Ch. 6. Polarization Optics

Ch. 6. Polarization Optics

Byoungho Lee

School of EE, Seoul National University byoungho@snu.ac.kr

Fall Semester, 2008

( ) ⎭ ⎬ ⎫

⎩ ⎨

⎧ ⎥⎦ ⎤

⎢⎣ ⎡

⎟ ⎠

⎜ ⎞

⎝ ⎛ −

= c

t z j

z,t Re A exp 2 πν

Ε

y x ˆ ˆ Α = A _x + A _y

( ) _{x y y} ( ) _y

x

x a j , A a j

A = exp ϕ = exp ϕ

x 축, y 축 두개의 직교 편광 성분으로 모든 파동방정 식의 해를 기술할 수 있다.

Polarization (I)

( ) ⎭ ⎬ ⎫

⎩ ⎨

⎧ ⎥⎦ ⎤

⎢⎣ ⎡

⎟ ⎠

⎜ ⎞

⎝ ⎛ −

−

= c

t z j

z,t Re A exp 2 πν

Ε

( _x ) _{y y} ( _y )

x

x a j , A a j

A = exp − ϕ = exp − ϕ

Different notation

Polarization (II)

( ) z,t = Ε x ^ˆ + Ε y ^ˆ

Ε x y

cos 2

x x x

E a t z

πν c ϕ

⎡ ⎛ ⎞ ⎤

= ⎢ ⎣ ⎜ ⎝ − ⎟ ⎠ + ⎥ ⎦

cos 2

y y y

E a t z

πν c ϕ

⎡ ⎛ ⎞ ⎤

= ⎢ ⎣ ⎜ ⎝ − ⎟ ⎠ + ⎥ ⎦

Polarization (III)

Ε ϕ ϕ Ε

Ε Ε

y x

y x 2

y 2 y 2

x 2 x

a 2 a

a

a + − cos = sin

x

y ϕ

ϕ ϕ = −

Polarization Ellipsoid

π ϕ = 0 or

Linear Polarization

2 ϕ = π : Polarized Circularly

Right

2 ϕ = − π :

Polarized Circularly

Left

Circular Polarization

다른 표기

a _x → a ₁ , a _y → a ₂ , ϕ = ϕ _y − ϕ _x → δ = δ ₂ − δ ₁

a b a

a = m

= χ

α , tan tan

1 2

( 2 : 장축의 길이, 2 : 단축의 길이 )

a b

Elliptical Polarization

x y x y x y

Comparison of Three Polarization Types

Types of Polarization

• 빛이 액정을 통과하면서 복굴절 현상으로 인해 빛의 진동축이 회전됨.

• LCD는 이러한 빛의 편광 변화를 통해 빛의 세기를 control하는 장치

polarizer

Liquid Crystal Display

ON OFF

LCD Cell

Polarization Filter

( )

( )

x

x

a j , A a j

A = exp ϕ = exp ϕ

⎟⎟ ⎠

⎜⎜ ⎞

⎝

= ⎛

y x

A J A

R. Clark Jones (1916-2004)

Jones Vector

Jones Vectors

⎟⎟ ⎠

⎞

⎜⎜ ⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝

= ⎛

⎟⎟ ⎠

⎞

⎜⎜ ⎝

⎛

y 1

x 1 22

21

12 11

y 2

x 2

A A T

T

T T

A A

1

2 TJ

J =

Jones Calculus

⎟⎟

⎠

⎜⎜ ⎞

⎝

= ⎛

0 0

0 T 1

Jones Matrix for Linear Polarizer

( )

exp ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛

= −

Γ j 0

0 T 1

retarder wave

- Quarter

: 2 Γ ₌ π

retarder wave

- Half

: Γ = π

Jones Matrix for Wave Retarders

cos sin

sin

cos ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛ −

=

θ θ

θ T θ

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝

⎛ −

⎟⎟ =

⎠

⎜⎜ ⎞

⎝

⎛

2 1 2

2

s

s

θ θ θ

θ

θ θ

θ θ

sin co cos

sin

sin cos

sin co

θ θ

θ ₂ = ₁ +

Jones Matrix for Polarization Rotators

( ) ^J

R J ' = θ

( ) _⎟⎟

⎠

⎜⎜ ⎞

⎝

= ⎛

θ θ

θ θ θ

cos sin

-

sin R cos

( ) ( ) ^{θ − θ}

= R TR T'

( ) ( ) ^{θ T R θ}

R

T = − '

Coordinate Transformation

2 2 2

1

0 a a

s = +

δ

2 cos

1

2 2 a a

s =

δ

2 sin

1

3 2 a a

s =

2 2 2

1

1 a a

s = −

2 3 2

2 2

1 2

0 s s s

s = + +

Stokes Parameters

⎟⎟

⎟ ⎟

⎟

⎠

⎞

⎜⎜

⎜ ⎜

⎜

⎝

⎛

3 2 1 0

S S S

S

S₁ = Power through LH polarizer – power through LV polarizer S₂ = Power through L+45 polarizer – power through L-45 polarizer S₃ = Power through RC polarizer – power through LC polarizer

2 3 2

2 2

1

S S

S

P

= + +

Stokes Parameters

(Polarized + Unpolarized) (I)

Stokes Parameters

(Polarized + Unpolarized) (II)

DOP

d unpolarize polarized

polarized

P P

P

= +

0

2 3 2

2 2

1

S

S S

S + +

= DOP

2 2 2

1 2 3

DOP = s + s + s

Degree of Polarization

Poincaré Sphere (I)

ψ χ 2 2

s

s ₁ = ₀ cos cos ψ χ 2 2

s

s ₂ = ₀ cos sin χ

2 s

s ₃ = ₀ sin

Poincaré Sphere (II)

반사와 굴절 (I)

TE polarization

TM polarization

( r

)

n t n

n n

n

r n = +

+

= − , 1

cos cos

cos cos

2 1 2

1 1

2

2 1

1

2

θ θ

θ θ

Reflection and refraction at the boundary between two dielectric media.

x x

x

t r

n n

n

r n = +

+

= − , 1

cos cos

cos cos

2 2

1 1

2 2

1

1

θ θ

θ θ

1

3

θ

θ =

반사와 굴절 (II)

반사와 굴절 (III)

Magnitude and phase of the reflection coefficient for internal

임계각 (Critical Angle)

⎟⎟ ⎠

⎜⎜ ⎞

⎝

= ⎛

>

−

1 1 2

2 1

sin For

n n n n

θ c

반사와 굴절 (IV)

Magnitude and phase of the reflection coefficient for external

Brewster Angle

⎟⎟ ⎠

⎜⎜ ⎞

⎝

= ⁻ ⎛

1 1 2

tan

waves, TM

For

material, c

nonmagneti In

n

B n

θ

반사와 굴절 (V)

Magnitude and phase of the reflection coefficient for internal