plasmon guides parti (1)

Plasmonics and WaveguidesEngineering plasmon-polaritons in metallic nanostructures

Luca Dal Negro ECE & Photonics Center, Boston University, USA

[email protected]://www.bu.edu/nano/

Plasmonics: engineering charge density oscillations bound at metal-dielectric interfaces

Design Parameters

200nm

Size

Walsh et al., Nano Lett., 13 (2), 786 (2013)

Hanke et al., Nano Lett., 12 (4) 2037 (2012)

Shape

Yan et al., Opt. Mat. Exp., 1, 8, 1548 (2011) http://www2.mpip-mainz.mpg.de/groups/bonn/research/menges_bernhard/cleanroom

Suh, Nano Lett., 12, 1 269 (2012)

Near-Field Coupling

Hanke et al., Nano Lett., 12 (4) 2037 (2012)Toroghi et al, Appl. Phys. Lett. 100, 183105 (2012)

Y. Chu, et. al. Applied Physics Letters 93, 181108 (2008).

Photonic / Diffractive CouplingMaterial

Nanoplasmonics

• Electron density oscillations coupled to EM waves • Strong field intensity enhancement at metal surfaces • Optical fields nanoscale localized (sub-)• Largely tunable optical properties

Key Features:

• Enormous polarizability

• Resonant behavior

• Classical EM description

Why metals:

Engineering polarization bound charges in nanomaterials (=Polaritonics) enables the control of strongly enhanced radiative and non-radiative fields

Example: Plasmon Resonances in Metallic Nanostructures

Localized Surface Plasmons=

Collective electron oscillations

• Field enhancement • Resonant response

Example: Surface modes of small particles

3

2

d

NP md

NP m

f a

f

Collective, in-phase motion of electrons

Quadrupole plasmon resonance

For larger particles

Half of the electron clouds moves parallel to the applied field and half moves anti-parallel

3

3 / 2

q

NP mq

NP m

f a

g

Enhanced light-matter coupling

•Optical sensing / spectroscopy•Nanoscale nonlinear optics •Field concentrators•Nanoscale light sources•Solar energy harvesting•Imaging / optical detection•Singular optics Hao, E.; Schatz, et al., J. Chem. Phys, 120, 357 (2004)

Molecular Emission

Chen et al., Sci. Rep., 3, 1505 (2013)

Solar

Atwater & Polman, Nat. Mat.. 9, 205 (2010)

Sensors

Semrock, www.semrock.com

COBRA DANE phased-array radar34,000 antennas, Siberia's Kamchatka Peninsula

Radiation engineering with plasmonic coupled arrays

Near Field Coupling in Clusters

Near-field of one particle induces additional polarization in close particles

Diffractive Coupling between Clusters

-Propagating scattered light from one particle can be scattered by another

d

d

Photonic-plasmonic coupling regimes in arrays

Transverse field 1/rRadiative

2

ˆ ˆ( ) ( )4

jkrk eE r r p r

r

• Propagating;

• Diffraction-limited;

• Transverse;

Radiative vs Non-radiative fields: always think of Hertzian dipoles first

Longitudinal fields 1/r2, 1/r3

ˆ ˆ1 1 3 ( )( )

4

jkrr r p p eE r jk

r r r

• Non-propagative (reactive);

• Sub-wavelength localized;

• Large nanoscale intensity;

Principles of Nano-Optics, L. Novotny, .B. Hecht, Cambridge (2006)

Quasi-static near-field coupling

Heisenberg principle for photons

x xp x k x

1

x

xk

Using evanescent (non-propagative) fields the spatial bandwidth of photons can be dramatically enhanced.

Copyright 2008, Boston University

Theory of Optical Constants

• Why polished aluminum cooking pan is shiny but opaque transmitted light ?• Why glass windows are transparent but weakly reflecting ?• Why all materials behave like metals at high frequencies ?• Why radio-waves do not escape from the atmosphere while satellite waves do ?

Models are needed to explain the (linear) opticalresponse of materials

Optics of conducting media

extJt

DH

0

Bt

BE

extD

extJDjH

For monochromatic waves:

Assuming:EJ

EED

)1(00

EjH eff

jjeff )1(00

Where we defined the: Effective electric permittivity:

jt /

r

,, are generally complex!

We can write:

tjerEtrE )(Re),(

Physical meaning

Where the complex permittivity is defined as:

0 (1 ) i

“Bound” charge current density

Free charge current density

Im( ) Im( ) Re

Both conductivity and susceptibility contribute to the imaginary part of the permittivity:

If the imaginary part of the material coefficients µ or ε is nonzero, the amplitudeof a plane wave will decrease as it propagates through the medium absorption

Complex phenomenological coefficients of a medium are equivalent to a phase difference between P and E (or H and B) and are manifested by absorption

0

2exp expc

zE E i nkz t

Complex refractive indexUsing the ansatz: 0 ( )E E f k r t we can obtain the dispersion relation:

2 2

2( )

c k

complex dielectric function

( ) ( )n n i Where we can define:

Complex refractive index

0

( )n

0

4( ) ( )

Snell’s law refractive index (real dispersion)

Intensity extinction(energy dissipation)

Re( ) ( )ph

g

cv

k n

vk

)(~)(

n

cc

k

Absorption coefficient

General relations

2 ( )n ( )

( ) ' ''

n n i

i

2 2'

'' 2

n

n

2 2

2 2

' '' '

2

' '' '

2

n

Or, equivalently: ( , ) ( ', '')n

are connected and describe the intrinsic optical properties of matter

1

)( in

ck

Both the real and imaginary parts of contribute to the attenuation of the optical field!

)(

Metal Skin Depth)(

inc

k

22

21

1

2

1

2

)(0 )(),( tkzierEtrE

zk

/)/(0 )(),( ztczni eerEtrE

2

c

21 iin

Skin depth – determines the attenuation length of the field in metals!

/20

zeII

For the intensity attenuation:

Useful only when the distances associated with spatial changes of the field are large compared to the mean free path l of the conduction electrons

0)( k

Skin depth data

Element Na Al Cu Ag Au Hg

δ(2eV) 38 13 30 24 31 255

δ(3eV) 42 13 30 29 37 141

δ(4eV) 48 13 29 82 27 115

lBulk 34 16 42 52 42 11*

All the lengths are expressed in nanometers and are measured at RT

*measured at 77K


The Drude-Sommerfeld (DS) model of free electrons – lossless metals

• Physical picture: Treat the metal as a gas of electrons

2

2

de ma m

dt

rF E Re expt i t E E

e2

2

dm e

dt

rE

( )t e tp r

22

2

dm e

dt

pE

Re expt i t p p 2

2

1e

m

p E

22

2 20 0

11 1 1 p

r

N Ne

m

p

E

ENpP 0

r

p m

Nep

0

2

Plasma frequency:Drude dielectric function – negligible damping

Dipole moment:


• Only conduction e’s contribute to: r

2

21 p

r

• Last Page

• becomes 1 at high r

Metal becomes transparent

Aluminum

r

n’ n’’

Photon energy (eV)

0

-2

Die

lect

ric f

unct

ions

-4

-6

-8

0 5 10 15

-10

0

-2

-4

-6

-8

-10

ħp

The dielectric function is negative at optical frequencies!

Optical response of simple Drude metals

• becomes 0 at =pr

For most metals, the plasma frequency is in the ultraviolet range (5-15eV)


Drude Model

Real and imaginary part of the dielectric constant for gold according to the Drude-Sommerfeld free electron model. The blue solid line is the real part, the red, dashed line is the imaginary part. Note the different scales for real and imaginary part.

114

115

10075.1

108.13

s

sp


General Drude model (with losses)

Optical response of a collection of free electrons

Lorentz model with “clipped” springs: 00 ( 0)K

2

2

2

2 2

2

2 2

1

' 1

''

p

p

p

i

Drude-Sommerfeld model = Lorentz model without springscollisionless gas of free electrons moving against A fixed background of positive ions

20/p Ne m

Free electron plasma frequency:

Density of free electrons

Effective electron mass

Collision timeLow T: impurity, imperfections

RT: electron-phonon scattering

1

l

vF

Fermi velocity

Electron mean free path


At the plasma frequency electrons undergo longitudinal oscillations. The quantum quasi-particle associated with these oscillations is called a plasmon

:p Natural frequency of oscillation of the free electron “sea”

Due to their longitudinal nature, volume plasmons do not couple to transverse electromagnetic waves, and can only be excited by particle impact.

Longitudinal electron waves can be excited at the plasma frequency!

ED )(0 at 0)( pp

0)()(0 0 EikD p

0 0 if k // E, the wave is longitudinal!

Longitudinal (bulk) waves in metals


Physical interpretation of plasma frequency

++

+

+

+

++

++

+

+

+

++

+

+- -

--

--

--

-

--

--

-

-

Equilibrium, neutrality

-

Non equilibrium, applied field E

++

+

+

+

++

++

+

+

+

++

+

+- -

--

--

--

-

--

--

-

--

- +

E

Homogeneous electron density

Non-homogeneous electron density

N

N N

0

e NE

Induced electric field:

u eE

t m

Continuity equation:

Nu

t N

Equation of motion:2

22

0p

N N

t N N

Plasma oscillationsCollective oscillations of the electron gas


Plasma frequencyN (cm-3) ωp λp Type of

plasma

1024 5.7 x 1016 33 nm metals

1022 5.7 x 1015 330 nm metals

1020 5.7 x 1014 3.3 µm Doped semiconductors

1018 5.7 x 1013 33 µm Doped semiconductors



106 5.7 x 107 33 m Ionosphere

105 1.8 x 107 105 m Ionosphere

p

Opaque Transparent


Aluminum

2

2

2

3

' 1

''

p

p

( )

Identical to the high frequency limit of the Lorentz model!

Plasma frequencies of metals are in the VIS and UV

3 eV 20 eVp

p

Opaque Transparent

Luca Dal Negro, ECE Department, Boston University

The resonant medium: Lorentz model

NpP

exp

Dipole moment

Polarization density

Lorentz model of materials

K

,m e“There is not a single granule of light which is not the fruit of an oscillating charge” A. Lorentz


The Lorentz model

Lorentz model of materials

K

,m e

i tc

i tloc c

x x e

E E e

Ansatz:

2 20

( / ) cc

e m Ex

i

20 /

/

K m

b m

Complex representation of the real time-harmonic quantities

2

2 loc

d x dxm b Kx eE

dt dt

Driving force

General solution: transient + oscillatory

Most general classical response model

The proportionality function between the field and the electron displacement is complex!


ConsequencesIf 0 the proportionality factor between F and x is complex

( / )ix Ae eE m where:

1/ 222 2 2 20

12 2

0

1

tan

A

A

0 In-phase response

0

0 180° out of phase response

0

Max(A) occurs at

FWHM

Max(A)

0

)(for 0

/1


Collection of oscillators

Induced dipole moment of an oscillator is: p exThe polarization of a medium containing N oscillators per unit volume is: P Np Nex

2

02 20

pP Ei

Plasma frequency: 2

2

0p

Ne

m

Particular example of the constitutive

relation: 0P E

2

2 20

1 1 p

i

0L


0

0 for 0

1 for 0L

n

n

n


Dielectric function of Lorentz oscillators

2

2 20

1 1 p

i

' ''i

2 2 20

22 2 2 20

2

22 2 2 20

' 1

''

p

p

1

L

0P E

Longitudinal Waves in Matter

0

1

P E

Longitudinal waves:

0

1( ) 0L E P

0

0)(

01

0

B

EkH

D

Pure polarization waves (not EM!)

ED )(0 at 0)( pp

0)()(0 0 EikD p

0 0 if k // E, the wave is longitudinal!

Klingshirn, Semiconductor opticsSpringer Verlag (2004)

Polarization mechanisms in dielectrics

Debye relaxation: alignment of permanent dipoles along E against thermal buffeting

Lattice vibrationsvibrational oscillators

Electronic transitionselectronic oscillators

ECE Department, Boston University

Stop band (Reststrahlbande)

0

0 for 0

1 for 0L

n

n

n

No wave propagation possible exponentially decaying wave amplitude

In this range matter is optically thinner than vacuum (n<<1) !!

cr

1 2n n

vacuum

2

1

sin cr

n

n

For:

2 0 0crn

Total internal reflectionat normal incidence!!

1n

medium


Important remarks

For metals:

00

T

LP

Surface polaritons in metals (called surface-plasmon polaritons) can be excited in the wide range:

P 0

Metals are ideal materials for broadband engineering of surface-polariton waves


Free and bound electrons in metals: Ag

Silver cannot be explained within the simple Drude model, why?

Bound charge effect : 4eV plasma frequency for Ag, 2 eV for Cu

2 2

2 2 21 pe pj

je j ji i

Free electrons Bound electrons

Bound (d electrons) and free charges are competing:• shift (decrease) of plasma frequency• extra reflectance peaks

To describe noble metals (Au,Ag, Cu, etc) in the visible, more general Lorentz-Drude models are needed!

Scattering losses/heat Radiation damping


Mixed excitations in a solid: polaritons

0 ( ) 1P E The electric field in matter is always accompanied by polarization waves

Waves traveling in solids are always a mixture of an electromagnetic wave and a mechanical polarization wave (as long as

( ) deviates from 1)

Light in matter: mixture of photons and quanta of the polarization field. The mixed state is quantized in polaritons

2 2

2( )

c k

Classical Polaritons equationFrom:

)( c

k

Material responseOptical response

( ) 1


Dispersion of waves

The fact that the eigenfrequency ω0 of some excitations of a solid depends on k is called “spatial dispersion”.

The term “dispersion relation” means the relation E(k) or ω(k) for all wave-like excitations. It can be a simple horizontal-line, a linear or parabolic relation, or something

more complicated. Every excitation which has a wave-like character has a dispersion relation.The detailed shape of ω(k) depends on the physical nature of the oscillators and the coupling mechanism.

The term “spatial dispersion” means that the eigenfrequency ω0 of one of the elementary excitations in a solid depends on k

The term “spatial dispersion” means that the eigenfrequency ω0 of one of the elementary excitations in a solid depends on k

Bulk plasmon: dispersion relation (Drude metal)

Dispersion relation for

bulk plasmons

2

22 2

,,r

tt

c

E r

E rt

• The wave equation is given by:

• For a propagating wave solution: , Re , expt i i t E r E r k r

2 2 2r c k

2

21 p

r

• Dielectric constant:

22 2 2 2 2

21 p

p c k

k

L p

No allowed propagating modes into the metal (imaginary k)

ck

2 2 2p c k

• Dispersion relation:

Note1: Solutions lie above light line

Note2: Metals: ħp 10 eV; Semiconductors ħp < 0.5 eV (depending on dopant conc.)

n2


Volume (bulk) Plasmons: Drude dispersion

Bulk Polaritons Dispersion

2 2

2 2 20

b

c k f

i

Using the general Lorentz model (or Kramers-Heisenberg) dielectric function we can write the polariton equation as:

Implicit representation of ( )k for polaritons.

Re k

Photon-like

0

LGap

LPB

UPB

Phonon/Exciton-like

Simple example: uncoupled oscillators at vanishing damping

plasmon guides parti (1)

Documents

boston universitycopyright

boston universityoptics

optical field

boston university cobra

nonradiative fields

attenuation length

intensity attenuation

satellite waves