director [deg.]

6. Polarization and crystal Optics

6. Polarization and crystal Optics

Spatial evolution of a plane wave vector: helicoidal trajectory

http://sar.kangwon.ac.kr/polsar/Tutorial/Part1_RadarPolarimetry/1_What_Is_Polarization.pdf

The electric field may be represented in an orthonormal basis (x, y, z) defined so that the direction of propagation in z-axis.

http://sar.kangwon.ac.kr/polsar/Tutorial/Part1_RadarPolarimetry/

증명!!!

Polarization ellipse

Polarization ellipse

The polarization ellipse shape may be characterized using 3 parameters :

- A is called the ellipse amplitude and is determined from the ellipse axis as

is the ellipse orientation and is defined as the angle between the ellipse major axis and x.

Is the ellipse aperture, also called ellipticity, defined as

τ

Sense of rotation : Time-dependent rotation of

The sense of rotation may then be related to the sign of the variable τ

By convention, the sense of rotation is determined while looking in the direction of propagation.

Right hand rotation :

Left hand rotation :

Left hand rotation : Right hand rotation :

Quick estimation of a wave polarization state

A wave polarization is completely defined by two parametersderived from the polarization ellipse

- its orientation,

- its ellipticity

^{with sign(τ}) indicating the sense of rotation

Three cases may be discriminated from the knowledge of Æ the polarization is linear since τ = 0

Æ the orientation angle is given by

Æ the polarization is circular, since τ = ±π/4 Æ the sense of rotation is given by sign(δ).

Æ If δ<0, the polarization is right circular, whereas for δ>0 the polarization is left circular.

Æ If δ<0, the polarization is right elliptic, whereas for δ>0 the polarization is left elliptic.

Jones vector

τ

Jones vector

A Jones vector can be formulated as a two-dimensional complex vector function of the polarization ellipse characteristics :

τ

This expression may be further developed

Jones vectors for linear polarizations

Jones vectors for circular/elliptical polarizations

Jones vectors

Æ Jones matrix

Coordinate transform of Jones vector/matrix

x y

x’

y’

θ

The Jones vector is given by

cos sin ' ( )

sin cos

J R J θ θ J

θ θ θ

⎡ ⎤

= = ⎢ ⎣ − ⎥ ⎦

The Jones matrix T is similarly transformed into T’

' ( ) ( )

( ) ' ( )

T R T R T R T R

θ θ

θ θ

= −

= − ^{(6.1-23) 증명!!}

Poincare sphere and Stokes parameters

Æ A characterization method of the wave polarization by power measurements

if we consider the Pauli group of matrices

Given the Jones vector E of a given wave, we can create the hermitian product as follows

where the parameters {g

₀

, g

₁

, g

₂

, g

₃

} receive the name of Stokes parameters.

http://sar.kangwon.ac.kr/polsar/Tutorial/Part1_RadarPolarimetry/

Representation of Stokes vectors: The Poincaré sphere

g ₁ g ₂

g ₃

2τ 2φ

The Stokes vectors for the canonical polarization states

6.2 Reflection and Refraction 6.2 Reflection and Refraction

TE pol.

TME pol.

Development of the Fresnel Equations

cos co

' ,

s co

:

s

i r t

i i r r t t

E E E

B B B

From Maxwell s EM field theory

we have the boundary conditions at the interface

Th tangential

components of both E and B are equal on both sides o

e above co

f the i

nditions imply that th for the T

e E case

θ θ θ

+ =

− =

G G

0

cos cos

. ,

c :

os .

i i r r t t

i t

i r t

We have also assumed that as is true for most dielectric materia

nterface

E E E

B B B

For the TM mode

ls

μ μ μ

θ θ θ

− + = −

+ =

≅ ≅

TE-case

TM-case

1

1 1

1 2

2

1

:

cos cos

:

cos cos c

v

s c

o os

i i r

i r t

i

r t t

i

i r r t t

Recall that E B c B n

Let n refractive index of incident medium n refractive index of refracting me

For the TM m

diu

ode

E

For the TE mode

E E E

n E n E

E E

n

n

E n

n

c

m

E

B E

θ θ θ

θ θ θ

⎛ ⎞ =

= = ⎜ ⇒

−

⎝ ⎠ ⎟

=

=

=

=

=

−

− +

+

+

2

r t

E = n E

TE-case

TM-case n

₁

n

₂

n

₁

n

₂

Development of the Fresnel Equations

2 1

cos cos

: cos cos

cos cos

: co

:

sin sin

cos 1

s cos

i t

r

i i t

i t

r i t

i

t

t

i

t

Eliminating E

n n n

n

n n

n TE case r E

E n

E n TM case r

E

from each set of equations

and solving for the reflection coefficient we obtain

where

We know that

n

θ θ

θ θ

θ θ

θ θ

θ θ

θ

= = −

+

= −

=

=

=

+

=

−

²

sin

²₂ ^{2 2}

sin

_t

n 1

ⁱ

n sin

_i

n

θ = − θ = − θ

TE-case

TM-case n

₁

n

₂

n

₁

n

₂

Development of the Fresnel Equations

TE-case

TM-case n

₁

n

₂

n

₁

n

₂

2 2

2 2

2 2 2

2 2 2

cos sin

:

cos sin

cos sin

:

cos

:

:

s in

i i

r

i i i

i i

r

i i i

transmission coefficient t TE

reflection coefficien Subst

c

ts r E n

TE case r

E n

n n

TM case r E

E n

as

ituting we obtain the Fresnel equations for

For the

e

n

θ θ

θ θ

θ θ

θ θ

− −

= =

+ −

− −

= =

+ −

2 2

2 2 2

2 cos :

cos sin

2 co

:

: s

1

: 1

cos sin

t i

i i i

t i

i i i

t E

E n

E n

TM case t

E n

TE t r

T nt r

n

M

θ

θ θ

θ

θ θ

=

= =

+ −

= =

+

+

= +

−

1 2

n n ≡ n

These mean just the boundary conditions

Development of the Fresnel Equations

TIR

TIR

TIR TIR

Power : Reflectance(R) and Transmittance(T) Power : Reflectance(R) and Transmittance(T)

.

1

, ,

: :

t r

i i

i r t

R and T are the ratios of reflected and transmit The quantities

The ratios respectively to

ted powers

P

R P T

P P

R T

r and t are ratios of electric field amplitudes

From conservation of ener

the incident power

P P P

We can

gy

= =

= +

= + ⇒

2 1 0 2

0 0 0

:

cos cos

cos cos cos

1

cos

1 c

2 2

i i i r r r t t t

i i r r

i i r r t

i i r r t

i t

t

t t

i

express the power in each of the fie

n terms of the product of an irradiance and area

P I A P I A P I A

I

lds

I A I A I A

But n c

I I

I n c

I A I I A

E A

E

θ θ θ

ε

θ θ θ

ε

=

⇒

= +

=

+

⇒

= = =

+

=

_{1 0 0}² _{2 0 0}²

2 2 2 2

0 2 0 0

2 2

0 2 0

2 2

0 0

0

2 2 2 2

0 1 0 0 0

1 1

os cos cos

2 2

cos cos

1

cos

cos cos

cos

co

s

s

co

i

r t t t

i i

r r t t

r t t r t t

i i i i

i

i i

i

n cE n cE

E n E E E

E E

R r T n

n R T

E

E

n E E

E n

E

θ ε θ ε θ

θ θ

θ

θ θ

θ θ

θ

⎛ ⎞ ⎛ ⎞

=

= +

⎛ ⎞

⇒ = + = + ⎜ ⎟ = +

⎝ ⎠

= = ⎜ ⎟ = ⎜

⎝ ⎝

⇒ ⎠ ⎠

t

2

⎟

²

2

cos

* cos cos

cos

*

t n

tt n

T

r rr R

i t i

t

⎟⎟

⎠

⎜⎜ ⎞

⎝

= ⎛

⎟⎟ ⎠

⎜⎜ ⎞

⎝

= ⎛

=

=

θ θ θ

θ

6.3 Optics of anisotropic media

6.3 Optics of anisotropic media

6.3 Optics of anisotropic media

6.3 Optics of anisotropic media

0

2

1

i

i n i

η ε

≡ ε = : for principal axes

Impermeability tensor

*Note, impedance

, for example,

Determination of two normal modes (with refractive indices n

_a

and n

_b

)

An index ellipse is defined.

E

S

Let’s start with

For uniaxial case

k₁ k₂

k₂ k₃

k₁ k₃

k₁ k₂

k₂ k₁ k₃

k₃

Optic axis

Optic axis Optic axis

k₁ k₂

k₂ k₁ k₃

k₃

k-surface obtained from dispersion relation

Determine the wavenumbers k and indices of two normal modes

u

Determine the direction of polarization of two normal modes

θ θ Z

k

D. Rays, wavefronts, and energy transport D. Rays, wavefronts, and energy transport

k surface

Equi-frequency surface

k

E. Double refraction = Birefringence E. Double refraction = Birefringence

AIR

6.4 Optical activity and faraday effect

6.4 Optical activity and faraday effect

6.5 Optics of liquid crystals

6.5 Optics of liquid crystals

Principles of LCD Optics

‰ Operation of TN LCD

V

_Lc

= 0V (off) V

_Lc

= 5V (on)

0 15 30 45 60 75 90

0 0.2 0.4 0.6 0.8 1

normalized depth

director [deg.] a (0V)

a (5V) a (8V) b (0V) b (5V) b (8V)

TNLC as a polarization rotator

(6.1-23)

6.6 Polarization devices

6.6 Polarization devices