- Photonic crystals - Surface plasmons

Summary

1 학기 2 학기

- Photonic crystals - Surface plasmons

Introduction to Optics, 3 ^rd Edition, Pedrottis.

Consider a EM wave propagating along the z-direction.

The complex amplitude is

Jones vector

2 2

, 1

o

E a a b

b

= ⎡ ⎤ ⎢ ⎥ + =

⎣ ⎦



Two-element matrix

14. Matrix treatment of polarization

15. Production of Polarized Light

Dichroic Materials

Polarization by Scattering

Polarization by Reflection from Dielectric Surfaces Birefringent Materials

Double Refraction

The Pockel’s Cell

16. Holography

• Holography = “whole recording”

– Records intensity & direction of light.

» Information in interference pattern.

» Reconstruct image by passing original light through hologram.

» Need laser so that light interferes.

Object Beam ^a ( ) ( ) ^{x , y = a x , y exp} [ ^{− j φ} ( ) ^{x , y} ]

Reference Beam ^A ( ) ^{x , y = A} ( ) ^{x , y exp} [ ^{− j ψ} ( ) ^{x , y} ]

Interference ^I ( ) ^{x , y} = ^A ( ) ^{x , y}

+ ^a ( ) ^{x , y}

+ ^{2 A} ( ) ( ) ^{x , y a x , y cos} [ ψ ( ) ( ) ^{x , y} − φ ^{x , y} ] ( ) ^{x y A a} ( ) ^{x y}

U

, = β '

, reconstruction

A A

a

a

Real time

holographic interferometry Double exposure

holographic interferometry

Detour-phase hologram (computer-generated

hologram : CGH)

17. Optical detectors and displays

평판 디스플레이(FPD)

PDP OLED

LCD FED

Operation of twisted nematic

field effect mode liquid crystal cell .

CCD = Charge Coupled Device.

CMOS = Complementary Metal Oxide Semiconductor

Chapter 18. Matrix Methods in paraxial optics

Cardinal points (planes) : focal (F), principal (H), and nodal (N) points (planes)

19. Optics of the eye

watt (W) W/m

W/sr W/(sr

m

)

lumen (lm) lux (lx)

candela (cd) Cd/m

Radiant flux : Irradiance : Radiant intensity : Radiance :

: Luminous flux : illuminance

: luminous intensity : luminance

Radiometry Photometry

555 nm

610 nm Luminous efficiency V( λ)

Radiant flux of 1 Watt at 555 nm is

the luminous flux of 685 lm (lumen)

Radiant flux of 1 Watt at 610 nm is

the luminous flux of 342.5 lm (lumen)

Photometric unit

=

685 x V( λ) x radiometric unit

Wien displacement law Planck’s blackbody radiation

Color temperature of light source ? : the blackbody temperature with

the closest spectral energy distribution -> the sun has a color temperature range

of 5000 K ~ 6000 K

Color temperature

20. Aberration Theory

Chromatic Aberration (색수차)

Monochromatic aberrations : Third-order (Seidel) aberration theory

9 Spherical aberrations, Coma, Astigmatism (Curvature of Field), Distortion

21. Fourier Optics

0 0

0 0

2

2

X

Y

X X

k k

r r

Y Y

k k

r r

π λ π λ

⎛ ⎞

= = ⎜ ⎟

⎝ ⎠

⎛ ⎞

= = ⎜ ⎝ ⎟ ⎠

Spatial frequency

Spectrum

( ) ( )

( ) ( )

ikx

ikx

f x g k e dk

g k f x e dx

= −

=

∫

∫

( ) ( )

( ) ( )

i t

i t

f t g e d

g f t e dt

ω

ω

ω ω

ω

= −

=

∫

∫

Object

2 _S k = π f 2 f _t

ω = π

1 1

S

f λ m

= ⎡ ⎤ ⎢ ⎥ ⎣ ⎦

1 1

s f t

t

= ⎡ ⎤ ⎢ ⎥ ⎣ ⎦

Object in Time Object in Space

Angular Frequency

Frequency

Temporal Frequency Spatial Frequency

k 2 π

= λ

wave number

Fourier and Inverse Fourier Transformation by Lenses

α α _β β

(

,

) F f f

(

,

)

F f f

22. Theory of Multilayer Films

Tangential components of E and B-fields are continuous

across the interface.

(a) (b)

0

0 11 0 12 21 22

0 11 0 12 21 22

0 11 0 12 21 22

2

s s

s s

s s

t m m m m

m m m m

r m m m m

γ

γ γ γ γ

γ γ γ γ

γ γ γ γ

= + + +

+ − −

= + + +

0 0 0 0 0

1 1 0 0 1

2 0 0 2

cos cos

cos

t

s t

n n n

γ ε μ θ

γ ε μ θ

γ ε μ θ

=

=

=

11 12

1

21 22

1

cos sin

sin cos m m i

m m

i δ δ

γ

γ δ δ

⎡ ⎤

⎡ ⎤ ⎢ = ⎥

⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎢ ⎣ ⎥ ⎦

1 1

0

2 π n t cos

δ θ

λ

⎛ ⎞

= ⎜ ⎟

⎝ ⎠

Reflectance at Normal Incidence: Two-Layer Quarter-Wave Films

0 1

2

n n n

n =

1 3

0 2

s

For the quarter - wave film, zero reflectance occurs when :

n n n n

n =

Reflectance at Normal Incidence: Three-Layer AR Films

2 2

2 2

2 2 2

2 2 2

cos sin

:

cos sin

cos sin

:

c :

os sin

r

r

E n

TE r

E n

E

r reflection coefficient

n n

TM r

E n n

θ θ

θ θ

θ θ

θ θ

− −

= =

+ −

− −

= =

+ −

2 2 2

2 cos :

cos sin

2 cos :

cos sin

:

t

t

t trans TE t E

E n

E n

TM t E

mission coeffi e

n ci nt

n

θ

θ θ

θ

θ θ

= =

+ −

= =

+ −

n ₂ E θ

E _t

E _r θ _r

θ _t

n ₁

1 2

n n ≡ n

23. Fresnel Equations

2 2

cos

* cos cos

cos

*

t n

tt n

T

r rr R

i t i

t

⎟⎟ ⎠

⎜⎜ ⎞

⎝

= ⎛

⎟⎟ ⎠

⎜⎜ ⎞

⎝

= ⎛

=

=

θ θ θ

θ

Reflectance and Transmittance

1 2

1 2 1

1 2

2

:

1 :

1

Internal reflection External reflect

n n

n n i n n

n on

n n

n

< = >

⇒ =

>

⇒

<

external reflection internal

reflection

θ _p θ _p θ _c

R

1 :

sin θ

= n n <

or n n

p

n

tan

θ

Brewster ‘s angle :

Critical angle :

External and Internal Reflection

external reflection

TE

TM

π

π

r _TM r _TE

t

Phase changes on External Reflection

1

,

,

.

80 0

0 0

( ) the phase shift is

When r is a real number as it always

the p

is for external reflection

then an

for r

hase shift is for r

d ° = π <

° >

' p

'

p c

2 2

1

2 c

180 ( ) <

0 <

2 tan sin <

cos

TM

i

n

n

π θ θ

φ θ θ θ

θ θ θ

θ

−

⎧ ⎪

⎪ ⎪⎪

= ⎨ <

⎪ ⎛ − ⎞

⎪ ⎜ ⎟

⎪ ⎜ ⎟

⎪ ⎝ ⎠

⎩

D

D

c c

0 <

: 0 >

TM TE

φ φ θ θ

θ θ

− ⎧= ⎨ ⎩ >

D D

c

2 2

1

c

0 <

2 tan sin >

cos

TE i

n

θ θ

φ θ

θ θ θ

−

⎧ ⎪⎪ ⎛ ⎞

= ⎨ ⎪ ⎪ ⎩ ⎜ ⎜ ⎝ − ⎟ ⎟ ⎠

D

Phase changes on Internal Reflection

Æ Important for understanding optical waveguides

2 3

0 1 0 2 0 3 "

Polarization : P = ε χ E + ε χ E + ε χ E +

24. Nonlinear optics

Second-order Nonlinear optics

Second-harmonic generation (SHG) and rectification (0) ),

2 ( )

(

2

P P

P ω ± ω = ω

Electro-optic (EO) effect (Pockell’s effect)

→

∝

→

= E ( ω ) P

E

( ω )

E

{ ^{but, ( 0 ) ( )} }

)

( )

0

(

E ω E E ω

E

E =

+

>>

Three-wave mixing

2

2 0 2

P = ε χ E

optical

optical

E

E

E = ( ω

) + ( ω

)

{ } { }

{ }

{ ^{( ) ( )} }

) (

, ) ( ) ( )

(

, ) ( )

(2 , ) ( )

(2

2 1

2 1 2

2 1

2 1 2

2 2 2

2 1 2 1

2

2 2

ω ω

ω ω

ω ω

ω ω

ω ω

ω ω

E E

P

E E

P

E P

E P

E P

∝

−

∝ +

∝

∝

→

∝

→

Æ SHG

Æ Frequency up-converter

Æ Parametric amplifier, parametric oscillator Æ Index modulation by DC E-field

ÆFrequency doubling

Æ SHG does not occur in isotropic, centrosymmetry crystals

Third-order Nonlinear optics

Third-harmonic generation (THG)

{ ^{( ) ( )} } ^{, ( 3 ) { ( ) }}

)

(

3

ω ^E ω ^E ω ^P ω ^E ω

P ∝ ∝

Optical Kerr effect

→

∝

→

= E ( ω ) P

E

( ω )

E

3

3 0 3

P = ε χ E

Æ Self-phase modulation

Æ Frequency tripling

) ( )

( ) ( )

( ) ( )

(

3

ω E ω E ω I ω E ω n I ω

P ∝ ∝ → Δ ∝ Æ Index modulation by optical Intensity

) (

)

(

0

n I k nL

n

n = + Δ → ϕ = ϕ + Δ ϕ = Δ

{ } { }

0

n I ( x ) n I ( x ) n

n

n = + Δ → Δ > Æ Self-focusing, Self-guiding (Spatial solitons)

{ } { }

0

n I ( x ) n I ( x ) n

n

n = + Δ → Δ < Æ Self-defocusing

Electro-optic (EO) Kerr effect

{ ^{but, ( 0 ) ( )} }

)

( )

0

(

E ω E E ω

E

E =

+

>>

2

DC , 2

DC

3

( ) E ( 0 )

E ( ) n E ( 0 )

P ∝ → Δ ∝

→ ω ω Æ Index modulation by DC E

2 2

1 1 2

n o E

n rE R

= + +

Second-order nonlinearity (P

)

Æ Linear electro-optic coefficient (r) Æ Pockels effect (E: DC field)

third-order nonlinearity (P

)

Æ Quadratic electro-optic coefficient(R) Æ Kerr effect

Nonlinearity of the refractive index

Phase conjugation by four-wave mixing

The equation of motion of the oscillat

( ) ( )

ing electron

( )

,

2 2

G G

G JJJG

G G G G G G

r E x

d r d r

m F v F E C r m e E

t r t

d F

γ γ d

= + + = − − −

Classical Electron Oscillator (CEO) Model = Lorentz model

25. Optical properties of materials

Metal

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

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

( : relaxation time )

2

14 2

0

1 10

G G G

JJG G

G JJG

e

e e e

C

d r m d r d v

m e E m m v e E s

dt dt

dt C r γ τ

τ γ

=

= − − − ⇒ + = − = ≈

Lorentz model

(Harmonic oscillator model)

Drude model (free-electron model) If

C = 0

γ τ

ω >> = ¹

2 2

2 3

( ) 1

/

p p

ω i ω

ε ω ω ω γ

⎛ ⎞ ⎛ ⎞

= − ⎜ ⎜ ⎝ ⎟ ⎟ ⎠ + ⎜ ⎜ ⎝ ⎟ ⎟ ⎠

2

0 2

( ) ^τ _γ ( ) 1 ω ^p

ε ω ε ω

ω

→∞

→

⎛ ⎞

⎯⎯⎯ → = − ⎜ ⎜ ⎟ ⎟

⎝ ⎠

Metal

/ ( ) / : electrostatic field by small charge separation exp( ) : small-amplitude oscillation

( )

( )

2 2 2

2 2

2

2

s o o o

o p

s p p

o

p

o

o

E Ne x x

x x i t

d x Ne Ne

m e E

m t m

e

m N

d

σ ε δ ε δ

δ δ ω

δ ω ω

ε

ω ε

ε

= =

= −

= − ⇒ − = − ⇒ =

=

: Plasma Frequency

2 2

2

2 2

n 1

1

( )

i

ω ω

ω γ

ω ω γ ω

= − ≈ − >>

+

: is complex

: is real

and radiation is attenuated.

and radiation is not attenuated(transparent).

p

p

n

n ω ω

ω ω

<

>

Æ EM waves with lower frequencies are reflected/absorbed at metal surfaces.

Æ EM waves with higher frequencies can propagate through metals.

26. Lasers : A brief introduction

absorption spontaneous emission stimulated emission LASER: Light Amplification by Stimulated Emission of Radiation.

population inversion

pumping

Gaussian beam

One simple solution to the paraxial Helmholtz equation : paraboloidal waves

Another solution of the paraxial Helmholtz equation : Gaussian beams

A paraxial wave is a plane wave e

^-jkz

The complex envelope A(r) must satisfy the paraxial Helmholtz equation

Transmission of Gaussian beams through a Thin Lens

29. Selected Topics : Surface plasmons

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

Surface plasmon dispersion relation:

2 / 1

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎛

= +

d m

d m

x

c

k ε ε

ε ε ω

ω

ω _p

d p

ε ω

+ 1

Re k _x

real k

_x

_z

imaginary k

_x

_z

real k

_x

imaginary k

_z

d

ck x

ε

Bound modes Radiative modes

Quasi-bound modes

Dielectric:

ε_d

Metal: ε_m = ε_m^'+ ε_m^"

x z

(ε'

_m

(−ε

_d

_m

(ε'

_m

_d

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

⎛ ⎞

= ⎜ ⎝ + ⎟ ⎠

λ π /

= 2

λ

^~ λ Λ

<< λ

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.

Air band

Dielectric band Band Gap

periodic structures with photonic band gaps (PBG)

and their lattice constants are comparable to wavelength

0 π/a

ω

k

29. Selected Topics : Photonic crystals