ECE280: Nano-Plasmonics and Its Applications

Week1Week1

Basics for PlasmonicsBasics for Plasmonics

Light Matter InteractionsLight Matter Interactions

Incident light

reflected light

transmitted lightpropagation through the medium

General Phenomena

refraction absorption and luminescence

scattering

Phenomena that can occur as a light beam propagates through an optical medium

• Crystalline insulators and semiconductors

• Glasses

• Metals

• Molecular Materials

• Doped glasses and insulators

Optical Materials Optical Materials

SilverSilver

Electron shell diagram

Among metals, pure silver has the highest thermal conductivity, the whitest color,and the highest optical reflectivity.

Major Applications

Photography

Jewelry

Industrial

Coins & medals

24%

33%

40%

3%

Based on data in 2001

Dielectric Constant for MetalsDielectric Constant for Metals

tieeEteEdtdxm

dtxdm ωγ −−=−=+ 002

2

0 )(

)()()( 2

0 γωω imteEtx+

=

Drude-Lorentz model:

tiexx ω−= 0

)( 20

2

000 γωωεεεε

imENeEPEED r +

−=+==

NexP −=

)(11)( 2

00

2

γωωεωε

imNe

r +−=

"i' , ,)(

1)( r00

2

p2

2

εεεε

ωγωω

ωωε +==

+−=

mNe

ip

r

ExamplesExamples

0.2 0.4 0.6 0.8 1-50

-40

-30

-20

-10

0

Wavelength (um)

Per

mit

tivi

tyAg

Real part

Imaginary part

EE--M wave at the interface (1)M wave at the interface (1)

Consider a p-polarized wave propagates in z-x plane:

Z>0 H2 = (0,Hy2,0)exp(ikx2x+ikz2z-iωt)E2 = (Ex2,0,Ez2)exp(ikx2x+ikz2z-iωt)

Z<0 H1 = (0,Hy1,0)exp(ikx1x+ikz1z-iωt)E1 = (Ex1,0,Ez1)exp(ikx1x+ikz1z-iωt)

1i i iH E

c tε ∂

∇× =∂

v v v

1i iE H

c t∂

∇× = −∂

v v v

0i iEε ∇ ⋅ =v v

0iH∇⋅ =v v

Maxwell’s equations: (μ=1)

Media 1

k2

k1

z

x

Media 2

ˆ ˆ ˆ1

ˆˆexp( ) exp( ) ( ,0, )exp( )

( ,0, )exp( ) ( ,0, )exp( )

x y z

x y x z z y x z x z x z

z y x y x z x z x z

x y z

Ex y z c t

H H H

ik H ik x ik z i t z ik H ik x ik z i t x i E E ik x ik z i tc

k H k H ik x ik z i t E E ik x ik z i tc

H ε

ωω ω ε ω

ωω ε ω

∂ ∂ ∂ ∂= =∂ ∂ ∂ ∂

⋅ + − − ⋅ + − = − + −

− + − = − + −

∇×v vuv

Boundary condition:Ex1=Ex2, Hy1=Hy2, ε1Ez1= ε2Ez2

2 22 2 2

2 2y x z

z x

H E Ek c k cε ω ε ω

= = −1 11 1 1

1 1y x z

z x

H E Ek c k cε ω ε ω

= = −

kx1=kx2=kx kz1/ ε1= kz2/ ε2

EE--M wave at the interface (2)M wave at the interface (2)

also, 2 2 2( )xi zi ik kcωε+ =

EE--M wave at the interface (3)M wave at the interface (3)

2 22 211 1

2 22 22 2

2

( )

( )

xz

zx

kk ck k

c

ωε εω εε

−= =

−

We have:

22

1 2 1 2 1 2 1 22( )( ) ( )xkcωε ε ε ε ε ε ε ε+ − = −

EE--M wave at the interface (4)M wave at the interface (4)

Case 1:

1 2k k=v v

1 2ε ε=

Media 1

k2

k1

z

x

Media 2

Means homogeneous media

kx= any allowed value

1 2( ) 0ε ε− =

EE--M wave at the interface (5)M wave at the interface (5)

12

1 2

1 2xk real number

cω ε ε

ε ε⎛ ⎞

= =⎜ ⎟+⎝ ⎠

1 2( ) 0ε ε− ≠ 1 2, 0ε ε >Case 2: and

1 2ε ε=

Media 1

k2

k1

z

x

Media 212 2

11

1 2zk

cω ε

ε ε⎛ ⎞

= ⎜ ⎟+⎝ ⎠

12 2

22

1 2zk

cω ε

ε ε⎛ ⎞

= ⎜ ⎟+⎝ ⎠

We have:

θ

21 2

1 11

tan( ) x

z

k nk n

εθ

ε= = =

The Brewster angle !

EE--M wave at the interface (6)M wave at the interface (6)

1 2 0ε ε <Case 3: and 1 2 0ε ε+ >

1122

1 21 2

1 2 1 2xk i imaginary number

c cε εω ε ε ω

ε ε ε ε⎛ ⎞⎛ ⎞

= = =⎜ ⎟⎜ ⎟+ +⎝ ⎠ ⎝ ⎠

We have:

Exponentially increase in -X direction !

Since the media in -x direction is infinite, the EM field will diverge, which means the solution is non-physical.

EE--M wave at the interface (7)M wave at the interface (7)

12

1 2

1 2xk real number

cω ε ε

ε ε⎛ ⎞

= =⎜ ⎟+⎝ ⎠

1 20, 0ε ε> <Case 4: and 1 2 0ε ε+ <

We have:

122

11

1 2zk i imaginary number

cω ε

ε ε⎛ ⎞

= − =⎜ ⎟+⎝ ⎠

122

22

1 2zk i imaginary number

cω ε

ε ε⎛ ⎞

= =⎜ ⎟+⎝ ⎠

EM wave propagates in x direction, but exponentially decays awayfrom the interface ! This is called the surface plasma.

Media 1

kx2

kx1

z

x

Media 2ε2<0

ε1>0

kz2

kz1

Surface Surface PlasmonsPlasmons

2/1

21

21⎟⎟⎠

⎞⎜⎜⎝

⎛+

=εε

εεωc

k x

21

21

'''εεεεω+

=c

k x 21

1

2/3

21

21

)'(2"

''

"εε

εεεεω

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=c

k x

• Shorter wavelength (comparing with excitation light)

• Bound to the surface• Propagation along the surface

• Shorter wavelength (comparing with excitation light)

• Bound to the surface• Propagation along the surface

SP Excitation using ElectronsSP Excitation using Electrons

Fast electrons are a good tool with which to study the dispersion relation at larger kx, e.g., measurements up to kx~0.3A have been performed.

However, it is not convenient to reach the region of small kx with electrons.

Coupling Light to Coupling Light to SPsSPs

dielectricmetal

prism

θc

SPmetaldielectric

prism

θc

SP

Otto configuration Kretschmann configuration

metal

dielectric

prism

SP

Grating configuration

n=ω/k n> ω/k

Prism couplingPrism coupling

-40 0 40 80 120 1600.0

0.1

0.2

0.3

0.4

0.5

0.6

Ral

ativ

e A

mpl

itude

Z Direction (nm)

Ez Ex

PRAl

0 1 2 3 4 50.0

0.2

0.4

0.6

Ex Ez

Ampl

itude

(a.u

.)X Direction (μm)

Wavelength: 365nm

2'

21

PR

PRAl

zk εεε

πλ +

=dm

dmsp kk

εεεε+

= 0Lz~1/Im(kz)

Lx~1/Im(ksp)

SP propagation length SP propagation length

y

metal

SPx

zz

SP coupling between interfaces SP coupling between interfaces

Thin film dispersion curveThin film dispersion curve

Plasmonics at nanoscalePlasmonics at nanoscale