P6-Polarization and Crystal Optics

7/27/2019 P6-Polarization and Crystal Optics

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6. Polarization and crystal Optics

6. Polarization and crystal Optics


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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/

!!!


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Polarization ellipse


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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


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Sense of rotation : Time-dependent rotation of

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

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 :


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Quick estimation of a wave polarization state

A wave polarization is completely defined by two parameters derived from the polarization ellipse

- its orientation,

- its ellipticity with sign() indicating the sense of rotation

Three cases may be discriminated from the knowledge of

the polarization is linear since = 0 the orientation angle is given by

the polarization is circular, since = /4

the sense of rotation is given by sign().

If0 the polarization is left circular.

If0 the polarization is left elliptic.


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Jones vector


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Jones vector

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

This expression may be further developed


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Jones vectors for linear polarizations


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Jones vectors for circular/elliptical polarizations


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Jones vectors


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Jones matrix


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Coordinate transform of Jones vector/matrix

x

y

x

y

The Jones vector is given by

cos sin' ( )sin cos

J R J J

= =

The Jones matrix T is similarly t ransformed into T

' ( ) ( )

( ) ' ( )

T R T R

T R T R

=

= (6.1-23) !!


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Poincare sphere and Stokes parameters

A characterization method of the wave polarization by power measurements

if we consider the Pauli group of matrices

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

where the parameters {g0, g1, g2, g3} receive the name ofStokes parameters.

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


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Representation of Stokes vectors: The Poincar sphere

g1

g2

g3

2

2


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Th St k t f th i l l i ti t t


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The Stokes vectors for the canonical polarization states


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6.2 Reflection and Refraction6.2 Reflection and Refraction

TE pol.

TME pol.


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Development of the Fresnel Equations

cos co

' ,

s co

:

s

i r t

i i r r t t

E E EB B B

From Maxwell s EM field theorywe 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

+ =

=

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

+ =

+ =

TE-case

TM-case


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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

cRecall that E B Bn

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

EB

= = =

=

=

=

=

=

+

+

+

2r tE n E=

TE-case

TM-case

n1

n2

n1

n2



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2

1

cos cos:

cos cos

cos cos:

co

:

sin sin

cos 1

s cos

i tr

i i t

i tr

i

t

i

t

t

i

t

Eliminating E

nn

n

n

n n

nETE case r

E n

nETM case r

E

from each set of equations

and solving for the reflection coefficient we obtain

where

We know that

n

= = +

=

=

=

=

+

=

2

2 2 2

2

sinsin 1 sini

t in n

n

= =

TE-case

TM-case

n1

n2

n1

n2



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TE-case

TM-case

n1

n2

n1

n2

2 2

2 2

2 2 2

2 2 2

cos sin:

cos sin

cos sin:

cos

:

:

s

in

i ir

i i i

i ir

i i i

transmission coefficient t

TE

reflection coefficienSubst

c

ts r

nETE case r

E n

n nETM case r

E n

as

ituting we obtain the Fresnel equations for

For the

e

n

= =

+

= =+

2 2

2 2 2

2cos:

cos sin

2 co

:

s:

1

: 1

cos sin

t i

i i i

t i

i i i

Et

E n

E nTM case t

E n

TE t r

T nt r

n

M

=

= =+

= =+

+

= +

1

2

n

nn

These mean just the boundary conditions



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TIR

TIR


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TIRTIR

P R fl t (R) d T itt (T)

Power : Reflectance(R) and Transmittance(T)


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

.

1

,,

:

:

tr

i i

i r t

R and T are the ratios of reflected and transmit

The quantities

The ratios

respectively to

ted powers

PPR 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 00

:

cos cos

cos cos cos

1

cos

1c

22

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

=

=

+

=

+

= = =

+

= 2 21 0 0 2 0 0

2 2 2 2

0 2 0 0

2 220 0

2 2

0 0

0

2 2 2 2

0 1 0 0 0

1 1os cos cos

2 2

cos cos1

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 ER r T n

n R TE

E

n E E

nE

E

=

= +

= + = + = +

= = =

2t 2

2

cos

cos*

cos

cos

*

tnttnT

rrrR

i

t

i

t

=

=

==


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6.3 Optics of anisotropic media6.3 Optics of anisotropic media

6 3 Optics of anisotropic media

6 3 Optics of anisotropic media


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6.3 Optics of anisotropic media6.3 Optics of anisotropic media


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0

2

1i

i in

=: for principal axes

Impermeabili ty tensor

*Note, impedance


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, for example,


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Determination of two normal modes (with refractive indices na and nb)

An index ellipse is defined.


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E

S

Lets start with


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For uniaxial case

k-surface obtained from dispersion relation


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k1

k2

k2

k3

k1

k3

k1

k2

k2

k1

k3

k3

Optic axisOptic axisOptic axis

k1

k2

k2

k1

k3

k3

Determine the wavenumbers k and indices of two normal modes


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u

Determine the direction of polarization of two normal modes

Z


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k

D. Rays, wavefronts, and energy transport

D. Rays, wavefronts, and energy transport


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k surface

Equi-frequency surface

k


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E. Double refraction = Birefringence

E. Double refraction = Birefringence


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g

AIR


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6.4 Optical activity and faraday effect

6.4 Optical activity and faraday effect


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6.5 Optics of liquid crystals

6.5 Optics of liquid crystals


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Principles of LCD Optics


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Principles of LCD Optics

Operation of TN LCD

VLc = 0V (off) VLc = 5V (on)

0

15

30

45

60

75

90

0 0.2 0.4 0.6 0.8 1normalized depth

director

[deg.

] a (0V)

a (5V)

a (8V)

b (0V)

b (5V)

b (8V)


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TNLC as a polarization rotator


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(6.1-23)


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6.6 Polarization devices

6.6 Polarization devices


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P6-Polarization and Crystal Optics