chapter 6. polarization opticsoptics.hanyang.ac.kr/~choh/degree/[2014-1] photonics_graduated... ·...

$: Chapter 6. Polarization Opticsoptics.hanyang.ac.kr/~choh/degree/[2014-1] photonics_graduated... · Chapter 6. Polarization Optics 6.1 Polarization of light 6.2 Reflection and refraction$
Nonlinear Optics Lab. Hanyang Univ.

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


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”.


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


6. 1 Polarization of light

A. Polarization

Monochromatic plane wave traveling in the z direction :

where, : complex envelope

Polarization ellipse ,

where,


where, : phase difference

Parametric equation

xy


: 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 2222

yxyx aaAA


Linear polarized light

or0 (+: 0, −: )

Circularly polarized light

0and2 aaa/ yx

2/

2/

: right circular

: left circular


Poincare sphere and Stokes parameters

State of light polarization can be described by

1) Complex polarization ratio :

2) Poincare sphere : (𝜓, 𝜒) spherical coordinate 𝑟, 𝜃, 𝜙 = (1, 𝜋/2 − 2𝜒, 2𝜓)

),exp( jr where, 𝑟 = 𝑎𝑦/𝑎𝑥, 𝜑 = 𝜑𝑦 − 𝜑𝑥

Each point on

the sphere represents

a polarization state

But, no information about the intensity


3) Stokes vector : Set of four real numbers (𝑆0, 𝑆1, 𝑆2, 𝑆3) contain about the intensity

- 𝑆0 = 𝑎2𝑥 + 𝑎2

𝑦 : proportional to the intensity

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

𝑆0 = 𝑎2𝑥 + 𝑎2

𝑦 = 𝐴𝑥2 + 𝐴𝑦

2

𝑆1 = 𝑎2𝑥 − 𝑎2

𝑦 = 𝐴𝑥2 − 𝐴𝑦

2

𝑆2 = 2𝑎𝑥𝑎𝑦 cos 𝜑 = 2Re 𝐴𝑥∗ 𝐴𝑦 Stokes parameters (6.1-9)

𝑆3 = 2𝑎𝑥𝑎𝑦 sin 𝜑 = 2Im 𝐴𝑥∗ 𝐴𝑦

They satisfies the condition, 𝑆12 + 𝑆2

2 + 𝑆32 = 𝑆0

2

, and

Report


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


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.


Report) Matrix representations

Linear polarizers (horizontal)

Wave(Phase) retarders (fast axis along the x-direction)

- Quarter wave plate :

- Half wave plate :

i0

01

10

01

- Polarization rotators :

cossin

sin-cos

Cascaded polarization devices

… T1 TN T2 Ttot

11NNtot T...TTT


6. 2 Reflection and refraction

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

yx , rr : Reflection coefficients for the TE and TM polarizations.


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


Total internal reflection (n1>n2)

TE TM

c c

)(sin 12

1 n/nc

Critical angle:

Brewster angle (TM polarization)

external

B

B B

internal

C

)(tan 12

1 n/nB

Brewster angle:


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)


A. Refractive indexes

Permittivity tensor

333231

232221

131211

: Permittivity tensor

Geometrical representation of vectors and tensors

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

- Vectors (a tensor of 1st 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 2nd 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.


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) :

AAAΑ zyx 22A

: Quadric representation

In the principal coordinate system, ij is diagonal, and the ellipsoid is given by simple form,

1

1112

3

2

2

2

2

2

1

2

/

z

/

y

/

x


Principal axes and Principal refractive indexes

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

of ij 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


Biaxial, uniaxial, and isotropic crystals

1) Isotropic : 321 nnn

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

2) Uniaxial : 321 nnn

(1) Positive uniaxial : oe nn

ex) Ice, Quartz, ZnS, …

(2) Negative uniaxial :

ex) KDP, ADP, LiIO3, LiNbO3, BBO, …

)::( ordinary ary,extraordin 013 nnnn e Fast/Slow axis

3) Biaxial : 321 nnn

ex) LBO, Mica, NaNO2, …

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

oe nn


Impermeability tensor

: Impermeability tensor

Index ellipsoid

By the quadric representation of the impermeability tensor,

In the principal coordinate system,


C. Propagation in an arbitrary direction

u : propagation direction of a plane wave

ba n,n : refractive indexes of the two normal modes

Crystal behaves as a wave retarder with the refractive

indexes na, nb along the major and minor axes

of the index ellipse, respectively.


Special case: Uniaxial crystals (positive uniaxial)

12

2

3

2

0

2

2

2

1

en

x

n

xx

)sin,cos,0( ee nn

u

1x

2x

3x

)0,,0( 0n

)0,0,( 0n

),0,0( en

B

A

0

propagation direction


Intersection of the index ellipsoid

u

2x

3x

A

0

)(en

0n

2

2

2

3

2 xx)(ne

12

2

3

2

0

2

2 en

x

n

x

cos)(nx,sin)(nx ee 23

)(

1sincos22

2

2

0

2

ee nnn

Birefringence : |)(| 0nne

000 |)90(|,0|)0(| nnnnnn eee

(6.3-15)


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 (ne>no)

3x

2x

en0n

0n

2) negative uniaxial (ne<no)

0n

en

0n

3) biaxial ( )

yn

xnzn

321 nnn

3x3x

2x2x


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.

je

je jj

1

2

11

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:

)2sin(

)2cos(1

2

11

2

10

/

/e

jee

jee

jjjjj

where, 00 /)(2 ),(2

1 d-nn-

Rotation angle of the polarization : 0/)(2 d-nn/ -

sin

cos

(n+, n-)

d

2/


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

0

-nn-

(6.4-1)

- Dextrorotatory ( ) : clockwise rotation nn-

- Levorotatory ( ) : counter-clockwise rotation nn-

Optical active materials :

Se(selenium), Te(tellurium), TeO2, quartz (a-SiO2), HgS(cinnabar),

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

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

(Dextrorotatory case)


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 (1st 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 !!


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 ,

00000

0

0

0

0

0

2

0

2

00

0

2

2

2

0

3

2

1

jD

D

E)Gn(j

E)Gn(

jE

E

n

njG

jGn

D

D

D

where,

ED2

0 n

where, (6.4-7)


Rotatory power

2

000

1-)(

nn

G-nn-

0

1

kG

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.


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(yttrium iron garnet), TGG(terbium gallium garnet),

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


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

00

-

Rotatory power,

Verdet constant

(6.4-12)


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.


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, …


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 ne>no)


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

- zm = mDz, m = mD (m=1,2,…N, DaDz)

- Jones matrix :

where,

can be ignored because it is a constant phase factor


Overall Jones matrix:

(6.1-22)

For a<<b, R(D) identity matrix, and


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

2

dRee

edR

'A

'A/dj

/dj

/dj

y

xaa b

b

b

Phase shift : rotates the polarization angle by ad


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


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)


B. Wave retarders

Phase retardation : 0/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

])[( 20 En,,dfn G

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


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 45o angle with respect to each other. The magnetic flux density

applied to the rotator is adjusted so that it rotates the polarization by 45o in the direction

of right-handed screw pointing in the z direction.


- 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


Nonreciprocal polarization rotation

A combination of a 45o Faraday rotator followed by a half-wave retarder also can

perform a function of optical isolator.

chapter 6. polarization opticsoptics.hanyang.ac.kr/~choh/degree/[2014-1] photonics_graduated... ·...

Documents