Dispersion relation of surface plasmons

“Metal Optics”Prof. Vlad Shalaev, Purdue Univ., ECE Department, http://shay.ecn.purdue.edu/~ece695s/

“Surface plasmons on smooth and rough surfaces and on gratings”Heinz Raether(Univ Hamburg), 1988, Springer-Verlag

“Surface-plasmon-polariton waveguides”Hyongsik Won, Ph.D Thesis, Hanyang Univ, 2005.

Metal Optics: An introduction

Photonic functionality based on metals?!

Note: SP is a TM wave!

What is a plasmon?

Plasmon-Polaritons

Plasmons in the bulk oscillate at ωp determined by the free electron density and effective mass

Plasmons confined to surfaces that can interact with light to form propagating “surface plasmon polaritons (SPP)”

Confinement effects result in resonant SPP modes

in nanoparticles (localized plasmons)

+ + +

- - -

+ - +

k

Plasma oscillation = density fluctuation of free electrons

0

2

εω

mNedrude

p =

0

2

31

εω

mNedrude

particle =

22

22

)1()(

pd

dpspx c

kkωωεεωωω

−+

−==

Local field intensity depends on wavelength

(small propagation constant, k) (large propagation constant, k)

Ekmel Ozbay, Science, vol.311, pp.189-193 (13 Jan. 2006).

Some of the challenges that face plasmonics research in the coming years are

(i) demonstrate optical frequency subwavelength metallic wired circuitswith a propagation loss that is comparable to conventional optical waveguides;

(ii) develop highly efficient plasmonic organic and inorganic LEDs with tunable radiation properties;

(iii) achieve active control of plasmonic signals by implementing electro-optic, all-optical, and piezoelectric modulation and gain mechanisms to plasmonic structures;

(iv) demonstrate 2D plasmonic optical components, including lenses and grating couplers,that can couple single mode fiber directly to plasmonic circuits;

(v) develop deep subwavelength plasmonic nanolithography over large surfaces;

(vi) develop highly sensitive plasmonic sensors that can couple to conventional waveguides;

(vii) demonstrate quantum information processing by mesoscopic plasmonics.

To derive the Dispersion Relation

of Surface Plasmons,

let’s start from the Drude Model

of dielectric constant of metals.

Dielectric constant of metal

Drude model : Lorenz model (Harmonic oscillator model) without restoration force(that is, free electrons which are not bound to a particular nucleus)

Linear Dielectric Response of Matter

The equation of motion of a free electron (not bound to a particular nucleus; ),

( : relaxation time )

The conduction current density

2

2

14

0

1 10

C

d r m dr dv mm Cr eE m v eEdt dtdt

s

J

τ τ

τγ

−

=

= − − − ⇒ + = −

= ≈

=

{ }{ }

.

For a harmonic wave ( ) Re ( ) ,

the current desity varies at the same rate ( ) Re ( )

( ) ( ) ( )

( / )( ) (( / )

2

2

2

1

1

1

i to

i to

Nev

d J NeJ Edt m

E t E e

J t J e

Nei J Em

ne mJ Ei

ω

ω

τ

ω

ω

ω ω ωτ

ω ωτ ω

−

−

−

⎛ ⎞+ = ⎜ ⎟

⎝ ⎠

=

=

⎛ ⎞− + = ⎜ ⎟

⎝ ⎠

=−

)

Lorentz model(Harmonic oscillator model)

If C = 0, it is called Drude model

Drude model: Metals

Local approximation to the current-field relation

those localized in interband(modified Drude model)

Maxwell Equations( )( ) o B E tD tH J Jt t

ε ε∂∂∇× = + = +

∂ ∂

( ) ( )( ) ( ) ( ) ( ) ( ) ( )o Bo B o B

o

E tH J i E E i Et i

ε ε σ ωωε ε ω ω σ ω ω ωε ε ω ωωε

⎡ ⎤∂∇× = + = − + = − −⎢ ⎥∂ ⎣ ⎦

: Static (ω =0) conductivity

2 2

2 2 2 2

2 2 3 3

( )( )

(1 )11 (1 )

1

EFF Bo

o oB B

o o

p pB

i

ii ii

i

σ ωε ω εε ω

σ σ ωτε εε ω ωτ ε ω ω τ

ω τ ω τε

ω τ ωτ ω τ

= +

+⎛ ⎞= + = +⎜ ⎟− +⎝ ⎠

⎛ ⎞ ⎛ ⎞= − +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠

22where, : bulk plasma frequency ( 10 for metal)op

o o e

ne eVm

σωε τ ε

= = ∼

Dielectric constant of metal : Drude model

Plasma frequency

From Microscopic origin of ω-response of matter

Al13 : 1S2 2S2 2P6 3S2 3P1

Aluminum

Ag47 : 1s2 … 4s2 4p6 4d10 5s1

Au79 : 1s2 … 5s2 5p6 5d10 6s1

Ag47 : 1s2 … 4s2 4p6 4d10 5s1

The fully occupied 4d band of Ag can influence the dynamical surface response in two distinct ways: • First, the s-d hybridization modifies the single-particle energies and wave functions.

As a result, the nonlocal density-density response function exhibits band structure effects.• Second, the effective time-varying fields are modified due to the mutual polarization of the s and d

electron densities.

Here we focus on the second mechanism for the following reason: At the parallel wave vectors of interest, the frequency of the Ag surface plasmon lies below the region of interband transitions. Thus, the relevant single-particle transitions all occur within the s-p band close to the Fermi energy where it displays excellent nearly-free-electron character.

A. Liebsch, “Surface Plasmon Dispersion of Ag,” PRL 71, 145-148 (1993).

Measured data and model for Ag:

γωω

εωω

ε 3

2

2

2

",1' pp =−=

γωω

εωω

εε 3

2

2

2

",' pp =−= ∞

Drude model:

Modified Drude model:

Contribution of bound electrons

Ag: 4.3=∞ε200 400 600 800 1000 1200 1400 1600 1800

-150

-100

-50

0

50

Measured data: ε' ε"

Drude model: ε' ε"

Modified Drude model: ε'

ε"

ε

Wavelength (nm)

ε'

-εd

Bound SP mode : ε’m < -εd

( )ωε

ε

Wavelength (nm)

Ideal case : metal as a free-electron gas

• no decay (γ 0 : infinite relaxation time)• no interband transitions (εB = 1)

2 2 2 2 2

12 2 3 3 2

( )( )

11 B

EFF Bo

p p pB EFF

i

i τε

σ ωε ω εε ω

ω τ ω τ ωε ε

ω τ ωτ ω τ ω→∞=

= +

⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − + ⎯⎯⎯→ = −⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠ ⎝ ⎠

2

21 pr

ωε

ω= − 0

Dispersion relation of surface-plasmon for dielectric-metal boundaries

Dispersion relation for EM waves in electron gas (bulk plasmons)

( )kω ω=

• Dispersion relation:

Dispersion relation for surface plasmons

TM waveεm

εd

xm xdE E= ym ydH H= m zm d zdE Eε ε=• At the boundary (continuity of the tangential Ex, Hy, and the normal Dz):

Z > 0

Z < 0

Dispersion relation for surface plasmon polaritons

zm zd

m d

k kε ε

=

xm xdE E=

ym ydH H=

xmmymzm EHk ωε=

ydd

zdym

m

zm HkHkεε

=

),0,( ziixii EiEi ωεωε −−),0,( yixiyizi HikHik−

xiiyizi EHk ωε=

xddydzd EHk ωε=

Dispersion relation for surface plasmon polaritons

• For any EM wave:2

2 2 2i x zi x xm xdk k k , where k k k

cωε ⎛ ⎞= = + ≡ =⎜ ⎟⎝ ⎠

m dx

m d

kc

ε εωε ε

=+

SP Dispersion Relation

1/ 2

' " m dx x x

m d

k k ikc

ε εωε ε

⎛ ⎞= + = ⎜ ⎟+⎝ ⎠

x-direction:

For a bound SP mode:

kzi must be imaginary: εm + εd < 0

k’x must be real: εm < 0

So,

1/ 22

' izi zi zi

m d

k k ikc

εωε ε

⎛ ⎞= + = ± ⎜ ⎟+⎝ ⎠

z-direction:2

2 2zi i xk k

cωε ⎛ ⎞= −⎜ ⎟⎝ ⎠

Dispersion relation:Dispersion relation for surface plasmon polaritons

' "m m miε ε ε= +

'm dε ε< −

2 22 2 zi i x x i x ik k i k k

c c cω ω ωε ε ε⎛ ⎞ ⎛ ⎞ ⎛ ⎞= ± − = ± − ⇒ >⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

+ for z < 0- for z > 0

( )( )

( )( )

( )

21

2"42

2"21

2"2'

"

21

2"4221

2"2'

'

2)(

2)(

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎭⎬⎫

⎩⎨⎧ ++⎥

⎥⎦

⎤

⎢⎢⎣

⎡

++=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡ ++

⎥⎥⎦

⎤

⎢⎢⎣

⎡

++=

dmee

dm

mdm

dx

dmee

mdm

dx

ck

ck

εεεε

εεεεε

εω

εεεε

εεεεω

( ) ( ) '2"2'2 , mdmmewhere εεεεε ++=

1/ 2' " m d

x x xm d

k k ikc

ε εωε ε

⎛ ⎞= + = ⎜ ⎟+⎝ ⎠

metals, ofmost in and , ,0 "'''mmdmm εεεεε >>><

,

( )2'

"2/3

'

'"

2/1

'

''

2 m

m

dd

dmx

dm

dmx

ck

ck

εε

εεεεω

εεεεω

⎟⎟⎠

⎞⎜⎜⎝

⎛+

≈

⎟⎟⎠

⎞⎜⎜⎝

⎛+

≈

' "m m miε ε ε= +

Plot of the dispersion relation

2

2

1)(ωω

ωε pm −=

m dx

m d

kc

ε εωε ε

=+

• Plot of the dielectric constants:

• Plot of the dispersion relation:

d

psp

dm

ωωε

ωεε

+=≡∞→⇒

−→•

1 ,k

, When

x

22

22

)1()(

pd

dpspx c

kkωωεεωωω

−+

−==

Surface plasmon dispersion relation:

2/1

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=dm

dmx c

kεεεεω

ω

ωp

d

p

ε

ω

+1

Re kx

real kx real kz

imaginary kx real kz

real kx imaginary kz

d

xckε

Bound modes

Radiative modes

Quasi-bound modes

Dielectric: εd

Metal: εm = εm' +

εm"

xz

(ε'm > 0)

(−εd < ε'm < 0)

(ε'm < −εd)

2 2 2 2p xc kω ω= +

Surface plasmon dispersion relation 1/ 22

izi

m d

kc

εωε ε

⎛ ⎞= ⎜ ⎟+⎝ ⎠

Very small SP wavelength

λvac=360 nm

X-ray wavelengths at optical frequencies

AgSiO2

2/1

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=dm

dmx c

kεεεεω

2 2 2 2

2 2 3 311

p pm i

ω τ ω τε

ω τ ωτ ω τ⎛ ⎞ ⎛ ⎞

= − +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠

Dispersion relation for bulk and surface plasmons

d

psp

d

pdpd

pm ωωω

εω

εω

εωεωω

ε+

=⇒+

=⇒−=−−=−=11

, 1When 2

2222

2

Cut-off frequency of SP

Measured data and model for Ag:

γω

ωε

ω

ωε 2

2

2

2

",1' pp =−=

γω

ωε

ω

ωεε 2

2

2

2

",' pp =−= ∞

Drude model:

Modified Drude model:

Contribution of bound electrons

Ag: 4.3=∞ε200 400 600 800 1000 1200 1400 1600 1800

-150

-100

-50

0

50

Measured data: ε' ε"

Drude model: ε' ε"

Modified Drude model: ε'

ε"

ε

Wavelength (nm)

ε'

-εd

Bound SP mode : ε’m < -εd

( )ωε

ε

Wavelength (nm)

( )3/ 2' "

1" " 1 2 1' ' 21 2 1

The length after which the intensity decreases to 1/e :

2 , where 2( )i x xL k k

cε ε εωε ε ε

− ⎛ ⎞= = ⎜ ⎟+⎝ ⎠

Propagation length

2 2 2 2' "

2 2 3 31p p

m m m Bi iω τ ω τ

ε ε ε εω τ ωτ ω τ

⎛ ⎞ ⎛ ⎞= + = − +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟+ +⎝ ⎠ ⎝ ⎠

Propagation length : Ag/air, Ag/glass

Penetration depth

'1 2

1At large ( ), z .ixx

kk

ε ε→ − ≈ Strong concentration near the surface in both media.

'1At low ( 1),xk ε >> Larger Ez component'

1 in air :z

x

Ei

Eε= −

( : , : - )z xE iE air i metal i= ± +

Smaller Ez component'1

1 in metal :z

x

Ei

E ε=

Gooood waveguide!

Another representation of SP dispersion relation

ck pp

mmm

/

2"'

ωνεεε

=−=+=

Excitation of surface plasmons

ddm

dmx c

kc

k εωεεεεω

=>⎟⎟⎠

⎞⎜⎜⎝

⎛+

= 2/1

'

''

sinx spp

c kω θε

=od

c kωε

=

'xk

ω

excitation

' sinx p sp spk kcωε θ⎛ ⎞= =⎜ ⎟

⎝ ⎠

1p

spd

ωω

ε=

+

Excitation of surface plasmon by prism

op

c kωε

=

εp

εd

1p

spp

ωω

ε=

+

pω

2 2 2 2p c kω ω= +

Generalization : Surface Electric Polaritons and Surface Magnetic Polaritons

: Energy quanta of surface localized oscillation of electric or magnetic dipoles in coherent manner

Surface Electric Polariton (SEP) Surface Magnetic Polariton (SMP)

+q -q +q -q SNSN

E H

Coupling to TM polarized EM wave Coupling to TE polarized EM wave

Common Features

- Non-radiative modes → scale down of control elements

- Smaller group velocity than light coupling to SP

- Enhancement of field and surface photon DOS

Generalization : Surface Electric Polaritons and Surface Magnetic Polaritons

1 2 2 1 1 202 2

2 1

' ' ( ' ' ' ' ) " ",' ' ' 'SEP k Oε ε ε μ ε μ ε μβ

ε ε ε μ⎛ ⎞−

= + ⎜ ⎟− ⎝ ⎠

,1 ,21 1 2 2, 02 2

2 1 1 2

' ( ' ' ' ' ) " ", ,' ' ' ' ' '

SEP SEPiSEP i k O

γ γε ε μ ε μ ε μγε ε ε μ ε ε

⎛ ⎞−= + =⎜ ⎟− ⎝ ⎠

Dispersion Relation & Decay Constants

( )"/ ' , "/ ' (1,1) ,For ε ε μ μ << {

J. Yoon, et al., Opt Exp 2005.

In Summary

1/ 2

d mSPP

d m

kc

ε εωε ε

⎛ ⎞= ⎜ ⎟+⎝ ⎠

Dispersion relations

2 2

2 2 2 2

2 2

( ) 1

1 /

p pm

p

iω ω γε ω

ωω γ ω γω ω

⎛ ⎞= − + ⎜ ⎟+ + ⎝ ⎠≈ −

Permittivity of a metal

Type-A

- Low frequency region (IR)

- Weak field-confinement

- Most of energy is guided in clad

- Low propagation loss

► clad sensitive applications

► SPP waveguides applications

Type-A : low k

H. Won, APL 88, 011110 (2006).

Type-B

- Visible-light frequency region

- Coupling of localized fieldand propagation field

- Moderated field enhancement

► Sensors, display applications

► Extraordinary transmission of light

Nano-hole

Type-B : middle k

Type-C

- UV frequency region

- Strong field confinement

- Very-low group velocity

► Nano-focusing, Nano-lithography

► SP-enhanced LEDs

n-GaN

p-GaN (20nm, 120nm)

Ag (20nm)

Λ

Light emission

QW

QW

2

0

1 1 ( )( ) 2

R f i ρ ωτ ω ε

= = ⋅p E

SE Rate :

Electric field strengthof half photon (vacuum fluctuation)

Photon DOS(Density of States)

Type-C : high k

Importance of understanding the dispersion relation :Broadband slow and subwavelength light in air

Mode conversion: negative group velocity

Re kx

SiO2

Si3N4

21 SiO

p

εω+

431 NSi

p

εω+

SiO2

Si3N4

Al

0=dkdω

0<dkdω

AppendixAppendix

If the electrons in a plasma are displaced from a uniform background of ions, electric fields will be built up in such a direction as to restore the neutrality of the plasma by pulling the electrons back to their original positions. Because of their inertia, the electrons will overshoot and oscillate around their equilibrium positions with a characteristic frequency known as the plasma frequency.

Plasma frequency

(surface charge density) / ( ) / : electrostatic field by small charge separation

exp( ) : small-amplitude oscillation

( ) ( ) 2 2 2

2 22

s o

o

o p

s p po o

ENe x x

x x i t

d x Ne Nem e E mmdt

εδ ε δ

δ δ ω

δ ω ωε ε

=

=

= −

= − ⇒ − = − ⇒ ∴ =