Chapter 6. Polarization Optics

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Chapter 6. Polarization Optics

6.1 Polarization of light

6.2 Reflection and refraction 6.3 Optics of anisotropic media

6.4 Optical activity and magneto-optics 6.5 Optics of liquid crystals

6.6 Polarization devices

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Polarization of light : determined by the time course of the electric-field vector, (r, t) - In general, complex-amplitude vector, E(r) traces an ellipse since two orthogonal components vary sinusoidally with time and have different amplitudes and phases in a plane tangential to the wavefront. And the plane, the orientation, and the shape of the ellipse also vary with position because the wavefront has different directions at different positions.

- For a plane wave, the polarization ellipses are the same everywhere, and therefore

the plane wave is described by a single ellipse, and is said to be “Elliptically polarized”.

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Polarization dependences :

- Reflectance at the boundary between two materials - Absorption coefficients of materials

- Scattering from matter

- Refractive index of anisotropic materials

 Measurement of optical properties of matter

 Manipulations of polarization state and transmittance of light

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6. 1 Polarization of light

A. Polarization

Monochromatic plane wave traveling in the z direction :

where, : complex envelope

Polarization ellipse

,



where,

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where, : phase difference

Parametric equation

x

y 



  

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 : determines the direction of the major axis

c : determines the ellipticity (the ratio of the minor to major axes of the ellipse b/a) Intensity of the wave  A_x ²  A_y ²  a_x² a²_y

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Linear polarized light



  0 or  (+: 0, −: )

Circularly polarized light

and 0

2 a a a

/ _x  _y 



 





2

 /

  

2

 /

  

: right circular

: left circular

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Poincare sphere and Stokes parameters

State of light polarization can be described by 1) Complex polarization ratio :

2) Poincare sphere : (𝜓, 𝜒)  spherical coordinate

), exp( j 

r

Each point on

the sphere represents

a polarization state

But, no information

about the intensity

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3) Stokes vector

- (𝑆₁, 𝑆₂, 𝑆₃) : Cartesian coordinates of the point on the Poincare sphere multiplied by 𝑆₀

𝑆₀ = 𝑎²_𝑥 + 𝑎²_𝑦 = 𝐴_𝑥 ² + 𝐴_𝑦 ² 𝑆₁ = 𝑎²_𝑥 − 𝑎²_𝑦 = 𝐴_𝑥 ² − 𝐴_𝑦 ²

𝑆₂ = 2𝑎_𝑥𝑎_𝑦 cos 𝜑 = 2Re 𝐴_𝑥^∗𝐴_𝑦 Stokes parameters (6.1-9) 𝑆₃ = 2𝑎_𝑥𝑎_𝑦 sin 𝜑 = 2Im 𝐴_𝑥^∗𝐴_𝑦

They satisfies the condition,

𝑆

+ 𝑆

+ 𝑆

= 𝑆

, and

Report

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B. Matrix representation

The Jones vector

: Complex envelopes of the E-field vector where,

(Put _x=0)

Report ) Exercise 6.1-1

: describes the polarization state

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Matrix representation of polarization devices

The system is assumed to be linear:

Principle of superposition of optical filed is obeyed.

A₁ A₂

Matrix form

Jones matrix : describes the optical system

Jones Calculus (1940, R.C. Jones) :

- The state of polarization is represented by a two-component Jones vector - Each optical element is represented by a 2 x 2 Jones matrix.

*) The overall Jones matrix for the whole system is obtained by multiplying all the individual element matrices.

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Report) Matrix representations

Linear polarizers (horizontal)

Wave(Phase) retarders (fast axis along the x-direction) - Quarter wave plate :

- Half wave plate :



 





i 0

0 1



 





1 0

0 1

- Polarization rotators : _



 













cos sin

sin - cos

Cascaded polarization devices

T

T

… T

T

1 1

N N

tot

T T ... T

T  

 

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6. 2 Reflection and refraction

y x, t

t : Transmission coefficients for the TE and TM polarizations, respectively;

y x, r

r : Reflection coefficients for the TE and TM polarizations.

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By applying the boundary conditions:

- Tangential components of the electric fields for TE case

- Tangential components of the magnetic fields for TM case Should be continuous

: Fresnel equations

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Total internal reflection (n

>n

)

TE TM

_c _c

) (

sin

n

/ n

c





Critical angle:

Brewster angle (TM polarization)

external

_B

_B

_B

internal

_C

) (

tan

n

/ n

B





Brewster angle:

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6. 3 Optics of anisotropic media

A medium is said to be anisotropic if its macroscopic optical properties depend on

direction.  Microscopic properties (the shape and orientation of the individual

molecules and the organization of their centers)

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A. Refractive indexes Permittivity tensor



















33 32

31

23 22

21

13 12

11



















 : Permittivity tensor

Geometrical representation of vectors and tensors

- Scalar (a tensor of 0^th rank) is described by a single number.

- Vectors (a tensor of 1^st rank) is represented by 3 numbers. The magnitude and direction of the vector are independent of the coordinate system although the components depend on the coordinate system.

- Tensors (a tensor of 2^nd rank) is a rule that relates two vectors, and represented by 9

numbers. The rule is independent of the coordinate system although the components depend on the coordinate system.

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The magnitude and direction of a vector are independent of the coordinate system :

The rule of a tensor independent of the coordinate system (for example dielectric tensor) :

A A

Α_x^  A_y²  _z² 

A

: Quadric representation In the principal coordinate system, _ij is diagonal, and the ellipsoid is given by simple form,



 

 



2 2

2 2 2

1

2   





 /

z /

y /

x

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Principal axes and Principal refractive indexes

A coordinate system can always be found for which the off-diagonal elements of 

vanish (diagonalization!). Then,

where, and,

The new axes 1,2,3 are defined as the principal axes :

if E points in the x direction, then so too must D.

: Principal refractive indexes

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Biaxial, uniaxial, and isotropic crystals

1) Isotropic : n

 n

 n

ex) CdTe, NaCl, Diamond, GaAs, Glass, …

2) Uniaxial : n

 n

 n

(1) Positive uniaxial :

n

 n

ex) Ice, Quartz, ZnS, … (2) Negative uniaxial :

ex) KDP, ADP, LiIO₃, LiNbO₃, BBO, …

) :

:

( n

 n

n

 n



3) Biaxial : n

 n

 n

ex) LBO, Mica, NaNO₂, …

- The z axis is called the optic axis. (The „c-axis‟ in solid state physics)

o

e

n

n 

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Impermeability tensor

: Impermeability tensor

Index ellipsoid

By the quadric representation of the impermeability tensor,

In the principal coordinate system,

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C. Propagation in an arbitrary direction

uˆ

b a

, n

n

Crystal behaves as a wave retarder with the refractive indexes n_a, n_b along the major and minor axes

of the index ellipse, respectively.

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Special case: Uniaxial crystals (positive uniaxial)

2

1

2 3 2

0 2 2 2

1

  

n

x n

x x

) sin ,

cos ,

0

(  n

 n



uˆ

x

x

x



) 0 , ,

0

(  n

) 0 , 0 , ( n

) , 0 , 0

( n

B A

0

propagation direction

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Intersection of the index ellipsoid

uˆ

x

x

 A

0



2 2 2

3

2

( ) x x

n

  

2

1

2 3 2

0 2

2

 

n

x n

x







 ) sin , x n ( ) cos (

n

x





) ( 1 sin

cos

2 2

2 2

0 2







e

e

n

n

n  



Birefringence : | n

(  )  n

|

0 0

0

| 0 , | ( 90 ) | )

0 (

| n

 n  n

 n  n

 n

(6.3-15)

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Normal index surface

: The surface in which the distance of a given point from the origin is equal to the index of refraction of a wave propagating along this direction.

1) Positive uniaxial (n

>n

)

x

x

n

n

n

2) negative uniaxial (n

<n

) n

n

n

3) biaxial ( ) n

n

n

3 2

1

n n

n   x

x

x

x

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6. 4 Optical activity and magneto-optics

A. Optical activity

Optical active medium has different refractive indexes (n

, n

) to the right- and

left-circular polarizations (circular birefringence), which acts as a polarization rotator.



 





 



 



 



 



 _

e j

e ^j j ^j 1

2 1 1 2

1 sin

cos _{ }





Circular representation for the incident linearly polarized wave:

After propagating a distance d through the medium, the phase shifts of the right and left circular polarized waves are

𝜑₊ = 2𝜋𝑛₊𝑑/𝜆₀, 𝜑₋ = 2𝜋𝑛₋𝑑/𝜆₀, resulting in a Jones vector:



 







 



 





 



 



 _{ }



 _{ }

) 2 sin(

) 2 cos(

1 2

1 1 2

1 ₀

/ e /

e j j e

e

e ^{j j j j j}







 

 

 

where, ₀ ( ), 2 ( ) / ₀

2

1       

  _  _  _  _  n_--n_ d Rotation angle of the polarization :

 / 2   ( n

-n

) d / 





 







 sin cos

(n⁺, n^-)

d

2

/

 

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Rotatory power (rotation angle per unit length) : ( )

0

 n_--n



  (6.4-1)

- Dextrorotatory ( ) : clockwise rotation n_-  n_

- Levorotatory ( ) : counter-clockwise rotation n_-  n_

Optical active materials :

chiral molecules [chitosan(^키토산)], amino acids (mostly levorotatory),

sugars [dextrose(^포도당) : dextrorotatory, levulose(fructose, ^과당) : levorotatory].

(Dextrorotatory case)

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Material equations

Time varying magnetic flux density B(t) induces a circulating current that set up an electric dipole moment proportional to 𝑗𝜔𝑩 = −𝛻 × 𝑬.

For a plane wave 𝑬 𝒓 = 𝑬 exp −𝑗𝒌 ∙ 𝒓 , −𝛻 × 𝑬 = −𝑗𝒌 × 𝑬

An optically active medium can be described by (1^st order approximation)

Linear Optical activity

where, 𝐆 = 𝜉𝒌 : gyration vector

𝜉 : pseudoscalar (changes sign depending on the handedness od the coordinate)

Dielectric permittivity tensor depends on the wave vector k !!

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Normal modes of the optically active medium

Wave propagating in the z direction, 𝒌 = (0,0, 𝑘) and thus 𝑮 = (0,0, 𝐺) (6.4-5) 

where, 𝑛² = 𝜀/𝜀₀ Consider two circularly polarized waves 𝐸 = 𝐸₀, ±𝑗𝐸₀, 0 ,













































































 



















0 0

0 0

0

0 0

0 0 0

2

0 2

0 0

0

2 2

2

0 3

2 1

jD D E

) G n

( j

E ) G n ( jE

E

n n

jG

jG n

D D D



 where,

E D  

n

where, (6.4-7)

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Rotatory power

2 0 0

0

- 1 )

(  





 

n n

-n G

n

 



0

1

 

 k G

Example)

- quartz : 31 deg/mm @ 500 nm, 22 deg/mm @ 600 nm

- AgGaS₂(silver thiogallate) : 700 deg/mm @ 490 nm, 500 deg/mm @ 500 nm.

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B. Magneto-Optics: The Faraday effect

Faraday effect : Polarization rotating effect as like polarization rotator in the presence of a “static” magnetic field.

Rotatory power :

(베르데 상수)

In contrast to optical activity, the sense of rotation does not reverse with the reversal of the direction of propagation of the wave

Twice the rotation, 2𝜑 !!

(applicable to optical isolator[6.6])

Faraday materials :

YIG

TbAlG(terbium aluminum garnet, ℬ ≈ −1.16 min/Oe − cm @ 500 nm)

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Material equations

In magneto-optic materials, static magnetic field interacts with the motion of electrons in the material in response to an optical electric field. This induces the changes in the electric permittivity tensor.

with,

 : magnetogyration coefficient

In contrast to optical activity, G does not depend on k but B !!

n B n

G

0 0

- 





    

Rotatory power,

Verdet constant

(6.4-12)

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6. 5 Optics of liquid crystals

LC comprises a collection of elongated molecules(typically cigar-shaped).

The molecules lack positional order(liquids) but possess orientational order(crystals).

Liquid crystals

- Nematic LC : The orientations tend to be the same, but the positions are totally random.

- Smectic LC : The orientations are the same, but the centers are stacked in parallel layers within which they have random position (positional order only in one dimension).

- Cholesteric LC : Distorted form of its nematic cousin in which the orientations undergo helical rotation about an axis.

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Molecules in LC are able to change orientation when subjected to a force (usually given by rubbing).

Twisted nematic LC :

Twist (exists naturally in the cholesteric LC) is externally imposed by placing a thin layer of nematic LC between

two glass plates that are polished in perpendicular directions.

Applications:

LC displays, Optical modulators and switches, LC lasers, …

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Optical properties of twisted nematic Liquid crystals

Each layer acts as a uniaxial crystal~

Assume that the twist angle varies linearly with z,

Phase retardation coefficient(retardation per unit length),

(Typically n_e>n_o)

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In practice, b>>a The phase retardation is much faster than the rotation of the optic axis).

Divide the cell width d into N incremental layers of equal width Dz=d/N. Then,

where,

m-th layer

- z_m = mDz, _m = mD (m=1,2,…N, DaDz) - Jones matrix :

where,

can be ignored because it is a constant phase factor

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Overall Jones matrix:

(6.1-22)

For a<<b, R(D) identity matrix, and

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Finally,

: wave retarder with retardation bd, followed by a polarization rotator with rotation angle ad.

Ex) Input wave is linearly polarized along the x direction,

 

 



 

 



 



 

 



 









1 ) 0 1 (

0 0

) 0

(

2 2

d R

e e d e

' R A

'

A

/ d j /

d j

y

x

a

a

Phase shift

: rotates the polarization angle by ad

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6. 6 Polarization devices

A. Polarizers

Polarization by selective absorption (Dichroism)

- Polaroid H-sheet : Iodine-impregnated polyvinyl alcohol sheet that is heated and stretched.

- Wire-grid polarizer : Closely spaced fine wires stretched in a single direction (IR region).

: Polarization dependent absorption

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Polarization by selective reflection

At the Brewster angle of incidence, the reflectance of TM-polarized wave vanishes so that it is totally refracted(transmitted).

_B TM(external)

 Reflector serves as a (TE) polarizer

 Brewster window serves as a TM-polarization selector in laser cavity

Polarization by selective refraction (polarizing beamsplitters)

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B. Wave retarders

Phase retardation :



/

 d

Ex) mica (biaxial) : Dn=0.005 (1.599-1.594) @ 633 nm  G/d~15.8 rad/mm

Thickness of half-wave plate (G=) : 63.3 mm (~hair diameter!!)

Transmittance :

Light intensity control via wave retarder and two polarizers

]) [ ( d ,

, n

E fn 

 G

- Thickness monitor, frequency(wavelength) filter, electro-optics modulator

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C. Polarization rotators

D. Nonreciprocal Polarization devices

Reciprocal devices : A devices whose effect on the polarization is invariant to reversal of the direction of propagation.

 The polarization state of round tripped wave through a reciprocal device is the very same to the polarization state of initial wave. Most dielectric devices are reciprocal, with the exception of the Faraday rotator (nonreciprocal).

Optical isolator (optical diode)

Useful for preventing reflected light from returning back to the source (optical feedback).

Such optical feedback can have deleterious effects on the operation of certain devices, such as semiconductor lasers.

An optical isolator is constructed by placing a Faraday rotator between two polarizers whose axes make a 45^o angle with respect to each other. The magnetic flux density

applied to the rotator is adjusted so that it rotates the polarization by 45^o in the direction of right-handed screw pointing in the z direction.

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- Isolator composed of YIG(yttrium iron garnet), TGG(terbium gallium garnet) offer attenuation of backward wave of up to 90 dB, over a relatively wide

wavelength range.

- Compact optical isolator : Thin film, Fiber

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Nonreciprocal polarization rotation

A combination of a 45^o Faraday rotator followed by a half-wave retarder also can perform a function of optical isolator.