Simulation of EM wave propagating Simulation of EM wave propagating p p g gp p g gin a in a nanocylindernanocylinder--base localized base localized surface plasma resonance senorsurface plasma resonance senorsurface plasma resonance senorsurface plasma resonance senor

Po-Han Chen, and Bing-Hung Chen

Institute of Electronic Engineering, National Dong Hwa University TaiwanNational Dong Hwa University , Taiwan

OutlinesIntroduction of surface plasmon resonanceE i lExperimental setupelectronic resonance in single or pair of cylindersField enhancement in Cylinder array Sensitivity enhancementMagnetic resonance conclusionsconclusions

IntroductionDispersion Relations of SPsDispersion Relations of SPs

SPs at metal-dielectric interface

Introduction

From Maxwell’s equations & E and H fields continuity relations

P-polarized: Ex, Ez, Hy

Dispersion relation (DR)

)](exp[0 tzkxkiEE zx ω−±+=±

0,0 ≤−≥+ zforzfor

imaginarykz :

z

Plasmon dispersive relation

Excitation of SPsExcitation of SPsATR (attenuated total

reflection) coupler

SP sensor

Experimental SystemExperimental System

1⎡ ⎤

Jones vectors

0i t11P= e12

ω⎡ ⎤⎢ ⎥⎣ ⎦

1 11 ⎡ ⎤t

1 11analyzer AN =1 12⎡ ⎤⎢ ⎥⎣ ⎦

pir e 0φ⎡ ⎤s

p

is

r e 0SPR=

0 r e φ⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

tω⎡ ⎤ti2

t-i2

e 0EO=

0 e

ω

ω

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦t tE =AN SPR EO P⋅ ⋅ ⋅ t t tI =E E∗ ⎣ ⎦

( )2 2

p st p s p s

r + r1I = + r r cos t+ -4 2

ω φ φ⎡ ⎤⎢ ⎥⎢ ⎥⎣ ⎦

In general

1 constantφ⎣ ⎦ sr =1 s constantφ ∼

( )tan expp rp ips rs is

r E Ei

r E Eρ ψ= = = Δ

ModeNanoscale cylinder arrayNanoscale cylinder array

Interaction between localized and delocalized surface plasmaon polarition modes in a metal photonic crystal (A. Christ etal, Phys. Stat. sol. (b) 243. No.10 2344- Corning 1737 glass

2.5cm

0.1cm

2348)

Plasmon resonance coupling in metallic nanocylinders(Kottmann etal, Optics Express Vol.8 No.12 P655

2.5cm

Nano-cylinder array

(Kottmann etal, Optics Express Vol.8 No.12 P655 2001, Sburlan etal, Physical Review B 73 035403 2006)

N i b d h t f l li d f

BK7 prismIndex match oil

Coring glass slideNanowire-based enhancemant of localized surface plasmon resonance for highly sensitive detection (K.Kim, Optics Express Vol.14, No.25, 12419 2006)

Coring glass slide

gold nano-cylinderDielectric (alalyte )

Resonance light interaction with plasmonicnanocylinder system (ref: Viktor A. Podolskiy etal J. of Optics A 7 P32-37 2005)

Macroscopic permittivities

Maxwell’s equation in the metallic cylinderMaxwell s equation in the metallic cylinder

( )4 4D =4 = J= E E =0i iπ πσπρ εω ω

⎛ ⎞∇ ⋅ ∇ ⋅ ∇ ⋅ ⇒∇⋅⎜ ⎟⎝ ⎠

4-iπσε εω∞

≡i iω ω⎝ ⎠

( ) ( ) ( ) ( )3 31 1 11= - ( ) -

2 4 8i iM d r r r J r D r d r r r E r

cV cω ωθ ε θπ π

⎡ ⎤⎛ ⎞× = ×⎡ ⎤⎜ ⎟⎢ ⎥ ⎣ ⎦⎝ ⎠⎣ ⎦∫ ∫2 4 8cV cπ π⎝ ⎠⎣ ⎦

( ) ( ) - 4cB k E e E H B Mε μ μ μ π= × = × ≡ = E Eε ε≡( ) ( )

( )

- 4

-1 -14 4

e e k e e

e ee e k

B k E e E H B M

M B e E

ε μ μ μ πω

μ μ ε μ

= × = × ≡ =

⇒ = = ×

eE Eε ε≡

2p

e 21- iω

ε ε∞⎛ ⎞

≡ ⎜ ⎟⎜ ⎟

Enhanced E-field make contribution εe, but μe limited.

( )4 4π π e 2 -iω ωγ∞ ⎜ ⎟⎝ ⎠

Mathematic Description

Single cylinderz L=

Single cylinderL>>a, λ>L, the standing waves are good approximation to the eigenstates: L

ε1 , k1ε2 , k2

M 1≥

( ) ( ) ( )-2 22 -1

-cos( ) ( ) i M qM z z q M qq

E r k z q signM k k e J e ϕαρ=±

= +∑

( ) ( )2+ i 2 iMk k k i M J ϕ0z =

( ) ( )22 0+sin 2 iMz z Mk z k k signM e J e ϕα αρ

22 2 2 1 2x y

e iek k iω

±

B.C.

EM = 0 at z = 0, L2 2 21 0 2, ; , 1,22

x yz i i i ze e e k k ic

ε α± ≡ = ≡ = + =∓

, zk L nπ=

2 2

,/ , / 1, ...

z

n k z k zπ π= +

TM Mode TE Mode

TM d ( ) fi ld h t i f ti f li dTM mode (p-wave) field enhancement is a function of cylinder dimension

S.E. Sburlan Physical Review B 73. 035403 2006

Two cylindersρ  a

( ) ( ) ( )-2 2( ) ( ) Y i M qE k i M k k ϕ∑( ) ( ) ( )-2 22 - 21

-cos( ) ( ) Y i M qM z z q M qq

E r k z q signM k k e e ϕα ρ=±

= +∑

( ) ( )2+sin 2 Y iMk z k k signM e e ϕα α ρ( ) ( )2 2 z 2+sin 2 Yz z Mk z k k signM e e ϕα α ρ

Coupled system has the same magnitude resonance as for the individual cylinder, but the coupling is much stronger and depends on the incident field

Main resonance is red-shifted with d i di t ddecreasing distance d

Cylinder array3.00

2.00

2.25

2.50

2.75

強度TM Mode

1 00

1.25

1.50

1.75

電場

強

y

0 100 200 300 400 500 600 7000.75

1.00

(nm)

xz

TE Mode

Local Surface Plasmon Enhance Sensitivity

Sensitivity Definition: Mathematic formula

Local Surface Plasmon Enhance Sensitivity

= hsn

ΔΔ

:nΔ

:hΔ

Refraction index variation

Physical parameter variation (θ or λ)

Resolutionσ :σ I t t R l ti

Present

RI = inssσσ

:insσ Instrument Resolution

Interrogation Angular Wave Intensity

PhasePresent y

σins 0.00010 0.02nm 0.2% 0.010

Prism  5×10‐7 2×10‐5 5×10‐5 4 6×10‐7RI :σ Prism coupling

5×10 7 2×10 5 5×10 5 4.6×10 7

Grating coupling

2×10‐6 6×10‐5 2×10‐4

RI

Can reach ~ 10-7 for using heterodyne interferometer p g

J. Homola, S.S.Yee and G.Gauglitz “Surface plasma resonance sensors: Review” Sens. Actuators B 54, 3-15 (1999)

Local surface plasma resonance is accompanied by the broadening in dispersion relation which increase the damping term γ *Surface plasmon resonance consists of gold thin film part and nanocylinderpart, angular width is

{ }Im k γ γ+ ⎛ ⎞{ }Imcos cos

sp i rsp

s sp s sp

k

n nc c

γ γθ ω ωθ θ

+Δ = = 2p ' ''

e 21- -i ca cai

ωε ε ε ε

ω ωγ∞⎛ ⎞

≡ = +⎜ ⎟⎝ ⎠

{ }( )

''' '

2' ' 'Im AuAu caspk

εω ε ε⎛ ⎞≈ ⎜ ⎟

⎝ ⎠

' ''Au Auε ε '' 0caε ≈{ }

( )2'2sp Au ca Auc ε ε ε⎜ ⎟+⎝ ⎠ Metallic photonic

crystal

' 'Au caε ε∼

( )''' '

2' ' '

2=cos 2

AuAu casp

s sp Au ca Aun

εε εθθ ε ε ε

⎛ ⎞Δ ⎜ ⎟+⎝ ⎠ ( )p Au⎝ ⎠

* Kyujung Kim Optics Express Vol 14, No 25 12419 (2006)

Magnetic ResonanceTE Mode:

Anti-symmetric polarition modeAnti-parallel current in the two cylinders induce magnetic dipole moment resonanceinduce magnetic dipole moment resonance

TM Mode: Symmetric polarition will induce the y pelectric resonance, but magnetic resonance is limited

1.672

1.128

1.357

y

0.232

0.464

0.714

0.696

x 0.000

H in TE mode H in TM modeE in TE mode E in TM mode

ConclusionsConclusionsSignificant sensitivity increase is associated with larger field enhance.

Field enhancement is achieved by nanocylinder based localized plasmonresonance mediated by nanocylinder and their coupling.

Coupling has more contribution on the field enhance.

Significant magnetic field enhancement is happen in TE mode due to anti-symmetric polarition mode.