Review of Basic Polarization

Optics for LCDsModule 4

Review of Basic Polarization

Optics for LCDsModule 4

1

0

1 11 1

12 2

iV

i i

( ) Re Re xi ti tx x xE t J e A e

Module 4 GoalsModule 4 Goals

• Polarization• Jones Vectors• Stokes Vectors• Poincare Sphere• Adiabadic Waveguiding

Objective: Model the polarization of light through an LCD.

Assumptions: • Linearity – this allows us to treat the

transmission of light independent of wavelength(or color).

• We can treat each angle of incidence independently.

Transmission is reduced to a linear superposition of the transmission of monochromatic (single wavelength) plane waves through LCD assembly.

Polarization of Optical WavesPolarization of Optical Waves

A monochromatic plane wave propagating in isotropic and homogenous medium:

( ) ( )cos( )t t tE A k r

= angular frequency

k = wave vector

A = constant amplitude vector

Monochromatic Plane Wave (I)Monochromatic Plane Wave (I)

k is related to frequency2

k n nc

n = index of refractionc = speed of light

= wavelength in vacuum

For transparent materials Re, ( )n n n Dispersion relation

• The E-field direction is always to the direction of propagation

0k E

exp ( )i tE A k r

• Complex notation for plane wave: (Real part represents actual E-field)

• Consider propagation along Z-axis, E-field vector is in X-Y plane:

cos( )x x xE A t kz

cos( )y y yE A t kz

,x yA A independent amplitudes

,x y two independent phases ,x y

Monochromatic Plane Wave (II)Monochromatic Plane Wave (II)

Y-axis

X-axis

Ex

EY

• There is no loss of generality in this case.

• Finally, we define the relative phase as

y x

• Now in a position to look at three specific cases.

1. Linear Polarization2. Circular Polarization3. Elliptical Polarization

Monochromatic Plane Wave (III)Monochromatic Plane Wave (III)

• Occurs when 0y x

or y x

Linear polarized or plane polarized are used interchangeably

Linear PolarizationLinear Polarization

• In this case, the E-field vectorfollows a linear pattern in the X-Y plane as either time orposition vary.

• Important parameters:1. Orientation2. Handedness3. Extent

Y-axis

X-axisAx

AY

tan y

x

A

A

2 22

x yE E A

2

(-) CCW rotation = RH, (+) CW rotation = LH

Circular PolarizationCircular Polarization

• In this case, the E-field vectorfollows a circular rotation in the X-Y plane as either time orposition vary.

x yA A

and2y x

• Occurs when

• Important parameters:1. Orientation2. Handedness3. Extent

Y-axis

X-axis

Ax

AY

Circular PolarizationCircular Polarization

2 2 2 2

2 2 2 2

2 2 2 2

cos( )

cos( / 2)

cos ( ) cos ( / 2)

cos ( ) sin ( )

(cos ( ) sin ( ))

x

y

x y

E A t kz

E A t kz

E E A t kz A t kz

A t kz A t kz

A t kz t kz A

Equation of a circle

• This is the most generalrepresentation of polarization. The E-field vector follows an elliptical rotation in the X-Y plane as either time or position vary.

be

a

Elliptic Polarization StatesElliptic Polarization States

Y-axis

X-axisAx

AY

• Important parameters:1. Orientation2. Handedness3. Extent of Ellipticity

y x • Occurs for all values of

a

b

2 2

2tan2 cosx y

x Y

A A

A A

Elliptic Polarization StatesElliptic Polarization States

cos( )

cos( )x

y

E A t kz

E A t kz

eliminate t22

2cos2 sinyx

x yx y x y

EEE E

A A A A

X-axisAx

ab

x’y’

22

1yxEE

a b

2 22 2 2

2 22 2 2

2 2

cos sin 2 cos cos sin

sin cos 2 cos cos sin

2tan2 cos

x y x y

x y x y

x y

x y

a A A A A

b A A A A

A A

A A

Transformation:

=3/4 =/2 =/4 =0

=/4 =/2 =3/4 =

=3/4 =/2 =/4 =0

=/4 =/2 =3/4 =

Review Complex NumbersReview Complex Numbers

-2+2i

3-4i

Im

Re

= 3 – 4i

= ei = cos + i.sin

= e-i = cos (-) + i.sin (-) = cos - i.sin

Remember the identities:

ex ey = ex+y

ex / ey = ex-y

d/dz ez = ez

Polarization can be described by an amplitude and phase anglesof the X-Y components of the electric field vector. This lendsitself to representation with complex numbers:

( )tan y xiyi

x

Ae e

A

Complex Number RepresentationComplex Number Representation

Im

Re

0

(cos0 sin(0))

x y

y y

x x

linear

A Ai

A A

on x axis

/ 2

,2

cos sin2 2

y x

i

circular

A A

e i i

on y (imaginary axis)

Convenient way to uniquely describe polarization state of aplane wave,using complex amplitudes as a column vector.

xix

i yy

A eJ

A e

��������������

J��������������

is not a vector in real space, it is a mathematical abstraction incomplex space.

( ) Re exp Re expx x x xE t J i t A i t

Jones Vectoramplitudephases

electric field

Polarization is uniquely specified

Jones Vector RepresentationJones Vector Representation

Jones Vector Representation (II)Jones Vector Representation (II)If you are only interested in polarization state, it is most convenient to normalize it.

1 J J

A linear polarized beam with electric field vector oscillating alonga given direction can be represented as:

cos

sinJ

sin

cosJ

For orthogonal state, 1

2

Jones Vector Representation (III)Jones Vector Representation (III)

Normalize Jones Vector

x

y

ix

iy

A eJ

A e

* *

* * 2 2

* ( , )

* | | | | 1

yx iix y

x x y y x y

J A e A e

J J A A A A A A

Take x

y

ix

iy

A eJ

A e

2 2 2 2

1

| | | | | | | |

cos

sin

xx

y

iix x

iiyyx y x y

i

A e AeA eA eA A A A

e

When =0 for linear polarized light, the electric field oscillates along coordinate system, the Jones Vectors are given by:

cos 1 sin 0

sin 0 cos 1x y

For circular polarized light:/ 2

/ 2

cos 11

sin 2

cos 11

sin 2

i

i

Re i

Le i

Mutually orthogonal condition 0R L

Jones Vector Representation (IV)Jones Vector Representation (IV)The Jones matrix of rank 2, any pair of orthogonal Jones vectors can be used as a basis for the mathematical space spanned by all the Jones vectors.

Polarization Ellipse Jones Vector () () Stokes

1

0

0

1

11

12

11

12

0,0 0,0

0, / 2 / 2,0

0, / 4 / 4,0

, / 4 / 4,0

1

1

0

0

1

1

0

0

1

0

1

0

1

0

1

0

Polarization RepresentationPolarization Representation

Polarization Ellipse Jones Vector () () Stokes

21

5 i

0, / 4 / 2, / 4

1

0

0

1

11

25 i

11

2 i

11

2 i

21

210

i

i

0, / 4 / 2, / 4

1/ 2, tan 2 1 1/ 2, tan2

1 1/ 2, tan2

1 10, tan2

1 1/ 4, tan2

1 4tan , / 43

1

3/ 5

0

4 / 5

1

3/ 5

0

4 / 5

1

0

3/ 5

4 / 5

1

0

0

1

Polarization RepresentationPolarization Representation

Jones is powerful for studying the propagation of plane waveswith arbitrary states of polarization through an arbitrary sequence of birefringent elements and polarizers.

Limitations:• Applies to normal incidence or paraxial rays only• Neglects Fresnel refraction and surface reflections• Deficient polarizer modeling• Only models polarized light

Other Methods:• 4x4 Method – exact solutions

(models refraction and multiple reflections)• 2x2 Extended Jones Matrix Method

(relaxes multiple reflections for greater simplicity)

Jones Matrix LimitationsJones Matrix Limitations

We discussed monochromatic/polarization thus far.

If light is not absolutely monochromatic, the amplitude and relative phase between x and y components can vary with time, and the electric field vector will first vibrate in one ellipse and then in another. The polarization state of a polychromatic wave is constantly changing.

If polarization state changes faster than speed of observation,the light is partially polarized or unpolarized.

Optics – light of oscillation frequencies 1014s-1

Whereas polarization may change 10-s (depending on source)

Partially Polarized & Unpolarized LightPartially Polarized & Unpolarized Light

Consider quasi monochromatic waves (<<)

Light can still be described as: ( )expt i tE A k rProvided the constancy condition of A is relaxed. denotes center frequency

A denotes complex amplitude

Because (<< ), changes in A(t) are small in a time interval1/ (slowly varying).

If the time constant of the detector d>1/, A(t) can change originally in a time interval d.

Partially Polarized & Unpolarized LightPartially Polarized & Unpolarized Light

To describe this type of polarization state, must consider time averaged quantities.

S0 = <<Ax2+Ay

2>>

S1 = <<Ax2-Ay

2>>

S2 = 2<<AxAy cos>>

S3 = 2<<AxAy sin>>

Ax, Ay, and are time dependent<< >> denotes averages over time interval d that is the characteristic time constant of the detection process.

These are STOKES parameters.

Partially Polarized & Unpolarized LightPartially Polarized & Unpolarized Light

Note: All four Stokes Parameters have the same dimension of intensity.

They satisfy the relation:

2 2 2 20 1 2 3S S S S

the equality sign holds only for polarized light.

Stokes ParametersStokes Parameters

Example: Unpolarized light

No preference between Ax and Ay (Ax=Ay), random

S0 = <<Ax2+Ay

2>>=2<<Ax2>>

S1= <<Ax2-Ay

2>>=S2,3=2<<AxAy cos>>=2<<AxAy sin>>=

since is a random functionof time

if S0 is normalized to 1, the Stokes vector parameter isfor unpolarized light.

1

0

0

0

Example: Horizontal Polarized Light Ay=0, Ax=1

S0=<<Ax2>>=1

S1=<<Ax2>>=1

S2,3=2<<AxAy cos>>=2<<AxAy sin>>=

1

1

0

0

Stokes ParametersStokes Parameters

Example: Vertically polarized light Ay=1, Ax=0

S0 = <<Ax2+Ay

2>>=<<Ay2>>=1

S1 = <<Ax2-Ay

2>>=<<-Ay2>>=-1

S2,3 = 2<<AxAy cos>>=2<<AxAy sin>>=

1

1

0

0

Example: Right handed circular polarized light (=-1/2) Ax=Ay

1

0

0

1

S2 = 2<<AxAy cos>> =

S0 = <<Ax2+Ay

2>> = 2<<Ax2>>

S1 = <<Ax2-Ay

2>> =

S3 = 2<<AxAy sin>> = -1

Stokes ParametersStokes Parameters

Example: Left handed circular polarized light (=1/2) Ax=Ay

S2 = 2<<AxAy cos>> =

S0 = <<Ax2+Ay

2>> = 2<<Ax2>>

S1 = <<Ax2-Ay

2>> =

S3 = 2<<AxAy sin>> = 1

Degree of polarization:

12 2 2 2

1 2 3

0

S S S

S0 1

Unpolarized S12 = S2

2 = S32 =

Polarized S12+S2

2+S32 = 1

useful for describing partially polarized light

Stokes ParametersStokes Parameters

1

0

0

1

Jones Matrix Method (I)Jones Matrix Method (I)

Z-axis

Y-axiss

X-axis

f

• The polarization state in a fixed lab axis X and Y:

x

y

VV

V

• Decomposed into fast and slowcoordinate transform:

cos sin

sin cosx xs

y yf

V VVR

V VV

(notation: fast (f) and slow (s) component of the polarization state)

rotation matrix

• If ns and nf are the refractive indices associated with the pro-pagation of slow and fast components, the emerging beam has the polarization state: 2

exp 0

20 exp

ss s

ff f

in dV V

VV in d

Where d is the thickness and isthe wavelength

• For a “simple” retardation film, the following phase changes occur:

2

s fn n d

1 2

2 s fn n d

/ 2

/ 2

2exp 0

20 exp

2exp 0

2 2

20 exp

2 2

0

0

ss s

ff f

s f s f

s

s f s f f

isi

i

in dV V

VV in d

n n n ni d

V

n n n n Vi d

Vee

e V

f

(relative phase retardation) (mean absolute phase change)

Jones Matrix Method (II)Jones Matrix Method (II)

• Rewriting previous retardation equation:

Jones Matrix Method (III)Jones Matrix Method (III)

•The Jones vector of the polarization state of the emerging beam in the X-Y coordinate system is given by transforming back to the S-F coordinate system.

cos sin

sin cosx s

y f

V V

V V

• By combining equations, the transformation due to the retarder plate is:

0x x

yy

V VR W R

VV

where W0 is the Jones matrix for the retarder plate and R() is thecoordinate rotation matrix.

cos sin

sin cosR

/ 2

0 / 2

0

0

ii

i

eW e

e

(The absolute phase can often be neglected if multiple reflections can be ignored)

A retardation plate is characterized by its phase retardation and its azimuth angle , and is represented by:

0W R W R

Jones Matrix Method (IV)Jones Matrix Method (IV)

Polarizer with transmissionaxis oriented to X-axis

0

1 0

0 0iP e

’ is due to finite optical thicknessof polarizer.

If polarizer is rotated by about 0P R P R

ignoring ’ polarizers transmittinglight with electric field vectors tox and y are:

1 0

0 0xP

0 0

0 1yP

Jones Vector

cos

sin

11

2 i

11

2 i

cos sin

sin cos

a ib

a ib

cos sin

sin cos

a ib

a ib

Y-axis

X-axis

E

ab

ab

Polarization State

ExamplesExamples

¼ Wave Plate

2

and the thickness

4t

n

045 and and incident beam is

vertically polarized:

exp 01 1 1 141 1

1 1 1 12 20 exp

4

11

12

i

Wi

iW

i

1

0

1 11 1

12 2

iV

i i

IncidentJones Vector

1

0

11

2 i

Y-axis

X-axis

E

Polarization State

ExamplesExamples

cos

sin

Emerging Jones Vector

cos cos sin

s

11

in sin co

1

1 2 s2

i

i

iV

i

11 11

1 02

iV

i i

y

xc-axis

c-axis

c-axis

c-axis

450

Jones MatricesWave Plates

2

e on n d

/ 2

/ 2

0

0

i

i

eW

e

/ 2

/ 2

0

0

i

i

eW

e

cos sin2 2

sin cos2 2

iW

i

/ 2

/ 2

0

0

i

i

eW R R

e

/ 2

/ 2

cos sin cos sin0

sin cos sin cos0

i

i

eW

e

Remember:

Ingeneral:

y

x

450

transmissionaxis

1 0

0 0W

0 0

0 1W

1/ 2 1/ 2

1/ 2 1/ 2W

2

2

cos sin 1 0 cos sin

sin cos 0 0 sin cos

cos cos sin

sin cos sin

W

Jones MatricesPolarizers

transmissionaxis

transmissionaxis

transmissionaxis

Ingeneral:

1 0

0 0W R R

Remember:

Birefringent PlatesBirefringent Plates

45 45

Parallel polarizers Cross polarizers

2 2

cos sin1 0 0 sin12 2' 20 0 12 2 0sin cos

2 2

1 1 ( )sin sin2 2 2

e o

ii

E

i

n n dI

2 2

cos sin0 0 0 cos1 12 2' 20 1 12 2 0sin cos

2 2

1 1 ( )cos cos2 2 2

e o

iE

i

n n dI

Poincare’s Representatives Method

Poincare’ Sphere:Poincare’ Sphere:

Linear Polarization States

Linear Polarization States

Poincare’ Sphere:Poincare’ Sphere:

Elliptic Polarization States

Elliptic Polarization States

Polarization Conversion:Polarization Conversion:

Polarization Conversion:Polarization Conversion:

Z-axis

Y-axis

s

X-axis

f

Some Examples

TN LCD Formulations

General Matrix For LCDGeneral Matrix For LCD

e – component || director

o – component director

sin sincos

2sin sin

cos2

e e

oo

X XX iV VX X

X X VV X iX X

22

2X

Twist angle Phase retardation

• Consider light polarized parallel to the slow axis of a twisted LC twisted structure:

1 mode

0e

o

VE

V

• Then, the output polarization will be:

sincos

2sin

e

o

XX iV X

XVX

22with

2X

Adiabatic WaveguidingAdiabatic Waveguiding

90° Twist90° Twist

• Notice that for TN displays since << (twist angle muchsmaller than retardation ):

• Then the outputpolarization reduces to:

/ 2sin

cos2

sin 0

ie

o

XX iV eX

XVX

which means that the electric field vector “follows” the nematicdirector as beam propagates through medium – it rotates –

Adiabatic WaveguidingAdiabatic Waveguiding

2 0.23 202

18.40.5

mnd

m

• Consider twisted structure between a pair of parallel polarizersand consider e-mode operation.

• The transmission after the second polarizer:

2 2

2

sin 11

2 1

uT

u

2

2 2

u nd

90º Twisted Nematic (Normal Black)

90º Twisted Nematic (Normal Black)

sin sincos

0 0 1 0 11 20 1 sin sin 0 0 12 cos

2

e

o

X XX iV X X

X XV X iX X

e-mode input

Transmission of Normal BlackTransmission of Normal Black

0

0.1

0.2

0.3

0.4

0.5

0 2 4 6 8 10 12 14u

T (

%)

2d nu

first minimum

3u 15

35

second minimum

third minimum

• Consider twisted structure between a pair of parallel polarizersand consider e-mode operation.

• The transmission after the second polarizer:

2 2

2

sin 11 1

2 2 1

uT

u

2

2 2

u nd

sin sincos

1 0 1 0 11 20 0 sin sin 0 0 12 cos

2

e

o

X XX iV X X

X XV X iX X

e-mode input

Normal White Mode (I)Normal White Mode (I)

0

0.1

0.2

0.3

0.4

0.5

0 2 4 6 8 10 12 14u

T (

%)

2d nu

3u 1535

Normal White Mode (II)Normal White Mode (II)

En

(n)

n n

n n

n n n

n n

E

E E

E E E

E E E

E

0z /10z d

6 /10z d

5 /10z d4 /10z d3 /10z d

2 /10z d

9 /10z d

8 /10z d7 /10z d

z d

Y-axis

X-axis

z

d

A

F B D

C

1n

1n

e

o

Phase Retardation at Oblique

Incidence: Complicating Matters

Phase Retardation at Oblique

Incidence: Complicating Matters

Vital to understanding LCD’s and their viewing angle solutions:

• Linear, circular, elliptical polarization

• Jones Vector

• Stokes Parameters

• Jones Matrixes

• Adiabatic Waveguiding

• Extended Jones and 4x4 Methods

Summary of OpticsSummary of Optics