pulse propagation in photonic crystals and nonlinear media

Pulse Propagation in Photonic Crystals andNonlinear Media

Victor Kimberg

Theoretical Chemistry

Royal Institute of Technology

Stockholm, 2005

c© Victor Kimberg, 2005

ISBN 91-7178-023-8

Printed by Universitetsservice US AB, Stockholm, 2005

i

Abstract

The present thesis is devoted to theoretical studies of light pulse propagation through dif-

ferent linear and nonlinear media. One dimensional holographic photonic crystals and one

dimensional impurity band based photonic crystals are investigated as linear media. The

effects of angular dependence of the band structures and pulse delay with respect to the

light polarization are analyzed. A strict theory of nonlinear propagation of a few strong

interacting light beams is presented and applied in the field of nonlinear optics. The key

idea of this approach is a self-consistent solution of the nonlinear wave equation and the

density matrix equations of the material beyond the so-called rotating wave approximation.

The results of numerical studies led to a successful interpretation of recent experimental

data [Nature, 415:767, 2002]. A theoretical study of the NO molecule by means of two-color

infrared – X-ray pump probe spectroscopy is presented. It was found that the phase of the

infrared field strongly influences the trajectory of the nuclear wave packet, and hence, the

X-ray spectrum. The dependence of the X-ray spectra on the delay time, the duration and

the shape of the pulses are studied.

ii

Preface

The work in this thesis has been carried out at the Laboratory of Theoretical Chemistry,

Department of Biotechnology, Royal Institute of Technology, Stockholm, Sweden.

List of papers included in the thesis

Paper I V. Kimberg, F. Gel’mukhanov, H. Agren, E. Pen, A. Plekhanov, I. Kuchin, M.

Rodionov, and V. Shelkovnikov, Angular properties of band structures in one-dimensional

holographic photonic crystals , J. Opt. A: Pure Appl. Opt. 6, 991-996 (2004).

Paper II V. Kimberg, F. Gel’mukhanov, and H. Agren, Angular anisotropy of delay time

of short pulses in impurity band based photonic crystals , J. Opt. A.: Pure Appl.Opt. 7,

118-122 (2005).

Paper III A. Baev, F. Gel’mukhanov, V. Kimberg, and H. Agren, Nonlinear propagation

of strong multi-mode fields, J. Phys. B: Atomic, Molecular & Optical Physics 36, 1-14

(2003).

Paper IV A. Baev, V. Kimberg, S. Polyutov, F. Gel’mukhanov, and H. Agren, Bi-directional

description of amplified spontaneous emission induced by three-photon absorption , J. of Opt.

Soc. of Am. B 22, 385-393 (2005).

Paper V F. F. Guimaraes, V. Kimberg, V. C. Felicıssimo, F. Gel’mukhanov, A. Cesar

and H. Agren, IR - X-ray pump-probe spectroscopy of the NO molecule, Phys. Rev. A 00,

000 (2005).

Comments on my contributions to the papers

• I was responsible for theory, calculations, writing and editing of manuscript in Papers I

and II.

• I was responsible for theory, and some calculations in Papers III, IV and V. I was also

involved in editing of manuscript, discussion and analysis of theoretical results in Papers

III, IV and V.

iii

Acknowledgments

I would like to express my great thanks to my supervisor Prof. Faris Gel’mukhanov. I thank

him for everything I learn from him, for his guidance, advice, and inspiration during the

course of each project, for his ability to explain any complicated theory in a simple way, to

show ’physics behind the formulas’. I am very thankful to Prof. Hans Agren for inviting

me to work in the Theoretical Chemistry group, his guidance and a very good organization

of the every-day work in the group. I wish to acknowledge my collaborators Alexander

Baev, Sergey Polyutov, Freddy Guimaraes and Viviane Costa Felicıssimo for their help,

tenderness, patience and warm atmosphere during the work.

I am sincerely thankful to the people without whom I could not imagine my presence and

work in the Theoretical Chemistry group: Prof. Yi Luo, Dr. Pawe l Sa lek, Dr. Fahmi Himo,

Prof. Boris Minaev, Oscar Rubio-Pons, Ivo Minkov, Elias Rudberg, Barbara Brena, Laban

Pettersson and all other members of our laboratory.

I would like to express my warmest thanks to the members of my family and my close friends

whom being near and far supported me very much in every respect during my Licentiate

studies in Stockholm.

Victor Kimberg, Stockholm, 2005.

Contents

1 Introduction 3

1.1 Pulses in linear periodical media . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Light propagation through nonlinear media . . . . . . . . . . . . . . . . . . . 5

1.3 Ultrafast processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Photonic Crystals 9

2.1 Basic theory of photonic band gaps . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Holographic photonic crystals . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3 Impurity band based photonic crystals . . . . . . . . . . . . . . . . . . . . . 14

3 Interaction of pulses with nonlinear media 19

3.1 Density matrix equations for multi-mode beam propagation . . . . . . . . . 19

3.2 Dynamics of amplified spontaneous emission . . . . . . . . . . . . . . . . . . 21

3.3 X-ray spectroscopy of molecules driven by an IR field . . . . . . . . . . . . . 22

3.3.1 Preparation of nuclear wave packets . . . . . . . . . . . . . . . . . . . 23

3.3.2 Phase sensitive pump-probe spectroscopy of the NO molecule . . . . 25

4 Summary of results 27

1

2 CONTENTS

Chapter 1

Introduction

“The most incomprehensible

thing about the universe is

that it is comprehensible.”

Albert Einstein

From the Newton time until the middle of the 1800’s the particle picture of light was the

generally accepted theory. Light was considered to be a stream of tiny particles. However,

in the late 1800’s Maxwell created the Theory of Electromagnetism, which replaced the

particle picture. Many phenomena associated with light,such as refraction, diffraction and

interference,were successfully explained by the wave picture. Visible light is just one particu-

lar type of electromagnetic radiation. Other types of radiation outside the visible spectrum

with wavelengths longer than red light are, for example, infrared (heat), microwave and

radio waves. Radiations with wavelength shorter than blue light are ultraviolet radiation,

X-rays and γ -rays.

However, in the early 20th century experiments showed many phenomena which could only

be explained by a particle picture of light. For example, experiments such as blackbody ra-

diation, the photoelectric effect, and Compton scattering can be explained using the photon

picture of light, but not with the wave picture. But, as it was mentioned, the experiments

on diffraction and interference need the wave picture to be explained. Thus, light as it is

now understood, has attributes of both particles and waves. Both pictures are needed in

different circumstances. One can say, that light has a dual nature: in some cases it behaves

as a wave, and in other cases it behaves as a photon. This wave-particle duality is the

basis of the quantum theory of light, and has some profound physical and philosophical

implications which are still being debated today.

3

4 CHAPTER 1. INTRODUCTION

The present thesis deals with light propagation through linear and nonlinear media. Many

scientific and technological problems are related to propagation of rather weak light which

does not change the properties of the medium. For example, this is the case of the light

propagation through waveguides and photonic crystals. In the case of a photonic crystal, or

periodical dielectric medium with the periodicity of the order of the wave length of the light,

low intense light does not affect the susceptibilities of matter. All phenomena appear due

to diffraction and interference of the scattered waves and can be described in the framework

of Maxwell’s equations with a periodical boundary condition (see Ch.2). However, in case

when strong coherent radiation interacts with nonlinear media we face the problem of solving

Maxwell’s equations coupled to the wave equation of the medium (see Ch.3). In this case

the interaction of the electromagnetic wave with different matter waves, like electronic, spin

or vibrational wave packets, becomes very important.

1.1 Pulses in linear periodical media

In 1913, the English physicists Sir W.H. Bragg and his son Sir W.L. Bragg explained why

the cleavage faces of crystals appear to reflect X-ray beams at certain angles of incidence.

They introduced the so-called Bragg’s law of diffraction, which is an interesting property

of periodical structure that has attracted much of attention ever since. There were many

detailed analyses of the phenomena of propagation of optical waves in crystals and layered

structures1 long time before Eli Yablonovitch introduced the term “Photonic Crystal” (PC)

in 1987. The pioneer works of Yablonovitch2 and John3 gave birth to a whole new area

of optical physics and technology which deals with molding the flow of light in PCs. PCs

manipulate optical or microwave photons in very similar ways as it occurs when X-rays

propagate through crystals or how semiconductors influence the motion of electrons. Many

ideas and terms in PC physics, such as photonic band gap, defect mode, impurity band,

were borrowed from solid state physics.

Nowadays4 PCs are discussed widely as basis for high-speed computers of the next gener-

ation. Progress has already been made towards replacing the slow copper connections in

computers with ultra-fast optical interconnects. Then photons, rather than electrons, will

pass signals from board to board, from chip to chip and even from one part of a chip to

another. In the future, it will be possible to implement optical manipulation, signal pro-

cessing and electronic circuitry all in one chip. In this field, to get shorter operation time

one needs to use a shorter signal, or light pulse.

The band structure of PCs defines the main properties of the light propagation through the

device. Propagation of short pulses through PCs looks different in many aspects comparing

1.2. LIGHT PROPAGATION THROUGH NONLINEAR MEDIA 5

to propagation of monochromatic CW light. The main reason for this is that a short light

pulse is a superposition of many harmonics according to the uncertainty relation:

∆ω ≥ 1/τ. (1.1)

These harmonics correspond to different parts of the band structure and therefore propagate

with different velocities. This results in undesirable effect, namely, the distortion of the

pulse.

One of the important application of PCs is the fabrication of delay lines for optical pulses.

The simplest idea of pulse delay is related to the fact that the group velocity approaches

zero near the band edge. However the large physical distances are needed in this case. An

alternative way is forcing the light to traverse a small distance many times making use of

the defect mode (optical cavity). These problems are discussed in Sec.2.3.

1.2 Light propagation through nonlinear media

A light wave consists of two components: electric and magnetic fields. These fields oscillate

sinusoidally with high frequencies. Interaction of light and matter results in some displace-

ment of the charge distribution inside the atoms or molecules of the material. This effect

can be easily understood with the help of the model of the forced harmonic oscillator. The

atoms, constructing the material, are seen as charge distributions pushed away from their

equilibrium state when exposed to the electro-magnetic field. When the field is weak light

interacts linearly with the matter. In the case of high intensity of the light, electro-magnetic

field modifies the optical properties of the medium strongly, which effects the propagation

of the radiation. Broadly speaking, nonlinear matter is a medium whose optical response

depends on the intensity of the optical field that propagates through it. One of the typi-

cal nonlinearties is the change of the dielectric constant due to the Kerr effect. The Kerr

nonlinearity enriches optics in general, and, in particular, the light propagation through

nonlinear PCs.5 Nonlinear PCs and nonlinear PC-based fibers were successfully applied in

the field of the frequency mixing and harmonic generation,6,7 efficient phase matching of

several wavelengths;8 as optical diodes,9 optical switchers and limiters,9,10 to mention just

a few examples.

Nowadays a lot of nonlinear effects are discovered and studied in different materials (gases,

liquids and solids). The area of applications of these nonlinear phenomena is vast. They

are applied in medicine, biology, telecommunications, military and industrial applications.

One can mention a few of them, like, nonlinear photoabsorption, harmonic generation, self-

focusing, nonlinear photo ionization and dissociation, phase conjugation, etc. There are two


qualitatively different nonlinear interactions. The first one occurs when the light frequency

is close to a resonant frequency of the media. In this case nonlinear effects appear at rather

small intensities (≈W/cm2). The nonlinear optical effects are accompanied in this case by

undesirable strong photoabsorption. To avoid absorption, one can induce nonlinearity by a

nonresonant field. However, much higher intensities are demanded in that case.

Ordinary light sources have rather low intensity and interact with matter linearly. The

creation of lasers in the 1960’s gave birth to a new area in optics – nonlinear optics – by

opening the possibilities to observe and manipulate the nonlinear effects mentioned above.

The nonlinear effects can be classified by the order of nonlinearity they exhibit. The

quadratic electric field term in the Taylor expansion of the polarization gives rise to many

important effects such as frequency doubling or second-harmonic generation, sum- and

difference-frequency generation (if instead of monochromatic two fields with different fre-

quencies are applied) and many others. Generally, the quadratic term involves the mixing

of three different fields (it multiplies two fields to generate a third). The cubic term is im-

portant for materials with inversion symmetry (isotropic materials, such as glasses, liquids

and gasses). There are no even powers of the field in the expansion of the polarization for

symmetry reasons11 and the lowest-order nonlinearity is then the cubic term. Four-wave

mixing processes and intensity dependent refractive index effects, stimulated Raman and

stimulated Brillouin scattering, third-harmonic generation and the quadratic electro-optic

effect are results of third-order polarization. The effects due to higher-order nonlinearities

are usually very small and not so widely studied as the quadratic and cubic terms.

In the case when the intensity of the laser field is so high that the conventional expansion

of the nonlinear polarization in a Taylor series ceases to be a good approximation the

corresponding language of susceptibilities breaks down. A typical example is two-photon

absorption which is accompanied by saturation effects. Such high intensities are widely

used in current experiments. The strict solution of the density matrix equations of the

material is needed in these cases for the explicit treatment of the nonlinear polarization

without addressing a Taylor expansion. In Sec.3.1 we review a generalized formalism for the

nonlinear propagation of a multi-mode field without any restriction on the field strength.

The theory was applied to studying amplified stimulated emission induced by three-photon

absorption in an organic chromophore molecule (Sec.3.2 and Paper IV).

1.3 Ultrafast processes

Investigation of ultrafast processes in solids, biomolecules and chemical reactions12 is a hot

topic of modern science. The characteristic times of nuclear and electronic dynamics are

1.3. ULTRAFAST PROCESSES 7

femto- and attoseconds, respectively. Ultrashort optical and X-ray pulses are needed to

study such fast processes. Much attention is paid nowadays to investigate the interaction

of ultrashort intense light pulses with matter.13 On the other hand, strong short pulses

give unique opportunities to modify the optical properties of matter. For example, laser

radiation can orient molecules,14 it can create a coherent superposition of the electronic,

spin, and nuclear quantum states.15

With the help of such coherent superpositions of quantum states, or so-called wave packets,

it is possible to monitor molecules at various stages of vibrational distortion, recording

”stop-action” spectroscopic events corresponding to well-defined molecular geometries far

from equilibrium16,17 including stretched and/or bent unstable transient structures. The

recent development of ultrafast control of coherent molecular dynamics has stimulated many

theoretical and experimental efforts18,19 to realize logical gates and quantum computing

algorithms in atoms and molecules. The analogs of several quantum bits within the shape

of a single wave packet were defined based on the wave packet symmetries.20 Recently it was

recognized that the trajectory of the wave packet is very sensitive to the phase of the pump

radiation. This constitutes the background of phase sensitive pump-probe spectroscopy21

(see Sec.3.3 and Paper V).

Chapter 2

Photonic Crystals

Photonic crystals (PCs) are periodic dielectric struc-

Figure 2.1: Butterfly wings are the

natural example of photonic crys-

tals.

tures that controls the propagation of light. The pe-

riodicity of the dielectric constant removes the degen-

eracies of the free-photon states at Bragg planes and

produces a range of forbidden energies for the pho-

tons. This leads to the concept of photonic band gaps

(PBGs) which underscores the analogy between elec-

trons in semiconductors and photons in a PC. Any

material which exhibits spatial periodicity in the re-

fractive index is a PC. There are some examples of

PCs in the nature; the sharp colors of natural opals

and butterfly wings are manifestations of the period-

ical structure of these objects (Fig.2.1). Periodicity

defines the dimensionality of a PC (Fig.2.2). Usually, one dimensional (1D) PCs pose a

PBG for only one particular direction of the incident light, two dimensional PCs are able

to stop the light beams in one particular plane, while some dielectric structures with a

three-dimensional periodicity, have no propagation modes in any direction for a range of

frequencies, giving rise to a complete PBG.

In 1987 a theoretical description of PBGs2,3 was proposed for the first time. In 1991 the

first three dimensional PC was fabricated,22 based on face-centered cubic structure. Since

that time numerous structures which poses PBG have been invented.4 By proper design of

the PC (for example, lattice type, dielectric constants of the material, size of the unit cell,

introduction of defects or impurity bands, choosing the angle of incidence of the light) one

can manipulate the PBG of the PC and, hence, control the light propagation through the

9

10 CHAPTER 2. PHOTONIC CRYSTALS

Figure 2.2: Illustration of the structure of one-, two- and three dimensional PCs.

PC device. PCs are successfully used in many applications now: optical fibers,23 nanoscopic

lasers,24 perfect mirrors, radio-frequency antennas25 and reflectors and many others. Pho-

tonic integrated circuits is the hot topic of modern developments for PC applications.26,27 In

1968 Victor Veselago28 invented the concept of materials with negative refractive index and

predicted a wide variety of new optical phenomena based on these so-called ’left-handed’

materials. The fabrication of PCs based on left-handed materials is one of the important

directions in the physics of PCs.29,30

In the present chapter we will discuss the properties and applications of 1D periodical

structures, namely 1D holographic PCs (see Sec2.2) and 1D impurity band based PCs (see

Sec.2.3).

2.1 Basic theory of photonic band gaps

The usual way to understand of the properties of a PC is to analyze solution of Maxwell’s

equations in the periodic system. Using Maxwell’s equation we can set up a wave equation

or, in the stationary case, a Helmholtz equation:

(

∆ + (ω/c)2ε(x))

Ek(x) = 0, ε(x) = ε0 + ε1(x) (2.1)

where ε1(x) is a periodic function: ε1(x) = ε(x + R). Because of the periodicity of the

medium we can expand the function ε1(x) in a Fourier series on the reciprocal lattice vector:

ε1(x) =∑

G

UGeiGx, (2.2)

here UG is the amplitude of the oscillation of the dielectric constant, and G is the reciprocal

lattice vector. Also the electric field can be expand in a Fourier series on the vectors G:

Ek(x) =∑

α

∑

G

Eαk−G

exp[i(k − G)x], (2.3)

2.1. BASIC THEORY OF PHOTONIC BAND GAPS 11

a) b) c)

Figure 2.3: Multilayer film - 1D photonic crystal (a). Illustration of photonic band gap

phenomena (b,c).

where α is the index of photon polarization. Any polarization can be expanded on two

components: α = (x, y) for linear polarization, or (+,−) for circular photon polarization.

Due to space inhomogenety the harmonics Eαk−G

are coupled. Through introducing (2.2,2.3)

into (2.1) we obtain approximate amplitude equations for the harmonics:∑

α

Eαk

[

k2 − (ω/c)2ε0

]

=∑

α

(ω/c)2ε1

(

UGEαk−G

+ U−GEαk+G

)

,

∑

α

Eαk−G

[

(k − G)2 − (ω/c)2ε0

]

=∑

α

(ω/c)2ε1

(

U−GEαk

+ UGEαk−2G

)

. (2.4)

which are valid when ε1(x) is rather small.

Let us discuss the 1D PC (Fig.2.3(a)). For the case of normal incidence (k‖G), all polar-

izations are decoupled and eqs.(2.4) become scalar. Near the band edge (k ≈ G/2) only the

scattering from the 1st Bragg plane is important:

Eαk

[

k2 − (ω/c)2ε0

]

= (ω/c)2ε1UGEαk−G

Eαk−G

[

(k − G)2 − (ω/c)2ε0

]

= (ω/c)2ε1U−GEαk . (2.5)

These equations result in the following dispersion relation:

[

k2 − (ω/c)2ε0

] [

(k − G)2 − (ω/c)2ε0

]

− (ω/c)4ε21UGU−G = 0. (2.6)

When k = G/2 and UG = U−G we get

ω±/c = k/√

ε0 ± ε1UG. (2.7)

Thus the forbidden band gap appears at the edge of the Brillouin zone (Fig. 2.3(b)) ∆ω =

ω− − ω+ ≈ ω|ε1UG|/ε0. The picture changes drastically when the light propagates parallel

to the layers. In this case a photonic band gap does not appear (Fig.2.3(c)).


The above example shows that properties of a PC depends strongly on the direction of

the incident light beam. In the next sections of this chapter we will review the angular

properties of 1D photonic crystals of different kinds, stressing the effects connected with

incident light polarization.

2.2 Holographic photonic crystals

There are many methods of creating a PC: mechanical techniques (drilling holes etc.), chem-

ically assisted ion beam etching,31 self-assembled aggregation of small particles,32 thin films

or multilayer methods. One promising way to fabricate PCs is holographic lithography.33

With the help of several intercrossed laser beams, one can create a periodical interference

pattern, which can be saved employing the photoresist recording medium (Fig.2.4(a),(b)).

The main advantage of this method is that one can make the interference pattern by ad-

justments of the writing laser beams, and create a PC with the desired properties. The

disadvantage is that existing photopolymers exhibit rather small contrast in the refraction

index after light exposure. (The bigger dielectric contrast, the wider the PBG.)

c) z

n(z)

n1 n

2 n

3

nj

a

nmin

nmax

Figure 2.4: Holographic lithography: (a) Laser beam configuration; (b) Photoresist media

after recording;33 (c) Dielectric constant distribution of 1D HPC continuous and step-like

numerical approximations.

The holographic photonic crystal (HPC) we discuss in the present section is a 1D analog of

the structure shown in Fig.2.4(b). This structure was fabricated as a 1D reflection hologram

by focusing of two intercrossed laser beams on a photoresist sample (Paper I). The principle

difference of the HPC from an ordinary PC is that the refractive index changes continuously,

2.2. HOLOGRAPHIC PHOTONIC CRYSTALS 13

and not step-like (Fig.2.4(c)). Based on experimental observation of angular properties of the

1D HPC (Fig.2.5(a)) made by our co-authors, we analyzed such a system theoretically with

help of the conventional transfer matrix technique.34 For this we divided each elementary

unit cell of the 1D HPC into sublayers with approximately constant value of refractive index

and then solved Maxwell’s equations for each sublayers (Fig. 2.4(c)). A comparison of the

theory with experimental data (Fig.2.5(b)) allowed us to define the parameters of the HPC

– the dielectric contrast and the shrinkage of the hologram.

a) b)

Figure 2.5: (a) Experimental transmission spectra versus the incident angles of the probe

beam (θa). (b) Dependence of the position of the BG center on the incident angle in air

(θa) (experiment and theory).

a)

θB θB

(n )1

p−, s− s−polarized polarized

polarized

isotropicdielectric

air

p−, s−2(n )

b)0 30 60 90

0.5

1

1.5

Internal angle (degree)

Fre

quen

cy (ω

a/2

π c)

9060300Incident angle (degree)

Figure 2.6: (a) The Brewster angle, ΘB = arctan(n2/n1). (b) Angular dependence of the

band structure of the 1D HPC. Refraction index varies from nmax = 2.5 to nmin = 2.1. The

shaded regions show the band gaps.


As it could be seen from the equations of Sec.2.1, when the incident light enters the PC

at angle other then 0 or 90 the degeneracy of light polarization is removed. Then light

with p− and s−polarization (or TM and TE modes) interacts differently with the PC. In

this context it is important to notice the presence of a Brewster angle for the TM mode

(Fig.2.6(a)). It is an angle at which the reflection of this light component equals zero

and the PBG disappears (Fig.2.6(b)). A theoretical analysis (Paper I) showed that the

value of the Brewster angles for different bands are different in the case of a (Fig.2.6(b)).

This differs qualitatively from the conventional two layer Bragg reflector1 with a step-like

distribution of the dielectric constant where the Brewster angle is the same for all bands,

ΘB = arctan(n2/n1). The band gap of the TE does not have such a particularity and

increases monotonically with increase of the incident angle (see Paper I).

2.3 Impurity band based photonic crystals

As well known from solid state physics an impurity creates a localized mode inside of the

electronic band gap. Similarly, one can obtain a very narrow defect mode inside the photonic

band gap4 by introducing a defect in a perfect PC structure (see Fig.2.7). By varying

the parameters of the defect and the photonic crystal one can change position and width

of this defect mode. That technique was applied for creating tunable wavelength filters

for continuous radiation35 and optical delay lines.36 Optical delay lines provide the delay

time of an optical signal. These devices have many applications, for example they are

essential elements for all-optical switching nodes in photonic network systems and quantum

computers. An enhanced delay is also desirable in the nonlinear optical systems as it

increases an effective interaction time.

A short pulse has a wide frequency dispersion (1.1). When such radiation enters the PC with

a narrow defect mode the pulse is damaged, and part of the radiation (which corresponds

to the defect mode frequency region) goes through the PC, while the rest is reflected. It

means that to manipulate of the propagation of the pulses, it is necessary to satisfy, to some

extent, two contradictory requirements: a small spectral width of the defect mode and a

rather broad spectrum of the short pulse. A method to overcome that contradiction was

shown in the papers of Lan et al,37 devoted to theoretical and experimental studies of pulse

propagation through coupled cavity waveguides. The key idea for reducing pulse distortion

in such delay lines is based on the possibility of creating a quasi-flat transmission spectrum in

the impurity band.38 Impurity bands, in analogy with semiconductors, can be implemented

in 1D PCs by periodically introducing N defect layers (see Fig.2.8). Due to the overlap,

different defect modes interact with each other and form an impurity band. Figure 2.9(A)

2.3. IMPURITY BAND BASED PHOTONIC CRYSTALS 15

a) b)

Figure 2.7: (a) Two dimensional photonic crystal with defect made by removing a hole from

a perfect PC structure. (b) An example of PC spectra with a defect. Sharp defect modes

inside the band gap are seen clearly.

shows the forming of an impurity band inside the photonic band gap frequency region. The

IB spectra depends strongly on the number of defects in the PC, N, and the number of

layers which form the cavities resonators, m and n.

. . . E

E

E in out

ref

d d d1 2 Na

nmnA

Figure 2.8: A one-dimensional PC with N defects (shown as black strips).

It is well established that transmission spectra of 1D PCs is very sensitive to the angle of

incidence of the light.1 The possibility of altering properties of 1D PCs with the help of the

incidence angle has become apparent and has generated substantial interest in this field. In

Paper II we presented a theoretical study of the angular dependence of the delay time of

IB based PCs. The main goal of our study was to establish the angular anisotropy of the

delay time.

The reason for the angular anisotropy of the delay time is the angular anisotropy of the

group velocity of the impurity band. We have shown that the group velocity of the TE

and TM modes of the impurity band has qualitatively different angular behavior: the group

velocity of the TM mode increases with increasing incidence angle up to the Brewster angle

(Fig.2.6a)), where the delay time for the TM mode equals zero (Fig.2.9(B)). The group

velocity decreases again when the incident angle exceeds the Brewster angle. The group

velocity of the TE mode decreases monotonically with increasing incident angle (Fig.2.9(B)).


A)

0.315 0.3160

0.5

1

0.3156 0.3158 0.3160

0.5

1

0.305 0.315 0.3250

0.5

1

Tra

nsm

issi

on

0.315 0.3160

0.5

1

Frequency (2π c/a)

N=1

N=2

N=5

n=20

m=40

N=5 n=10 n=8

m=10 m=20

N=5

N=5 n=10 m=20

a) b)

c) d)

B)0 25 50 75

10−6

10−4

10−2

100

log(

v gr/c

)

TE−modeTM−mode

0 30 60 90Incident angle (degree)

Internal angle (degree)

Figure 2.9: A) Transmission spectra of the impurity band versus N, m and n. B) The

angular dependence of the maximal group velocity of the impurity band (c is the light speed

in vacuum).

0.3774 0.3775 0.37760

0.5

1

Frequency (2π c/a)

Tra

nsm

issi

on

IinT

s

0 100 200 300 400 500 600 7000

1

2

Time (ps)

Inte

nsity

(ar

b. u

nits

) IoutIinIref200 500 800

0

0.1

Time (ps)

a)

b)

Iin

Inte

nsity

(ar

b. u

nits

)

Figure 2.10: (a) Transmission spectra of the impurity band (solid) and spectral distribution

of an incident Gaussian pulse (dashed). (b) Time dependence of incident TE-polarized

(dashed), transmitted (solid) and reflected (dash-dotted) pulses. The carrier frequency is

ω0 = 0.37752(2πc/a), angle of incidence is θ0 = 60.

Thus the TM and TE modes are split up in the time domain due to the qualitatively different

angular dependence of the group velocity, and hence, of the delay time for these modes.

2.3. IMPURITY BAND BASED PHOTONIC CRYSTALS 17

Moreover, the edges of the impurity band spectrum have narrow defect peaks, while at the

central part of the spectrum they are broader (Fig.2.10(a)), and hence the group velocity

dispersion is large for different spectral components of the incident pulse. This results in

a distortion of the TE-polarized incident pulse (Fig.2.10(b)). It splits in few transmitted

pulses with different delay times (see insets in (Fig.2.10(b))).

The role of the incident angle, θ, is important in applications, because the delay time can

be changed easily by varying θ without changing the optical device itself (see Paper II for

further details).

Chapter 3

Interaction of short light pulses with

nonlinear media

Nonlinear optics39 constitutes one of the most important areas of modern science and tech-

nology. It has already revolutionized telecommunications and is expected to soon make

a similar impact on computer technology. Nonlinear techniques cover a wide region of

different applications now, such as sum frequency and harmonic generation,40 frequency

upconververted lasing,41 optical power limiting,42,43 optical data storage44 and photody-

namic therapy.45 However, it is a common view that for future use more sophisticated

understanding of the basic mechanisms underlying non-linear phenomena will be required.

The scientific problems of this kind motivate us to explore non-linear propagation of light

pulses through complex media, where details of the quantum structure of the molecular

units become essential.

3.1 Density matrix equations for multi-mode beam prop-

agation

Real laser experiments deal with many factors which act simultaneously, something that can

make special models inadequate for use in proper analysis of experimental measurements.

For example, the standard theoretical way to explore the nonlinear propagation of strong

interacting electromagnetic fields is based on the solution of the nonlinear equation for

special nonlinearity (quadratic, cubic, etc.). But in many experiments the intensity of the

laser fields are so high that the conventional expansion of the nonlinear polarization in a

Taylor series in powers of the electrical field strength ceases to be a good approximation

19

20 CHAPTER 3. INTERACTION OF PULSES WITH NONLINEAR MEDIA

and the corresponding language of susceptibilities breaks down. These cases require a strict

solution of the density matrix equations of the material for the explicit description of the

nonlinear polarization without addressing a Taylor expansion. The character of many-

photon absorption in condensed phases is often complicated by the competition between one-

step and sequential absorption channels. In Paper III we presented a generalized formalism

for the nonlinear propagation of a multi-mode field without any restriction on the field

strength.

In the case of multi-mode interaction we seek for a solution of the density matrix equations46

(

∂

∂t+ Γ

)

ρ =i

~[ρV ] (3.1)

making use of the Fourier transform:

ραβ = eiωαβt∑

n

rnαβei(nk·r−nωt−nφ), (3.2)

where Γ is the relaxation operator, V is the interaction between molecule and electromag-

netic field, ωαβ = (Eα−Eβ)/~ is the frequency of the transition α → β. Here we introduced

the N -dimensional vector n and the N -dimensional scalar product

n ≡ (n1, n2, ..., nN), n ω ≡

N∑

j=1

njωj, n k ≡

N∑

j=1

njkj, (3.3)

with nj = 0,±1,±2, ...,±∞ as the numbers of photons of the field E j; ω = (ω1, ω2, ..., ωN),

and k = (k1,k2, ...,kN ).

The substitution of the expansion (3.2) into (3.1) give the following equations:

∂

∂t− i[n (ω − k · v − ωαβ] + Γαβ

rnαβ = δαβ

∑

γ>α

Γαγ rn

γγ

+i

N∑

i=1

∑

γ

[

(

rn−1jαγ + rn+1j

αγ

)

G(j)γβ − G(j)

αγ

(

rn−1j

γβ + rn+1j

γβ

)]

. (3.4)

Where G(j)αβ = E j ·dαβ/2~ is the Rabi frequency of the jth field on the transition α → β. The

origin of the the upper indices of the density matrix in the field term n±1j ≡ (n1, n2, ..., nj±

1, ..., nN) is the absorption or emission of one photon of the jth mode due to the molecular

field interaction. This representation is a key to solve the problem of propagation of strong

pump and probe fields through a nonlinear many-level medium. The nonlinear polarization

can be expressed in terms of the density matrix elements as:

Pj =∑

βα

dβαr1j

αβ. (3.5)

3.2. DYNAMICS OF AMPLIFIED SPONTANEOUS EMISSION 21

Here r1j

αβ means rnαβ with n = (0, 0, ..., 1j, ..., 0). Equations (3.4,3.5) together with the

paraxial wave equation39

(

k · ∇ +1

c

∂

∂t−

i

2kj

∆⊥

)

E j =ikj

ε0Pj. (3.6)

solves the problem of propagation of the strong multi-mode field through a nonlinear many-

level medium without restrictions on the mode intensities.

3.2 Dynamics of amplified spontaneous emission

In this section we will discuss the application of the theory described above to the amplified

spontaneous emission (ASE) induced by three-photon absorption of an organic stilbenchro-

mophore (APSS) (Fig.3.1(a)) dissolved in a solvent (see Paper IV). Our work was initiated

by recent a experimental study of this system.41 The scheme of transitions is shown in

Fig.3.1(b). The pump laser populates vibrational level 4 by three-photon absorption. A

competitive channel of population of this level exists: one-photon off-resonant photoabsorp-

tion. Level 4 decays non-radiatively to the lowest vibrational level 3 giving rise to population

inversion between levels 3 and 2. The lasing transition appears between vibrational levels 3

and 2, which corresponds to the vertical transition S1 → S0.

In this work we concentrated on the evolution of counter propagating ASE pulses in con-

nection with the efficiency of nonlinear conversion. Accounting for ASE in both forward

and backward directions with respect to the pump pulse results in a smaller efficiency of

non-linear conversion for the forward ASE compared with the case when forward emission

is considered alone,47 something that results from the partial re-pump of the absorbed en-

ergy to the backward ASE component; the overall efficiency is nevertheless higher than

for the forward emission considered alone (Fig.3.2). The efficiency of non-linear conversion

of the pump energy to the counter-propagating ASE pulses is strongly dependent on the

concentration of active molecules so that a particular combination of concentration versus

cell length optimizes the conversion coefficient. Under certain specified conditions the ASE

effect is found to be oscillatory. The origin of self-pulsation is the dynamical competition

between the stimulated emission and off-resonant absorption. This result of ours can be

considered as one of the possible explanations of the temporal fluctuations of the measured

ASE pulse.41 (See Paper IV for details).


a) b)

mµ1.3

mµ1.3

mµ1.3

4

3

5

2

1

S

S

0

1

553 nm

S2

∆ E

ASE

Figure 3.1: Structure of APSS and reduced molecule47 (a). Energy level scheme for ASE

lasing (a).

Figure 3.2: ASE intensity dynamics.

3.3 X-ray spectroscopy of molecules driven by a strong

infrared field

In previous sections we studied the propagation of light pulses in different media. However,

short and brilliant pulses of coherent radiation can disturb the medium. This opens new

3.3. X-RAY SPECTROSCOPY OF MOLECULES DRIVEN BY AN IR FIELD 23

opportunities and experimental tools for studies of different properties of matter.

This section is devoted to time and frequency resolved X-ray spectroscopy of molecules

driven by strong and coherent infrared (IR) pulses. Studies of IR-induced nuclear dynamics

require quite short X-ray pulses (< 100 fs). As reviewed in48 several kinds of such sources

are available already today. In fact, ultrashort X-ray pulses as short as ≈ 250 attosecond

have recently been reported.49 This indicates that X-ray pump-probe spectroscopy is able

to explore nuclear dynamics even by the use of current light sources.

3.3.1 Preparation of nuclear wave packets

In year 1926 Erwin Schrodinger introduced the concept of wave packets in order to make a

bridge between the classical and quantum description of nature.50 For a long time the wave

packets, which follow a classical trajectory, had no practical use because their preparation

seemed impossible. However, recent advances in the physics and chemistry of laser inter-

action with atoms and molecules have brought wave packets and their dynamics into the

limelight.51 In the 1990’s it became possible to prepare electronic wave packets by exciting

atoms to Rydberg states52 with short pulses and to create nuclear wave packets in X-ray

Raman scattering.53 A whole new realm of phenomena opened when femtosecond pulse

technology emerged. With ultrashort pulses one can prepare a molecular wave packet and

probe its evolution, observe molecular reactions in the real time scale (femtoseconds and

tens of femtoseconds). The possibility to manipulate the reaction in order to achieve precise

control over the output products appeared. In the field of molecular reactions it gave birth

to the field of femtochemistry.12

X-ray pump-probe spectroscopy together with recent development of femtosecond lasers

brings new possibilities into time-resolved X-ray spectroscopy.13 It becomes possible to

investigate the nuclear dynamics with the time scale of the period of molecular vibrations.

The principle of IR – X-ray pump-probe spectroscopies illustrated in Fig.3.3(a) using the

NO molecule as example (see Paper V). The coherent pump IR pulse resonantly excites the

vibrational levels of the ground electronic state, thereby creating a coherent superposition

of vibrational states or wave packet. The short probe X-ray pulse exposes the molecule

at different moments of time, and excites the wave packet to a certain point at the core-

excited potential. A change of the delay time between the probe and pump pulses allows

to investigate the dynamics of nuclear wave packets in the ground state, as well as to map

the core-excited potential surface.

The nuclear wave packet created in the ground state (Fig.3.3(a)) by the coherent IR field

with the strength EL(t), frequency ωL and phase ϕL, obeys to the time-dependent Schrodinger


a) b)0.0 0.5 1.0 1.5

⟨Ω⟩ (eV)

1000

1500

2000

∆t (

fs)

Trev

0.01.02.0⟨Ω⟩ (eV)

0.01.02.0⟨Ω⟩ (eV)

1330

1340

1350

1360

2.1 2.4⟨r⟩ (a.u.)

2.1 2.4⟨r⟩ (a.u.)

ϕL= 0 ϕ

L = π

Figure 3.3: (a) Scheme of OK X-ray absorption of NO driven by a strong IR field. (b)

Trajectory of the WP, 〈r(t)〉, and the center gravity of the OK X-ray spectrum, 〈(Ω(∆t)〉,

of NO (solid lines) (coincide). Filled and dashed bands at the right-hand side display,

respectively, the WP and the X-ray spectrum. IL = 2.3 × 1012 W/cm2, τL = 100 fs,

τX = 4 fs.

equation:

ı∂

∂tφ(t) = [H0 − (d · EL(t)) cos(ωLt + ϕL)] φ(t), (3.7)

where H0 is the nuclear Hamiltonian of the ground electronic state. The interaction with

the coherent IR field makes the dynamics of the nuclear WP sensitive to the phase of the

IR field, φL. It is seen from the perturbative solution of the Schrodinger equation (3.7)

|φ(t)〉 ≈

∞∑

ν=0

|ν〉cν(t) e−ı(νϕL+ενt) (3.8)

where the coefficients cν(t) do not depend on ϕL in the rotating wave approximation. The

numerical simulations show that the trajectory of the center of gravity of the WP,

r(t) = 〈φ(t)|r|φ(t)〉, (3.9)

is very sensitive to the phase of the IR light (see Fig. 3.3(b)). This effect has nothing to do

with the role of the phase in few-cycle experiments;54 in the present case the duration of the

IR pulse is longer than the inverse frequencies of the IR field. The WP performs fast back

and forth oscillations with the vibrational frequency. These oscillations are modulated due

the anharmonicity of the ground state potential with a period, Ta = π/ωexe (see the insert

in Fig. 3.3(b)). The measurement of the revival time Ta using X-ray absorption55 gives a

3.3. X-RAY SPECTROSCOPY OF MOLECULES DRIVEN BY AN IR FIELD 25

new opportunity to determine the anharmonicity constant ωexe. Here we face a paradoxical

situation, namely that the ultrafast X-ray measurements allow to determine fine structure

of the spectrum (anharmonicity).

3.3.2 Phase sensitive pump-probe spectroscopy of the NO molecule

The absorption of X-ray probe radiation by laser excited molecules is defined as the expec-

tation value of the interaction of the X-ray field with the system

P (Ω) = −=m

∞∫

−∞

Tr [ρ(t1)(EX(t1) · D)] dt1 = 〈φc(−Ω)|φc(−Ω)〉, (3.10)

where ρij(t, Ω) = ai(t)a∗

j(t) is the density matrix of the molecule, and

|φc(−Ω)〉 =

∞∫

−∞

dt e−ıΩt EX(t) |φc(t)〉, (3.11)

|φc(t)〉 = eıHctζ|φ(t)〉 is the nuclear wave packet in the potential surface of the core excited

state. ζ = (EX · D)/2EX, EX(t) is the strength of the X-ray field, D is the dipole moment

of the X-ray transition, Ω = ωX −ωc0 is the detuning of the frequency of the X-ray photons

from the adiabatic excitation energy ωc0. The calculated X-ray spectra show strong phase

sensitivity (Fig.3.4), caused by the phase dependence of the wave packet (Fig. 3.3(b)). An

alternative interpretation of the effect is given in ref.21

-1.0 0.0 1.0 2.0 3.0 4.0Ω (eV)

0.00

0.02

0.04

0.06

0.08

P0 (

arb.

uni

ts)

ϕL = 0

ϕL = π/2

ϕL = π

ϕL = 3π/2

⟨ϕL(0)⟩

⟨ϕL(π/2)⟩

(A)

-1.0 0.0 1.0 2.0 3.0 4.0Ω (eV)

0.00

0.02

0.04

0.06

0.08

P0 (

arb.

uni

ts)

ϕL = 0

ϕL = π/2

ϕL = π

ϕL = 3π/2

⟨ϕL(0)⟩

⟨ϕL(π/2)⟩

(B)

Figure 3.4: Phase sensitivity of the OK X-ray absorption spectra of NO versus different

intensities of the pump field. The spectra averaged over molecular orientations are marked

by 〈φL〉 with φL = 0 and φL = π/2. τL = 100 fs, τX = 3 fs, the delay time is 610 fs. A)

IL = 1.5 × 1012 W/cm2, B)IL = 2.3 × 1012 W/cm2.


Contrary to the wave packet dynamics the populations of vibrational levels do almost not

depend on the phase at all. Fig.3.5 shows only a weak phase sensitive modulation of the

population on the double frequency of the laser field. This modulation corresponds to non-

resonant interaction of the field and appears only when the intensity of the pump pules is

strong enough to violate the rotating wave approximation (see insert in Fig.3.5(A)).

0.0

0.2

0.4

0.6

0.8

1.0

ρ ν

TL/2 = 8.58 fs

ν = 0ν = 1ν = 2ν = 3

400 600 800 1000 1200t (fs)

-1.0-0.50.00.51.0

ζ L(t

)

(A)

0.0

0.2

0.4

0.6

0.8

1.0

ρ ν

TR

ν = 0ν = 1ν = 2ν = 3

600 1200 1800 2400 3000 3600t (fs)

-1.0-0.50.00.51.0

ζ L(t

)

∆T

(B)

Figure 3.5: Populations of the vibrational levels of the ground electronic state versus time

for different durations of switching off the pump field.

Fig. 3.5(A) shows the case when the duration of the IR pulse, τL = 100 fs, is much shorter

than the Rabi period ≈ 750 fs. This corresponds to the limiting case of a sudden switching

of the pump field: The IR field is shut off faster than the Rabi period and the molecule

remains in the vibrationall excited state after the pulse leaves the system (Fig. 3.5 (A)).

The life time of this state is very long (∼ 1 msec) in a diatomic molecule. During this time

the wave packet remains to be coherent and keeps the memory about the pump pulse and,

in particular, the phase memory. The X-ray absorption profiles (Fig. 3.4) “measured” after

the IR pulse display clearly the effect of phase memory. The scenario changes drastically

when the time for the switching-off of the IR pulse, ∆T ≈ 715 fs, is long and becomes

comparable with the Rabi period, TR ≈ 750 fs (see Fig. 3.5 (B)). In this case the system

follows adiabatically the slow decrease of the light intensity up to zero where only the

lowest vibrational level is populated (Fig. 3.5 (B)). In this case the molecule keeps no phase

memory. Moreover, the X-ray spectrum coincides, in this case, with the X-ray spectrum of

the molecules that are not exposed to an IR field.

Chapter 4

Summary of results

Photonic crystals

• The angular dependence of light propagation through one-dimensional photonic crystals

with continuous distribution of the dielectric constant, fabricated by the use of photopolymer

materials and laser holography, was studied theoretically.

• It is shown that the Brewster angles for different bands are different in contrast to the

conventional two-layer Bragg reflector with a step-like distribution of the dielectric constant.

• Comparison of the theory results with experimental data allowed to define the parameters

of the hologram - the dielectric contrast and the shrinkage of the structure.

• We found a strong dependence of the delay time of short pulses in one-dimensional impurity

band based photonic crystals on the angle of incidence.

• The delay time is larger for larger incident angles for the transverse electrical mode,

while the delay time of the transverse magnetic mode has a qualitatively different angular

dependence, especially in the region below the Brewster angle.

• The strong anisotropy of the delay time is traced to the anisotropy of the group velocity

which is directly related to the angular dependence of the impurity band structure.

Interaction of short light pulses with nonlinear media

• A strict theory of nonlinear propagation of few interacting strong light beams was devel-

oped: The self-consistent solution of the nonlinear wave equation and the density matrix

27

28 CHAPTER 4. SUMMARY OF RESULTS

equation of the material beyond the rotation wave approximation.

• The strict solution for the nonlinear interaction between different strong light beams was

found and used in a study of the role of saturation in three-photon absorption by an organic

chromophore in solution.

• An accounting for the amplified spontaneous emission in both forward and backward

directions results in a smaller efficiency of non-linear conversion for the forward amplified

spontaneous emission (ASE) compared with the case when forward emission is considered

alone. The overall efficiency is nevertheless higher than for the forward emission considered

alone.

• The efficiency of non-linear conversion of the pump energy to the counter-propagating ASE

pulses is strongly dependent on the concentration of active molecules so that a particular

combination of concentration versus cell length optimizes the conversion coefficient.

• Under certain specified conditions the ASE effect is found to be oscillatory. The origin of

self-pulsation is the dynamical competition between the stimulated emission and off-resonant

absorption.

IR – X-ray Pump-Probe spectroscopy

• Two color IR – X-ray pump-probe spectroscopy of the NO molecule was studied theoret-

ically and numerically.

• It is found that the phase of the infrared field strongly influences the trajectory of the

nuclear wave packet, and hence, the X-ray spectrum.

• The trajectory of the nuclear wave packet experiences fast oscillations with the vibrational

frequency and with a modulation due to the anharmonicity of the potential.

• It is shown that the X-ray spectrum keeps memory about the infrared phase after the

pump field left the system. This memory effect is sensitive to the time of switching-off the

pump field and to the Rabi frequency.

• The phase effect takes maximum value when the duration of the X-ray pulse is one fourth

of the infrared field period, and can be enhanced by a proper control of the duration and

intensity of the pump pulse.

• The manifestation of the phase is different for oriented and disordered molecules and

depends strongly on the intensity of the pump radiation.

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pulse propagation in photonic crystals and nonlinear media

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