ECE 5616Curtis

Polarization of Light

• Polarization states• Stokes Vectors• Mueller Matrices• Jones Matrices• Components• Examples

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Polarization statesLinear states of polarization

x

y

x

y

0 ,0 =≠ yx EELinear x polarized

0 ,0 ≠= yx EELinear y polarized

πφφ =−= xyyx EE , Linear -45o polarized

πφφ ,0 , =−≠ xyyx EELinear θ polarized

O’Shea 1.6, Saleh & Teich 6

x

y

x

y

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Polarization statesElliptical states of polarization

Right-hand circular 2 , πφφ =−= xyyx EE

Left-hand circular 2 , πφφ −=−= xyyx EE

Elliptical

2

and/or

,

πφφ ±≠−

≠

xy

yx EE

x

y

x

y

x

y

x

y

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Stokes vectorsComplete description of polarization state

Perform 6 irradiance measurements with ideal polarizers:Ex Horizontal linearEy Vertical linearE45 45o linearE135 135o linearER Right circularEL Left circular

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−+

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

LR

VH

VH

EEEEEEEE

SSSS

S13545

3

2

1

0

r

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Stokes vectorsComplete description of polarization state

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

0011

H

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡−

=

00

11

V

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

0101

45

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−=

01

01

135

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

1001

R

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

=

1001

L

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

6.8.01

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

6.8.

01

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Poincarè sphereUseful visualization of Stokes parameters

• Surface is polarized, center is unpolarized• Equator is linear polarized• North hemisphere is right elliptical, south is left elliptical• Orthogonal polarizations are on opposite points of sphere

332211 jijijiji SSSSSSSS ++=⋅

S1

S2

S3

R

L

H

V

45

135

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Stokes vectorsDegree of polarization

0

23

22

21

SSSS

DoP++

= Degree of polarization

0

3

SSDoCP = Degree of circular polarization

Degree of linear polarization0

22

21

SSS

DoLP+

=

Stokes vectors are polychromatic on addition:

- 1 - 0 . 5 0 . 5 1

- 1

- 0 . 5

0 . 5

1

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡−

+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0002

00

11

0011 H + V at different

frequencies give Lissagous figure with no

average polarization state

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡−

+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0013

00

11

0022

- 2 - 1 1 2

- 1

- 0 . 5

0 . 5

1

2H + V = H polarized DoP = 1/3

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Mueller MatricsComplete polarization modeling

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0011

0011

21

21

21

21

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡−

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

0000

00

11

21

21

21

21

E.g. Linear polarizer

4x4 matrix of real values describing transformation of polarizations.Can describe de-polarizing elements.Eigenvectors are eigenpolarization of system (unchanged).

Moral: We are going to use Jones vectors/matrices to do designs, but you should know their limitations. Jones vectors deal only with systems with perfect temporal and spatial coherence. Systems with finite coherence can be partially polarized and then you must use Mueller matrices and Stokes parameters.

ECE 5616Curtis

Jones vectorsSimplified for fully polarized systems

⎥⎦

⎤⎢⎣

⎡≡

y

x

AA

Jr where Ax and Ay are the complex amplitudes of the x and y

polarized electric fields.

*21

*2121 yyxx AAAAJJ +≡⋅ Inner product

⎥⎦

⎤⎢⎣

⎡≡

01

H ⎥⎦

⎤⎢⎣

⎡ −≡

11

V

⎥⎦

⎤⎢⎣

⎡≡

11

2145 ⎥

⎦

⎤⎢⎣

⎡−

≡1

12

1135

⎥⎦

⎤⎢⎣

⎡≡

jR

12

1⎥⎦

⎤⎢⎣

⎡−

≡j

L1

21

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Table of Jones and Mueller matrices

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Rotation matricesComponents not aligned with x or y

( )JRJrr

θ=′

( ) ⎥⎦

⎤⎢⎣

⎡ −=

θθθθ

θcossin

sincosR

( ) ( )θθ −=′ RTRT

Coordinate Transform Matrix

Transform for Jones Vectors

Transform for Jones Matrices

ECE 5616Curtis

Three types of physics1. Diattenuation (polarization dependent loss)

– Transmission is polarization dependent– “Polarizers”– A.k.a.: polarization dependent loss (PDL), dichroism

2. Retardance– Optical path length is polarization dependent.– “Wave plates”, optical activity, electro-optic– A.k.a.: polarization mode dispersion (PMD)– Poincarè sphere geometry

3. Depolarization– The degree of polarization may decrease depending on input

polarization– A.k.a.: polarization scrambling

ECE 5616Curtis

Polarizers⎥⎦

⎤⎢⎣

⎡≡

0001

xP Jones matrix of linear polarizer passing H

( ) ⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡⎥⎦

⎤⎢⎣

⎡==

0cos

sincos

0001 θ

θθ

θLPJ xOut

Power transmission of analyzer and arbitrary linear polarization:

θ2cos=⋅= OutOut JJT Malus’ Law

Dichroism: polarization dependent absorption

⎥⎦

⎤⎢⎣

⎡≡

−

−

z

z

x ee

P

00

β

αE.g.: Sheet polarizer, polarizing sunglasses,

PolarcorTM, wire-grid

Crystal polarizersReflective polarizers: Brewster’s angle, thin-film coatings

E.g.: Wollaston, Rochon, Sénarmont E.g.: Glan-Focault

ECE 5616Curtis

Jones matrix exampleCascaded polarizers

Crossed polarizers:

0 0 1 0 0 00 1 0 0 0 0

⎡ ⎤ ⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦ ⎣ ⎦ ⎣ ⎦y xA A

x

y z

1 0y xE E= A A

so no light leaks through.

Rotation tolerance

( )0 0 1 0 00 1 0 0

εε

ε ε⎡ ⎤ ⎡ ⎤ ⎡ ⎤

= =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

y xA A

( )00 0

0x x

y y x

E EE E E

εεε

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤= =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥

⎣ ⎦ ⎣ ⎦⎣ ⎦ ⎣ ⎦y xA A So Iout ≈ ε2 Iin,x

0E1E

x-pol

y-pol

0E 1E

rotatedx-pol

y-pol

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Retarders⎥⎦

⎤⎢⎣

⎡≡ Γ− je

R0

01 Jones matrix of retarder with fast axis in x

242 ππ

==Γ Quarter-wave plate. Converts linear into circular.

ππ==Γ

22 Half-wave plate.

Converts linear into linear.

( ) ⎭⎬⎫

⎩⎨⎧

+= 2

2

2

2

22

SinC,11

eoo nnos

nnθθ

θ

( )

n

oe

o

o

L

Lnn

Δ=

=−=Γ

2

2

λ

λπ π

E.g. Half-wave plate.

Quartz QWP is only 48 microns thick at 1550 nm.Crossing plates of differing dispersion can reduce λ, θ dependence

ECE 5616Curtis

Power walk-offConsider a Fourier propagation problem in which the k-surface is a tilted line:

Fourier propagator

Shift theorem

x

z

kr

- 2 - 1.5 - 1 - 0.5 0.5 1 1.5 2nt

- 0.5

0.5

1

1.5

2nn sr

kx

kz

θ

( ) ( ){ }

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

=−

−

0,

0,, 11

zdkdkxE

exEFFzxE

x

z

zkdkdkj

xx

xx

z

Poynting vector is normal to k surface

ECE 5616Curtis

Beam displacing polarizer

Separates polarizations with very high isolation (limited only by crystal scatter). Emerging polarizations are parallel to a very high degree since the crystal can be flat to an arc-second.

L

Real space

ks

θ

Fourier space

α

Wave in crystal is going “sideways”

S

ECE 5616Curtis

Beam displacing polarizer

( )2

sincos

arctan

2221

2

2

2

2eo

eomaxBDe

o

emaxBD

nnnn

n

nn

+=⎥

⎦

⎤⎢⎣

⎡+=

⎟⎟⎠

⎞⎜⎜⎝

⎛=

−θθθ

θ Crystal orientation for maximum walk-off

Phase velocity at this propagation angle

Differential optical path length

1045.09447.12

1486.29447.1

2

22

22

=⎟⎟⎠

⎞⎜⎜⎝

⎛−

+=

⎟⎟⎠

⎞⎜⎜⎝

⎛−

+=Δ o

eo nnnn

YVO4 at 1.55 μm

θθα −⎥⎦

⎤⎢⎣

⎡= tanarctan 2

2

e

o

nn Walk-off angle vs. crystal cut angle

ECE 5616Curtis

Poincarè and retarders• Retarders rotate the polarization state on the Poincarè sphere.• Axis of rotation connects eigenpolarizations of the retarder.

• Example1: QWP with horizontal axis converts 45o linear into RHC• Example2: Optically active (or Faraday) rotator converts H to 45o

S1

S2

S3

R

L

H

V

45

135

ECE 5616Curtis

Useful combinationsQWP

@-45o

QWP@45o

Pol@0o

Circular polarizer

HWP@θ

Pol@0o

Adjustable linear polarizer

Pol@0o

Adjustable transmission

HWP@θ

Babinet-Soleil adjustable retarder

ECE 5616Curtis

Optical activityRetarders with circular eigenstates

⎥⎦

⎤⎢⎣

⎡−

≡22

22

cossinsincos

δδ

δδ

OA

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Wollaston• Wollaston polarizer is made of two birefringent material prisms that are

cemented together. The deviations of the ordinary and extraordinary beams are nearly symmetrical about the input beam axis. The separation angle exhibits chromatic dispersion, as shown in the blow. Any separation angle can be designed upon the requirement. Typical separation angle vs wavelength is shown in the plot below for Calcite.

Extinsion Ratio: 5x10 -6Beam Separation Angle:16.7-22.5 at 980nm

ECE 5616Curtis

Faraday Rotator• A Faraday rotator is an optical device that rotates the polarization of light due to

the Faraday effect, which in turn is based on a magneto-optic effect.• The Faraday rotator works because one polarization of the input light is in

ferromagnetic resonance with the material which causes its phase velocity to be higher than the other.

• The plane of linearly polarized light is rotated when a magnetic field is applied parallel to the propagation direction (permanent magnets). The empirical angle of rotation is given by:

• Where β is the angle of rotation (in radians).B is the magnetic flux density in the direction of propagation (in teslas). d is the length of the path (in metres) where the light and magnetic field interact.Then V is the Verdet constant for the material. This empirical proportionality constant (in units of radians per tesla per metre, rad/(T·m)) varies with wavelength and temperature and is tabulated for various materials.

• Note that direction of rotation given by sign of B Rotation is independent of propagation direction (NOT like waveplate)

• Faraday rotators are used in optical isolators to prevent undesired back propagation of light from disrupting or damaging an optical system.

VBd=β

ECE 5616Curtis

Polarization Dependent Isolator• The polarization dependent isolator, or Faraday isolator, is made of

three parts, an input polarizer (polarized vertically), a Faraday rotator, and an output polarizer, called an analyzer (polarized at 45 degrees)

• Light traveling in the forward direction becomes polarized vertically by the input polarizer. The Faraday rotator will rotate the polarization by 45 degrees. The analyser then enables the light to be transmitted through the isolator.

• Light traveling in the backward direction becomes polarized at 45 degrees by the analyzer. The Faraday rotator will again rotate the polarization by 45 degrees. This means the light is polarized horizontally (the rotation is insensitive to direction of propagation). Since the polarizer is vertically aligned, the light will be extinguished

ECE 5616Curtis

Nonreciprocal opticsaka Faraday rotators

HWP @ 22.5o 45o FRWollaston Wollaston

Example: Polarization independent isolator

Similar optics can form a pseudo-circulator:

in

out

FBG

ECE 5616Curtis

Design notation for polarization manipulation

HWP @ 22.5o 45o FRWollaston Wollaston

View of polarization as if looking into laser. This is the EE convention and yields RHC with thumb pointing into viewers eyes, along with laser. To use the Physics convention, consider view as with laser.

Angle of propagation

Position in system

FORWARD

REVERSE

In Out

OutIn

Collimator Collimator

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Polarization Independent Isolator

ECE 5616Curtis

Another exampleSpatial phase-shifting interferometer

PBS HWP @ 22.5O Vertical beam displacer

Combine Rotate Interfere

0

π

Hei

ght o

f pro

paga

tion

Many thanks to Bill Chang for teaching me this notation.

ECE 5616Curtis

PBS Example

Newport Catalogue

2 kW/cm2 CW, 1 J/cm2 with 10 nsec pulses, typicalDamage Threshold

Non-abrasive method, acetone or isopropyl alcoholon lens tissue recommended (see Care and Cleaning of

Optics)Cemented optic, do not immerse in a solvent

Cleaning

MIL-C-675CDurability

-50 °C–90 °CTemperature Range

Broadband, multilayer coating, Ravg <1.0% per surfaceAntireflection Coating

±0.25 mmDimensions Tolerance

0° ±5°Angle of Incidence

90° ±5 arc minReflected Beam Deviation

≤5 arc minTransmitted Beam Deviation

Tp/Ts >500:1, 1000:1 averageExtinction Ratio

Tp >80%, >90% average, Rs >99.5% averageEfficiency

20-10 scratch-digSurface Quality

Central diameter, >80% of dimensionClear Aperture

≤λ/4 at 632.8 nm over the clear apertureWavefront Distortion

SF 2, NSSK grade, precision annealed optical glassMaterial

ECE 5616Curtis

Polarizers

Newport Catalogue

ECE 5616Curtis

Zero Order Waveplates

Newport CatalogueBlack anodized aluminumHousing

500 W/cm2 CW, 0.3 J/cm2 with 10 nsec pulses, visibleDamage Threshold

Non-abrasive method, acetone or isopropyl alcoholCleaning

Broadband, multilayer coating, Ravg <0.5%Antireflection Coating

-20 °C to 50 °CTemperature Range

6.2 mmHousing Thickness

25.4 ±0.13 mmHousing Diameter

3.56 mmThickness

± 7°Acceptance Angle

≤1 arc minTransmitted Beam Deviation

40-20 scratch-digSurface Quality

10.2 mmClear Aperture

≤λ/4 at 632.8 nm over the full apertureWavefront Distortion

±λ/100Retardation Accuracy

λ/4 or λ/2Retardation

BK 7, grade A, fine annealed optical glassSubstrate Material

Birefringent polymer film stackRetarder Material

ECE 5616Curtis

Multiple order Waveplates

Newport Catalogue

2 MW/cm2 CW, 2 J/cm2 with 10 nsec pulsesDamage

Threshold

Non-abrasive method, acetone or isopropyl alcohol

on lens tissue recommended, caution: fragile, thin optic

Cleaning

Single wavelength: 1 mm, nominalDual wavelength: 0.5–4 mm, nominalThickness

Single wavelength: R <0.5% total, or 0.25% per surface

Dual wavelength: Ravg <1.5%

Reflectivity per Surface

+0/-0.25 mmDiameter Tolerance

<0.5 arc secWedge

10-5 scratch-digSurface Quality

≥central 90% of diameterClear Aperture

≤λ/10 at 632.8 nm over the full apertureWavefront Distortion

±λ/300 at 20°C ±1°CRetardation Accuracy

λ/4 or λ/2Retardation

Single plateConstruction

Quartz, schlieren gradeMaterial

Must order for laser lineIncident angles must be close to normalMultiple nulls with polarizer