¾ Dispersion relation of surface plasmons

¾ Basics :

¾ Dispersion relation of surface plasmons

¾ References

“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.

Surface plasmons

50nm

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 ω

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

+ + +

- - -

+ - +

k

Plasma oscillation = density fluctuation of free electrons

0 2

ω ε

m

drude

Ne

p

=

0 2

3 1 ω ε

m Ne

drude

particle

=

Local field intensity depends on wavelength

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

Dielectric constant of metal : Drude model

2 2

2 2 2 2

2 2 3 3

( ) ( )

(1 )

1

1 (1 )

1

EFF B

o

o o

B B

o o

p p

B

i

i i i

i i ε ω ε σ ω

ε ω

σ σ ωτ

ε ε

ε ω ωτ ε ω ω τ

ω τ ω τ

ε ω τ ωτ ω τ

= +

⎛ ⎞ +

= + ⎜ ⎝ − ⎟ ⎠ = + +

⎛ ⎞ ⎛ ⎞

= ⎜ ⎜ ⎝ − + ⎟ ⎟ ⎠ + ⎜ ⎜ ⎝ + ⎟ ⎟ ⎠

2

where,

: bulk plasma frequency ( 10 for metal)

o o e

ne eV

m ω σ

ε τ ε

= = ∼

Plasma frequency

Ideal case : metal as a free-electron gas

• no decay (γ Æ 0 : infinite relaxation time)

• no interband transitions (ε

= 1)

2 2 2 2 2

1

2 2 3 3 2

( ) ( )

1

1

EFF B

o

p p p

B EFF

i

i

ε ω ε σ ω

ε ω

ω τ ω τ ω

ε ε

ω τ ωτ ω τ ω

→∞

=

= +

⎛ ⎞ ⎛ ⎞ ⎛ ⎞

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

2

1 ^p 2 r

ε ω

= − ω ⁰

Surface-plasmons

on dielectric-metal boundaries

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

( ) k

ω ω =

• Dispersion relation:

Dispersion relation for surface plasmons

TM wave

ε _m ε _d

xm xd

E = E H

= H

ε

E

= ε

E

• At the boundary (continuity of the tangential E

, H

, and the normal D

):

Z > 0

Z < 0

Dispersion relation for surface plasmon polaritons

zm zd

m d

k k

ε ⁼ ε

xm xd

E = E

ym yd

H = H

xm m ym

zm

H E

k = ωε

yd d

zd ym

m

zm

k H

k H

ε

ε ⁼

) ,

0 ,

( − i ωε

E

− i ωε

E

)

, 0 ,

( − ik

H

ik

H

xi i yi

zi

H E

k = ωε

xd d yd

zd

H E

k = ωε

Dispersion relation for surface plasmon polaritons

• For any EM wave:

2

2 2 2

i x zi x xm xd

k k k , where k k k

c ε ^{⎛ ⎞} ω

= ⎜ ⎟ = + ≡ =

⎝ ⎠

m d x

m d

k c

ε ε ω

ε ε

= +

SP Dispersion Relation

Surface plasmon dispersion relation:

2 / 1

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

= +

d m

d m

x

c

k ε ε

ε ε ω

ω

ω

d p

ε ω

+ 1

Re k _x

real k

real k

imaginary k

real k

real k

imaginary k

d

ck x

ε

Bound modes Radiative modes

Quasi-bound modes

Dielectric:

ε_d

Metal: ε_m= ε_m^'+ ε_m^"

x z

(ε'

> 0)

(−ε

< ε'

< 0)

(ε'

< −ε

)

2 2 2 2

p

c k

ω = ω +

Surface plasmon dispersion relation

2 1/ 2 i zi

m d

k c

ε ω

ε ε

⎛ ⎞

= ⎜ ⎝ + ⎟ ⎠

Very small SP wavelength

λ

=360 nm

X-ray wavelengths at optical frequencies !!

Ag

SiO

Excitation of surface plasmons

Surface-plasmon-polariton waveguides

Ssurface plasmon polaritons

propagating along very-thin(~ 10 nm) metal strips

Dielectric – ε

Dielectric – ε

Metal – ε

When the film thickness becomes finite.

mode

overlap

Possibility of Propagation Range Extension

frequency

in-plane wavevector

Long-Range SP:

weak surface confinement, low loss

Short-Range SP:

strong surface confinement, high loss

Several 1 cm long, 15 nm thin and 8 micron wide gold stripes guiding LRSPPs 3-6 mm long control electrodes

low driving powers (approx. 100 mW) and high extinction ratios (approx. 30 dB) response times (approx. 0.5 ms)

total (fiber-to-fiber) insertion loss of approx. 8 dB when using single-mode fibers

18nm 20nm

14nm 16nm

T

Fundamental symmetric mode of a metal stripe : thickness (T)

W=10um

LR-SP WG

P. Berini, PhotonicWest 2005.

(Spectalis Co.)

Tunable Wavelength filter by direct heating a metal wire

1 5 4 0 1 5 4 5 1 5 5 0 1 5 5 5 1 5 6 0

- 7 0 - 6 5 - 6 0 - 5 5 - 5 0 - 4 5 - 4 0

Transmittance (dB)

W a v e l e n g t h ( n m )

1544.1 1558.3

INPUT OUTPUT

Polymer 1 Substrate

+ -

Polymer 2

Vertical directional couplers

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

Variable optical attenuator based on LR-SPP

Submitted to EL, S. Park & S. Song

Localized Surface Plasmons :

Nanofocusing and Nanolithography

'

1 2

At large (

), z

1 .

x

k k

ε → − ε  ≈

Strong confinement at the interface Æ Nano focusing

( : , : - )

z x

E = ± iE air + i metal i

0 20 40 60 80 100

0 2 4 6 8

ω

=c k

λ

=337 nm;

1

= -1

ω (10

s

)

k

(

m

)

Broad dispersion

Beam radius -> zero!

Propagation Loss (asymmetric mode) High

Propose metal nanowires.

Asymmetric mode : field enhancement at a metallic tip

E _r ^E

E _z

E

* See MOVIES : SPP propagation through a metallic tip

M. I. Stockman, “Nanofocusing of Optical Energy in Tapered Plasmonic Waveguides,” Phys. Rev. Lett. 93, 137404 (2004)]

50nm

2007/5/1 ~

an optical range resonator based on single mode metal-insulator-metal plasmonic gap waveguides.

A small bridge between the resonator and the input waveguide can be used to tune the resonance frequency.

FDTD with the perfectly matched layer boundary conditions

Plasmonic Crystal Demultiplexer and Multiports

the realization of two-dimensional optical wavelength demultiplexers and multiports for surface plasmons polaritons (SPPs) based on plasmonic crystals, i.e., photonic crystals for SPPs.

Slow Propagation, Anomalous Absorption, and Total External Reflection of Surface Plasmon Polaritons in Nanolayer Systems

n=0 n=2 n=1

we show how the dispersion relation of surface plasmon polaritons (SPPs) propagating along a perfectly conducting wire can be tailored by corrugating its surface with a periodic array of radial grooves.

Importantly, the propagation characteristics of these spoof SPPs can be controlled by the surface geometry, opening the way to important applications such as energy concentration on cylindrical wires and superfocusing using conical structures.

Plasmonics: the next chip-scale technology

Plasmonics is an exciting new device technology that has recently emerged.

A tremendous synergy can be attained by integrating plasmonic, electronic, and conventional dielectric photonic devices on the same chip and taking advantage of the strengths of each technology.

Plasmonic devices,

therefore, might interface naturally with similar speed photonic devices and similar size electronic components. For these reasons, plasmonics may well serve as the missing link between the two device

technologies that currently have a difficult time communicating. By increasing the synergy between these technologies, plasmonics may be able to unleash the full potential of nanoscale functionality and

become the next wave of chip-scale technology.

Summary : Plasmonic Photonics