Summary

Introduction to Optics, 3rd Edition, Pedrottis.

1 학기 2 학기

- Photonic crystals
- Surface plasmons
 14. Matrix treatment of polarization

Consider a EM wave propagating along the z-direction.

The complex amplitude is

Two-element matrix

 ⎡a ⎤
Eo = ⎢ ⎥ , a + b =1
2 2
Jones vector
⎣b ⎦
 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.

a
a

A A

Object Beam a( x, y ) = a( x, y ) exp[− jφ ( x, y )]

Reference Beam A( x, y ) = A( x, y ) exp[− jψ ( x, y )]

I ( x, y ) = A( x, y ) + a( x, y ) + 2 A( x, y ) a( x, y ) cos[ψ ( x, y ) − φ ( x, y )]
2 2
Interference

U 3 ( x, y ) = β ' A a ( x, y )
2
reconstruction
 Double exposure
holographic interferometry

Detour-phase hologram
Real time
(computer-generated
hologram : CGH) holographic interferometry
 17. Optical detectors and displays

평판 디스플레이(FPD)

LCD PDP FED OLED

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
Radiometry Photometry

Radiant flux : watt (W) lumen (lm) : Luminous flux

Irradiance : W/m2 lux (lx) : illuminance
Radiant intensity : W/sr candela (cd) : luminous intensity
Radiance : W/(sr.m2) Cd/m2 : luminance

555 nm Radiant flux of 1 Watt at 555 nm

is
the luminous flux of 685 lm (lumen)

Luminous efficiency V(λ)

610 nm 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
Color temperature

Planck’s blackbody radiation

Wien displacement law

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
 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

Spatial frequency
X ⎛ 2π ⎞ X
kX = k =
r0 ⎜⎝ λ ⎟⎠ r0
Y ⎛ 2π ⎞ Y
kY = k =
r0 ⎜⎝ λ ⎟⎠ r0
Object in Time Object in Space

f ( t ) = ∫ g (ω )e − iω t d ω Object f ( x ) = ∫ g ( k )e − ikx dk

g (ω ) = ∫ f ( t )e iω t dt Spectrum g ( k ) = ∫ f ( x )e ikx dx

ω = 2π f t Angular Frequency k = 2π f S wave number

2π
k=
1 ⎡1⎤ 1 ⎡1⎤ λ
ft = Frequency fS =
t ⎢⎣ s ⎥⎦ λ ⎢m⎥
⎣ ⎦

Temporal Frequency Spatial Frequency

Fourier and Inverse Fourier Transformation by Lenses

F ( fx , f y )

αβ
β

F ( fx , f y )
 22. Theory of Multilayer Films

2γ 0 Tangential components
t= of E and B-fields are continuous
γ 0 m11 + γ 0 γ s m12 + m21 + γ s m22 across the interface.
γ 0 m11 + γ 0 γ s m12 − m21 − γ s m22 (a) (b)
r=
γ 0 m11 + γ 0 γ s m12 + m21 + γ s m22

⎡ i sin δ ⎤
⎡ m11 m12 ⎤ ⎢ cos δ
⎢m = γ1 ⎥
⎣ 21 m22 ⎥⎦ ⎢ ⎥
⎢⎣γ 1 i sin δ cos δ ⎥⎦

γ 0 = n0 ε 0 μ0 cos θ 0
γ 1 = n1 ε 0 μ0 cos θt1
γ s = n2 ε 0 μ0 cos θt 2

⎛ 2π ⎞
δ=⎜ ⎟ n1 t cos θt1
λ
⎝ 0 ⎠
Reflectance at Normal Incidence: Two-Layer Quarter-Wave Films

n2 ns
=
n1 n0
 Reflectance at Normal Incidence: Three-Layer AR Films

For the quarter - wave film,

zero reflectance occurs when :
n1 n3
= n0 ns
n2
 23. Fresnel Equations
r : reflection coefficient
Er cos θ − n2 − sin 2 θ
TE : r = = E Er
E cos θ + n2 − sin 2 θ
θ θr
Er n2 cos θ − n2 − sin 2 θ
TM : r = =
E n2 cos θ + n2 − sin 2 θ
n1
t : transmission coefficient
n2
E 2cos θ
TE : t = t =
E cos θ + n2 − sin 2 θ
E 2n cos θ
TM : t = t =
E n2 cos θ + n2 − sin 2 θ θt
Et
Reflectance and Transmittance
n2
n≡
R = rr* = r
2
n1
⎛ cos θ t ⎞ ⎛ cos θ t ⎞ 2
T = ⎜⎜ n ⎟⎟tt* = ⎜⎜ n ⎟⎟ t
⎝ cos θ i ⎠ ⎝ cos θ i ⎠
External and Internal Reflection
θc
n2
External reflection : n1 < n2 ⇒ n = >1
n1
n θp θp
Internal reflection : n1 > n2 ⇒ n = 2 <1
n1
R

Brewster ‘s angle : tan θ p = n : n > 1 or n < 1 external

internal
reflection
reflection
Critical angle : sin θ c = n :n < 1
Phase changes on External Reflection

tTE,TM external
When r is a real number , reflection
as it always is for external reflection, rTM

then the phase shift is 0° for r > 0, rTE

and the phase shift is 180°(= π ) for r < 0.

π
TE π
TM
 Phase changes on Internal Reflection

⎧
⎪ 180 D
(π ) θ < θ '
p
⎪
⎪⎪
φTM = ⎨ 0D θ p' < θ <θ c
⎪
⎪ ⎛ sin 2 θ − n 2 ⎞
⎪ 2 tan −1
⎜ i
⎟ θ <θ c
⎜ n cos θ ⎟
2
⎩⎪ ⎝ ⎠

⎧ 0D θ <θ c
⎪⎪
φTE =⎨ ⎛ sin 2 θ − n 2 ⎞
⎪ 2 tan ⎜⎜ ⎟ θ >θ c
−1 i

cos θ ⎟
⎪⎩ ⎝ ⎠

⎧ = 0D θ <θ c
φTM − φTE : ⎨ D
⎩> 0 θ >θ c

Æ Important for understanding optical waveguides

 24. Nonlinear optics

Polarization : P = ε 0 χ1 E + ε 0 χ 2 E 2 + ε 0 χ 3 E 3 + "
Second-order Nonlinear optics P2 = ε 0 χ 2 E 2
Second-harmonic generation (SHG) and rectification
E = E (ω ) optical → P2 ∝ E 2 (ω ) → P2 (ω ± ω ) = P2 ( 2ω ), P2 (0) ÆFrequency doubling
Æ SHG does not occur in isotropic, centrosymmetry crystals

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

E = E (0) electrical , DC + E (ω ) optical {but, E (0) >> E (ω ) } Æ Index modulation by DC E-field

Three-wave mixing

E = E (ω1 ) optical + E (ω2 ) optical

→ P2 ∝ E 2
{ } {
→ P2 (2ω1 ) ∝ E 2 (ω1 ) , P2 (2ω2 ) ∝ E 2 (ω2 ) , } Æ SHG
P2 (ω1 + ω2 ){∝ E (ω1 ) E (ω2 )},
P2 (ω1 − ω2 ){∝ E (ω1 ) E (ω2 )} Æ Frequency up-converter

Æ Parametric amplifier, parametric oscillator

Third-order Nonlinear optics P3 = ε 0 χ 3 E 3
Third-harmonic generation (THG)
{ }
E = E (ω ) optical → P3 ∝ E 3 (ω ) → P3 (ω ) ∝ E (ω ) 2 E (ω ) , P3 (3ω ){∝ E 3 (ω )}
Æ Frequency tripling

Electro-optic (EO) Kerr effect

E = E (0) electrical , DC + E (ω ) optical {but, E (0) >> E (ω ) }
→ P3 (ω ) ∝ E (0) electric , DC E (ω ) → Δn ∝ E (0) electric , DC
2 2
Æ Index modulation by DC E2

Optical Kerr effect

P3 (ω ) ∝ E (ω ) E (ω ) ∝ I (ω ) E (ω ) → Δn ∝ I (ω )
2
Æ Index modulation by optical Intensity

n = n0 + Δn( I ) → ϕ = ϕ 0 + Δϕ ( = k 0 ΔnL) Æ Self-phase modulation

n = n0 + Δn{I ( x)} → Δn{I ( x)} > n0 Æ Self-focusing, Self-guiding (Spatial solitons)

n = n0 + Δn{I ( x )} → Δn{I ( x )} < n0 Æ Self-defocusing

Nonlinearity of the refractive index

1 1
= + rE + R E 2
n 2 no2

Second-order nonlinearity (P2) third-order nonlinearity (P3)

Æ Linear electro-optic coefficient (r) Æ Quadratic electro-optic coefficient(R)

Æ Pockels effect (E: DC field) Æ Kerr effect

Phase conjugation by four-wave mixing
 25. Optical properties of materials

Classical Electron Oscillator (CEO) Model = Lorentz model

The equation of motion of the oscillating electron,

G G
d r G
2 G G G G G G dr JJJG
m 2 = Fr (r ) + Fγ (v ) + FE ( E ) = −C r − mγ − e Ex
dt dt
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; C = 0),
G G G
2
d r G m e dr
JJ
G dv G JJG 1
me 2 = − C r − − eE ⇒ me + meγ v = − eE (τ = : relaxation time ≈ 10−14 s)
dt τ dt dt γ
If
Lorentz model C=0 Drude model
(Harmonic oscillator model) (free-electron model)

1 ⎛ ω p2 ⎞ ⎛ ω p2 ⎞
ω >> γ = ε (ω ) = ⎜⎜1 − 2 ⎟⎟ + i ⎜⎜ 3 ⎟⎟
τ ⎝ ω ⎠ ⎝ω /γ ⎠

τ →∞
⎛ ω 2
⎞
ε (ω ) ⎯⎯⎯
γ →0
→ ε (ω ) = ⎜1 − 2 ⎟
⎜ ω ⎟
p

⎝ ⎠
Metal : Plasma Frequency

E s = σ o / ε o = Ne (δ x ) / ε o : electrostatic field by small charge separation δ x

δ x = δ xo exp(− iω p t ) : small-amplitude oscillation
d 2 (δ x ) Ne 2 Ne 2
m = (− e ) E s ⇒ − mω 2p =− ⇒ ω 2p =
dt 2
εo mε o
Ne 2
ωp =
mε o

ω p2 ω p2
n = 1− 2
2
≈ 1 − 2 (ω >> γ )
ω + iω γ ω

ω < ω p : n is complex and radiation is attenuated.

Æ EM waves with lower frequencies are reflected/absorbed at metal surfaces.
ω > ω p : n is real and radiation is not attenuated(transparent).
Æ EM waves with higher frequencies can propagate through metals.
 26. Lasers : A brief introduction
LASER: Light Amplification by Stimulated Emission of Radiation.

absorption spontaneous emission stimulated emission

population inversion
pumping
 Gaussian beam
A paraxial wave is a plane wave e-jkz modulated by a complex envelope A(r)
that is a slowly varying function of position:

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

One simple solution to the paraxial Helmholtz equation : paraboloidal waves

Another solution of the paraxial Helmholtz equation : Gaussian beams

Transmission of Gaussian beams through a Thin Lens
 29. Selected Topics : Surface plasmons

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

 Surface plasmon
Surface plasmon dispersion relation:
dispersion relation
1/ 2 1/ 2
ω⎛ ε ε ⎞ ω⎛ ε i2 ⎞
k x = ⎜⎜ m d ⎟⎟ k zi = ⎜ ⎟
c ⎝ εm + εd c ⎝ εm + εd ⎠
⎠
ω ω 2 = ω p2 + c 2 k x2 ck x
Radiative modes real kx
εd (ε'm > 0) real kz
ωp
Quasi-bound modes imaginary kx
(−εd < ε'm < 0) real kz
ωp
1+ εd
Dielectric: Bound modes real kx
εd
z imaginary kz
x (ε'm < −εd)
Metal: εm = εm' +
εm"

λx~λ Λx<<λ Re kx = 2π / 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.
 29. Selected Topics : Photonic crystals

periodic structures with photonic band gaps (PBG)

and their lattice constants are comparable to wavelength
ω

Air band

Band Gap

Dielectric band

k
0
π/a