

p, p = 1, 2, . . . P1P2,

[

E(j,−)2,q

]

p, p = P1P2 + 1, P1P2 + 2, . . . 2P1P2,

, (3.111)

[

R3(j)]

pq=

[

N(j,+)1,q

]

p, p = 1, 2, . . . P1P2,

[

N(j,+)2,q

]

p, p = P1P2 + 1, P1P2 + 2, . . . 2P1P2,

, (3.112)

[

R4(j)]

pq=

[

N(j,−)1,q

]

p, p = 1, 2, . . . P1P2,

[

N(j,−)2,q

]

p, p = P1P2 + 1, P1P2 + 2, . . . 2P1P2.

(3.113)

In other words,

[

R1(j) R2(j)

R3(j) R4(j)

]

=

E(j,+)1 E

(j,−)1

E(j,+)2 E

(j,−)2

N(j,+)1 N

(j,−)1

N(j,+)2 N

(j,−)2

, (3.114)

where E(j,+)1 is a matrix of size P1P2 × 2P1P2 with its columns given by E

(j,+)1,q . The

size of the column vector E(j,+)1,q is P1P2 × 1.

Using the notations introduced above, we can write the continuity relation at theinterface between layers j and j + 1. Using Eqs. (3.108) and (3.109) we get

[

R1(j) R2(j)

R3(j) R4(j)

] [

u(j)

d(j)

]

=

[

R1(j+1) R2(j+1)

R3(j+1) R4(j+1)

] [

Fj+1,+− 0

0 Fj+1,−+

]

×[

u(j+1)

d(j+1)

]

, (3.115)

where[

Fj+1,±−

]

pq= δpq exp

[

−iγj,±p h3

]

,[

Fj+1,±+

]

pq= δpq exp

[

iγj,±p h3

]

, (3.116)

and h3 = x3,(j+1)− x3,(j). The modal amplitudes in the j + 1th and the J + 1th layersare connected by the S-matrix (S(j+1)( J+1)) relation

[

u( J+1)

d(j+1)

]

= S(j+1)( J+1)

[

u(j+1)

d( J+1)

]

= (3.117)

[

S(j+1)( J+1)uu u(j+1) + S

(j+1)( J+1)ud d( J+1)

S(j+1)( J+1)du u(j+1) + S

(j+1)( J+1)dd d( J+1)

]

, (3.118)

65

Now we can write the expressions for the elements of the S-matrix as we did in thecase of thin-film stack in section 2.17.

S(j)( J+1)uu = S

(j+1)( J+1)uu F

j+1,++ (3.119)

×[

Z(j)( J+1)uu R1(j) + Z

(j)( J+1)ud R3(j)

]

,

S(j)( J+1)ud = S

(j+1)( J+1)ud

−S(j+1)( J+1)uu F

j+1,++

[

Z(j)( J+1)du R2(j+1) + Z

(j)( J+1)dd R4(j+1)

]

, (3.120)

S(j)( J+1)du = Z

(j)( J+1)du R1(j) + Z

(j)( J+1)dd R3(j), (3.121)

S(j)( J+1)dd = −

[

Z(j)( J+1)du R2(j+1) + Z

(j)( J+1)dd R4(j+1)

]

Fj+1,−+

×S(j+1)( J+1)dd , (3.122)

where

Z(j)( J+1) =

[

Z(j)( J+1)uu Z

(j)( J+1)ud

Z(j)( J+1)du Z

(j)( J+1)dd

]

[

R1(j+1) + R2(j+1)Fj+1,−+ S

(j+1)( J+1)du F

j+1,++ −R2(j)

R3(j+1) + R4(j+1)Fj+1,−+ S

(j+1)( J+1)du F

j+1,++ −R4(j)

]−1

. (3.123)

Equations (3.119)–(3.122) can be used to derive the complex amplitudes of the diffrac-tion orders. Noting d( J+1) = 0 as before, we can find the following expressions ofthe fields in regions j = 0 and j = J + 1 respectively

E(j)σ (x1, x2, x3) = ∑

m∑n

(

E(j,+)σ,mn exp

ik(j,+)3,mn

[

x3 − x3,(j)]

+E(j,−)σ,mn exp

ik(j,−)3,mn

[

x3 − x3,(j)])

exp i(

k1mx1 + k2nx2)

, (3.124)

N(j)σ (x1, x2, x3) = ∑

m∑n

(

N(j,+)σ,mn exp

ik(j,+)3,mn

[

x3 − x3,(j)]

+N(j,−)σ,mn exp

ik(j,−)3,mn

[

x3 − x3,(j)])

exp i(

k1mx1 + k2nx2)

, (3.125)

where x3,( J+1) = x3,( J).

k(j,±)3,mn =

1g33

[

±k3,(j)mn − g13k1m − g23k2n

]

, (3.126)

and

k3,(j)mn =

(

g33

[

k(j)]2

− k21mg11 − k2

2ng22 − 2k1mk2ng12

+(

k1mg13 + k2ng23))1/2

(3.127)

are the third covariant and contravariant tensors components respectively of thewave vectors of the diffraction orders.

66

3.8 STAIRCASE APPROXIMATION

In the previous sections, while formulating the eigenvalue problems inside the j-thgrating layer, we have always assumed the layer to be x3 or z independent. To solvethe multilayered grating problems i.e. to connect the fields in several z- independentlayers, we have used recursive S-matrix algorithm. If the grating profile is arbitraryalong x3 direction, we slice the grating region into many layers parallel to the grat-ing plane and in each layer the medium boundary is locally substituted with anx3-invariant boundary. This type of approximation is known as the staircase approx-imation. As the number of slices tends to infinity and the thickness of a single slicetends to zero, the modified structure tends the original one. Hence, the approxima-tion seems to be reasonable.

However, several numerical experiments have demonstrated that for 1-D periodicgratings, the staircase approximation produces accurate results only if the incidentlight is TE polarized. For TM polarization and especially for highly conductingmetallic gratings, this approximation produces large numerical errors. At the sharpedge of a wedge shaped structure, the electric field component parallel to the edgedirection is finite but the component transverse to the edge direction becomes infi-nite [171]. In staircase approximation, we artificially create many such edges. Hence,for TM polarized incident field, the electric near field which should be finite becomesinfinite at the edges of the staircase boundary. This alters the total near as well as thefar fields. For one-dimensional gratings in conical mounting and crossed gratingsunder any incidence condition, such type of numerical errors are unavoidable.

Also in terms of computing efficiency, staircase approximation makes the so-lution of the grating problem inefficient as the computation time as well as thecomputing resources grow almost linearly with the total number of z-independentslices. On the contrary, use of a larger number of slices produces more accurateresults. Thus we need a trade-off between these two. Nevertheless, the advantageof the staircase approximation is that if it produces accurate numerical results, it iseasy to implement algorithmically.

3.9 SUMMARY

This chapter covers from the basic diffraction grating principles to the detailed rig-orous mathematical formulations needed to treat these. Though several rigourousmethods exist to date for modeling diffraction gratings accurately, in this chapterwe have restricted ourselves only to the detailed mathematical formulations of theFMM for both one-dimensionally periodic structures and two-dimensionally peri-odic structures with anisotropic materials (which is the most general FMM). Theone-dimensional formulation will be helpful to explain the theory of form birefrin-gence in Chapter 5. The most general anisotropic FMM formulation will be usefulfor the next chapter where we shall talk about modeling optical Kerr nonlinearityby the linear FMM. Inclusions of the effects of periodicity along the third directionis out of scope for this thesis.

67

4 Light propagation in Periodic media with optical Kerr

nonlinearity

After the discovery of lasers in 1960 [172], it had become possible to study thebehavior of light in optical materials at higher intensities than previously possible.This gave birth to the field of nonlinear optics, where the term ’nonlinear’ denotesthat the response of a medium to an externally applied field is a nonlinear functionof the field. Since the discovery of second harmonic generation in 1961 [40], the fieldof nonlinear optics has flourished rapidly.

Certain nonlinear processes arise when the response of a material varies withthe incident electric field to the third power (a cubic relationship). These are knownas third-order processes. The optical Kerr effect, where the refractive index of amaterial depends on the intensity of light used to measure it, is one of these third-order nonlinear optical processes [134].

Besides the linear optical properties, nonlinear optical properties of a materialcan also be tailored by nanostructuring. In the last few decades, a wide variety ofdevice applications of optical Kerr nonlinearity have emerged, especially in the con-text of integrated optics (IO) [173]. Some of these include power-dependent gratingand prism couplers [174–176], directional couplers [177], Mach-Zehnder interferom-eters [178], and all-optical switching devices [179,180], couplers, and gates [177,178].Many of these nonlinear IO devices use periodically varying refractive-index pro-files. Examples include optically-tunable filters, multiplexers [181], and distributed-feedback bistable optical devices [85, 182, 183]. However, free-space diffractive ele-ments employing Kerr materials have also been proposed [184] which can be strictlyperiodic (gratings) or non-periodic, such as Fresnel zone plates and other diffractivelenses.

In this Chapter, we introduce a FMM based technique for modeling light prop-agation in periodic structures with optical Kerr nonlinear media and verify the ac-curacy of the proposed technique by comparing our results with those obtainedby other standard numerical approaches such as the differential method and theFDTD [185, 186]. After that, we discuss about the procedure to increase the compu-tational efficiency of our method by use of symmetries. Finally, we include severalnumerical examples to demonstrate the versatility of the FMM based technique.These examples include the roles of surface plasmon resonance and waveguide res-onance on the enhancement of the optical Kerr effect in nanostructured materials.

4.1 LIGHT PROPAGATION IN ISOTROPIC THIRD ORDER NONLIN-

EAR MATERIALS

In section 2.3, we have seen that an optical medium responds in a complicated wayif the incident field is strongly intense. Assuming the medium to be lossless andnon-dispersive (hence also instantaneous), the ith component, where i stands for thecartesian indices 1,2, or 3 (alternatively x,y, or z), of time-dependent macroscopic

69

electric polarization can be written as a power series expansion

Pi(t) = ε0

[

χ(1)ij Ej(t) + χ

(2)ijk Ej(t)Ek(t)

+χ(3)ijklEj(t)Ek(t)El(t) + . . .

]

= P(1)i (t) + P

(2)i (t) + P

(3)i (t) + . . . (4.1)

where χ(n) (the nth order susceptibility) is a tensor of rank (n + 1) which contains3(n+1) elements in general. In particular

P(1)i (t) = ε0χ

(1)ij Ej(t), P

(2)i (t) = ε0χ

(2)ijk Ej(t)Ek(t), P

(3)i (t) = ε0χ

(3)ijklEj(t)Ek(t)El(t)

are the linear, the second-order and the third-order nonlinear macroscopic polar-izations respectively. Usually, the physical processes which occur as a result of thesecond-order polarization are distinct from those resulting from the third-order po-larization.

In general, the third-order susceptibility χ(3)ijkl(r) is a fourth-rank tensor contain-

ing 81 distinct elements. In an isotropic medium, the axes 1, 2, or 3 are equivalent.As a result, the tensor contains only 21 non-zero elements. It is easy to show thatfor an isotropic material the components of the susceptibility tensor possess thefollowing symmetry conditions:

χ1111 = χ2222 = χ3333,

χ1122 = χ1133 = χ2211 = χ2233 = χ3311 = χ3322,

χ1212 = χ1313 = χ2323 = χ2121 = χ3131 = χ3232,

χ1221 = χ1331 = χ2112 = χ2332 = χ3113 = χ3223. (4.2)

Also, one can show, by requiring that the values of the nonlinear polarization arethe same when calculated in two different cartesian coordinate systems which arerotated with respect to one another, that the tensor components χ1111, χ1122, χ1212,and χ1221 are related by the following equation

χ1111 = χ1122 + χ1212 + χ1221. (4.3)

Finally, the third-order susceptibility of an isotropic third-order nonlinear materialcan be conveniently written in the following compact form

χijkl = χ1122δijδkl + χ1212δikδjl + χ1221δilδjk, (4.4)

which shows that when the field frequencies are arbitrarily chosen, the third-ordersusceptibility tensor contains only three independent elements.

4.2 THEORY OF THE OPTICAL KERR EFFECT IN ISOTROPIC ME-

DIA

The refractive index of a third-order nonlinear material strongly depends on theintensity of the light used to measure it. This phenomenon is termed as the opticalKerr effect by analogy with the traditional Kerr electrooptic effect discovered in1875 by a Scottish physicist John Kerr [187], where the refractive index change of amaterial is proportional to the square of the static electric field applied across it.

70

Let us now consider that an intense wave of frequency ω is propagating in anisotropic third-order nonlinear medium. If we assume the simple case in which theapplied field is monochromatic and is given by

E(t) = U cos ωt, (4.5)

the time dependent third-order contribution to the nonlinear polarization can bewritten as

P(3)(t) = ε0χ(3)E(t)3. (4.6)

Now, using the well-known trigonometric identity cos3 ωt = (1/4) × cos 3ωt +(3/4) cos ωt, we can write Eq. (4.6) in the form

P(3)(t) =14

ε0χ(3)U3 cos 3ωt +34

ε0χ(3)U3 cos ωt.

The first term in Eq. (4.7) describes nonlinear response at frequency 3ω and leadsto third-harmonic generation. The second term describes a nonlinear effect at thefrequency of the incident field. Hence, this term leads to a nonlinear contributionto the refractive index experienced by the incident wave of frequency ω, which isknown as the optical Kerr effect. The two cases mentioned above are physicallydistinct from one another though might be exhibited in the same material at thesame time. The choice of the frequencies to find expressions for the respectivesusceptibility tensors for these two instances can be conventionally described by

χ(3)ijkl(3ω = ω + ω + ω), for third-harmonic generation,

χ(3)ijkl(ω = ω + ω − ω), for the optical Kerr effect, (4.7)

where for the case of third-harmonic generation, one may imagine that three pho-tons at frequency ω are destroyed and a single photon at frequency 3ω is created.

We now go back to Eq. (4.4). For the second instance of Eq. (4.7), if we usethe intrinsic permutation symmetry of the cartesian indices, which states that theproduct of Ej(ω) and Ek(ω) is commutable, we obtain

χ(3)1122(ω = ω + ω − ω) = χ

(3)1212(ω = ω + ω − ω). (4.8)

Hence, Eq. (4.4) reduces to

χ(3)ijkl(ω = ω + ω − ω) = χ

(3)1122(ω = ω + ω − ω)× (δijδkl + δikδjl)

+χ(3)1221(ω = ω + ω − ω)× (δilδjk). (4.9)

For the optical Kerr effect, the nonlinear polarization can be written as

Pi(ω) = 3ε0 ∑jkl

χ(3)ijkl(ω = ω + ω − ω)Ej(ω)Ek(ω)El(−ω), (4.10)

where 3 is the degeneracy factor which indicates the number of distinct permuta-

tions of the frequencies ω, ω, and −ω. Now, substituting χ(3)ijkl from Eq. (4.9) into

Eq. (4.10) we obtain

Pi = 6ε0χ1122Ei(E · E∗) + 3ε0χ1221E∗i (E · E), (4.11)

71

where the superscript ∗ denotes the complex conjugate as before. Equation (4.11)can also be written in the following vector form

P = 6ε0χ1122(E · E∗)E + 3ε0χ1221(E · E)E∗. (4.12)

Clearly the first term on the right hand side of Eq. (4.12) has the same handedness asthe incident electric field vector E, whereas the second term has the opposite hand-edness due to complex conjugation. Usually, one-photon-resonant contributions tothe nonlinear coupling as depicted in Fig. 4.1 (a), leads to the first term. Whereas,two-photon-resonant processes as shown in Fig. 4.1 (b) may lead to the first and thesecond terms both.

(a) (b)

Figure 4.1: Energy level description for (a) one-photon-resonant contributions and(b) two-photon-resonant contributions to Kerr nonlinearity. The solid lines in thefigures represent atomic ground state whereas the dashed lines represent virtuallevels. These energy levels usually represent combined energy of one of the energyeigenstates of the atom and of one or several photons in the radiation field [134].

Let us now combine both the linear and the Kerr nonlinear terms and write theelectric polarization in the following form

Pi(ω) = ε0

[

χ(1)ij (ω)Ej + 3χ

(3)ijkl(ω = ω + ω − ω)EjEkE∗

l

]

. (4.13)

In isotropic media, the linear susceptibility is a diagonal tensor, i.e. χ(1)ij = (n2

0 −1)δij, where n0 is the linear refractive index of the medium. This allows one torewrite the constitutive Eq. (4.13) in terms of the “Effective” linear susceptibility,

Pi(ω) = ∑j

ǫ0χ(eff)ij Ej, (4.14)

where,

χ(eff)ij = [n2

0 − 1 + A|E|2]δij + Bℜ

EiE∗j

, (4.15)

and

A = 6χ1122 − 3χ1221,

B = 6χ1221. (4.16)

Here we introduce the coefficients A and B by following the notation introducedby Maker and Terhune [188]. Equation (4.15) indicates that in an isotropic medium,

72

Kerr nonlinearity may result in optical anisotropy. We also note that while writingEq. (4.15), we have omitted the r-dependence of the susceptibility tensor for the sakeof brevity. Clearly, the first term on the right hand side of Eq. (4.15) is independenton the polarization state of the incident field and creates the diagonal elements ofthe effective susceptibility tensor. The second term is polarization-dependent andmust be retained if the intense light wave propagating in the medium is ellipticallypolarized. For linearly polarized light we have a simpler expression for the nonlinearrefractive index which is written as

n = n0 + n2 I, (4.17)

where n0 is the linear refractive index and n2, known as the nonlinear refractiveindex, gives the rate at which the refractive index changes with a change in theincident light intensity I. One can derive the mathematical expression relating n2

and χ(3) using the approach used in Chapter 4 of Ref. [134]. In SI units, they arerelated by

n2

(

m2

W

)

=283n2

0χ(3)

(

m2

V2

)

. (4.18)

The ratio of the coefficients A and B depend on the nature of the physical processwhich produces the Kerr nonlinearity. Some of the standard physical processeswhich produce Kerr nonlinearity are listed below

Nonresonant electronic response: This occurs as a response of the bound electronsto the applied intense field. This type of nonlinearity is extremely fast (re-sponse time ∼femto second) but is believed to be relatively weak and is presentin all dielectric materials. For Kerr nonlinearity arising as a result of the non-resonant electronic response, A/B = 2.

Molecular orientation: In liquids comprised of anisotropic molecules, an externalelectric field tends to align the molecules along the direction of the appliedfield. This alters the average microscopic polarizability per molecule. The inci-dent intense wave then experiences a modified refractive index which resultsin Kerr nonlinearity. The response time of nonlinearity in this case is usually∼pico second and the ratio of the coefficients is A/B = −3.

Electrostriction: Electrostriction is defined as the tendency of a material to becomecompressed in the presence of an external electric field. This alters the opticaldensity of the material and hence also the effective susceptibility. Kerr nonlin-earity caused by electrostriction is believed to be rather weak and the ratio ofA and B equals to zero.

Throughout this chapter, we shall assume that the origin of the optical Kerr nonlin-earity is due to nonresonant electronic response of the bound electrons.

4.3 MODELING LIGHT-INDUCED ANISOTROPY WITH THE LINEAR

FMM

We proceed to extend the existing implementations of the anisotropic FMM, as dis-cussed in section 3.7, to treat the optical Kerr effect (OKE) in a rigorous manner.However, we assume that the origin of the anisotropy in FMM is purely light-induced.

73

Let us consider a two-dimensionally periodic structure as shown in Fig. 4.2, withperiods d1 and d2 along x and y directions (i.e. parallel to the cartesian coordinateaxes), respectively. In cartesian system, φ = Θ = ξ = 0 and the x3 axis is parallelto the z axis. As we saw before, any field component Ei or Hi, i = 1, 2, 3, in anyz-independent layer can be expressed as a superposition of z-invariant modes, viz.

Ui(x, y, z) = ∑mnp

apUi,pmn exp[i(k1,mx + k2,ny + γpz)], (4.19)

where ap, γp, and Uipmn represent the complex amplitude, the propagation constant,and the transverse distribution of the mode p, respectively, and

k1,m = k1,0 + m2π/d1, k2,m = k2,0 + n2π/d2. (4.20)

Here k1,0 and k2,0 are the x and y components of the incident-field wave vectorrespectively. The matrix M which contains the material properties in the Fourier-space, along with the wave-vector components can be written in the new coordinatesystem as

M =

−s1ε31 −s1ε32 s1ε33s2 I − s1ε33s1−s2ε31 −s2ε32 −I + s2ε33s2 −s2ε33s1

−ε21 − s1s2 −ε22 + s1s1 −ε23s2 ε23s1ε11 − s2s2 ε12 + s2s1 ε13s2 −ε13s1

. (4.21)

Here the elements of the matrices s1 and s2 are s1,mnpq = k1,mδmpδnq/k0, and s2,mnpq =k2,nδmpδnq/k0 respectively, I = δmpδnq, and εij is the ij-th element of the matrix

ε = l−3 l+2 F2l−2 l+1 F1l−1 ε(x, y), (4.22)

where ε(x, y) is the x and y dependent (complex) relative permittivity tensor. Asbefore Fj stands for the Toeplitz operation along the direction j, and l±τ operatorsare defined by Eq. (3.91) in section 3.7. Once we solve the eigenproblem, the zcomponent of the electric field can be obtained from [170]

E3,p = ν−133 (s2H1,p − s1H2,p), (4.23)

where ν = l+2 F2l−2 l+1 F1l−1 ε(x, y). The matrix eigenvalue equation from which we cansolve the propagation constants and the shape of the grating modes can be writtenas

MFp =γ

k0Fp, (4.24)

where k0 is the vacuum wave number and the eigenvector Fp contains the x and ycomponents of the electric and magnetic fields of mode p as follows:

Fp =

Ex

Ey

Hx

Hy

p

. (4.25)

Equation (4.15) shows that in case of light-induced anisotropy, the permittivitytensor components become field-dependent. Hence, we need to extend the exist-ing implementation of the anisotropic linear FMM summarized above to problems

74

dealing with field-dependent permittivity tensors. This is done by following an iter-ative approach and assuming the medium to be isotropic and Kerr nonlinear, wherethe only source of anisotropy is optical nonlinearity. The algorithm we follow is anextended version of the algorithm introduced by Laine and Friberg for linear grat-ings [189], but we also consider the polarization effects in a rigorous manner. Ourapproach is to some extent similar to the iterative approach with fast Fourier fac-torization in the differential method [185,190]. The steps of the modeling algorithmare as follows:

1. In each z-independent layer, we solve the eigenvalues γ and the eigenvectorsFp for modes p using the linear permittivity. Then use the S-matrix algo-rithm [129,130] to connect the modes in different layers and solve the complexamplitudes ap of the grating modes in Eq. (4.19) as described in section 3.7.

2. We solve the electric-field components using Eq. (4.19) and the Fast-FourierTransform (FFT) algorithm in a 3D grid of nx × ny × nz points as shown inFig. 4.3.

3. Using the solved field components, we can evaluate the effective linear sus-ceptibility from Eq. (4.15).

4. We go back to the eigenvalue problem in Eq. (4.24), but now using the materialparameters from the previous step.

This iterative process is continued until the computed field converges. It is worth tomention that the method may not converge if the field change is too large betweentwo subsequent steps. In such a case the electric-field strength may be increasedgradually during the iteration, which ensures better convergence in most situations.

x

y

z

d1

d2

Figure 4.2: Geometry of a crossed grating. The grating structure is periodic alongx and y directions. The pillars in this figure are assumed to be made with isotropicKerr nonlinear material.

75

Also, like in all Fourier domain approaches, the number of Fourier coefficientsused in the computation should be high enough. A poor convergence may be asa consequence of the Gibbs phenomenon as described in section 3.5.1. This maystrongly affect the effective linear permittivity tensor through the space-domainfield. Additionally, the sampling of the field in the real space must be dense enough,i.e., one must use a sufficient number of layers in FMM even if the structure is in-variant in the z direction.

Here we also want to emphasize the fact that since we make use of the FFT algo-rithm to compute the field in the (x, y, z) space, and since the number of samplingpoints is retained in FFT, we must pad the Fourier-domain field by zeros before theFFT, which in turn increases the resolution in the (x, y, z) space. This is a standard”trick” in FFT. However, it does not affect the results, but it just allows (almost)arbitrary sampling in Eq. (4.19). One obtains identical results with explicit summa-tion over all included Fourier coefficients in Eq. (4.19), but FFT with zero-paddingis significantly faster.

4.4 SYMMETRIES IN LIGHT-INDUCED ANISOTROPY

In FMM, one can employ structural symmetries to reduce the computation timeand also to increase the computational efficiency by reformulating the whole eigen-value problem into a more compact form [191–198]. The number of floating-pointoperations required to solve an eigenvalue problem of size L × L is asymptoticallyproportional to L3. Hence halving the matrix by use of the symmetries reduces therequired workload to 1/8. Such type of reductions of computational efforts are ofgreat importance especially in the context of nonlinear optics as the size of the matrixM in Eq. (4.21) is 2 × 2 times larger than that in the corresponding linear problem,where one can use the isotropic FMM [150]. Also, as we saw in the previous sec-tion, the nonlinear case requires several iteration steps and each of these steps thusrequires eight times more resources than the same problem in linear optics.

Here as an example, we study the reduction of the eigenvalue problem at normalincidence of the incoming field if the structure possess C2v symmetry (i.e. rotationof the structure through 180 yields a structure which is indistinguishable from theoriginal one) such that ε(−x, y, z) = ε(x, y, z) and ε(x,−y, z) = ε(x, y, z). In the linear

x

y

z

Figure 4.3: Sampling of the structure in the real space. Such a sampling producesa 3D grid of nx × ny × nz points.

76

limit i.e. when we deal with the isotropic problem, we can easily show that [191]

Ej(−x,−y, z) = Ej(x, y, z), (4.26a)

Hj(−x,−y, z) = Hj(x, y, z), (4.26b)

where j = 1 or 2. Using Eq. (4.26a) and the symmetry of the permittivity we getthe following symmetry condition for the transverse components of the electric dis-placement vector D(x, y, z)

Dj(−x,−y, z) = Dj(x, y, z), (4.26c)

where again j = 1 or 2, and, consequently,

∂

∂xD1(−x,−y, z) = − ∂

∂xD1(x, y, z), (4.27a)

∂

∂yD2(−x,−y, z) = − ∂

∂yD2(x, y, z). (4.27b)

Maxwell’s divergence equation ∇ · D(x, y, z) = 0 then implies that

D3(−x,−y, z) = −D3(x, y, z). (4.28a)

This with the help of the symmetry condition of the permittivity function, leads to

E3(−x,−y, z) = −E3(x, y, z). (4.28b)

Let us first assume that these symmetry conditions also hold for the nonlinearcase. Our assumption remains valid at least in the first step of the iteration as westart the process from the unmodified linear permittivity tensor, which is isotropic,for calculating the field inside the structure. It follows immediately from Eqs. (4.15),(4.26), and (4.28b) that the effective linear relative permittivity tensor

ε(eff)ij = δij + χ

(eff)ij /ε0 (4.29)

obeys the symmetry relations

ε(eff)ij (−x,−y, z) = ±ε

(eff)ij (x, y, z). (4.30)

Here the − sign is chosen if either i or j (but not both) is 3, and + is chosen other-wise.

Now we proceed to apply the symmetry conditions to Eq. (4.30). Recalling theoperator relation in Eq. (3.91) of section 3.7, we can write

l±τ Aρσ =

(Aττ)−1, ρ = τ, σ = τ,(Aττ)−1Aτσ, ρ = τ, σ 6= τ,Aρτ(Aττ)−1, ρ 6= τ, σ = τ,Aρσ ± Aρτ(Aττ)−1Aτσ, ρ 6= τ, σ 6= τ,

(4.31)

where Aρσ is any matrix-form element of tensor A. We find from Eqs. (4.31) and

(4.30) that if Aij(x, y) = l−1 ǫ(eff)ij (x, y), then (we drop the z-dependence for brevity)

Aij(−x,−y) = ±Aij(x, y). (4.32)

77

Let us now denote the Fourier coefficient of a function f (x, y) with respect to x byf (m)(y), i.e.,

f (m)(y) =1d1

∫ d1/2

−d1/2f (x, y) exp(−i2πmx/d1) dx. (4.33)

It follows from Eqs. (4.32) and (4.33) that

A(−m)ij (−y) = ±A

(m)ij (y). (4.34)

If Bij(y) is a matrix with its elements Bij,mn(y) given by A(m−n)ij (y), we have Bij(y) =

F1 Aij(x, y). Clearly from Eq. (4.34) we get

Bij,−m−n(−y) = ±Bij,mn(y). (4.35)

Now denoting C(y) = l−2 l+1 B(y), and making use of Eq. (4.31) and (4.35), we findthat

Cij,−m−n(−y) = ±Cij,mn(y). (4.36)

Here we have made use of the fact that if the elements of matrix N follows theidentity N−m−n = Nmn, then also the elements of Q = N−1 obey the same identity,i.e., Q−m−n = Qmn. In addition, we have used the identity that if N−m−n = sN Nmn

and P−m−n = sPPmn, where sN and sP can be independently either −1 or +1, thenthe product matrix R = NP obeys the relation R−m−n = sPsN Rmn. The two resultsmentioned above follow at once from the definitions of the inverse matrix and thematrix product, respectively.

In a similar fashion, one can denote the Fourier coefficients of any y- dependentfunction F(y) by F[p], i.e.,

F[p] =1d2

∫ d2/2

−d2/2F(y) exp(−i2πpy/d2) dy. (4.37)

Denoting Gij,mnpq = C[p−q]ij,mn , we have Gij = F2Cij(y). It follows from Eqs. (4.36) and

(4.37) that

C[−p]ij,−m−n = ±C

[p]ij,mn (4.38)

and, consequently,

Gij,−m−n−p−q = ±Gij,mnpq. (4.39)

One can now verify using Eqs. (4.31) and (4.39) (which is a rather straightforwardtask) that ε = l−3 l+2 G obeys the identity

ε ij,−m−n−p−q = ±ε ij,mnpq. (4.40)

From Eq. (4.21) we see that if we denote the blocks of matrix M by Mrs, r, s =1, 2, 3, 4, their elements obey the identity

Mrs,−m−n−p−q = Mrs,mnpq. (4.41)

78

We now focus on the symmetry conditions of the eigenvectors. First we writeEq. (4.24) in the form

γ

k0Ur,mn = ∑

pq

Mrs,mnpqUs,pq, (4.42)

where U1 = E1, U2 = E2, U3 = H1, and U4 = H2. Since m, n, p, and q are dummyindices, we may express Eq. (4.42) as

γ

k0Ur,−m−n = ∑

pq

Mrs,−m−n−p−qUs,−p−q

= ∑pq

Mrs,mnpqUs,−p−q, (4.43)

where to derive the above equation, we have used Eq. (4.41). As a result, the vectorswith elements Ur,−m−n i.e. the ‘flipped’ vectors form the eigenvector of M, with thesame eigenvalues as the original one having the elements Ur,mn. To be more specific,we are dealing with a degenerate situation in which also vectors U+

r and U−r with

elements

U±r,mn =

12(Ur,mn ± Ur,−m−n) (4.44)

form the eigenvectors of M. These eigenvectors obey the condition U±r,−m−n =

±U±r,mn. However, as we are dealing with the normal incidence, the antisymmet-

ric part with − sign can be omitted. Hence we may assume that all eigenvectorsobey the symmetry condition

Ur,−m−n = Ur,mn. (4.45)

Thus we have the following symmetry conditions

E1,−m−n = E1,mn, E2,−m−n = E2,mn,

H1,−m−n = H1,mn, H2,−m−n = H2,mn (4.46)

which are identical to those in the linear case. We can now conclude that the fieldsymmetry is not changed due to the nonlinear light-matter interaction. The obtainedresult may sound surprising, but it follows from the fact that all materials are in-trinsically isotropic. The effective anisotropy is due to the field itself and there isnothing in the structure that can break the existing symmetry.

Let us now consider the case in which the index m takes on values from −Mto M, and n between −N and N. Hence, the total size of the matrix M is 4(2M +1)(2N + 1)× 4(2M+ 1)(2N + 1). Making use of Eqs. (4.24) and (4.21), and the iden-tities in Eqs. (4.41) and (4.46), we may now replace the original eigenvalue equation(4.24) with the reduced problem in which m takes on values only from 0 to M, and

M =

−s1ε(+)31 −s1ε

(+)32 s1ε

(−)33 s2 I − s1ε

(−)33 s1

−s2ε(+)31 −s2ε

(+)32 −I + s2ε

(−)33 s2 −s2ε

(−)33 s1

−ε(+)21 − s1s2 −ε

(+)22 + s1s2 −ε

(−)23 s2 ε

(−)23 s1

ε(+)11 − s2s2 ε

(+)12 + s1s2 ε

(−)13 s2 −ε

(−)13 s1

, (4.47)

79

where

ε(±)ij,mnpq =

ε ij,mn0q, p = 0ε ij,mnpq ± ε ij,mn−p−q, p 6= 0 . (4.48)

All matrices in blocks of M are of size (M + 1)(2N + 1), and the total size of thematrix M is 4(M + 1)(2N + 1)× 4(M + 1)(2N + 1). Thus the required number offloating-point operations is asymptotically 1/23 = 1/8 of the number of operationsin the original eigenvalue problem which results in a reduction of the work-load by8 times.

In the above calculations, we chose to reduce the dimension of the matrix in thex direction. Analogously we can proceed in the y direction. However, we mustnote that reductions in both of these directions at the same time is not possible.This is because of the fact that the field is not symmetric in either direction, but issymmetric only when we compare the field values at (−x,−y, z) and (x, y, z).

Now, we shall find out that if the incident field is polarized either in the x ory direction, the problem may be reduced further. Let us assume that we have y-polarized incident field. In the linear case, the symmetry conditions lead to [199]

E1(x, y, z) = −E1(−x, y, z) = −E1(x,−y, z),

E2(x, y, z) = E2(−x, y, z) = E2(x,−y, z),

H1(x, y, z) = H1(−x, y, z) = H1(x,−y, z),

H2(x, y, z) = −H2(−x, y, z) = −H2(x,−y, z). (4.49)

With the help of Maxwell’s divergence equation, we find that

E3(x, y, z) = E3(−x, y, z) = −E3(x,−y, z). (4.50)

It then follows from Eqs. (4.15) and (4.49) that

ε(eff)jj (x, y, z) = ε

(eff)jj (−x, y, z) = ε

(eff)jj (x,−y, z),

ε(eff)12 (x, y, z) = −ε

(eff)12 (−x, y, z) = −ε

(eff)12 (x,−y, z),

ε(eff)13 (x, y, z) = −ε

(eff)13 (−x, y, z) = ε

(eff)13 (x,−y, z),

ε(eff)23 (x, y, z) = ε

(eff)23 (−x, y, z) = −ε

(eff)23 (x,−y, z), (4.51)

and we may use the identity ε(eff)ji (x, y, z) = ε

(eff)ij (x, y, z) to derive the remaining

three elements.Proceeding analogously to the case for the arbitrarily polarized light, we find

that

ε jj,mnpq = ε jj,−mn−pq = ε jj,m−np−q,

ε12,mnpq = −ε12,−mn−pq = −ε12,m−np−q,

ε13,mnpq = −ε13,−mn−pq = ε13,m−np−q,

ε23,mnpq = ε23,−mn−pq = −ε23,m−np−q (4.52)

and hence Eq. (4.21) implies that

Mrs,mnpq = ±Mrs,−mn−pq = ±Mrs,m−np−q, (4.53)

80

where + is chosen for rs = 11, 14, 22, 23, 32, 33, 41, 44, and − otherwise. Similarlyto the previous derivation for the symmetry of the eigenvectors, we find that theeigenvectors obey the following conditions

E1mn = −E1−mn = −E1m−n

E2mn = E2−mn = E2m−n

H1mn = H1−mn = H1m−n

H2mn = −H2−mn = −H2m−n. (4.54)

The results in Eq. (4.54) are in full agreement with the linear case in Eq. (4.49).Finally, in the eigenvalue problem, Eq. (4.24), we have

M =

−s1ε(oo)31 −s1ε

(ee)32 s1ε

(eo)33 s2 I − s1ε

(eo)33 s1

−s2ε(oo)31 −s2ε

(ee)32 −I + s2ε

(eo)33 s2 −s2ε

(eo)33 s1

−ε(oo)21 − s1s2 −ε

(ee)22 + s1s2 −ε

(eo)23 s2 ε

(eo)23 s1

ε(oo)11 − s2s2 ε

(ee)12 + s1s2 ε

(eo)13 s2 −ε

(eo)13 s1

, (4.55)

where, for any matrix Ω,

Ω(oo)mnpq = Ωmnpq − Ωmn−pq − Ωmnp−q + Ωmn−p−q,

Ω(ee)mnpq =

Ωmn00, p = 0, q = 0Ωmn0q + Ωmn0−q, p = 0, q 6= 0Ωmnp0 + Ωmn−p0, p 6= 0, q = 0Ωmnpq + Ωmn−pq

+Ωmnp−q + Ωmn−p−q, p 6= 0, q 6= 0,

Ω(eo)mnpq =

Ωmn0q − Ωmn0−q, p = 0,Ωmnpq + Ωmn−pq

−Ωmnp−q − Ωmn−p−q, p 6= 0.(4.56)

The superscripts e and o are the abbreviations for ’even’ and ’odd’ respectively. Theyrepresent symmetries of either even type or of odd type. In matrix M, the indices mand n run from 1 to M and N, respectively, for rows 1 and 4, since the correspondingfield components E1 and H2 are antisymmetric. Similarly, both indices begin from0 for rows 2 and 3 due to the symmetry of E2 and H1. An equivalent rule appliesto the column indices p and q, which begin from 1 for columns 1 and 4 of M, andfrom 0 for other two columns. Hence the total size of M is 2(2M + 1)(2N + 1) ×2(2M + 1)(2N + 1). Finally, we may conclude that the eigenvalue problem requiresasymptotically just 1/64 of the original problem. Similar result can be obtained byassuming the incident field to be x- polarized.

Unlike in linear problems, one cannot split the eigenproblem for an arbitrarilypolarized incident field into two smaller eigenproblems that can be solved separately(which would mean a remarkable further improvement in computation efficiency).This is because of the fact that the permittivity is affected by the total field, andEq. (4.52) does not hold unless the incident field is y polarized [199].

4.5 NUMERICAL EXAMPLES

We now apply the algorithm developed in the previous sections to periodic struc-tures composed of isotropic nonlinear media.

81

4.5.1 One dimensional metallic gratings with grooves filled with χ(3) media

At first, we model a metallic linear grating with its grooves filled with Kerr nonlinearmaterial. In our numerical simulations, we use the same parameters as used byBonod et al. [185]. In this example, we assume that TM polarized light is normallyincident from air on the lamellar grating side. The grating pillars are made of ametal with relative permittivity ǫr = −182.4 + i43.52. The 494 nm deep groovesare assumed to be filled with amorphous silicon. The substrate is assumed to becomposed of the same metal as the pillars. The grating period is 1 µm.

Figure 4.4 shows the efficiency in direct reflection (reflected zeroth diffractionorder) as a function of the wavelength of the incident light. The absolute value ofthe incident electric field is chosen to be 106 V/m. One may readily compare oursimulation result i.e. Fig. 4.4 with Fig. 2 in Ref. [185], and conclude that the resultsare essentially identical.

1400 1420 1440 1460 1480 1500 1520 1540 1560 1580 1600

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

η0

λ [nm]

Figure 4.4: Efficiency in direct reflection as a function of the wavelength for ametallic linear grating.

4.5.2 1-D binary grating with TiO2 as the Kerr nonlinear material

As a second example, we simulate a 1-D periodic nonlinear binary grating possess-ing Kerr nonlinearity in the material that fills the pillars. The substrate materialis assumed to be SiO2 with linear refractive index n0 = 1.46. The SiO2 gratingpillars are filled with amorphous (isotropic) TiO2 (linear refractive index= 2) withχ(3) = 2.1 × 1020 m2/V2. The fill factor of the grating is f = 0.5 and the period ofthe structure is d = 2.5 × λ0, where λ0 = 633 nm is the wavelength of the incidentlight in vacuum. The results obtained with FMM are compared with those obtainedwith the SF-FDTD as implemented in Ref. [186, 200]. For the nonlinear simulations,the intensity of the incident field was taken to be 0.5 MW/µm2 in both numericalmethods. Clearly, from Fig. 4.5, there is a high correlation between both analysis.The small differences can be due to the difficulties in triggering the same amountof nonlinearity which follows from the difference in defining the intensity in theSF-FDTD cell [186, 200, 201] and the FMM approach [199].

82

Linear

(a)

(b)

η0

h/λ0

η±

1

h/λ0

Figure 4.5: Diffraction efficiencies of a nonlinear binary grating computed bymeans of SF-FDTD and FMM. h is the structure depth.

4.5.3 Crossed gratings with the pillars made with Si3N4

Next example deals with a crossed diffraction grating composed of cylindrical shapedsilicon nitride (Si3N4) pillars. The linear refractive index of Si3N4 (which is amor-

phous in nature) is assumed to be n0 = 2, and the value of χ(3)Si3N4

is taken to be

3.39 × 10−21 m2/V2 as in Ref. [69]. Also, we assume that the silicon nitride pillarsare surrounded by air. The substrate material is fused silica with linear refractiveindex n = 1.5. Furthermore, we presume that the surrounding media as well asthe substrate material are optically linear. This assumption remains valid as thethird-order nonlinearity of silicon nitride is two-orders of magnitude higher thanthat of fused silica. Consequently, we do not get nonlinear signal from the substrateat the used intensity level of the incident field. We first assume that Left CircularlyPolarized (LCP) light is normally incident from the substrate side, and then repeatthe simulation for 45 linearly polarized light. The grating period is assumed to bed1 = d2 = 2.5λ and the height of the pillars is h = 1.25λ, where λ denotes the wave-length in vacuum. The discretization of the pillars to treat the circular boundarywas done in a similar way as in Fig. 4 of Ref. [150].

In Figure 4.6, we plot the convergence of the nonlinear FMM when the in-cident field intensity is fixed at 5 × 1014 W/cm2. The sampling parameters arenx = ny = 175 and nz = 25. Clearly, for this specific example, the method convergeswith M = 8 i.e. about 17 diffraction orders are needed in this specific case.

Figures 4.7 and 4.8 show the convergence as a function of the transverse (nz)and the longitudinal sampling (nx and ny), respectively, where all other parametersare kept fixed to their maximum values. The transverse sampling can be chosen

83

1 2 3 4 5 6 7 8 9 10 110.2

0.3

0.4

0.5

0.6

η00

M

Figure 4.6: Efficiency in direct transmission η00 as a function of the maximumdiffraction order M. The maximum order is taken to be the same both in x and ydirections.

to be large as it does not affect the computation time significantly (solution insidea single z-independent layer). On the other hand, the computation time increaseslinearly with an increase of the value of nz. Hence, it should be chosen as low aspossible but of course without sacrificing neither accuracy nor convergence. Figure4.9 illustrates a typical convergence curve during the iteration process. Though notabsolutely necessary, in this example we increased the incident intensity from zeroto its maximum value during the iteration within the first 20 steps. Then we keptthe intensity fixed at its final value (i.e. the value at the 20-th iteration step) for afew rounds (here five) to ensure the convergence of the method.

Figure 4.10 illustrates the effective-permittivity tensor in the middle of the

60 80 100 120 140 160 180 2000.49

0.5

0.51

0.52

0.53

η00

nx = ny

Figure 4.7: Zero-order transmitted efficiency η00 plotted against the number oftransverse sampling points nx = ny.

structure (z = h/2) and during the final round of the iterations.

84

5 10 15 20 25 300.43

0.46

0.49

0.52

η00

nz

Figure 4.8: Zero-order transmitted efficiency η00 plotted against the number oflayers nz.

5 10 15 20 250.2

0.3

0.4

0.5

45° LP

LCPη00

qg

Figure 4.9: Zero-order transmitted efficiency η00 plotted against the iteration stepsfor LCP and 45 linear polarization respectively. qg is the number of iteration steps.

4.5.4 Si3N4 resonance waveguide-grating

As a final example, we illustrate the effect of the optical Kerr nonlinearity on the0-th order reflected signal from a waveguide-resonance structure or guided moderesonance filter (GMRF). The structure geometry is shown in Fig. 4.11. The substrateis made of fused silica (FS) and the grating layer as well as the waveguide layer (un-etched part) is taken to be Si3N4. Light is normally incident on the structure fromair. The grating periods are d1 = d2 = 300 nm, Height of the pillars is 100 nm, thethickness of the unetched part (waveguide layer) is 150 nm, and the linewidths ofthe pillars are 120 nm in both transverse directions. Here, the waveguide-gratingstructure is designed to act as a narrow-band reflector. We investigate the influenceof the incident field intensity and the state of polarization of the normally incidentlight on the resonance peak in reflection. Figure 4.12 shows the red-shift of the res-

85

1

2

3

4

−0.6

−0.4

−0.2

0

−0.1

0

0.1

1

2

3

4

−0.1

0

0.1

1

2

3

4

Figure 4.10: Elements of the effective relative-permittivity-tensor ǫ(eff)ij (x, y) in the

middle of the cylindrical pillars. The tensor elements in the left-hand column are 11(top), 22 (middle), and 33 (bottom), and in the right-hand side 12 = 21 (top), 23 = 32(middle), and 31 = 13 (bottom).

onance peak i.e. shift towards the longer wavelengths when the value of the fieldintensity increases. This is in agreement with the theory of the optical Kerr effect.As the nonlinear refractive index of Si3N4 increases with increasing field intensity,the resonance peak shifts in wavelength scale due to the increase (n2 of silicon ni-tride is positive) in mismatch between the refractive indices of the grating materialand the surrounding material.

Figure 4.13 shows how the incident state of polarization affects the resonancepeak: clearly the shift is larger with the linearly polarized light than that with thecircularly polarized light. The reason is that the nonlinear refractive index is, ingeneral, larger with linearly polarized states [134].

Resonance waveguide-grating structures illustrated in Fig. 4.11 can producestrong nonlinear effects for reasonably low values of field intensities due to thestrong enhancement of local field inside the waveguide layer. This property of GM-RFs can be exploited to construct novel devices especially in relation to sensingand all-optical computing. In Chapter 6, we shall investigate these silicon nitridewaveguide-grating structures in a more detailed manner.

86

Grating layer

Waveguide layerFS substrate

Incident field0-th order reflected field

Figure 4.11: Guided-mode resonance filter (GMRF) structure.

496 496.4 496.8 497.2 497.6 498

0

0.2

0.4

0.6

0.8

1

I=1014

I=1015

I=2 x1015

W/ m2

W/ m2W/ m2

η00

λ [nm]

Figure 4.12: Red shift of resonance peak due to the change in the incident fieldintensity. The incident field is assumed to be x polarized.

87

496 496.4 496.8 497.2 497.6 4980

0.2

0.4

0.6

0.8

1

45 °LPy-pol

LCPη

00

λ [nm]

Figure 4.13: Resonance peak shift due to the change of polarization states of theincident light. In these plots, the field intensity is kept fixed at 1015 W/m2.

4.6 SUMMARY

In this Chapter, we have developed an efficient numerical tool for rigorous modelingof optical Kerr nonlinearity in 2D periodic structures with isotropic third-order non-linear media having instantaneous nonlinear response. For structures possessingC2v symmetry i.e. two-fold rotational symmetry, we demonstrate that the compu-tational workload can be reduced to 1/8-th as compared to the original nonlineareigenvalue problem. Furthermore, for TE or TM polarized incident field, we canreduce the computational effort to 1/64-th as compared to that in the most generalcase.

Our numerical technique produces well-converging results even if there is struc-tural resonance and nonlinearity generated inside the structure is large as a resultof strongly confined local field. Hence we may conclude that this nonlinear FMMcan be used as a powerful numerical modeling tool for designing grating based nextgeneration all-optical devices as well as novel nonlinear metamaterials.

88

5 Theory of form birefringence in Kerr-type media

The term birefringence is used to denote having two different refractive indices ofa medium depending on different propagation directions and different polariza-tion states of light [202]. Though birefringence is commonly observed in opticallyanisotropic media, it can also arise in isotropic media under some special condi-tions. It was David Brewster who first observed that isotropic media can becomebirefringent under application of mechanical stresses [203]. This type of inducedbirefringence is called ’Stress birefringence’. However, there are several other waysto make a medium optically anisotropic for example by applying an external staticelectric field (D.C Kerr effect) [187], or magnetic field (Faraday effect) [113] etc. Inall the cases mentioned above, the origin of birefringence may be explained in termsof the molecular anisotropy. However, it might be surprising to know that birefrin-gence may also arise due to anisotropy which is larger in scale than the dimensionsof a molecule though much smaller in scale than the wavelength of light. Such typeof birefringence is termed as form birefringence.

This chapter starts with the theory of light propagation in optical crystals. Aftersome general discussions, we proceed to derive the wave equations governing lightpropagation in a uniaxial crystal. Next we demonstrate that subwavelength grat-ings (SWG) with isotropic materials behave as uniaxial crystals and exhibit (form)birefringence in the quasistatic limit. After that, we extend the theory of form bire-fringence in a SWG to gratings with optical Kerr nonlinear media [204]. Finally,we include several numerical examples to elucidate our developed theory and alsoto verify it by comparing with the results obtained by the nonlinear FMM [199]developed in Chapter 4.

5.1 PROPAGATION OF LIGHT IN CRYSTALS

In section 2.13, we discussed light propagation in an electrically anisotropic mediumand found that it is possible to write the expression for the electric energy densityin a form which is equivalent to the mathematical expression for an ellipsoid. Also,we found that we can always choose a coordinate system with axes parallel to theprincipal axes of the ellipsoid to rewrite the ellipsoid equation in a more compactform i.e.

D2x

ǫx+

D2y

ǫy+

D2z

ǫz= constant, (5.1)

withDx = ǫxEx, Dy = ǫyEy, Dz = ǫzEz. (5.2)

Here the cartesian coordinate axes x, y, and z are parallel to the principal axes andǫj, where j = x, y, z, are the principal dielectric constants of the crystal as before.Let’s now substitute the expression of D from Eq. (2.128) into Eq. (2.138) to yield aset of three homogeneous linear equations in Ex, Ey, and Ez. This set of equationscan be written as

µǫjEj =n2

c2

[

Ej − uj (E · u)]

, (5.3)

89

where uj s are the cartesian components of the unit vector u. Equation (5.3) can alsobe written in the following form

Ej =(n2/c2)uj(E · u)

(n2/c2)− µǫj. (5.4)

Multiplying Eq. (5.4) with uj, adding the resulting three component-form equations,and finally dividing both sides by the common factor E · u, we obtain

u2x

(n2/c2)− µǫx+

u2y

(n2/c2)− µǫy+

u2z

(n2/c2)− µǫz=

c2

n2 . (5.5)

Some simple algebra with Eq. (5.5) yield

u2x

c2

n2 − 1µǫx

+u2

y

c2

n2 − 1µǫy

+u2

zc2

n2 − 1µǫz

= 0. (5.6)

Let us now define the terms ’principal velocities of propagation’ by the formulae

vx =1√µǫx

, vy =1

√µǫy

, vz =1√µǫz

. (5.7)

Recalling the definition of phase velocity from section 2.13 and using Eq. (5.7), wecan rewrite Eqs. (5.4) and (5.6) in the following forms respectively

Ej =v2

j

v2j − v2

p

uj (E · u) , (j = x, y, z), (5.8)

u2x

v2p − v2

x+

u2y

v2p − v2

y+

u2z

v2p − v2

z= 0. (5.9)

Equation (5.9) is quadratic in v2p. We may now conclude that to every direction u,

there exists two phase velocities vp (we count only the positive roots as the negativeroots correspond to the negative u direction). For each of these two values of vp, Ej

and Dj can be solved from Eqs. (5.8) and (5.2) respectively. Also, we see that the ra-tio of the electric field components as well as the electric displacement componentsare real. Hence E and D are linearly polarized. Finally, we have the most impor-tant conclusion regarding light propagation in a crystalline medium with electricalanisotropy which can be stated as: the structure of an anisotropic medium allowstwo monochromatic linearly polarized plane waves (where the states of polarizationare different for these two plane waves) with two different velocities to propagate inany given direction. By some straightforward calculus it is easy to prove [113] thatthe directions of vibrations of the two electric displacement vectors D which corre-sponds to the two plane waves are in general orthogonal to each other as shown inFig. 5.1(a). We denote the two electric displacement vectors by D′ and D′′. Clearly,D′, D′′ and u form an orthogonal triplet. The ellipsoid in Fig. 5.1(a) is governed bythe equation

(Dx/√

c)2

ǫx+

(Dy/√

c)2

ǫy+

(Dz/√

c)2

ǫz= 1, (5.10)

90

which directly follows from Eq. (5.1) and is known as the ellipsoid of wave normals.Clearly, the semi-axes of this ellipsoid are equal to the square roots of the principaldielectric constants ǫx, ǫy, and ǫz and coincides with in directions with the principaldielectric axes. In the special case, when light propagates in the direction of one ofthe principal axes of the ellipsoid of wave normals, say x axis, the phase velocitybecomes equal to the principal velocities of propagation vy and vz. The optic axesof wave normals can be determined by constructing two circular sections C1 andC2 passing through the center of the ellipsoid as shown in Fig. 5.1(b) (an ellipsoidcan have only two circular cross sections). The normals N1 and N2 to the surfacesC1 and C2 are known as the optic axes of the crystal. Since the sections C1 and C2are circular, the directions N1 and N2 allow only one velocity of propagation alongthem and D can take any direction perpendicular to the wave normal.

5.2 BIREFRINGENCE OF A UNIAXIAL CRYSTAL

In a uniaxial crystal, two or more crystallographically-equivalent directions in oneplane exist. Usually, the plane containing these equivalent directions is perpendicu-lar to the axis of rotational symmetry of the crystal. For a uniaxial crystal, we maychoose one of the principal axes as the distinguished direction while the other twoaxes can be chosen such that they are perpendicular to the distinguished axis. Ifwe take z- axis as the distinguished direction, the principal dielectric constant ǫx

becomes equal to ǫy and we have the following relation

ǫx = ǫy 6= ǫz. (5.11)

(a) (b)

D′D′′

u

C1C2

N2 N1

Figure 5.1: The ellipsoid of wave normals. Construction of the direction of vibra-tions of the D vectors corresponding to the propagation direction u (a) and construc-tion of the optic axes (b).

91

Clearly, for an uniaxial crystal the ellipsoid in Fig. 5.1(a) reduces to a spheroid withtwo equal axes. Also, unlike in Fig. 5.1(b), we have only one circular cross-sectionpassing through the center of the spheroid. Hence, uniaxial crystals possess onlyone optic axis.

Let’s us now go back to the equation of wave normals (also known as Fresnel’sequation) i.e. Eq. (5.9). We can write this in the following form

u2x(v

2p − v2

y)(v2p − v2

z) + u2y(v

2p − v2

z)(v2p − v2

x) + u2z(v

2p − v2

x)(v2p − v2

y) = 0,

If we now assume the direction of the optic axis along z-axis and write vx = vy = vo,and vz = ve, Eq. (5.12) for an uniaxial crystal reduces to

(v2p − v2

o)[

(u2x + u2

y)(v2p − v2

e) + u2z(v

2p − v2

o)]

= 0, (5.12)

where the new subscripts ’e’ and ’o’ stand for the ordinary wave and the extraor-dinary wave respectively which will be explained later in this section. Let us nowassume that the wave normal u forms an angle θ with the z-axis as shown in Fig. 5.3.Thus we have

u2x + u2

y = sin2 θ, u2z = cos2 θ. (5.13)

Using Eq. (5.13), we can write Eq. (5.12) in the following form

(v2p − v2

o)[

(v2p − v2

e) sin2 θ + (v2p − v2

o) cos2 θ]

= 0. (5.14)

Equation (5.14) is a quadratic equation with roots

v′2p = v2o, (5.15a)

v′′2p = v2o cos2 θ + v2

e sin2 θ. (5.15b)

Hence we see that one of the two shells of the spheroid (surface of wave normals)is a sphere of radius v′p = vo. The other shell is an ovaloid. The wave with phasevelocity v′p which corresponds to the wave normal u propagates inside the crystalanalogous to an ordinary wave as the velocity is direction independent. This waveis termed as ordinary wave. The other wave that corresponds to the wave normalu propagates with a phase velocity v′′p which is direction dependent as can be seenfrom Eq. (5.15b). To emphasize the contradiction with the previous case, this waveis termed as extraordinary wave. Phase velocities of these two waves are equal onlyif θ = 0 i.e. when the wave normal directs along the optic axis (here z- axis) of thecrystal. When the wave normal is perpendicular to the optic axis (here x- axis), thedifference between the phase velocities of these two waves is maximum and fromEq. (5.15b) we have,

v′′2p = v2e. (5.15c)

Due to the different phase velocities of the wave propagation inside the uniaxialcrystal, we can also define two different refractive indices by

no = c/vo, and ne = c/ve, (5.15d)

which are known as the ordinary and the extraordinary refractive indices respec-tively. Now we can easily understand that light propagation inside a uniaxial crystalgives rise to birefringence.

92

There are situations when the ordinary wave can travel faster than the extraordi-nary wave inside the crystal. Such type of uniaxial crystals, example include quartz,are termed as positive uniaxial crystals. The opposite situations are observed in neg-ative uniaxial crystals for example in Feldspar. Light propagation geometries thatcorrespond to these two cases are illustrated in Figs. 5.2(a) and 5.2(b) respectively.

The directions of vibrations of D′ and D′′ can be found from Fig. 5.3. The planecontaining u and the optic axis OZ is termed as the principal plane. The spheroid issymmetrical about this plane. Clearly from the figure, the elliptical section throughO by the plane perpendicular to the wave normal is also symmetrical about theprincipal plane. Hence, the principal axes are perpendicular and parallel to theprincipal plane. Finally, the vector D of the ordinary wave i.e. D′ vibrates at rightangles to the principal plane, whereas the vector D of the extraordinary wave i.e.D′′ vibrates in the principal plane as illustrated in Fig. 5.3.

5.3 THEORY OF FORM BIREFRINGENCE

Form birefringence can be exhibited by subwavelength gratings with grating peri-ods much smaller than the wavelength of the incident light. Even if the materialwhich fills such a subwavelength grating is isotropic, due to anisotropy arising fromstructuring one can observe strong birefringence. Hence, these gratings behave ashomogeneous effective birefringent media. This type of birefringence is usuallymuch stronger than the birefringence in optical crystals [205]. Also the magnitudeof form birefringence in a SWG can be tailored by varying the grating period, dutycycle, and depth [143, 206–208].

Let’s now examine the geometry in Fig. 5.4 i.e. we consider conical illumina-tion. This is identical to the geometry in Fig. 3.3. The name conical arises from

xx

zz

(a) (b)

v′pv′p

veve

uu

vovo v′′pv′′p

Figure 5.2: Normal surfaces of (a) positive uniaxial crystal and (b) negative uniaxialcrystal.

93

o D′

D′′

z

u

θ

Figure 5.3: Directions of vibrations inside a uniaxial crystal.

x

y

z

φin

uin,σ kin

θin

u(0,0)r,σ

θ(0,0)r

u(0,0)r,π

k(0,0)r

φ(0,0)r

uin,π

Figure 5.4: Wave vectors of input plane wave and zero-order reflected wave as wellas directions of vibrations of TE-electric field (σ) and TM-magnetic field (π) in caseof conical illumination.

94

the fact that the wave vectors of the reflected and the transmitted orders in sucha situation lie on a conical surface. As described in section 3.6.1, assuming a y-invariant grating geometry we can derive the expressions for the grating modes inany z- independent layer (say layer number j) starting from the space-frequency do-main Maxwell’s equations in the j-th layer which are identical to Eqs. (2.33)–(2.36).Though the structure is y- invariant, due to the light-matter interaction geometry asshown in Fig. 5.4, everywhere the fields are y- dependent and this dependence is ofthe form exp

(

ikyy)

which directly follows from Bloch’s theorem. Thus the electricand the magnetic fields in the j-th layer can be written in the following forms

E(j)(r) = exp(

ikyy)

E(j)(x, z), B(j)(r) = exp(

ikyy)

B(j)(x, z). (5.16)

However, it can be shown [209] that even in such a situation one can decomposeMaxwell’s equations in the j-th layer into two sets as described in section 2.6. The

first set corresponds to E(j)x (x, z) = 0 and the second set corresponds to B

(j)x (x, z) =

0. The other field components are usually non-zero. However, we must note thatthese sets are independent only inside the layers but they may be coupled at theinterfaces between the layers. Let us use the superscript ’o’ for the first set and thesuperscript ’e’ for the second set from now on for reasons that will become obviouslater in this section. The eigenvalue equations in matrix form for these two sets canbe derived in a similar fashion as described in section 3.6.1. These equations takethe forms

[

Jε(j)Kq−m − Lx

]

A(j) = A(j)[

Λ(j,o)/k0

]2, (5.17a)

Jζ(j)K−1q−m

[

k20 I − LxJε(j)K−1

q−mLx

]

− Y

C(j) = C(j)[

Λ(j,e)]2

, (5.17b)

where the elements of the matrix Lx are now Lx,q,m = (k2x,q + k2

y)δq,m. All the othermatrices remain the same as in section 3.6.1. For the second set, now we have a newmatrix Y with elements defined as Yi,j = k2

yδi,j.Now if the grating period is much smaller in scale as compared to the wavelength

of the incoming light (usually 1/10-th or less), we reach the quasistatic limit. Undersuch circumstances one may retain only the zeroth-order mode in Eqs. (5.17a) and(5.17b) and the higher-order modes do not contribute to the final solution. Assum-ing plane wave illumination and the quasistatic limit one may write the eigenvalueequations in the following forms

k2x,0 + k2

y +[

γ(j,o)0

]2− k2

0 ε(j)0 = 0, (5.18a)

k2x,0/ξ

(j)0 + k2

y ε(j)0 +

[

γ(j,e)0

]2ε(j)0 − k2

0 ε(j)0 /ξ

(j)0 = 0. (5.18b)

Clearly, Eq. (5.18a) is of the form of the fundamental wave vector equation for aplane wave propagating in a medium with relative complex permittivity ε(j,o) =

ε(j)0 where the z- component of the wave vector is

[

γ(j,o)0

]2. The same holds for

Eq. (5.18b) under normal incidence. For non-normal incidence we get from Eq. (5.18)

k2x,0 + k2

y +[

γ(j,o)0

]2− k2

0 ε(j,o) = 0, (5.19a)

k2x,0ε(j,e) + k2

y ε(j,o) +[

γ(j,e)0

]2ε(j,o) − k2

0 ε(j,o)ε(j,e) = 0, (5.19b)

95

where we define ε(j,e) = ξ(j)0 . Equation (5.19b) is analogous to the wave equation for

an extraordinary wave in a uniaxial crystal i.e. Eq. (5.12). Whereas, as mentionedbefore, Eq. (5.19a) is similar to the wave equation of a ordinary wave. Finally, wemay conclude that the subwavelength gratings in the quasistatic limit behave asuniaxial crystals. This property of the subwavelength gratings can be employed toconstruct wave plates, and phase retarders on demand [210–212].

5.4 FORM BIREFRINGENCE IN KERR MEDIA: ANALYTICAL FORMU-

LATION

In the previous section we found that due to structural anisotropy, subwavelengthgratings (SWGs) behave like uniaxial optical crystals in the quasistatic regime, i.e.when the grating period Λ ≪ λ, where λ is the wavelength of the incident light [113,210] and one may treat these subwavelength gratings as homogeneous media witheffective optical properties. The effective medium theory (EMT) provides two dif-ferent effective refractive indices for these SWGs. For the incident light with electricfield polarized along the grating period the SWG can be treated as a homogeneousslab with refractive index given by the ordinary refractive index, noo =

√ε(oo) and

for the light with electric field polarized perpendicular to the grating period theSWG assembly can be treated as a slab with refractive index equals to nee =

√ε(ee)

i.e. the extraordinary refractive index.In this section, we extend the classical form-birefringence theory (in linear op-

tics) to the nonlinear optical domain by considering gratings with Kerr nonlinearmaterials. We first proceed to derive analytical formulae for the theory of formbirefringence in Kerr media in the framework of the first-order EMT.

When a plane electromagnetic wave is normally incident on a lamellar 1D SWGsandwiched between two homogeneous media, the form birefringence of the gratingintroduces additional phase difference between the components of the transmittedwave polarized parallel (TM) and perpendicular (TE) to the grating vector (Λ) [113].Let us consider the lamellar grating geometry in Fig. 5.5. We assume that the inci-dent medium is linear, homogeneous, and isotropic with refractive index ni). Themodulated region located at 0 ≤ z < h is assumed to have a real-valued permittivitydistribution given by

ε(x) =

ε1 if 0 ≤ x < lε2 if l ≤ x < Λ

(5.20)

The medium with index no in the half-space z > h is defined as the substrate.Also, we assume that both lamella are made with anisotropic materials. Let us nowemploy EMT to describe the fields in the modulated region.

Assuming the validity of the plane wave limit and the quasistatic limit, the elec-tric field E and the electric displacement D can be considered constants in eachlamellae 0 ≤ x < l and l ≤ x < Λ within the grating period Λ. From the elec-tromagnetic boundary conditions we know that the tangential electric field compo-nents, and the normal components of the electric displacement remain continuousacross the interface between two materials. Hence, inspecting Fig. 5.5, we may con-clude that Dx and Ey are continuous across the interfaces at x = 0 and x = l.Consequently, Dx and Ey have constant values within the entire structure. Hence it

96

x

z

0

0

Λ

ε1ε1 ε2ε2

hεo = ε0n2

o

εi = ε0n2i

l

Figure 5.5: Geometry of the form birefringent subwavelength-period grating. Thegrating material as well as the surrounding medium are assumed to be anisotropic.

is implied from the material constitutive relation D(x) = ε(x)E(x) that

Dx = εxx(x)Ex(x) + εxy(x)Ey, (5.21)

Dy(x) = εyx(x)Ex(x) + εyy(x)Ey (5.22)

and, consequently, Ex and Dy become binary functions that are discontinuous atx = 0 and x = l. We denote the fill factor of the grating by f = l/Λ, and define theaverage value of a binary function across x

g(x) =

g1 if 0 ≤ x < lg2 if l ≤ x < Λ

(5.23)

as

〈g〉x =1Λ

∫ Λ

0g(x)dx = f g1 + (1 − f )g2. (5.24)

From now on we use a similar shorthand notation for averages over products andlinear combinations of functions. Solving Ex(x) from Eq. (5.21) and averaging overa grating period across x we get

〈Ex〉x = 〈ε−1xx 〉xDx − 〈εxyε−1

xx 〉xEy. (5.25)

from Eq. (5.25) we obtain

Dx = 〈ε−1xx 〉−1

x 〈Ex〉x + 〈ε−1xx 〉−1

x 〈εxyε−1xx 〉xEy, (5.26)

which is identical to Eq. (5.21) though all the quantities in Eq. (5.26) are averagedquantities over a grating period. Analogously, substituting the expression for Ex(x)from Eq. (5.21) into Eq. (5.22), we get the following expression

〈Dy〉x = 〈εyxε−1xx 〉xDx + 〈εyy − ε2

xyε−1xx 〉xEy. (5.27)

97

Now by substituting the expression for Dx from Eq. (5.26) into Eq. (5.27), we arriveat

〈Dy〉x = 〈ε−1xx 〉−1

x 〈εyxε−1xx 〉x〈Ex〉x

+(

〈ε−1xx 〉−1

x 〈εxyε−1xx 〉2

x + 〈εyy − ε2xyε−1

xx 〉x

)

Ey. (5.28)

Clearly, Eqs. (5.26) and (5.28), describe light-matter interactions in a SWG composedof optically anisotropic materials. These two equations can be conveniently writtenin the following matrix form

[

Dx

〈Dy〉x

]

= 〈ε〉x

[

〈Ex〉x

Ey

]

, (5.29)

where 〈ε〉x is a 2 × 2 matrix with elements defined in Eqs. (5.26) and (5.28). ForSWG with isotropic media we have ε ij(x) = δijε(x) (i, j = x, y stand for the cartesianindices, δij is the Kronecker delta) where

ε(x) =

ε1 if 0 ≤ x < lε2 if l ≤ x < Λ

. (5.30)

Hence the effective permittivity tensor 〈ε〉x becomes a diagonal tensor with the onlynon-zero elements εxx = 〈ε−1〉−1

x and εyy = 〈ε〉x.Let us now proceed to investigate the effect of Kerr nonlinearity on the form

birefringence of the structure defined in Fig. 5.5. We already saw in Chapter 4 thatin a homogeneous isotropic Kerr nonlinear medium, the electric displacement can

be conveniently written in terms of ‘effective’ linear susceptibility χ(eff)ij [134], i.e.,

Di(x) = ε0 ∑j

[

ε(x)δij + χ(eff)ij (x)

]

Ej(x). (5.31)

Here i, j = x, y stand for the cartesian indices, δij is the Kronecker delta as before, andε(x) = n2(x) is the relative (linear) permittivity. The effective linear susceptibility,that represents the Kerr effect in this case, can be written in the following form

χ(eff)ij (x) = A′(x)|E|2δij + B′(x)ℜ

[

Ei(x)E∗j (x)

]

. (5.32)

In a similar way, we can now define the corresponding effective relative permittivity

ε(eff)ij = ε(x)δij + χ

(eff)ij (x) that combines the linear and the Kerr nonlinear contribu-

tions. For the geometry in Fig. 5.5,

A′(x) =

6χ(3)1122 − 3χ

(3)1221 if 0 ≤ x < l

0 if l ≤ x < Λ, (5.33)

B′(x) =

6χ(3)1122 if 0 ≤ x < l

0 if l ≤ x < Λ, (5.34)

where as usual χ(3)ijkl are the elements of the general fourth-rank third-order suscep-

tibility tensor [134]. As we discussed in section 4.2, the optical Kerr effect creates

98

off-diagonal elements in the matrix ε in Eq. (5.29) and hence propagation of in-tense light in an isotropic medium results in light-induced optical anisotropy. Also,Eq. (5.29) shows that the elements of the matrix ε now become field-dependent.

Finally, we can employ equations (5.26), (5.28), and (5.32) to derive a mathe-matical expression for ε which strongly depends on the intensity as well as on thepolarization state of the incident field. However, for a subwavelength grating withfinite thickness, we must account for the interference effects arising from the reflec-tions at the interfaces located at z = 0 and z = h as these effects have non-negligibleimpact on the local field inside the modulated region located at 0 < z < h.

Although in the quasistatic limit, the field is independent of x and y coordinates,it strongly depends on the z coordinate. Consequently, ε also becomes z- depen-dent. Hence, in the nonlinear regime, we are dealing with an effective anisotropicstructure which behaves as a uniaxial crystal but the optic axis of the crystal likemedium varies as a function of z.

The thin-film problem must be solved iteratively, for example by increasing theintensity from a value corresponding to the linear-optics limit towards the targetvalue as described in section 4.2. For this, we prefer to split the modulated re-gion into a number of z-invariant layers, and then employ the recursive S-matrixapproach as described in section 2.17, to solve the field distribution inside theanisotropic-thin-film-stack and the interplay between the nonlinearity and the aniso-tropic-thin-film interference inside the component.

To get further insight to the interrelation between the elements of ε and theelectric-field components, let us now proceed to derive more transparent expressionsfor the mean permittivity-tensor components. Let us first combine Eqs. (5.21), (5.22),and (5.32) to obtain

−Ex = −ε−1D′x + ε−1[(A′ + B′)|Ex|2Ex + A′|Ey|2Ex

+ B′ℜ(ExE∗y)Ey], (5.35)

D′y = εEy + (A′ + B′)|Ey|2Ey + A′|Ex|2Ey

+ B′ℜ(ExE∗y)Ey, (5.36)

where D′j = Dj/ε0 and we have omitted the explicit x-dependence for the sake of

brevity. We first approximate Ex ≈ ε−1D′x in the right-hand sides of Eqs. (5.35) and

(5.36), and then average over the grating period to arrive at

D′x〈ε−1〉x =

⟨

A′ + B′

ε4

⟩

x

|Dx|2Dx +

⟨

A′

ε2

⟩

x

|Ey|2Dx

+ 〈Ex〉x +

⟨

B′

ε2

⟩

x

Re(DxE∗y)Ey, (5.37)

D′y = εEy + 〈A′ + B′〉x|Ey|2Ey +

⟨

A′

ε2

⟩

x

|Dx|2Ey

+

⟨

B′

ε

⟩

x

Re(DxE∗y)Ey. (5.38)

Next, we combine the relation D′x ≈ 〈ε−1〉x〈Ex〉x with Eqs. (5.37) and (5.38), to find

99

out at once that

〈ε(eff)xx 〉x ≈ 1

〈ε−1〉x+

⟨

A′ + B′

ε4

⟩

x

|〈Ex〉x|2〈ε−1〉4

x+

⟨

A′

ε2

⟩

x

|Ey|2〈ε−1〉2

x

, (5.39)

〈ε(eff)xy 〉x = 〈ε(eff)

yx 〉x ≈⟨

B′

ε2

⟩

x

Re(〈Ex〉xE∗y)

〈ε−1〉2x

, (5.40)

〈ε(eff)yy 〉x ≈ 〈ε〉x +

⟨

A′ + B′⟩x|Ey|2 +

⟨

A′

ε2

⟩

x

|〈Ex〉x|2〈ε−1〉2

x. (5.41)

In the next section we will see that although the expressions given by Eqs. (5.39)–(5.41) are approximate, they may lead to sufficiently accurate prediction of the non-linear form birefringence. However, we note that these expressions are also z- depen-dent in case we have subwavelength gratings with finite thickness. From Eq. (5.40)we find that the permittivity matrix has light-induced off-diagonal elements withtheir magnitudes depending also on the polarization state inside the SWG. Sincethe phase difference between Ex and Ey varies with the propagation distance insidethe structure, the direction of optic axis of the form-birefringent component variesas a function of z.


Let us first assume that the pillars i.e. the grating ridges in Fig. 5.5 are composedof a certain class of amorphous polymer material with negligible nonlinear opticalabsorption, linear refractive index n1 = 1.88 and the nonresonant third-order sus-

ceptibility χ(3)1122 = χ

(3)1221 = 1.19 × 10−17 m2/V2. The surrounding medium and the

medium after the component are assumed to be fused silica with n2 = no = 1.47,and the medium of incidence is air with ni = 1. Furthermore, we assume that the in-tensity of the incident field is I = 5 GW/cm2. The fill factor of the SWG is assumedto be f = 0.7. We compare the results obtained by the analytical formulae derivedin the previous section with those obtained with the nonlinear FMM as introducedin section 4.3. In the FMM simulations λ = 532nm and Λ = 50 nm are assumed.

Figure 5.6 elucidates the effect of Kerr nonlinearity as a function of the SWGthickness h. With the assumed very high value of third order nonlinearity as wellas high value of the incident field intensity, the light-induced form birefringence isseen to lead to a radical increase in the phase difference between the y and x com-ponents of the field. Also, we find that the Approximate Effective-Medium Theory(A-EMT), given in Eqs. (5.39)–(5.41), predicts the results quite well. Since the gratingperiod is much smaller than the wavelength of the incident light, the general EMT[Eqs. (5.26), (5.28), and (5.32)] is essentially identical to the results obtained by thenonlinear FMM.

In our second example, we study the effect of the polarization state of the in-cident field on the light-induced form birefringence. This time we use the EMTapproach to study the same structure as in the previous example, but now withfour different incident polarization states. Clearly from the results illustrated inFig. 5.7 we observe that the strongest effect is obtained with an incident field thatis linearly polarized in the y direction (for this y- invariant geometry one may ex-pect the opposite for a x- invariant geometry), and the weakest with the x-polarized

100

0 300 600 900 1200 15000

0.3

0.6

0.9

1.2

1.5

FMM/EMT(nl)A-EMT(nl)A-EMT/EMT/FMM(linear)

h [nm]

(Φy−

Φx)[

Rad

]

Figure 5.6: Comparison of EMT, A-EMT, and the rigorous Fourier Modal Method(FMM) with 45 linearly polarized incident field.

field.These results can be understood by noting that in the linear limit 〈ε(eff)

xx 〉x ≤〈ε(eff)

yy 〉x, where the equality holds only if l = Λ or l = 0. Inspecting Eqs. (5.39)–(5.41),it is easy to understand that with y- polarized field, the light-induced anisotropy is

larger for 〈ε(eff)yy 〉x than for 〈ε(eff)

xx 〉x. For the x- polarized case, the light-induced formbirefringence is rather weak because the x component contributes almost equally to

〈ε(eff)xx 〉x and 〈ε(eff)

yy 〉x. The examples included so far do not uncover the information

0 300 600 900 1200 15000

0.5

1

1.5

0 300 600 900 1200 15000

0.3

0.7

1

y-Polx-Pol

45° LPCP

45° LPCP

y-Polx-Pol

h [nm]

h [nm]

(Φy−

Φx)[

Rad

]∆(Φ

y−

Φx)

Figure 5.7: Effect of the state of polarization of the incident field on the formbirefringence (top) and the nonlinear form birefringence viz. the difference of thephase difference to the linear case (bottom). Here 45 LP means incident light thatis linearly polarized at ±45 with respect to the x and y axes and CP means left (L)or right-handed (R) circularly polarized light. The oscillatory behaviours are due tothin-film interferences which grow with film thickness.

about the polarization change upon propagation through the Kerr nonlinear form

101

birefringent component. In our third example, we study the effect of Kerr nonlin-earity on the polarization azimuth Θ and the ellipticity angle τ, which are definedin the inset of Fig. 5.8. Figure 5.8 shows the light-induced change of the state ofpolarization for the geometry considered in Fig. 5.5. We observe that the state ofpolarization of the incident light has a significant effect on the light-induced polar-ization change, as can be expected based on the results discussed in the previousexample.

h [nm]

h [nm]

∆Θ[R

ad]

∆τ[R

ad]

Figure 5.8: Change in the polarization azimuth (top) and the ellipticity (bottom) ofthe zero-order transmitted field. In the inset we show how we define the polarizationazimuth Θ and the ellipticity angle τ. Clearly, these are identical to the definitionsintroduced in section 2.9.

5.6 SUMMARY

In this Chapter, we have investigated the effects of intensity and the polarizationstate of the incident field on the form-birefringence properties of subwavelengthgratings. Simple expressions are derived for the permittivity-matrix elements bytaking into account the light-induced changes. We found that in contrast to thelinear case, in which a form-birefringent SWG always acts as a negative uniaxialcrystal, the nonlinear case is much more intricate because of the role of polarizationstate of light.

Although in the numerical examples, we assumed high values of the incidentintensity and the nonlinear susceptibility, which leads to extremely strong light-induced effects, we believe that observable light-induced changes can be realized inpractice especially using ultrashort laser pulses with high peak intensity. Since thedispersion of form birefringence is, in general, rather weak [208], we believe that it ispossible to design components for which the polarization properties stay essentiallyunmodified within the spectral bandwidth of the pulses.

102

To summarize, we have developed a theory that can be used to design all-optically tunable form birefringent components which may find applications in non-linear integrated photonics especially to construct tunable waveplates, retarders, orphase modulators.

103

6 All-optical modulation and optical bistability with a

Silicon Nitride waveguide grating

Nonlinear optical processes can be largely enhanced using resonant optical ele-ments. Strongly confined local fields inside these resonant structures can reach arather high level such that the nonlinearities become remarkable even at low in-put powers. To date, several structure geometries have been proposed and boththeoretical and experimental research are carried out to demonstrate all-opticallytunable functionalities by exploiting enhanced nonlinear optical responses of theseresonators. Examples include ring resonators [60–62,213], resonant cavities inducedby defect modes in photonic crystals [63–66], resonant microspheres for whispering-gallery modes [214], metallic nanostructures with surface plasmon resonances [215],waveguides [216], grating couplers [217], hybrid plasmonic-dielectric systems [218],resonant waveguide gratings [67, 68] etc.

Among numerous all-optically controllable functionalities, all-optical switchingis perhaps the most fascinating one, where an incoming switching beam redirectsother beams through nonlinear light-by-light scattering [134]. In all-optical rout-ing, it is required to control the state of an optical switch using past informationi.e. one needs to build a sequential logic circuit. One way to build such a logicelement is to employ optical bistability [219]. In an optically bistable system, thesystem output can be switched between two stable states optically [134]. Usually,bistability is observed with extremely intense light fields and is difficult to achievein solid state materials as the required intensity level is high enough to damage anymaterial. Hence, for practical applications, materials with high nonlinear opticalcoefficients are needed and large interaction length is required to increase the effec-tive nonlinearity. An alternative and more efficient way is to employ the resonantnanostructures cited in the previous paragraph.

Another important parameter one should carefully consider while constructinga bistable device is the response time of the nonlinear material, which may sig-nificantly affect the switching speed. Hence nonlinear materials with ultrafast re-sponse time are required as well to build fast optical switches. Since 1960’s, therehas been an extensive research focus on nonlinear optical materials. Third ordernonlinear materials (both centrosymmetric and non-centrosymmetric) such as pureSilicon (Si) [70], Titanium di-oxide (TiO2) [220], Silicon Carbide (SiC) [221], SiliconNitride (Si3N4) [71], Chalcogenide glasses [222], and Graphene-Silicon hybrid mate-rials [223,224] have been carefully examined in the past. Eventually, it was found outthat while some Si based devices suffer from low response time and high nonlinearoptical losses (large Two Photon Absorption (TPA), free-carrier effects etc.), Si3N4and chalcogenide glass based devices can offer large ultrafast third-order nonlinear-ities, and low TPA at the same time.

Recently, low power optical bistability in nonlinear silicon and silicon nitridephotonic crystals and ring resonators have been experimentally demonstrated [213,225, 226]. Besides devices that can be integrated on chip, free-space bistable deviceshave their own significance. These free-space devices have easy optical fan-in/outand hence can be considered as good candidates for large-scale parallel all-optical

105

signal processing. Possibility to achieve optical bistability in diffraction gratingswith nonlinear dielectric/semiconductor materials, metallic gratings with groovesfilled with nonlinear media, multilayered gratings, parallely stacked thin films andone dimensional slab waveguide gratings have been extensively studied theoreti-cally [179, 183, 227–231].

Among the commonly used free-space diffractive optical components, the reso-nant waveguide grating structure (RWG) or guided mode resonance filter (GMRF)is thought to be an excellent candidate for realizing all-optical sequential logic el-ements based on optical bistability [228, 229, 232]. A waveguide grating in its sim-plest form is composed of a relief grating layer, a waveguide layer and a substratelayer [141]. Depending on the structural dimensions of these RWGs and the geome-try of light-matter interaction, strong resonance phenomena can be observed whichyield reflection/transmission peaks/dips. The central frequency and linewidth ofthese resonance peaks also depend on the structural parameters. At resonance,a guided mode is excited by the incident wave which interferes with the directlytransmitted/reflected wave destructively and as a result 100 percent efficiency canbe achieved in reflection/transmission. Also, at resonance there is strong field con-finement inside the structure which enhances the nonlinear effects. This enhancedlocal field inside a RWG, might also be helpful to realize low energy optical bista-bility in practice.

Here, using the nonlinear FMM developed in Chapter 4, we perform numeri-cal experiments with GMRFs having different structural parameters. The results ofthese experiments demonstrate that low energy optical bistability in reflection/trans-mission for normally incident field can be observed by strong nonlinear light-matterinteractions in Silicon Nitride waveguide-gratings having 2-D periodicity. We fabri-cate the RWGs from PECVD synthesized Silicon Nitride thin films on top of quartzsubstrates with standard electron beam lithography and reactive ion etching tech-niques. Finally, we perform the experiments with a single wall Carbon nanotube(SWCNT) modelocked ultrafast fiber laser and demonstrate all-optical modulationof the transmission spectra of these SiN RWGs.

This Chapter starts with the working principle of a RWG. Then we define theterm optical bistability and briefly discuss about the theory of optical bistability of aFabry-Perot resonator. After that, discussions are made regarding the merits of us-ing Silicon Nitride as a material for constructing these RWGs. Next, we present thefabrication methodology for these nanogratings and then the numerical simulationresults. Finally, the experimental procedure is described and experimental resultsare included.

6.1 THEORY OF OPTICAL BISTABILITY - THE FABRY-PEROT RES-

ONATOR APPROACH

Certain nonlinear optical systems can possess more than one output states for agiven value of input. This phenomenon is termed as ‘Optical multistability’. Bista-bility refers to a case where there are two output states for a specific input state.

Let us now recall the very first experimental demonstration of optical bistability[233]. In that experiment, a Fabry-Perot resonator with nonlinear optical mediaembedded inside its resonant cavity was considered. Geometry of such a structureis shown in Fig. 6.1. In the sketch, M1 and M2 are two identical loss-less dielectricmirrors with reflection coefficient ρ, reflectance R, transmission coefficient τ, and

106

transmittance T. A, Ar, and At are the incident, reflected, and the transmittedfield amplitudes respectively. B and C are the amplitudes of the forward and thebackward propagating fields inside the resonator, where the field amplitudes aremeasured at the inner surface of M1. Now, we can represent the internal fields ofthe resonant cavity in terms of the incident field using the following relations

C = ρB exp (2inωL/c − 2αL) , (6.1)

B = τA + ρC. (6.2)

Here n is the refractive index and α is the absorption coefficient of the cavity whichare functions of the incident field intensity (I), and L is the length of the cavity.Here, we have assumed that n and α are spatially invariant. We eliminate C fromEquations (6.1) and (6.2) to yield

B =τA

1 − ρ2 exp[2inωL/c − αL]. (6.3)

Equation (6.3) can predict optical bistability if n and/or α have strong dependenceon I. In this context, we must mention that considering the strength of the inten-sity dependence of n or α, optical bistability may be classified into two categorieswhich are pure absorptive and pure refractive types. Here, we shall consider onlypure refractive bistability i.e. we consider nonlinear materials with negligible nonlin-ear absorption/two-photon absorption (TPA) is placed inside the resonator cavity.Furthermore, we assume that the linear loss of the cavity is negligible. Hence, inEq. (6.3), we put α = 0 to obtain

B =τA

1 − ρ2 exp[2inωL/c]. (6.4)

Again, making use of the relation ρ2 = R exp (iφ), in Eq. (6.4) and assuming TEpolarized light incidence we get,

B =τA

1 − R exp[i(φ + 2n0ωL/c + 2n2 IωL/c)]=

τA

1 − R exp[iδ]. (6.5)

Here, n0 and n2 are the linear and the nonlinear refractive indices of the cavitymaterial respectively, δ is the total phase shift acquired in a cavity round trip andI = IB + IC = |B|2 + |C|2 ≃ 2IB (ignoring the standing wave nature of the fieldinside the cavity). Next, we use the relation Iin = 2n1ǫ0c |A|2 for the incident fieldintensity, where n1 is the linear refractive index of the medium to the left of themirror M1, to readily obtain the following expression from Eq. (6.5)

IB

Iin=

1/T

1 + (4R/T2) sin2(δ/2). (6.6)

Equation (6.6) shows that for a particular value of the incident field intensity (Iin),there might exist more than one distinct values of the cavity intensity (IB). Under aspecial circumstance, when IB has three distinct solutions for the range of the inputintensities Iin, It = |At|2 vs. Iin plot looks like the curves in Fig. 6.2. Clearly from thefigure, any value of Iin between I1 and I2 yields two distinct values of It dependingon the previous state (in terms of the intensity) of the system.

107

A B

CAr

At

M1 M2

L

Figure 6.1: Geometry of a Fabry-Perot resonator with optical Kerr nonlinearmedium filling its cavity.

It

IinI1 I2Im

Figure 6.2: Typical input-output relationship of an optically bistable system. Bista-bility can be observed for a range of incident field intensities lying between I1 andI2.

6.2 WORKING PRINCIPLE OF A WAVEGUIDE GRATING

In this section, we present a simple ray picture model for describing the nature ofthe interaction of the grating-waveguide structure with an incident plane wave. Aresonance waveguide grating (RWG) or a guided mode resonance filter (GMRF) inits simplest form is shown in Fig. 6.3. It consists of three layers- a thin gratinglayer of thickness δ and linear refractive index n2, a waveguide layer of thickness tand having the same refractive index (n2) as the grating layer, and a substrate layer

108

A

B C

D E

F

θ

d

E1

ξO1

O2

n2

δ

t

a

a

c

c

Figure 6.3: Geometric interpretation of a resonant waveguide grating [141] in itssimplest form consisting of a grating layer of thickness δ, a waveguide layer ofthickness t and a substrate layer. θ is the angle of incidence of the incoming planewave and ξ is the angle of diffraction. The higher order diffracted rays are notshown in the figure.

(refractive index n3). When the structure is illuminated with a plane wave from thegrating side, part of the incident wave is directly transmitted through the structure(D) and part is diffracted by the grating layer and then might get trapped insidethe waveguide layer (B) [141]. Again, this trapped wave B might face total internalreflection at the interface between the waveguide layer and the substrate and emergeas C as shown in Fig. 6.3. Part of C gets diffracted out of the waveguide layer andemerges as E which is clearly collinear with D. On special conditions, which dependon the grating period (d), δ, t, the angle of incidence (θ) of the incoming plane wave,and the refractive indices n1, n2; D and E might interfere destructively resultingin complete loss of transmission. Again, part of F gets reflected from the interfacebetween the waveguide layer and the substrate and after getting diffracted again bythe grating layer as depicted in Fig. 6.3, might emerge as E1 which is also collinearwith D and E. The cycle is repeated and hence the system is analogous to a Fabry-perot resonator described in section 6.1.

We can describe the grating diffraction in B by the following relation

n12π

λinsin θ + m

2π

d= n2

2π

λincos(ξ). (6.7)

To understand the resonance mechanisms qualitatively, let us consider for now thatD and E are the only interfering transmitted waves. Here we emphasize that onemust retain all the higher order diffracted waves (Ei, where i = 1, 2, 3, . . . ) andshould use multiple interference model as implemented in Ref. [141] for an accuratedescription of resonance phenomenon and while calculating the spectral bandwidthof the resonance. Alternatively, one can use FMM (as described in Chapter 3) to

109

model the optical properties of a GMRF.The complete destructive interference of D and E is governed by certain phase dif-ference conditions between these two waves. We find that the total phase differencebetween D and E is (under the assumption that the grating layer is infinitely thin)

ΦTotal = Φp + Φr + 2Φd (6.8)

where, Φp is the phase difference gained due to the optical path length differencebetween the incident wave and the wave traveling in the waveguide, Φr is the phasedifference acquired by the total internal reflection at the interface, and 2Φd denotesthe phase difference attained through the phase shift associated with each of thediffractions. The leading edge of the incident wavefront a strikes the grating atpoint O1, where it is diffracted towards the bottom surface and then reflected tothe top surface as wave front c, whose leading edge strikes the grating at point O2,which corresponds to point O1, located an integer number of grating periods fromO1. By some geometrical constructions, using Eqs. (6.8) and (6.7) and assuming TEpolarized incident field we get,

ΦTotal = 2n32π sin(ξ)

λint + m

2πl

d+ 2Φ f + 2Φn − π. (6.9)

Here, l is the distance (l = q × d, where q is an integer) between the points O1 andO2 as illustrated in Fig. 6.3, φ f denotes the Fresnel phase shift at the interface, Φn

is the Fresnel phase shift acquired due to the refractive index mismatch betweenthe grating material (n2) and the surrounding medium (n1), and m can take anyinteger value. At resonance, the structure supports guided modes inside the waveg-uide layer (which eventually leaks into the superstrate). The following condition isrequired to support a guided mode

2n32π sin(ξ)

λint + mq2π + 2Φ f + 2Φn = m2π. (6.10)

From Eqs. (6.9) and (6.10), we see that for m = 1, i.e. for the first diffraction or-ders, the phase difference between the waves D and E is π. Thus, total destructiveinterference takes place between these two waves for a particular range of values(depends on the structure geometry) of λin (wavelength of the incident wave) andone can achieve 100 percent efficiency in direct reflection. Thus the structure acts asa narrow band reflector. With increasing value of n2, this resonance peak usuallygets shifted towards the longer wavelengths. Now, if the grating is made with Kerr-nonlinear materials, assuming TE polarized light input, its refractive index can bedescribed by the following equation

n2t = n2 + nnl I, (6.11)

where, I is the incident field intensity, and nnl is the nonlinear refractive index of thegrating material as defined in Chapter 4. A moderately intense field can change theeffective refractive index of the grating layer as well as the waveguide layer due tothe local field enhanced Kerr nonlinearity inside these layers and hence can tune thestructure out of resonance. As a result, 100 percent to 0 percent efficiency in zerothorder reflected field can be obtained.

110

6.3 SILICON NITRIDE VS. CRYSTALLINE SILICON AS A NONLINEAR

MATERIAL

Silicon-on-insulator (SOI) has already been established as a platform for linear pho-tonics and initiated the development of silicon photonic chip industry [234]. Someof the advantages of SOI are as listed below

• SOI platform can reap the benefits of exploiting already well established com-plementary metal-oxide-semiconductor (CMOS) infrastructure.

• It is possible to combine both electronics and photonics on the same chip.

• High refractive index of silicon allows tight light confinement.

• Silicon has high Kerr nonlinear coefficient (n2) and hence when combined withits high linear index, can be effective for low power nonlinear photonics.

However, in crystalline bulk silicon, the optical Kerr nonlinearity competes withparasitic nonlinear absorption effects which include two-photon absorption (TPA)effects, free-carrier effects, and defect-induced effects. TPA and TPA generated free-carrier effects are significant in all telecommunication bands with wavelengths (λ)shorter than 2000 nm. Though the free-carrier effects can be compensated by usinga p-i-n junction which sweeps out the carriers [235], there is no way to overcome theTPA in the telecommunication window not even by engineering waveguide dimen-sions. Hence, the band structure of silicon poses a fundamental limit for using it asa material for nonlinear optics in the near infrared region.

Numerous all-optical switching devices have been demonstrated lately with sili-con based resonant micro and nanostructures. It has been realized and also provedexperimentally that the switching operations in silicon occur mainly due to free-carrier induced nonlinearity [60,236]. The response time of these devices are limitedby the lifetime of these free carriers (∼ 100 ps). Clearly, this also limits the realiza-tion of ultrahigh speed all-optical operations. Hence, there have been a quest forfinding a material with either shorter carrier lifetime [237, 238] or negligible TPA.

Historically, Aluminium Gallium Arsenide (AlGaAs) is the first material pro-posed for nonlinear optics in the telecommunication band [239]. AlGaAs is a semi-conductor material with larger bandgap and it is possible to tune its nonlinearity bychanging the alloy composition. Another important platform is based on Chalco-genide glasses [240] which has very high Kerr nonlinearity. However, there arefabrication challenges associated with both of these platforms.

Very recently, a new platform based on Silicon Nitride (SiN) has been establishedfor nonlinear integrated photonics [71]. SiN is a CMOS compatible material and inthe past, it has been used as a platform for linear integrated optics. Historically, ithas been very challenging to achieve a low-loss, thick (> 250 nm) SiN layers due tothe associated tensile film stress which results in cracking of the films. However, inrecent years, it has been possible to grow thicker films using both plasma-enhancedchemical vapour deposition [PECVD] [241] and low-pressure chemical vapour depo-sition [LPCVD] [242] techniques. Perhaps, the first nonlinear effect in a SiN waveg-uide was demonstrated in 2008 [241], where nonlinearity induced resonance peakshift of a ring resonator was experimentally demonstrated. Here we list some of themerits of SiN as a material for nonlinear optics

111

• SiN has a larger refractive index (≃1.8-2) as compared to SiO2 which allowssmall waveguide dimensions.

• Large bandgap of SiN makes it optically transparent already in the visiblespectral regime (from λ = 400 nm).

• SiN has negligible TPA coefficient in the near infrared (communication wave-lengths) and the free carrier effects are believed to be minimal even if inputpower ∼100 mW is used.

• As SiN is grown by deposition process, multilayered structures can be grownat ease (layer stack flexibility).

• It has moderately high n2.

• Linear absorption losses in amorphous SiN can be extremely low.

• Thermo-optic coefficient of SiN is very low and hence thermal nonlinearitiesare minimal.

• Integration with CMOS electronics is possible.

Though numerous works based on SiN waveguides, where the SiN layers weregrown by LPCVD, have been reported [243–246], there are not much referencesavailable where the SiN films are grown by PECVD. PECVD as compared to LPCVD(process temperature about 800C) is a lower temperature (400C) process which isadvantageous for hybrid integration. Table 6.1 shows a comparison between the SiNfilms grown by these two techniques. In this context, we must mention that there is

Table 6.1: Comparison between PECVD and LPCVD grown SiN films.

PECVD LPCVDDeposition temp. 80-400C 800C

r.i. (λ =800 nm) 1.89 2.03

Film stress Manageable High

Absorption Water peaks at 1520 nm Low

Uniformity Medium High

Etch selectivity No Good vs. SiO2

Content Trace of H Stoichiometric SiN

a possibility to control the ratio of the silicon and the nitride contents [247] duringthe deposition process, which is crucial in optimizing the nonlinear properties of thedeposited films for case specific applications. Recently, strong third harmonic gen-eration in a silicon-rich nitride (PECVD) waveguide grating [69] has been reported.Also, a micro ring resonator based Kerr switch has been demonstrated experimen-tally where the silicon-rich nitride thin film is synthesized by PECVD [62].

112

6.4 FABRICATION OF THE WAVEGUIDE GRATING STRUCTURES

Before going into the detailed fabrication process, we shall first briefly describe theprinciples associated with some of the most important steps involved in lithographicfabrication of these SiN RWG samples.

6.4.1 Thin film deposition

In this subsection, we briefly describe the thin film coating techniques we used inthis work.

Evaporation

In thermal evaporation, both the material source and the substrate are placed insidea vacuum chamber to avoid reactions between vapour and atmosphere. There aretwo ways to heat the source material. These are resistive heating or heating by highenergy electron beam. As evaporation is a line-of sight technique, there is usually athickness gradient on the substrate [248]. One way to overcome this is to rotate thesample during deposition which in turn averages out the thickness variation. In ourwork, Kurt J. Lesker lab-18 evaporation unit was used.

Chemical vapour deposition

Chemical vapour deposition (CVD) is a well known and widely used material-processing technique [249]. Though the primary application of CVD is thin-filmcoating of solid state materials, it can also be used to produce bulk and compositematerials. In a CVD process, one or more precursor gases are flown into the reactorchamber which contains heated objects to be coated. Chemical reactions near andon these heated surfaces result in thin-film deposition. The chemical by-productsgenerated from the reactions and unused precursor gases are exhausted out of thechamber. Depending on the process parameters and reactor configurations, thereare many variants of CVD. It can be either hot-wall or cold-wall reactor type. Theprocess pressure can range from sub-torr level to above-atmospheric level. Depend-ing on the type of the CVD, process temperature can also vary in between 200-1600C. There also exists a variety of enhanced CVD processes which may involve theuse of plasmas, ions, lasers, hot filaments etc. to increase deposition rate and/ordecrease process temperature (economic process). Hot wall reactor CVDs are run atvery high temperatures and at low pressures (∼10 Torr). Example include LPCVD.

Some of the advantages of CVD are as listed below

• CVD films are conformal in nature (film thickness on the substrate side wallsequals to the film thickness on the top of the substrate).

• A wide range of material can be deposited with high purity.

• Deposition rate is high.

• High vacuum is not needed as compared to physical vapour deposition.

However, there are also some disadvantages which include

• CVD precursors need to be volatile at near-room temperature which is non-trivial for a large number of elements in the periodic table.

113

• Precursors can be highly toxic.

• Precursors can be costly.

• Chemical byproducts can be hazardous.

• High temperature deposition might not be suitable for many substrates.

• The stresses in the films (resulting from high temperature deposition) maycause mechanical instabilities.

An example for a relatively lower temperature CVD is plasma enhanced chemicalvapour deposition [250]. In PECVD, electrical energy is used to generate plasma(glow discharge). Due to the electrical energy transferred into the precursor gasmixture, reactive ions and radicals, neutral atoms and molecules etc. are formedwhich are in highly excited state. These reactive and energetic species, which areformed by collision in the gas phase, interact with the substrate and thin films aredeposited. The process temperature may lie in between 300-400C. Some of theproperties of PECVD films include good adhesion, and uniformity.

6.4.2 Electron beam lithography

Electron beam lithography (EBL) is a technique for creating small patterns in a high-energy sensitive chemical called resist. It was originally developed for the electronicsindustry [251] but later was adopted also by the optics community [252, 253]. Theresolution of EBL is far better as compared to optical lithography [254, 255] and itcan be used to generate more complex patterns. Another advantage is that a widevariety of substrates can be used. However, EBL system is relatively slow and hugelyexpensive. Hence, it is not suitable for industrial applications but more appropriatefor research purposes.

The first EBL was constructed in late 1960s by adding a beam blanking unit, apattern generator, and an interferometric stage controlling unit to a scanning elec-tron microscope. Schematic of the column in the Vistec EBPG5000+ES HR EBL tool(the system used in the present work) is presented in Fig. 6.4. It includes an elec-tron source, an alignment system, a blanker, magnetic lenses, a beam deflector, astigmator, apertures and a detector. The source generates an electron beam usinga thermal field emission gun, or by thermionic emission. The emitted electrons areaccelerated through the column by application of voltage. The alignment system isused to center the beam in the column and the lenses are used for focusing pur-poses. Apertures are used to limit the beam and also to block stray electrons. Theblanker is responsible to turn on and turn off the beam. By use of the deflector, thebeam is scanned on the sample surface. Any imperfection in beam alignment maycause beam astigmatism which is corrected by the stigmator. Electron detectors areused to help in focusing of the beam and to find the alignment marks. The sample isplaced on a high-precision movable stage (x-y) in a chamber placed underneath thecolumn. Clearly, a vacuum system is needed inside the column and the chamber.The area which can be patterned without moving the stage is called the main field.Structure with larger size (than the main field) is patterned by moving the x-y stageto the center of the next main field. This movement can be done with high precisionusing interferometric stage measurement system. Several EBL systems are availablein market. The Vistec EBPG5000+ES HR is a Gaussian beam system which providesbetter resolution as compared to other beam systems though the system throughput

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Lens

Anode

EmitterSuppressorExtractor

Focus

Gun alignment

Lens

Blanking

Blanking

Deflection

Final lens

Substrate

Figure 6.4: Schematic of the column of the Vistec EBPG5000+HR electron beampatterning tool. (Courtesy of Vistec Lithography Ltd.)

is not good enough for very-large-scale integration (VLSI) manufacturing industries.This system can operate with 20 kV, 50 kV or 100 kV voltage with up to 50 MHzpattern generator. With 50 kV and 100 kV operating voltages, the maximum mainfield sizes are 409.6 × 409.6 µm and 256 × 256 µm respectively. The beam exposure

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time (t) on a surface area S is determined from the following relation [256]

t = σsS/I, (6.12)

where σs is the surface charge density widely known as the dose and I is the beamcurrent.

6.4.3 Resist technology

Resists are sensitive to the energy of the electron beam. Due to absorption of energyfrom the beam, the molecular weight of the resist material in the exposed regionsis changed which enables dissolution of either the exposed or the unexposed areaby a chemical, called the developer [257]. Some of the sought properties of the elec-tron beam (e-beam) resists include high resolution, high sensitivity, and proper etchresistance (important while transferring the resist pattern to the substrate). E-beamresists are dissolved in a liquid solvent and can be sprayed on the substrate usingseveral techniques such as spray coating, roll coating, dip coating or spin coating.In most cases, spin coating technique is used as this provides better layer unifor-mity. After the coating, the solvent is evaporated by pre-heating or soft-baking. Thebaking time and the temperature affect the exposure and the development [258].After the e-beam exposure, the pattern created in the resist is developed by immer-sion, spray, or puddle method. The later two are automated processes with betterreproducibility.

Depending on the response to the energy of the electrons, resists can be dividedinto two categories-

• Positive resist- The exposed areas become more soluble in the developer.

• Negative resist- The exposed areas become less soluble in the developer.

Generally, positive resists are more suitable to create binary profiles [259] and thenegative resists for making continuous profiles [260]. However, there exist someexceptions [261, 262]. To fabricate the RWG samples, we use AR-P 6200 which is apositive tone resist.

6.4.4 Reactive ion etching

After the pattern is generated in the resist, it is required to transfer it to the substrate.This is done by etching. In this process, the material is removed selectively in the ar-eas defined by the etching mask. There are both dry and wet etching processes. Wetetching is a chemical process and usually isotropic in nature which means that thematerial is removed from all directions. On the contrary, dry etching is anisotropicin nature.

Reactive ion etching (RIE) [263] is a dry etching method, which uses plasmacontaining charged particles (initiated by a strong radio frequency (RF) field) to re-move material. The material removal can be due to processes which are physical orchemical in nature. These two processes can occur simultaneously. In physical etch-ing, also known as sputtering, the ions remove material by collisions. This processis highly directional and anisotropic. In chemical process, material is removed bychemical reactions with the etchant and the target material which generates volatileproducts [255]. This process is isotropic in nature as the chemical reactions occur

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along the horizontal direction. In order to make the etching process highly selective,one needs to control both the physical and the chemical parts of process carefully.

In all dry etching processes, both the material to be etched and the mask materialare removed. The ratio of their etching depths is termed as selectivity. For highervalues of selectivity, deeper structures can be fabricated.

Chromium etching

Chromium (Cr) etching is performed in chlorine atmosphere. One can add suitableamount of oxygen to increase the Cr etch rate without losing selectivity over theresist mask [264]. In Cr etching process, resists such as ZEP, ARP-6200, or HSQ canbe used as etching masks. However, SiO2 or TiO2 can also be used.

SiN etching

Nitrides can be etched using gases containing fluorine such as CHF3, CF4, or SF6.The physical part of the etching can be done by adding O2, He, or Ar. In our work,we used a gas mixture of CHF3 and O2. Addition of O2 suppresses formation offluorocarbon on nitrile layer which slows down the etch rate. It also increases theetch rate [265] of the resist hence decreases the selectivity. However, in our work,we needed to etch only a shallow part of Si3N4. Hence, decrease of etch selectivitywas not a crucial issue.

The fabrication process flow is shown in Fig. 6.5. The fabrication steps are listedas below-

Quartz

Silicon Nitride

Quartz

Silicon Nitride

Cr

Cr Deposition

Quartz

Silicon Nitride

Cr

ARP 6200 1:1

Resist Coating

Quartz

Silicon Nitride

Cr

E beam exposure

Quartz

Silicon Nitride

Cr Etching

Silicon Nitride

Quartz

Resist removalSilicon Nitride Etching

Quartz

Cr Removal

Figure 6.5: Fabrication process flow of the Silicon Nitride waveguide gratings.

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• Silicon Nitride thin film of thickness 400 nm (±10%) was deposited on top of aquartz substrate with 1 inch diameter and thickness 0.5 mm using PEVCD. Agas mixture of (2% SiH4/N2): NH3=100:3 was used. The process temperatureand pressure were 300C and 1000 mTorr, respectively.

• Cr layer of 50 nm layer thickness was deposited on top of Si3N4 film usinglab-18 evaporation unit.

• The Cr layer was coated with positive resist AR-P 6200 of layer thickness≃220nm and the sample was pre-baked at 150 for 3 minutes.

• Grooves of the grating were patterned using electron beam lithography tool.

• The exposed resist layer was developed with ethyl 3-ethoxypropionate (EEP).

• Cr was etched in Plasmalab100 with a gas mixture of Cl2 (54 sccm) and O2 (4sccm) for 3 minutes and 30 seconds. In this dry etching process, we used AR-P6200 as the etch mask.

• Residual resist layer was removed by means of O2 cleaning (30 sccm O2) for30 seconds.

• Si3N4 etching was performed in Plasmalab80 with a gas mixture of CHF3/O2=45/10 sccm. Suitable etching time for the desired grating depths were foundout during the process.

• Finally, chromium was removed by means of Cr wet etch solution and thesamples were cleaned in acetone and isopropanol.

Fig. 6.6 shows a side cut view (a) and a top view SEM (b) of a fabricated twodimensionally periodic structure (periodic square pillars).

(a) (b)

Figure 6.6: (a) side cut and (b) top view SEM images of a 2-D periodic waveguidegrating structure.

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6.5 NUMERICAL SIMULATION RESULTS

Most of the numerical experiments are performed with waveguide gratings whichare periodic along two mutually orthogonal directions (in this case x and y) withequal grating periods along x and y i.e. with square shaped pillars as shown inFig. 6.6.

The possibility to construct optical bistable devices based on all-dielectric waveg-uide gratings having only one dimensional periodicity have been numerically stud-ied before by Magnusson et. al. [232]. However, our numerical analysis reveal thatthe 2-D periodic gratings have better angular tolerance as compared to 1-D periodicwaveguide grating structures (in its simplest form) with similar spectral response.Also, to achieve high quality factor (a quantity used to measure the sharpness ofresonances) with a 1-D periodic GMRF, tight fabrication tolerances are needed.

To exemplify, let us compare the diffraction efficiencies in the 0-th order reflectedlight for two waveguide grating geometries as illustrated in Fig. 6.7. The grating pa-

SiN

SiN

SiO2

d

c δt

(a) (b)

Figure 6.7: (a) 1-D periodic and (b) 2-D periodic waveguide grating geometries.

rameters for the geometries in Fig. 6.7(a) and Fig. 6.7(b) are listed in Table 6.8. Therefractive indices of the grating material (i.e. silicon nitride) and the substrate areassumed to be ng = 1.934 and ns = 1.4442 respectively. Figure 6.8 (a) shows that thequality factors (Q) of the resonances, which are defined as the ratios of the resonancefull widths at half of maxima (∆λmax) to the resonance peak wavelengths (λmax), arealmost the same. Clearly, to attain same Q-resonance with 1-D periodic geometries,shallower grating layers are needed, which is very challenging in fabrication pointof view. Figure 6.8 (b) shows the 0-th order reflected efficiencies of the 1-D and the

Table 6.2: Grating parameters for the 1-D and 2-D periodic geometries in Fig. 6.7.

1-D 2-DGrating period (d) 942 nm 1035 nm

Grating layer thickness (δ) 20 nm 40 nm

Waveguide layer thickness (t) 369.3 nm 349.3 nm

Linewidth (c) 0.6 × d 0.6 × d

2-D periodic structures (with grating dimensions given in Table 6.8) as a function ofthe angle of incidence (θ) of the incoming TE polarized plane wave of wavelength

119

λmax = 1548.9 nm. From the plots, it is clear that the 2-D periodic geometry offersbetter angular tolerance. Hence, 1-D periodic waveguide gratings with very high Q-s are more prone to get tuned out of resonance. Figures 6.9(a) and (b) show electricfield intensity distributions inside and outside the 1-D periodic waveguide gratingstructure for θ = 0 and θ = 0.3 respectively at λmax = 1548.9 nm. Clearly, thestructure gets tuned out of resonance for angle of incidence θ = 0.3 and the lo-calized electric field intensity is much weaker as compared to the case with normalincidence of light i.e. for θ = 0.

In Figs. 6.10 (a) and (b), the reflected efficiencies in 0-th diffraction orders cor-responding to the 1-D and the 2-D periodic geometries respectively are plotted asfunctions of both θ and λ. Clearly from the plots, it is challenging to design andrealize a high Q 1-D periodic waveguide grating structure as a slight deviation fromthe normal incidence can tune the structure out of resonance. Also, a modest changein the grating dimensions of the fabricated samples can largely shift the resonanceacross the wavelength. On the contrary, 2-D periodic geometries have better angu-lar tolerance and any shift in resonance resulting from the fabrication errors can becompensated by varying the angle of incidence of the incoming light.

After initial design, the 2-D periodic grating structures are fabricated by the

1548 1548.4 1548.8 1549.2 1549.6 15500

0.2

0.4

0.6

0.8

1

1D RWG2D RWG

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

2D RWG1D RWG

λ [nm]

ηr0

0η

r00

θ [Degrees]

(a)

(b)

Figure 6.8: (a) Comparison between spectral responses in direct reflection of a 1-Dperiodic and a 2-D periodic waveguide grating. The grating parameters are listedin Table 6.8. (b) Efficiency in direct reflection plotted as a function of the angle ofincidence of the incoming TE polarized plane wave with wavelength λmax = 1548.9nm. With almost the same Q resonances, the 2-D periodic geometry offers betterangular tolerance.

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−200 0 200 400 600

−1500

−1000

−500

0

500

1000

1500

500

1000

1500

2000

−200 0 200 400 600

−1500

−1000

−500

0

500

1000

1500 1

2

3

4

5

6

7

z [nm]z [nm]

x[n

m]

x[n

m]

(a) (b)

Figure 6.9: Localized electric field intensity distributions inside and outside the 1-Dperiodic waveguide grating structures at λmax = 1548.9 nm. (a) and (b) correspondto the cases with θ = 0 and θ = 0.3 respectively. The grating geometries arehighlighted by the white lines. TE polarized light is assumed to be incident from airon the grating side.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

1548

1548.5

1549

1549.5

1550 0

0.2

0.4

0.6

0.8

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

1548

1548.5

1549

1549.5

15500.10.20.30.40.50.60.70.80.9

θ [Degrees]

θ [Degrees]

λ[n

m]

λ[n

m]

(a)

(b)

ηr00

ηr00

Figure 6.10: Efficiency in direct reflection as functions of θ and λ for (a) a 1-D and(b) a 2-D waveguide grating having about the same Q factors.

121

lithographic techniques as described in the previous section. Numerical simula-tions are performed with the values of the grating dimensions (grating periods dx

and dy, thickness of grating layer δ, thickness of waveguide layer t, fill factors fx

and fy) obtained from SEM and optical profilometer measurements. The refractiveindices of the SiN thin films and the substrates are measured with ellipsometerictechnique [266].

In Table 6.3, we present the measured values of the refractive indices, and thegrating dimensions for samples indicated by S2 and S3. Grating dimensions andrefractive indices for sample Ss presented in Table 6.3 are chosen arbitrarily. Fig-

Table 6.3: Specifications of the gratings, and the refractive indices of SiN films andsubstrates evaluated at λ = 1550 nm. Values corresponding to samples S2 and S3are evaluated by SEM and optical profilometric measurements, whereas the valuescorresponding to Ss are chosen arbitrarily.

S2 S3 Ss

dx = dy = d [nm] 1027 1030 1030

ng (λ =1550 nm) 1.95 1.95 1.95

ns (λ =1550 nm) 1.4442 1.4442 1.4442

δ [nm] 41 55 20

t [nm] 345 341 380

fx = fy = f 0.6 0.6 0.6

ures 6.11 (a)–(c) show the shifts of the resonance peaks in reflection towards thelonger wavelengths with growing intensity of the incident electric field (I0) for sam-ples S2, S3, and Ss respectively. These curves are plotted with the FMM based tech-nique [199] introduced in Chapter 4. Here, we have assumed normal incidence oflight from air on the grating sides. Also, we have used the experimentally measuredvalue of χ(3) = 3 × 10(−20) m2/V2 for the SiN thin films as reported in ref. [69]. TheSiN films used in ref. [69] and the films employed in our work were grown undersimilar PECVD conditions. The nonlinear absorption of SiN is neglected aroundλ = 1550 nm, the Kerr nonlinear response of SiN is assumed to be instantaneousand the substrate as well as the surrounding media are modeled as linear materials.It is clear from the plots that the gratings with lower δ/t value, yield resonanceswith higher Q-factors as with increasing value of t, the waveguide modes get lessleaky. As the field enhancements inside the gratings are directly proportional to theQ-factors, Kerr-nonlinearity induced resonance peak shift can be achieved at lowerfield intensities for gratings with higher Q-s. From the plots, we notice that at higherfield intensities, the resonance curves become asymmetric, which might result in op-tical bistability.

To investigate further, for each sample we choose a specific wavelength fromFigs. 6.11 (a)–(c), at which the reflection efficiency is about 10-20% of maximum ef-ficiency on the falling edge of the resonance curve (for the linear case). Intensitiesof 0-th order reflected fields at those specific wavelengths are plotted with respectto the incident electric field intensities (Iin). Figures 6.12(a) and (b) show such plotsfor S2 and S3, where the two branches in each plot (solid black and dotted redlines) correspond to the cases when the incident field intensities are increased and

122

1543 1543.5 1544 1544.5 1545 1545.5 1546 1546.5 15470

0.2

0.4

0.6

0.8

1

Linear

I 0 =50 MW/cm2

1563.2 1563.3 1563.4 1563.5 1563.6 1563.7 1563.80

0.2

0.4

0.6

0.8

1

Linear

I0=300 KW/cm

2

I0=500 KW/cm

2

I0=700 KW/cm

2

1540 1540.5 1541 1541.5 1542 1542.5 15430

0.2

0.4

0.6

0.8

1

Linear

I0=10 MW/cm2

I0=20 MW/cm2

λ [nm]

λ [nm]

λ [nm]

ηr0

0η

r00

ηr0

0

(a)

(b)

(c)

S2

S3

Ss

Figure 6.11: Efficiency in direct reflection plotted as function of λ for RWG samples(a) S2, (b) S3, and (c) Ss respectively with their specifications given in Table 6.3. Foreach sample, several plots corresponding to different incident field intensities areincluded.

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decreased respectively using staircase functions. Step-sizes for these staircase func-tions are chosen carefully to assure numerical convergence. Clearly, the plots forS2 and S3 show optical bistability at the chosen wavelengths λS2 = 1541.56 nm andλS3 = 1545.7 nm respectively for a range of values of Iin. From the figures, we seethat the cut-off intensities for observing bistable switching at λS2 and λS3 are 15MW/cm2 and 40 MW/cm2 for the samples S2 and S3 respectively.

Figures 6.13(a) and (b) show the reflected and the transmitted field intensities

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1 2.2

x 1011

0

0.5

1

1.5

2

2.5x 10

11

Increasing branch

Decreasing branch

2.5 3 3.5 4 4.5 5 5.5 6

x 1011

0

1

2

3

4

5

6x 10

11

Increasing branch

Decreasing branch

Iin [Watts/m2]

Iin [Watts/m2]

Ir out[W

atts

/m

2 ]Ir ou

t[W

atts

/m

2 ]

(a)

(b)

λS2 = 1541.56 nm

λS3 = 1545.7 nm

Figure 6.12: 0-th order reflected field intensities plotted as a function of the in-cident electric field intensities for (a) S2 and (b) S3. The solid black lines and thedotted red lines are the plots corresponding to the increasing and the decreasingfield intensities respectively and are marked by arrows.

respectively plotted against the incident field intensity for the sample Ss. The cut-offintensity required for observing bistable switching at λSs in this case is about 750

124

KW/cm2. Optical bistability at wavelengths which are slightly different than thoseused in our calculations, can also be observed for all the samples. Nevertheless,the cut-off intensity as well as the nature of the hysteresis loop will be different ifwe choose a different working wavelength. It is worth mentioning that in all thenumerical simulations presented in this section, we assumed monochromatic planewave incidence.

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

x 109

0

2

4

6

8

10x 10

9

Increasing branch

Decreasing branch

5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 10

x 109

0

1

2

3

4

5x 10

9

Increasing branch

Decreasing branch

Ir out[W

atts

/m

2 ]It ou

t[W

atts

/m

2 ]

(a)

(b)

λ = 1563.62 nm

λ = 1563.62 nm

Iin [Watts/m2]

Iin [Watts/m2]

Figure 6.13: (a) 0-th order reflected field intensity and (b) 0-th order transmittedfield intensity plotted as a function of the incident electric field intensity for thesample Ss. The solid black lines and the dotted red lines are the plots correspondingto the increasing and the decreasing field intensities respectively and are marked byarrows.

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6.6 EXPERIMENTAL RESULTS

Schematic of the experimental set-up is illustrated in Fig. 6.14. We use a fiber lasermodule providing transform-limited 495 fs optical pulses with pulse repetition rate17.7 MHz. The central wavelength (λp) of the pulses is 1560.5 nm and their 3 dB(decibel unit defined as 10 × log10 η, where η is the power efficiency) spectral band-width is 6.7 nm. Mode locking is performed using single wall carbon nanotube(SWNT) saturable absorber. The intracavity polarization controller (PC) is used toadjust polarization for mode-locking optimization. In this section, we do not gointo the details of constructing the fiber laser module. The interested reader is en-couraged to read ref. [267] for a detailed description. The fiber-integrated tunable

Fiber laser module Cavity PC

BP filter

Amplifier

PC

Chopper Attenuator

Sample

OSA

U-bench collimator

Figure 6.14: Schematic of the experimental set-up.

band pass (BP) filter TB-TWF-1550 (PriTel, Inc USA) is used to precisely tune thecentral wavelength of the pulses inside the optical communication C-band and theKOA 3000-Keopsys Erbium doped fiber amplifier (EDFA) unit is used to amplifythe optical pulses. The pulses after the tunable unit and the amplifier unit are mea-sured using a second-harmonic generation autocorrelator APE Pulse-check 50 andthe pulse train is measured by an oscilloscope connected with a photodetector (notshown in Fig. 6.14). The measured pulse duration of the amplified pulses is 0.84 psat λp = 1547.3 nm, the fundamental pulse repetition rate is 17.7 MHz, and the 3 dBspectral bandwidth of the pulses after amplification is > 50 nm. The polarizationcontroller after the amplifier unit is used to study the polarization dependence ofthe sample response (transmission). To ensure collimated beam, we use an opti-cal U-bench collimator. The tunable attenuator, and the sample mounted inside aholder that can be tilted across the horizontal and the vertical axes (to ensure normalincidence of light) are placed inside the U-bench collimator. Inside the collimatorbench, we also use an electronic chopper to reduce the average input power andhence to minimize the thermo optical effects. The output of the U-bench collimatoris connected to an optical spectrum analyzer (OSA) which measures the transmittedsignal from the sample.

Figures 6.15 (a)–(b) show the comparison between the numerical simulation re-

126

sults and the experimental results showing diffraction efficiencies (in dB units de-fined as 10 × log10 ηt00, where ηt00 is the transmission efficiency) in direct transmis-sion for the samples S2 and S3 respectively with TE polarized light at the input. Thereference signals used to plot the experimental curves, are the transmitted signalsfrom the respective samples when they are out of resonance.

Clearly, the experimental results show 97-99% maximum efficiency (−16-20 dB)in transmission inside a narrow spectral width. Broadening of the resonance widthsin the experimental results can be attributed to the fabrication defects, SEM measure-ment errors (for measuring the grating dimensions) and to various loss mechanisms(include material loss, waveguide loss etc.) in the linear optical regime. Resonancepeak position mismatch between the numerical simulation and the experimental re-sults can be ascribed to the uncertainties associated with the SEM, the profilometer,and the ellipsometer measurements of the grating dimensions and the refractive in-dices of the SiN films.

In Figs. 6.16 (a)–(b), we plot the transmission efficiency spectra for the samples

1536 1537 1538 1539 1540 1541 1542 1543 1544 1545 1546−60

−50

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−30

−20

−10

0

Experimental

Simulation

1540 1541 1542 1543 1544 1545 1546 1547 1548 1549 1550−40

−30

−20

−10

0

Experimental

Simulation

λ [nm]

λ [nm]

10×

log 10

[η00

t][d

B]

10×

log 10

[η00

t][d

B]

(a)

(b)

Figure 6.15: Numerical simulation results vs. experimental results showing effi-ciency spectra in direct transmission for samples (a) S2 and (b) S3 respectively withincident electric field intensity I = 100 W/cm2.

127

S2 and S3 respectively for several values of the incident electric field intensity. Thespot diameter Dbeam of the collimated beam is evaluated using the Knife-Edge mea-surement technique [268]. The measured value of the beam diameter Dbeam = 700µm meets the requirement for the minimum illumination area on the incident planeas described in ref. [232]. The tunable attenuator placed in front of the sample isused to tune the peak power of the pulses and hence to control the intensity levelof the incident field. Both Figs. 6.16 (a) and (b) show modulation of transmitted sig-

1540 1541 1542 1543 1544 1545 1546 1547 1548 1549 1550

−15

−10

−5

0

I=0.118 MW/cm2

I=0.725 MW/cm2

I=1.012 MW/cm2

1536 1537 1538 1539 1540 1541 1542 1543 1544 1545 1546−20

−15

−10

−5

0

I=0.11 MW/cm2

I=0.72 MW/cm

2I=0.41 MW/cm2

1539.5 1540 1540.5 1541 1541.5 1542 1542.5 1543−20

−15

−10

−5

0

λ [nm]

λ [nm]

10×

log 10

[η00

t][d

B]

10×

log 10

[η00

t][d

B]

(a)

(b)

Figure 6.16: Experimental results showing efficiency spectra in direct transmissionfor samples (a) S2 and (b) S3 respectively at different intensity levels of the incidentfield. Inset of (a) shows blue shift of the resonance peak by 0.16 nm at the incidentfield intensity of 0.41 MW/cm2.

nals from the samples with increasing field intensities. Shifts of the resonance peaksacross the wavelength are understood as results of the refractive index changes ofSiN. Using the chopper, we found out that these effects only depend on the peakpower levels of the pulses and reducing the average power does not have any effecton the resonance peak. Hence we may conclude that the refractive index changesare not associated with the thermo-optic effects. Also, we notice that at higher in-tensities, the resonance becomes wider and shallower. This can be understood by

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noting that the SiN material used in our work contains silicon nanocrystals of size1-2 nm [269]. At higher intensities, free-carriers are generated inside these SiN filmscontaining silicon nanocrystals which are responsible for the observed strong non-linear absorption. However, due to the quantum-size effects associated with thesilicon nanocrystals [62], the electronic Kerr effect in SiN is also enhanced. Besides,nonlinear absorption, free-carriers also cause blue-shift of resonance as the refractiveindex of SiN decreases with increased concentration of free carriers. On the contrary,the electronic Kerr effect causes red-shift of the resonances as the refractive indexof SiN increases with increasing light intensity. Usually, the electronic Kerr effect isinstantaneous having response time ∼fs, whereas the response time of free carriereffects in SiN can be ∼ps or sub-ps. At higher intensities, these two effects competewith each other. The blue-shifts of the resonance curves in Fig. 6.16 (a) is due tothe dominance of the free-carrier Kerr effects in sample S2 which has narrower res-onance (and hence greater field enhancement) as compared to S3. The red-shift ofthe resonance curves in Fig. 6.16 (b) is due to the dominance of the electronic Kerreffect. Clearly from Fig. 6.16 (a), for sample S2, free-carrier induced resonance shiftsare ≈ 0.16 nm (shown in the inset of Fig. 6.16 (a)) and ≈ 0.35 nm at the incidentpulse peak intensities of 0.41 MW/cm2 and 0.72 MW/cm2 respectively. For S2 atλ = 1540.7 nm, about 36% change in efficiency in direct transmission is observeddue to the free-carrier dominated optical Kerr effect.

6.7 SUMMARY

This Chapter covers numerical simulation and experimental results showing thepossibilities to achieve optical bistability and all-optical modulation of optical sig-nals inside the optical communication C band with silicon nitride resonance waveg-uide grating structures. While the numerical simulation results show prospects forachieving optical bistability, experimental results demonstrate all-optical modula-tion of signal by the optical Kerr effect.

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7 Modeling nanocomposite optical materials with FMM

To date, there exists a large variety of optical materials which are suitable for spe-cific applications. Nevertheless, still there is need for new materials with propertiessuperior to the existing ones. Hence, controlling the properties of optical materialsis a subject of eminent importance. One can tailor the properties of a material in sev-eral ways. Nanostructuring at subwavelength scale can yield unprecedented opticalproperties which gave birth to the field of optical metamaterials [3]. Another way tocontrol optical properties is by means of molecular engineering i.e. by intermixingtwo or more materials at the molecular level [270]. Best results can be produced bymerging the two aforementioned methods.

In this chapter, we shall first briefly discuss the theory of intermixing two ormore materials to form new media, which are widely known as nanocomposites.However, we shall restrict our discussions mainly to the optical properties of thenanocomposites. After that, we present analytical models for three common typesof nanocomposite geometries. Although accurate and efficient under certain con-ditions, these models have limitations. We propose an alternative technique foraccurate modeling of the nanocomposites which is based on FMM. Finally, wepresent several numerical examples produced with our FMM based model for vari-ous nanocomposite geometries containing both linear and Kerr nonlinear materials.These include porous silicon and glass-metal nanocomposites. In the quasi-staticlimit, some of these examples are validated using the standard effective mediummodels.

7.1 NANOCOMPOSITE OPTICAL MATERIALS

Nanocomposite media are formed by mixing two or more homogeneous materialsin nanoscale. Though the particles forming the nanocomposites are usually muchsmaller than the wavelength of light, each of the constituents in the mixture retaintheir individualities and can be characterized by their respective permittivities. Op-tical properties of the nanocomposite materials can be controlled by tailoring the ra-tio of the constituents, changing the mixing morphology, and controlling the shapesand sizes of the individual nanoparticles [271]. Upon proper tailoring, nanocom-posites can portray the best properties of the individual constituents. In some casesthey can show properties which even exceed those of their constituents.

In recent years, these new classes of artificial materials have emerged as efficientalternatives to conventional media in optical applications. For example, these mate-rials gave birth to the photonic crystals which enabled dispersion control to achievephase-matched nonlinear optical processes [272, 273]. Composite media where thequantum dots are embedded in dielectric matrix evolved as efficient media formode-locking of solid state lasers [274,275]. Also, nanoscale ceramic composite gainmedia for lasing [276,277] which possess improved optical properties as well as ther-mal properties [278, 279] have been reported. Besides the lasing properties, one canalso exploit the local field effects in nanocomposites to obtain improved nonlinearoptical properties as in the nonlinear optical regime, the material response scales

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as several powers of the local-field correction factor. Several experiments related tothese showed promising results [280–283].

With advancement of nanofabrication techniques, nanocomposite materials arebecoming more and more important. Among several types of composite media,glass-metal nanocomposites (GMNs) have remained at the focus of nanocompositeresearch for a long time. GMNs are glasses containing silver, copper, gold, nickel orother metal nanoparticles. Their optical properties are determined by surface plas-mon resonance (SPR) of the metal nanoparticles contained in these. In the vicinityof the metal nanoparticles, field is strongly enhanced at the SPR frequencies, whichin turn leads to enhanced optical nonlinearities of GMNs. This giant enhancementof optical nonlinearity can be used in optoelectronics applications [284]. Also, theenhancement of the photoluminescence [285, 286] and SERS [287, 288] signals in theimmediate neighbourhood of the individual metal nanoparticles, make GMNs ap-pealing for development of novel active media for solid state lasers and biophotonicsrespectively. GMNs can be useful also for data storage applications [289].

Another important class of artificial material is porous silicon (por-Si). Por-Si is acomposite medium containing nanometer-size silicon crystals separated by air poresof similar size [290] and can be formed by electrochemical etching of crystalline sil-icon [291, 292]. Porous silicon (Por-Si) is promising material for several practicalapplications including sensing, photovoltaic technology, and lasing [293–300]. Also,due to the air pores, there is strong local field confinement inside por-Si structureswhich may also be beneficial for nonlinear optical applications [301–303].

7.2 QUASI-STATIC APPROXIMATION AND ITS VALIDITY

Effective medium theories can be applied to describe optical properties of nanocom-posite materials. However, several characteristic length scales are associated withthe nanocomposites. The size (d) of the individual grains which form the nanocom-posites is the limit at which the individual constituents can still be characterized bytheir permittivities. Usually, the lower limit is 1-2 nm [304]. However, below d ∼ 1nm, quantum mechanical effects may dominate. The upper-limit is determined bythe validity of the quasi-static approximation when the wavelength of the incidentfield is infinite (as compared to the individual grain sizes) and the time-variationof the optical field can be neglected. Hence, in the framework of the quasi-staticapproximation, the individual particles forming the composite medium behave as ifthey were placed in a static electric field and they exhibit screening surface charge.Due to this, local field inside the medium varies. Under specific conditions, elec-tric field can get concentrated inside the grains with lower optical densities (lowerrefractive indices). Beyond the quasi-static limit, we already reach the region offinite-wavelength and waveguiding effects start to take place. In the guided waveregime, electric field tends to be confined within the regions with higher opticaldensities.

Aspens and Egan performed extensive theoretical and experimental studies [304–306] on the validity of the quasi-static limit in the context of nanocomposite mediaformed by pressed spherical Al2O3 particles. They made conclusions that if thesizes of the individual grains do not exceed 0.25λ (λ is the wavelength of the inci-dent light), effective medium theories can still be applied. Between d ∼ 0.25λ-0.5λ,quasi-static approximation starts to break and beyond d ∼ 0.5λ strong waveguidingeffects become dominant. However, in this context we should emphasize the fact

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that the threshold for the validity of the quasi-static limit is different for differentcomposite media. For silver nanoparticles in glass, the quasi static approximationalready starts to break if the size of the metal grains d ≅ 100 nm or if the volumeconcentration of the silver and glass in the composite becomes comparable.

In effective medium theories, one needs to perform spatial averaging to derivethe average material properties. This introduces another length scale. The lengthover which spatial averaging can be done should be much larger than the individ-ual grain size though much smaller than λ. In a work by LeBihan et al. [307], thislength was determined to be λ/4 but once again this might not be the same for allnanocomposites.

7.3 COMMON COMPOSITE GEOMETRIES

Mainly three different types of composite geometries have been extensively inves-tigated in the past which are Maxwell-Garnett composites [308, 309], Bruggemancomposites [310], and layered composites [282, 311]. These three composite geome-tries are illustrated in Fig. 7.1.

(a) (b) (c)

εh

εi ε1

ε2

εe

εg

Figure 7.1: Three types of common composite structures (a) Maxwell-Garnett type,(b) Bruggeman type, and (c) layered composites.

7.3.1 Maxwell Garnett geometry

In the Maxwell-Garnett geometry (shown in Fig. 7.1(a)), the composite medium isassumed to be formed by a collection of small spherical/ellipsoidal nanoparticlesdistributed in a host medium. The dimensions of these nanoparticles are assumedto be much smaller than the wavelength. Also, the separation distances between theinclusion particles are assumed to be much larger as compared to their characteris-tic size though much smaller than the wavelength. In this model, it is assumed thatthe host medium completely surrounds the inclusions. The effective dielectric con-stant (εeff) of the composite can be derived using Maxwell-Garnett effective mediumtheory which gives the following relation

εeff − εh

εeff + 2εh= fi

εi − εh

εi + 2εh, (7.1)

where, εh and εi are the dielectric constants of the host and the inclusion materialsrespectively, and fi is the volume fraction of the inclusion in the composite. TheMaxwell-Garnett model can predict the existence of plasmon resonances i.e. thecase when εi + 2εh = 0. However, due to asymmetrical treatment of the host and theinclusion, different results are obtained if we interchange εi and εh in Eq. (7.1). Also,

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this model is applicable only if the volume fraction of the inclusions is extremelysmall i.e. fi ≪ 1 and hence gives incorrect results when the host and the inclusionhave comparable volume fill fraction.

7.3.2 Bruggeman geometry

In the Bruggeman model, the host and the inclusion are treated symmetrically. Eachparticle of the constituents is assumed to be embedded in an effective medium withrelative permittivity εeff. The equation which defines εeff is given by

f1ε1 − εeff

ε1 + 2εeff+ f2

ε2 − εeff

ε2 + 2εeff= 0, (7.2)

where ε1 and ε2 are the relative permittivities of the constituent materials 1 and2 respectively, f1 and f2 are their volume fractions. Clearly, in the limit f1 ≪ f2,if we replace f1 by fi, f2 by fh, ε1 by εi, and ε2 by εh, Eq. (7.2) reduces to theMaxwell-Garnett relation i.e. to Eq. (7.1). Due to the symmetrical treatment ofthe host and inclusion, Bruggeman model can describe percolation (drastic increaseof the conductivity of a composite material at certain volume concentrations of itsconstituents). However, this model cannot predict surface plasmon resonance. Also,when fi and fh are comparable, this model starts to break.

7.3.3 Layered composite geometry

There exists a third kind of composite geometry which is shown in Fig. 7.1(c). Itconsists of alternating layers of two homogeneous materials e and g with differentoptical properties. Thicknesses of these layers are much smaller than the wave-length. This type of composites are artificially anisotropic.

If the electric field of the incident light is polarized parallel to the layer surfaces,the effective relative permittivity is given by the following expression

εeff = feεe + fgεg, (7.3)

which is a simple volume average of the relative permittivities of materials e andg. Electric field in this case stays uniform inside the composite as electromagneticboundary conditions require that the tangential electric field components should becontinuous across the interface of materials e and g. However, for incident light withelectric field polarized perpendicular to the layers, the effective relative permittivitycan be evaluated from the following expression

1εeff

=fe

εe+

fg

εg. (7.4)

Clearly in this case, electric field is distributed inside the layers in a nonuniformmanner resulting in inhomogeneous local fields.

7.4 OPTICAL PROPERTIES OF NANOCOMPOSITES CONTAINING

METAL NANOPARTICLES

In subsection 2.15.3, we saw that if a metal nanoparticle is exposed to an electricfield, the shift of the conduction electrons with respect to the metal’s ionic core

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induces surface charge on the other side of the metal particle. This surface chargein turn causes restoring force. Hence, in an alternating electric field, the shiftedconduction electrons together with the restoring force behaves as an oscillator. At aspecific frequency, strong resonance is observed, which is also known as localizedsurface plasmon resonance (LSPR). This resonance causes giant enhancement of thenear field.

We also saw that the resonance frequency depends on the free electron densityof the metal and also on the geometry of the particle. Mie proposed a solutionof Maxwell’s equations for spherical metal particles which can explain the originof LSP. According to Mie [312], different eigenmodes of the spherical particles aredipolar or multipolar in nature. In the quasi-static limit, i.e. when the size of themetal particles are extremely small as compared to the wavelength, the externalelectric field can be assumed to be static. Let us now recall Mie’s solution for themicroscopic polarizability from subsection 2.15.3

α = 4πr3 εi(ω)− εh

εi(ω) + 2εh, (7.5)

where r is the radius of the particle as before, εi(ω) is the frequency dependentdielectric constant of metal nanoparticles and εh is the dielectric function of the sur-rounding medium. Thus the microscopic polarization (p) of the metal nanoparticleembedded in a transparent dielectric matrix is given by

p(ω) = αε0E0(ω) = 4πε0r3 εi(ω)− εh

εi(ω) + 2εhE0. (7.6)

The absorption cross-section (σ) can be calculated by following the procedure de-scribed in Chapter 4 of Ref. [313]. The detailed derivation is out of scope for thisthesis. σ is evalauated by the following mathematical expression

σ(ω) = 12πr3 ω

cε(3/2)h

εi1(ω)

[εi1(ω) + 2εh]2 + εi2(ω)2

. (7.7)

Here E0 is the external electric field, εi1(ω) and εi2(ω) are the real and the imaginaryparts of the frequency-dependent dielectric constant of the metal particle whichaccording to Drude-Sommerfeld theory is given by

εi(ω) = εb + 1 − ω2BP

ω2 + iγω, (7.8)

where ωBP is the bulk plasmon frequency, εb is the complex permittivity associ-ated with the interband transitions of the core electrons inside the atom, and γ isthe damping constant as before. While deducing Eq. (7.7), we have assumed thatthe permittivity of the surrounding medium is real. Clearly, from Eqs. (7.5) and(7.7), resonance occurs when the denominator on the right hand side of Eq. (7.7) isminimum. This gives,

εi1(ωLSP) = −2εh. (7.9)

On fulfillment of the condition in Eq. (7.9), the local field and the dipole momentin the immediate neighbourhood of the nanosphere grow resonantly and can havemagnitudes enhanced by several orders. Furthermore, Eq. (7.9) requires the realpart of the dielectric constant of the metal to be negative which indeed is the case

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for noble metals in the visible frequency regime. Combining Eqs. (7.9) and (7.8), wecan describe the position of the LSP resonance using the following expression

ω2LSP =

ω2BP

ℜ[εb] + 1 + 2εh− γ2. (7.10)

Clearly from Eq. (7.10), the interband transitions of the core electrons have signifi-cant influence on the position of LSP resonance. Usually for silver (Ag) nanoparti-cles embedded in glass, LSPR is observed around λ = 415 nm. For gold (Au), andcopper (Cu) nanoparticles, LSPR is observed at slightly higher wavelengths (528 nmand 570 nm respectively). Resonance for interband transitions for Ag is observedfar away from the LSPR at about 310 nm. Equation (7.10) also describes the effect ofthe surrounding medium on the position of the resonance. Clearly, an increase of εh

shifts the resonance peak towards the longer wavelengths.Besides these, the size of the metal nanoparticle also strongly influences the LSPR

position. The position remains almost constant for nanoparticles with radii r < 10-15 nm. However, for particles with larger radii, LSPR starts to shift towards longerwavelengths. These effects are known as extrinsic size effects [314–318]. For smallerparticles (< 1 nm), spill-out of electrons from the particle surface may result in aninhomogeneous dielectric function. As a consequence, very broad plasmon bandscan be observed for extremely small nanoparticles.

Lastly, the shapes of the metal nanoparticles have strong impact on plasmonresonances. For ellipsoidal nanoparticles with semi major axes a, b, and c, three dif-ferent LSP modes are observed which are associated with the three principal axes.The microscopic polarizability in this case takes the following form

αj(ω) =4π

3abc

εi(ω)− εh

εh + [εi(ω) + εh] Lj, (7.11)

where Lj is known as the geometrical depolarization factor with ∑ Lj = 1. For aspherical particle La = Lb = Lc = 1/3 and hence we have only one LSP mode. Inthe most general case, when the propagation direction of the incoming light doesnot coincide with any of the axes of the ellipsoid, we obtain three separate LSPbands in the absorption spectra. These three bands correspond to the oscillationsof the free electrons along three axes [314]. In case, light is polarized along one ofthe principal axes of the ellipsoid, we observe only one LSP band. These dichroicproperties of elongated metal nanoparticles have been largely employed in the pastto construct broad-band high-contrast polarizers [319].

Let us now consider the situation when we have many metal nanoparticles em-bedded in a dielectric host medium. In the quasi-static limit, the effective dielectricconstant of such a medium is given by Eq. (7.1) i.e. by the Maxwell-Garnett theory.By rearranging Eq. (7.1), we can write it in the form

εeff(ω) = εh[εi(ω) + 2εh] + 2 fi [εi(ω)− εh]

[εi(ω) + 2εh]− fi [εi(ω)− εh], (7.12)

where all the parameters are defined as in subsection 7.3.1. However, the MG(Maxwell-Garnett) model cannot be used to explain the extrinsic size effects i.e.the multipolar effects. Also, when the metal concentration is high, i.e. when wehave small separation distance between two nanoparticles, strong collective dipolarinteractions between the nanoparticles may occur which results in strong enhance-ment of the linear and the nonlinear optical properties in the vicinity of the metal

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particles. These collective dipolar interactions also cause broadening and red-shiftof absorption spectra. MG theory fails to explain these effects also and one must em-ploy rigorous theories. In the next section, we shall present a method for modelingnanocomposite optical materials accurately which is based on the FMM.

7.5 RIGOROUS MODELING- METHODOLOGY

To model the nanocomposites with FMM, we create a window that serves as an ar-tificial period or a unit cell representing the whole composite medium. Figure 7.2shows such a unit cell which is repeated along the orthogonal cartesian coordinateaxes x and y. However, in the most general case, the coordinate axes can be non-orthogonal to each other. We divide the artificial period of size D × D, where sizeof D can be ∼ λ, into sub-cells (in this case square boxes) each of size d × d as illus-trated in Fig. 7.2. Material inside each small square box is assigned a distinct valueof permittivity. In the limit d ≪ D, boundaries of a particle with arbitrary geometrycan be modeled accurately. Also, effects arising from random arrangements of thenanoparticles and clustering of nanoparticles inside the unit cell can be taken intoaccount properly. In Fig. 7.2 we illustrate the way we model particles with arbitrarygeometries (marked as type ‘1’), with size approaching the wavelength of the incom-ing field (marked as type ‘2’), overlapping of two nanoparticles (marked as type ‘3’),and infinitesimal interspacing between two particles (marked as type ‘4’). Along thepropagation direction, we slice the structure into several thin layers as described inChapter 3. This way, any arbitrary profile along the propagation direction can alsobe well fitted. However, it is worth mentioning that although fineness of meshinginside the artificial period does not significantly affect the computation efficiency(sampling in xy plane does not increase the number of linear algebraic equations tobe solved by S-matrix approach), the computation workload increases linearly withan increase of the number of layers along the propagation direction. By averagingover ensembles of such structural realizations, we can determine optical propertiesof the nanocomposite; with a sufficiently large unit cell, results obtained for a singlerealization are good approximations of the ensemble average.

This FMM based model can correctly describe the collective dipolar oscillationsin case two or more nanoparticles form a cluster with small interspacing. Also, thismodel does not impose any restriction on the size of the individual nanoparticles(unless they are so small that quantum effects take place). Hence, it can also predictmultipolar effects. Furthermore, if the volume fill fraction of the host material andthe inclusion are comparable to each other, most of the effective medium modelsfail and we need to employ full-rigorous approaches. In case, the host (or the inclu-sion) is amorphous in nature and possess optical Kerr nonlinearity with real valuedχ(3), we employ the technique described in Chapter 4 to derive the effective Kerrnonlinearity of the composite medium.


The method described in the previous section is used to model the linear opti-cal properties of all-dielectric as well as metal-dielectric nanocomposites. Further-more, we investigate plasmon enhanced Kerr nonlinear optical properties of a metal-dielectric composite medium.

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d

d

x

y

(D, D )

(0, D )1

2

3

4

Figure 7.2: Unit cell representing a nanocomposite structure. Figure shows theway we model particles with arbitrary geometries (marked as type ‘1’), with sizeapproaching the wavelength of the incoming field (marked as type ‘2’), overlappingof two nanoparticles (marked as type ‘3’), and infinitesimal interspacing betweentwo particles (marked as type ‘4’).

7.6.1 Porous silicon nanostructures

In por-Si composite media, which are prepared by electrochemical etching of crys-talline silicon, the air pores are usually oriented along a particular crystallographicaxis and depending on their sizes (a), por-Si can be classified into three categories-microporous (a < 5 nm), mesoporous (a ≈ 5 − 100 nm), and macroporous (a > 100nm). Top view of a monolayer mesoporous silicon nanocomposite is shown inFig. 7.3. In the quasi-static limit i.e. when the pore sizes are much smaller than thewavelength of light, one can apply effective medium theories to derive the opticalproperties of por-Si. However, for the mesoporous and the macroporous samples,finite wavelength effects may come into play and for samples with porosity in therange 10-90 % effective medium theory fails as shown in Ref. [303]. Furthermore, forpore sizes < 1 nm, strong quantum mechanical effects are observed. In this thesis,we will treat porous-Si media from a classical perspective i.e. we assume that thepore sizes are at least few nanometers.

To model the porous-Si media using the full-wave numerical simulation tech-nique described in the previous section, we use the structural unit cells of size 200nm × 200 nm as shown in Figures 7.4 (a)–(c). These unit cells are used to mimicthe actual structures. Three different configurations of the unit cell are used to in-vestigate the effects of random pore arrangements on the optical properties. Here,we emphasize on the birefringence properties of these nanocomposites. First, wemodel microporous monolayer structures grown on top of crystalline silicon sub-strates where the largest pore dimension is 18 nm. Thickness of the porous layer is30 nm and refractive index of silicon is assumed to be 3.5. Volume fill fraction of

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Figure 7.3: Scanning electron microscopy (SEM) image showing top-view of asponge-like mesoporous silicon nanocomposite medium.

the air pores in the composite media, corresponding to the three configurations inFigs. 7.4(a)–(c) are the same.

Numerical simulation results reveal that these microporous composite media canbe described by the effective medium theory as the feature sizes are much smallerthan the wavelength of the incoming plane wave (λ = 1550 nm) and light does notget diffracted into higher orders. Results obtained with the FMM based rigorousanalysis agree well with those obtained with the effective medium theory. The ef-fective indices of these composites for y- and x- polarized incident fields are listed inTable 7.1 which are calculated by treating the porous layers as optically anisotropichomogeneous effective media. The tabulated effective index data show that the de-gree of birefringence (difference between Ny and Nx) for different configurationsis non-identical although the effective volumes of the pores are almost the same.Hence, rigorous modeling of the randomness in pore arrangements is necessary toaccurately describe any experimental result. Next, we study model mesoporous

(a) (b) (c)

Figure 7.4: Unit cells of microporous model structures. (a), (b), (c) show threedistinct pore arrangements inside the unit cell of size 200 nm × 200 nm. Volume fillfractions of the pores for these three configurations are the same.

silicon structures where all feature sizes are 10 times of those shown in Figs. 7.4 (a)–(c) i.e. the largest pore dimension is now 180 nm. However, we use configurationswith exactly the same pore arrangements inside the unit cell as in the microporous

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Table 7.1: Effective indices of the microporous silicon layers of thickness h = 30nm on top of silicon substrates. Wavelength of the incident light is λ = 1550 nm. Ny

and Nx represent effective indices for y- and x- polarized incident fields respectively.(a), (b), and (c) represent three distinct pore arrangements inside the unit cell of size200 nm × 200 nm as shown in Figs. 7.4 (a)-(c) respectively.

Configuration type Ny Nx

(a) 2.31 2.25

(b) 2.31 2.29

(c) 2.31 2.26

model structures for the sake of comparison. We find out that for λ = 1550 nm,these mesoporous structures can not be correctly described in terms of effective in-dices as the porous layers already start to diffract light into higher orders. Hence,rigorous numerical treatment is essential. Simulation results show that the sum ofthe diffraction efficiencies in direct reflection and direct transmission for these com-posites range between 92-96% and the rest 4-8% goes into higher diffraction orders.

In Table 7.2, we list degree of birefringence i.e. φx00 − φy00 (measured in terms ofthe phase differences introduced by these porous slabs between the x and y polar-ized transmitted electric field components) of these mesoporous composites corre-sponding to configurations (a), (b), and (c). In each case, thickness of porous layerwas assumed to be 300 nm. Clearly, the degree of birefringence strongly depends onthe pore arrangements. As silicon is a high index material, for structures with larger

Table 7.2: Degree of birefringence (φx00 − φy00) of the mesoporous silicon layers ofthickness h = 300 nm on top of silicon substrates. Wavelength of the incident lightis λ = 1550 nm. (a), (b), and (c) represent three distinct pore arrangements insidethe unit cell of size 2000 nm × 2000 nm as shown in Figs. 7.4 (a)-(c) respectively.

Configuration type (φx00 − φy00) in radians

(a) 0.4287

(b) 0.2943

(c) 0.3530

pore sizes, the effects of local electric fields, which govern the optical properties ofthese nanocomposites, become stronger. Hence, to understand the characteristics ofthese media, it would be helpful to locate these regions with strongly confined localelectrical fields. The strength of our FMM based technique lies in the fact that it canbe employed to locate the hot spots i.e. the regions with maximum field concentra-tions. Figures 7.5 (a)–(c) show the transmitted near field (electric) intensity maps forthe pore arrangements corresponding to Figs. 7.4 (a)–(c) respectively.

7.6.2 Silver nanospheres on glass substrate

One of the most common metal-dielectric nanocomposite media is silver nanopar-ticles arranged on top of a glass substrate and surrounded by a dielectric medium.Plasmon resonances of such composite media are observed around λ = 400 nm and

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the absorption spectra (at resonance) of these nanocomposites strongly depend onthe surrounding dielectric media, the shape and the size of the silver nanoparticlesand also on their volume fill fractions inside the composites.

We start with a test object in the form of silver spheres arranged on top of a glasssubstrate (monolayer) and arrayed into an infinite rectangular lattice. This is thesame object used to plot Fig. 2(a) in Ref [320]. The radius of the spheres is taken tobe a = 24 nm, the refractive indices of the surrounding medium and the substrateare nd = 1 and ns = 1.52 respectively, and the refractive index data of silver is takenfrom Ref [321]. Separation between two consecutive nanoparticles define the vol-ume fill fraction of silver in the monolayer composite medium. We first assume thatd = 120 nm i.e. the size of the unit cell in Fig. 7.2 is 120 nm ×120 nm and it containsonly one silver particle. The thickness of the composite monolayer is 48 nm. Fig-ure 7.6(a) shows the transmission spectra of the monolayer structure for normallyincident TE polarized light obtained with both the FMM based technique and theMaxwell-Garnett formula in Eq. (7.1).

Clearly, these two results are in close agreement. We also see that the resultobtained with FMM is in very good agreement with the FDTD result in Fig. (2 a)in Ref [320]. With d = 120 nm, volume fill fraction (Vf ) of silver in the compos-ite medium is Vf = 0.084 which is ≪ 1. Hence, the Maxwell-Garnett theory cancorrectly estimate the location of the plasmon peak though it slightly overestimatesthe peak amplitude. In Figure 7.6(b), we plot the absorption cross-section spectra(σabs) of the composite medium, which is defined as σabs = −(1/a) log[ηt00] (ηt00 isthe diffraction efficiency in 0-th order transmitted field), for different values of Vf .

To change Vf , we simply change the value of d i.e. the size of the unit cell in theFMM simulations. From the plots, we see that the mismatch between the resultsobtained with FMM and the Maxwell-Garnett theory increases as Vf increases. TheFMM result showing broadening of the absorption spectra at Vf = 0.48 is due tocollective plasmon oscillations of the silver particles. Finally, we may conclude thatthe quasi-static approximation breaks at higher values of Vf and one must applyrigorous techniques to accurately estimate the nature of plasmon resonance. Wenow proceed to investigate the effects of particle clustering on the absorption cross-section spectra of such a monolayer composite medium. For this, we consider a unitcell containing nine silver nanoparticles. Four different arrangements as shown inFigs. 7.7(a)-(d) are considered. These figures show the cross-sections of the particles

(a) (b) (c)

0.10.1 0.10.20.2 0.30.3 0.30.40.4 0.50.5 0.50.7

Figure 7.5: Transmitted near field intensity maps for mesoporous silicon struc-tures. (a), (b), and (c) correspond to the configurations shown in Figs. 7.4 (a)-(c)respectively.

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320 340 360 380 400 420 4400

0.02

0.04

0.06

0.08

0.1

Vf =0.084 FMM

Vf

Vf=0.3 FMM

Vf=0.3 MG

Vf=0.48 FMM

Vf=0.48 MG

=0.084 MG

(a)

(b)

λ [nm]

λ [nm]

−(1/a

)lo

g[η

t00]

ηt00

300 320 340 360 380 400 420 4400

0.2

0.4

0.6

0.8

1

MG theory

FMM

300

Figure 7.6: (a) Efficiency (ηt00) in direct transmission plotted as a function of thewavelength (λ) of the incident field where the volume fill fraction of silver nanopar-ticles inside the composite medium is Vf = 0.084 (corresponding to layer thicknessof 48 nm) and normal incidence of light is assumed, (b) Absorption cross-sectionspectra of monolayer composite media for different Vf -s plotted both with FMMand the Maxwell-Garnett theory.

Config. 0 Config. 1 Config. 2 Config. 3

Figure 7.7: Cross-sections of the unit cells across the mid planes of the particlesrepresenting Configuration 0, Configuration 1, Configuration 2, and Configuration3. Vf = 0.3 for all of these four configurations.

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across their mid plane. Vf = 0.3 for all of these i.e. we only change the locationsof the particles inside the unit cell without changing the volume fill fraction of sil-ver particles inside the composite media. The configuration in Fig. 7.7(a) is alreadysimulated in Fig. 7.6 where all the particles are equally spaced. In all other config-urations, the size of the period is the size of the unit cell containing nine particles.Figure 7.8 shows the absorption cross-section spectra for these four different con-figurations. The difference is notable for Configuration 1 where collective plasmonoscillations take place because of extremely small gaps (1.4 nm) between the clus-tered particles. For Configuration 2, the gap size is increased to a limit (5.2 nm)such that collective plasmon resonances (which are observed at relatively higherwavelengths) vanish. Collective plasmon oscillations disappear also for Configura-tion 3, where the clustered particles just touch each other along x and y directions.To illustrate the outset of collective plasmon oscillations, we plot the local electric

300 320 340 360 380 400 420 4400

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Config. 0Config. 1Config. 2

Config. 3

λ [nm]

−(1

/a)

log

[ηt00]

Figure 7.8: Absorption cross-section spectra of monolayer composite media forfour different configurations as shown in Figs. 7.7 (a)–(d) with Vf = 0.3008.

field intensity distributions across the mid-planes of the particles for Configurations0 and 1 at two different wavelengths λ1 = 363 nm and λ2 = 435 nm respectively.These two wavelengths are marked by arrows in Fig. 7.8. The field intensity plotsin Figs. 7.9 (a)–(d) show that the local electric field concentrations in the small gapsbetween the clustered particles for Configuration 1 are extremely high because ofthe collective plasmon resonance. For λ2 = 435 nm, field concentration in the gapsbetween the closely-packed particles in Configuration 1 is even higher which givesrise to additional absorption peak around λ2 = 435 nm as illustrated in Fig. 7.8.

7.6.3 Silver nanorods embedded in a Kerr nonlinear host

As a final example, we simulate a monolayer nanocomposite medium consistingof cylindrical shaped silver nanorods embedded inside a nonlinear polymer hostmedium with instantaneous Kerr nonlinearity. These nanorods are assumed to bearrayed in an infinite rectangular lattice with their axes along the z- direction. Thelinear refractive index and the third-order susceptibility of the host matrix are as-sumed to be nd = 1.7, and χ(3) = 10(−17) m2/V2 respectively. Radii and heights ofthese silver nano cylinders are taken to be a = 26 nm and h = 50 nm respectively.Separations between two consecutive silver particles in the array are assumed tobe d = 65 nm both along x and y directions. Furthermore, the substrate mediumis glass with refractive index ns = 1.47. Plots in Fig. 7.10 show the absorptioncross-section spectra of the composite layer with TE polarized (y-polarized) incident

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Config . 0, λ = 363 nm Config . 0, λ = 435 nm

Config . 1, λ = 363 nm Config . 1, λ = 435 nm

00

00

1010

1010

2020

2020

3030

3030

(a ) (b)

(c) (d)

Figure 7.9: Electric field intensity distributions inside the cross-sections of the unitcells across the mid planes of the silver spheres for (a) Configuration 0 at λ = 363nm, (b) Configuration 0 at λ = 435 nm, (c) Configuration 1 at λ = 363 nm, and (d)Configuration 1 at λ = 435 nm. Vf = 0.3.

wave. These two plots correspond to the linear case and the nonlinear case withthe intensity of the incoming plane wave I = 100 MW/mm2 respectively. Clearly,due to plasmon enhanced optical Kerr effect, the absorption peak gets enhancedand red-shifted. In this example, we have modeled only the host medium as a Kerrnonlinear medium with the FMM based technique introduced in Chapter 4. Theseresults can be understood by noting the fact that with increasing field intensity, therefractive index of the surrounding polymer material also increases.

350 380 410 440 470 500 530 5500.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

I=10 GW/cm2

Linear

λ [nm]

−(1

/h)

log[η

t00]

Figure 7.10: Light induced shift of absorption cross-section spectra of monolayercomposite media consisting of silver nano cylinders embedded in a nonlinear poly-mer host material with instantaneous Kerr nonlinearity. The linear refractive indexof the polymer is nd = 1.7 and its third-order susceptibility is χ(3) = 10(−17) m2/V2.Radii and heights of the nano cylinders are a = 26 nm and h = 50 nm respectively.

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7.7 SUMMARY

In this chapter, using the well-known Fourier Modal Method commonly used tomodel diffraction gratings or periodic structures, we have developed a techniquewhich can be used to mimic nanocomposite optical materials and describe their op-tical properties. Arbitrary particle geometries, random particle arrangements insidethe host medium as well as the effects of particle clustering can be accurately mod-eled using this method. Furthermore, finite wavelength effects, multipolar effects,and local field controlled effective Kerr nonlinear properties of the nanocompositescan be accurately described by this scheme. Hence, this serves as a unified approachfor modeling the optical properties of nanocomposite media and fills the gap wherethe most commonly used effective medium theories fail.

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8 Summary, conclusions and scope of future work

Within the scope of this thesis, we have investigated the role of local electric fieldsin controlling the linear and the Kerr nonlinear optical properties of subwavelengthperiodic structures. Most of the numerical studies included in this thesis are basedon the Fourier modal method (FMM) which has been discussed in detail in Chapter3. This Chapter includes the summary along with the main conclusions made onthe basis of the theoretical, numerical, and experimental results presented in thisdoctoral thesis and lastly a brief discussion regarding the possible future researchdirections.

8.1 SUMMARY WITH CONCLUSIONS

In the subwavelength regime, as the feature sizes approach the wavelength of light,quasi-static approximations start to break due to the dominance of waveguiding ef-fects and one needs to employ full-rigorous theories such as the FMM. In Chapter4, we have shown that the FMM can be used to model local field controlled opticalKerr nonlinearity in periodic structures containing isotropic nonlinear materials, ifan iterative approach is taken. The accuracy of this FMM based approach is veri-fied. Furthermore, we demonstrate the possibility of increasing the computationalefficiency of the proposed approach by use of symmetries in light-matter interactiongeometry. Numerical results included in Chapter 4 show the convergence of themethod, and its suitability for the design and analysis of local-field enhanced lowpower all-optical devices.

• Conclusion: The FMM can be applied to model optical Kerr nonlinearity inperiodic structures using an iterative approach. Symmetries in light-matterinteraction geometry can be employed to greatly reduce the computationalefforts of this FMM based approach. The developed numerical method canbe used to accurately estimate the local field controlled optical Kerr effect insubwavelength periodic structures.

In Chapter 5, we have developed an analytical model which can describe nonlinearlight-matter interactions inside a form birefringent Kerr nonlinear optical mediumand verified its accuracy by comparing the results obtained with this theoreticalmodel with those obtained by the full rigorous FMM based technique introduced inChapter 4. Theoretical results suggest that in the quasi-static regime, both the linearand the Kerr nonlinear optical properties of structured media can be accuratelydescribed using approximative theories as these theories can estimate the local fieldsprecisely. The results also show that in contrast to the linear case, in which a form-birefringent subwavelength grating always acts as a negative uniaxial crystal, thenonlinear case is much more intricate because of the role of the polarization stateof light. The developed analytical model may aid in design of all-optically tunableform birefringent wave plates and retarders.

• Conclusion: It is possible to develop an analytical model which can describethe theory of form birefringence in optical Kerr nonlinear media. Results based

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on this analytical model show that a form birefringent medium with opticalKerr nonlinearity still behaves like a uniaxial crystal. The developed analyticaltheory can be employed to design all optically tunable wave plates.

In Chapter 6, we discussed about the possibilities to achieve optical bistability andall-optical modulation of signals with resonance waveguide grating structures. Wehave chosen silicon nitride as a material to construct the resonant gratings. Numer-ical simulation results show optical bistable switching inside the optical communi-cation C band both in reflected and transmitted signals from these gratings undernormal incidence of light. The waveguide grating structures are fabricated by elec-tron beam lithography and reactive ion etching techniques from silicon nitride thinfilms grown on top of fused silica substrates. Experiments have been carried outusing single wall carbon nanotube modelocked ultrafast fiber laser together with anamplifier and a tunable unit. Experimental results show all-optical modulation oftransmitted signals by the local field enhanced optical Kerr effect in silicon nitrideresonance waveguide grating structures.

• Conclusion: Nonlinear light-matter interactions inside a silicon nitride reso-nant waveguide grating is greatly enhanced due to strongly confined localfield. These structures can be exploited to construct low energy optical bistabledevices as well as all-optical modulators which rely on the mechanisms of theoptical Kerr effect.

In Chapter 7, we have shown the influence of the local fields on the linear and theKerr nonlinear optical properties of nanocomposites. In this chapter, we demon-strate how the rigorous FMM can be employed to model nanocomposite optical ma-terials. Numerical results display that in the quasi-static regime, effective mediumtheories can be applied to accurately model the nanocomposites. Also, in case ofmetal-dielectric nanocomposites, Maxwell-Garnett effective medium theory may ac-curately predict the plasmon resonances if the volume fill fraction of the sphericalshaped inclusions inside the host medium is ≪ 1. However, to accurately model ar-bitrary particle geometries, random particle arrangements inside the host medium,effects of particle clustering, multipolar effects and finite wavelength effects onemust employ full rigorous approaches because the optical properties in the abovementioned cases strongly depend on the localized fields inside the composite media.

• Conclusion: Local fields can strongly influence the optical properties of nanoco-mposites. The FMM based full rigorous analysis introduced in Chapter 7serves as a unified and accurate approach for modeling the optical proper-ties of nanocomposite media and fills the gap where the most commonly usedeffective medium theories fail.

8.2 SCOPE OF FUTURE WORK

The results presented in this thesis suggest several possible extensions. For exam-ple, the FMM based approach presented in Chapter 4 which assumes the incidentwave to be continuous in nature and having infinite extent, can be extended to time-domain optical pulses and also to light beams having finite extent. This extensionwould be useful for studying optical solitons.

Furthermore, to develop the theory for light propagation in a form birefringentKerr medium in Chapter 5, we assumed normal incidence of light which simplified

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our mathematical model substantially. Nevertheless, it might be possible to gener-alize our analytical approach for oblique incidence of light. Also, in future it mightbe possible to experimentally demonstrate all-optical control of form birefringence.

Resonant waveguide gratings in their simplest form were analyzed in Chapter6. It might be possible to reduce further the threshold intensity required to observelight-induced changes using multilayered structures or by means of coupled reso-nant gratings. Another possibility would be to use materials having superior non-linear optical properties, viz. materials having higher electronic Kerr nonlinearityand ultrafast response time, to construct these gratings. The experiment presentedin this thesis was carried out using only one light beam. To demonstrate all-opticalswitching mechanisms in future, it will need to build a pump-probe system withtwo beams.

In Chapter 7, we have seen that remarkable optical properties can be attainedusing properly tailored nanocomposite optical materials. In future, periodically pat-terned nanocomposites having tailored unit cells can be designed and fabricated toattain striking optical properties with vast scope of applications in optical data stor-age, sensing, and photovoltaics.

In this thesis, most of our discussions remained confined to nonlinear modula-tion of amplitude and phase of the incident wave. It would be interesting also todesign and analyze structures that can give rise to light-induced polarization rota-tion i.e. nonlinear optical activity. Lastly, all examples included in this thesis werebased on free-space propagation of light. For certain applications, it would be nec-essary to build systems where all light beams are maintained within the integratedcircuitry. Hence, it would be interesting to extend our study for on-chip light prop-agation.

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ISBN 978-952-61-2441-4ISSN 1798-5668


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SUBHAJIT BEJ

LOCAL FIELD CONTROLLED LINEAR AND KERR NONLINEAR OPTICAL PROPERTIES OF

PERIODIC SUBWAVELENGTH STRUCTURES


In this book, local-field controlled linear and Kerr nonlinear optical properties of subwavelength periodic nanostructures

and nanocomposites are studied. Efficient numerical techniques and novel analytical models have been developed to aid in these

studies. In addition, prospect for achieving low energy optical bistability with a silicon nitride

guided mode resonance filter is examined numerically followed by an experimental

demonstration of all-optical modulation using such a structure.

SUBHAJIT BEJ

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