Polarization Light Intro

Polarization of Light:from Basics to Instruments

(in less than 100 slides)

N. Manset

CFHT

N. Manset / CFHT Polarization of Light: Basics to Instruments 2

Introduction

• Part I: Different polarization states of light

• Part II: Stokes parameters, Mueller matrices

• Part III: Optical components for polarimetry

• Part IV: Polarimeters

• Part V: ESPaDOnS


Part I: Different polarization states of light

• Light as an electromagnetic wave

• Mathematical and graphical descriptions of polarization

• Linear, circular, elliptical light

• Polarized, unpolarized light


Light as an electromagnetic wave

Light is a transverse wave,

an electromagnetic wave

Part I: Polarization states


Mathematical description of the EM wave

Light wave that propagates in the z direction:

y)t-kzcos(E)tz,(E

xt)-kzcos(E)tz,(E

0yy

0xx



Graphical representation of the EM wave (I)

One can go from:

to the equation of an ellipse (using trigonometric

identities, squaring, adding):

2

0y

y

0x

x

2

0y

y

2

0x

x sincosE

E

E

E2

E

E

E

E

y)t-kzcos(E)tz,(E

xt)-kzcos(E)tz,(E

0yy

0xx



Graphical representation of the EM wave (II)

An ellipse can be represented by 4 quantities:

1. size of minor axis

2. size of major axis

3. orientation (angle)

4. sense (CW, CCW)

Light can be represented by 4 quantities...



Vertically polarized light

If there is no amplitude in x (E0x = 0), there is only one component, in y (vertical).

y)t-kzcos(E)tz,(E

xt)-kzcos(E)tz,(E

0yy

0xx

Part I: Polarization states, linear polarization


Polarization at 45º (I)

If there is no phase difference (=0) and

E0x = E0y, then Ex = Ey

y)t-kzcos(E)tz,(E

xt)-kzcos(E)tz,(E

0yy

0xx



Polarization at 45º (II)



Circular polarization (I)

If the phase difference is = 90º and E0x = E0y

then: Ex / E0x = cos , Ey / E0y = sin

and we get the equation of a circle:

1sin cosE

E

E

E 22

2

0y

y

2

0x

x

y)t-kzcos(E)tz,(E

xt)-kzcos(E)tz,(E

0yy

0xx

Part I: Polarization states, circular polarization


Circular polarization (II)



Circular polarization (III)



Circular polarization (IV)

Part I: Polarization states, circular polarization... see it now?


Elliptical polarization

Part I: Polarization states, elliptical polarization

• Linear + circular polarization = elliptical polarization


Unpolarized light(natural light)

Part I: Polarization states, unpolarized light


A cool Applet

Electromagnetic Wave

Location: http://www.uno.edu/~jsulliva/java/EMWave.html


http://www.uno.edu/~jsulliva/java/EMWave.html


Part II: Stokes parameters and Mueller matrices

• Stokes parameters, Stokes vector

• Stokes parameters for linear and circular polarization

• Stokes parameters and polarization P

• Mueller matrices, Mueller calculus

• Jones formalism


Stokes parametersA tiny itsy-bitsy little bit of history...

• 1669: Bartholinus discovers double refraction in calcite

• 17th – 19th centuries: Huygens, Malus, Brewster, Biot, Fresnel and Arago, Nicol...

• 19th century: unsuccessful attempts to describe unpolarized light in terms of amplitudes

• 1852: Sir George Gabriel Stokes took a very different approach and discovered that polarization can be described in terms of observables using an experimental definition

Part II: Stokes parameters


Stokes parameters (I)

The polarization ellipse is only valid at a given instant of time (function of time):

εsinεcos(t)E

(t)E

(t)E

(t)E2

(t)E

(t)E

(t)E

(t)E 2

0y

y

0x

x

2

0y

y

2

0x

x

To get the Stokes parameters, do a time average (integral over time) and a little bit of algebra...



Stokes parameters (II)described in terms of the electric field

20y0x2

0y0x

220y

20x

220y

20x εsinEE2εcosEE2EEEE

The 4 Stokes parameters are:

εsinEE2V

εcosEE2U

E E Q

EEI

0y0x3

0y0x2

20y

20x1

20y

20x0

S

S

S

S



Stokes parameters (III)described in geometrical terms

2sin

2sin2cos

2cos2cos

V

U

Q

I

2

2

2

2

a

a

a

a



Stokes vectorThe Stokes parameters can be arranged in a Stokes vector:

LCPIRCPI

135I45I

90I0I

intensity

εsinEE2

εcosEE2

EE

EE

V

U

Q

I

0y0x

0y0x

20y

20x

20y

20x

• Linear polarization• Circular polarization• Fully polarized light• Partially polarized light• Unpolarized light 0VUQ

VUQI

VUQI

0V 0, U0,Q

0V 0, U0,Q

2222

2222

Part II: Stokes parameters, Stokes vectors


Pictorial representation of the Stokes parameters



Stokes vectors for linearly polarized light

LHP light

0

0

1

1

I0

LVP light +45º light -45º light

0

0

1

1

I0

0

1

0

1

I0

0

1

0

1

I0

Part II: Stokes parameters, examples


Stokes vectors for circularly polarized light

RCP light

1

0

0

1

I0

LCP light

1

0

0

1

I0

Part II: Stokes parameters, examples


(Q,U) to (P,)In the case of linear polarization (V=0):

I

UQP

22

Q

Uarctan

2

1

2 cos PQ 2 sin PU



Mueller matrices

If light is represented by Stokes vectors, optical components are then described with Mueller matrices:

[output light] = [Muller matrix] [input light]

V

U

Q

I

V'

U'

Q'

I'

44434241

34333231

24232221

14131211

mmmm

mmmm

mmmm

mmmm

Part II: Stokes parameters, Mueller matrices


Mueller calculus (I)

Element 1 Element 2 Element 3

1M 2M 3M

I’ = M3 M2 M1 I



Mueller calculus (II)

Mueller matrix M’ of an optical component with Mueller matrix M rotated by an angle :

M’ = R(- ) M R() with:

1000

02cos2sin0

02sin2cos0

0001

)R(



Jones formalism

Stokes vectors and Mueller matrices cannot describe interference effects. If the phase information is important (radio-astronomy, masers...), one has to use the Jones formalism, with complex vectors and Jones matrices: • Jones vectors to describe the polarization of light:

• Jones matrices to represent optical components:

(t)E

(t)E(t)J

y

x

2221

1211

jj

jjJ

BUT: Jones formalism can only deal with 100% polarization...

Part II: Stokes parameters, Jones formalism, not that important here...


Part III: Optical components for polarimetry

• Complex index of refraction

• Polarizers

• Retarders


Complex index of refraction

The index of refraction is actually a complex quantity:

iknm • real part

• optical path length, refraction: speed of light depends on media

• birefringence: speed of light also depends on P

• imaginary part

• absorption, attenuation, extinction: depends on media

• dichroism/diattenuation: also depends on P

Part III: Optical components


Polarizers

Polarizers absorb one component of the polarization but not the other. The input is natural light, the output is polarized light (linear, circular, elliptical). They work by dichroism, birefringence, reflection, or scattering.

Part III: Optical components, polarizers


Wire-grid polarizers (I)[dichroism]

• Mainly used in the IR and longer wavelengths

• Grid of parallel conducting wires with a spacing comparable to the wavelength of observation

• Electric field vector parallel to the wires is attenuated because of currents induced in the wires



Wide-grid polarizers (II) [dichroism]



Dichroic crystals [dichroism]

Dichroic crystals absorb one polarization state over the other one.

Example: tourmaline.



Polaroids [dichroism]

Made by heating and stretching a sheet of PVA laminated to a supporting sheet of cellulose acetate treated with iodine solution (H-type polaroid). Invented in 1928.

Part III: Optical components, polarizers – Polaroids, like in sunglasses!


Crystal polarizers (I) [birefringence]

• Optically anisotropic crystals

• Mechanical model:• the crystal is anisotropic, which means that the electrons are bound with different ‘springs’ depending on the orientation

• different ‘spring constants’ gives different propagation speeds, therefore different indices of refraction, therefore 2 output beams



Crystal polarizers (II)[birefringence]

The 2 output beams are polarized (orthogonally).

isotropiccrystal (sodiumchloride)

anisotropiccrystal(calcite)



Crystal polarizers (IV)[birefringence]

• Crystal polarizers used as:• Beam displacers,• Beam splitters,• Polarizers,• Analyzers, ...

• Examples: Nicol prism, Glan-Thomson polarizer, Glan or Glan-Foucault prism, Wollaston prism, Thin-film polarizer, ...



Mueller matrices of polarizers (I)

• (Ideal) linear polarizer at angle :

0000

0χ2sinχ2cosχ2sinχ2sin

0χ2cosχ2sinχ2cosχ2cos

0χ2sinχ2cos1

2

12

2



Mueller matrices of polarizers (II)

Linear (±Q) polarizer at 0º:

0000

0000

0011

0011

5.0

Linear (±U) polarizer at 0º :

0000

0101

0000

0101

5.0


Circular (±V) polarizer at 0º :

1001

0000

0000

1001

5.0


Mueller calculus with a polarizer

Input light: unpolarized --- output light: polarized

0

I-

0

I

5.0

0

0

0

I

0000

0101

0000

0101

5.0

V'

U'

Q'

I'

Total output intensity: 0.5 I



Retarders

• In retarders, one polarization gets ‘retarded’, or delayed, with respect to the other one. There is a final phase difference between the 2 components of the polarization. Therefore, the polarization is changed.

• Most retarders are based on birefringent materials (quartz, mica, polymers) that have different indices of refraction depending on the polarization of the incoming light.

Part III: Optical components, retarders


Half-Wave plate (I)

• Retardation of ½ wave or 180º for one of the polarizations.

• Used to flip the linear polarization or change the handedness of circular polarization.



Half-Wave plate (II)



Quarter-Wave plate (I)

• Retardation of ¼ wave or 90º for one of the polarizations

• Used to convert linear polarization to elliptical.



• Special case: incoming light polarized at 45º with respect to the retarder’s axis

• Conversion from linear to circular polarization (vice versa)

Quarter-Wave plate (II)



Mueller matrix of retarders (I)

• Retarder of retardance and position angle :

cosτ12

1Handcosτ1

2

1G :with

cosτcos2ψsinτsin2ψsinτ0

cos2ψsinτcos4ψHGsin4ψH0

sin2ψsinτsin4ψHcos4ψHG0

0001



Mueller matrix of retarders (II)

• Half-wave oriented at 0º or 90º

• Half-wave oriented at ±45º

1000

0100

0010

0001

k

1000

0100

0010

0001

k



Mueller matrix of retarders (III)

• Quarter-wave oriented at 0º

• Quarter-wave oriented at ±45º

0100

1000

0010

0001

k

0010

0100

1000

0001

k



Mueller calculus with a retarder

1

0

0

1

0

0

1

1

0010

0100

1000

0001

V'

U'

Q'

I'

kk

• Input light linear polarized (Q=1)• Quarter-wave at +45º• Output light circularly polarized (V=1)



(Back to polarizers, briefly)

Circular polarizers• Input light: unpolarized --- Output light: circularly polarized

• Made of a linear polarizer glued to a quarter-wave plate oriented at 45º with respect to one another.



Achromatic retarders (I)

• Retardation depends on wavelength• Achromatic retarders: made of 2 different materials with

opposite variations of index of refraction as a function of wavelength

• Pancharatnam achromatic retarders: made of 3 identical plates rotated w/r one another

• Superachromatic retarders: 3 pairs of quartz and MgF2

plates



Achromatic retarders (II)


=140-220º

not very achromatic!

= 177-183º

much better!


Retardation on total internal reflection

• Total internal reflection produces retardation (phase shift)

• In this case, retardation is very achromatic since it only depends on the refractive index

• Application: Fresnel rhombs



Fresnel rhombs

• Quarter-wave and half-wave rhombs are achieved with 2 or 4 reflections



Other retarders

• Soleil-Babinet: variable retardation to better than 0.01 waves

• Nematic liquid crystals... Liquid crystal variable retarders... Ferroelectric liquid crystals... Piezo-elastic modulators... Pockels and Kerr cells...



Part IV: Polarimeters

• Polaroid-type polarimeters

• Dual-beam polarimeters


Polaroid-type polarimeterfor linear polarimetry (I)

• Use a linear polarizer (polaroid) to measure linear polarization ... [another cool applet] Location: http://www.colorado.edu/physics/2000/applets/lens.html

• Polarization percentage and position angle:

)II(

II

IIP

max

minmax

minmax

Part IV: Polarimeters, polaroid-type

http://www.colorado.edu/physics/2000/applets/lens.html

http://www.colorado.edu/physics/2000/applets/lens.html


Polaroid-type polarimeterfor linear polarimetry (II)

• Advantage: very simple to make

• Disadvantage: half of the light is cut out

• Other disadvantages: non-simultaneous measurements, cross-talk...

• Move the polaroid to 2 positions, 0º and 45º (to measure Q, then U)



Polaroid-type polarimeterfor circular polarimetry

• Polaroids are not sensitive to circular polarization, so convert circular polarization to linear first, by using a quarter-wave plate

• Polarimeter now uses a quarter-wave plate and a polaroid

• Same disadvantages as before



Dual-beam polarimetersPrinciple

• Instead of cutting out one polarization and keeping the other one (polaroid), split the 2 polarization states and keep them both

• Use a Wollaston prism as an analyzer• Disadvantages: need 2 detectors (PMTs, APDs) or

an array; end up with 2 ‘pixels’ with different gain• Solution: rotate the Wollaston or keep it fixed and

use a half-wave plate to switch the 2 beams

Part IV: Polarimeters, dual-beam type


Dual-beam polarimetersSwitching beams


• Unpolarized light: two beams have identical intensities whatever the prism’s position if the 2 pixels have the same gain

• To compensate different gains, switch the 2 beams and average the 2 measurements


Dual-beam polarimetersSwitching beams by rotating the prism

rotate by 180º



Dual-beam polarimetersSwitching beams using a ½ wave plate

Rotated by 45º



A real circular polarimeterSemel, Donati, Rees (1993)

Quarter-wave plate, rotated at -45º and +45º

Analyser: double calcite crystal

Part IV: Polarimeters, example of circular polarimeter


A real circular polarimeterfree from gain (g) and atmospheric transmission () variation effects

• First measurement with quarter-wave plate at -45º, signal in the (r)ight and (l)eft beams:

• Second measurement with quarter-wave plate at +45º, signal in the (r)ight and (l)eft beams:

• Measurements of the signals:

rl SS 11 ,

rl SS 22 ,

)()(

)()(

22222222

11111111

VIgSVIgS

VIgSVIgSrrll

rrll



A real circular polarimeterfree from gain and atmospheric transmission variation effects

• Build a ratio of measured signals which is free of gain and variable atmospheric transmission effects:

1for 2

1

2

11

4

1

2

2

1

1

21211221

2112

1

2

2

1

VI

V

I

VF

VVVIVIII

VIVI

S

S

S

SF

r

r

l

l

average of the 2 measurements



Polarimeters - Summary• 2 types:

– polaroid-type: easy to make but ½ light is lost, and affected by variable atmospheric transmission

– dual-beam type: no light lost but affected by gain differences and variable transmission problems

• Linear polarimetry: – analyzer, rotatable– analyzer + half-wave plate

• Circular polarimetry:– analyzer + quarter-wave plate

2 positions minimum

1 position minimum

Part IV: Polarimeters, summary


Part V: ESPaDOnS

Optical components of the polarimeter part :

• Wollaston prism: analyses the polarization and separates the 2 (linear!) orthogonal polarization states

• Retarders, 3 Fresnel rhombs:– Two half-wave plates to switch the

beams around– Quarter-wave plate to do circular

polarimetry


ESPaDOnS: circular polarimetry

• Fixed quarter-wave rhomb

• Rotating bottom half-wave, at 22.5º increments

• Top half-wave rotates continuously at about 1Hz to average out linear polarization when measuring circular polarization

Part V: ESPaDOnS, circular polarimetry mode


ESPaDOnS: circular polarimetry of circular polarization

• half-wave

• 22.5º positions

• flips polarization

• gain, transmission

• quarter-wave

• fixed

• circular to linear

• analyzer



ESPaDOnS: circular polarimetry of (unwanted) linear polarization

• half-wave

• 22.5º positions

• gain, transmission

• quarter-wave

• fixed

• linear to elliptical

• analyzer• circular part goes through not analyzed and adds same intensities to both beams

• linear part is analyzed!

• Add a rotating half-wave to “spread out” the unwanted signal



ESPaDOnS: linear polarimetry

• Half-Wave rhombs positioned at 22.5º increments

• Quarter-Wave fixed

Part V: ESPaDOnS, linear polarimetry


ESPaDOnS: linear polarimetry

• Half-Wave rhombs positioned as 22.5º increments– First position gives Q– Second position gives U– Switch beams for gain and atmosphere effects

• Quarter-Wave fixed

Part V: ESPaDOnS, linear polarimetry


ESPaDOnS - Summary

• ESPaDOnS can do linear and circular polarimetry (quarter-wave plate)

• Beams are switched around to do the measurements, compensate for gain and atmospheric effects

• Fesnel rhombs are very achromatic

Part V: ESPaDOnS, summary



Credits for pictures and movies• Christoph Keller’s home page – his 5 lectures

http://www.noao.edu/noao/staff/keller/

• “Basic Polarisation techniques and devices”, Meadowlark Optics Inc. http://www.meadowlark.com/

• Optics, E. Hecht and Astronomical Polarimetry, J. Tinbergen • Planets, Stars and Nebulae Studied With Photopolarimetry, T.

Gehrels

• Circular polarization movie http://www.optics.arizona.edu/jcwyant/JoseDiaz/Polarization-Circular.htm

• Unpolarized light movie http://www.colorado.edu/physics/2000/polarization/polarizationII.html

• Reflection of wave http://www.physicsclassroom.com/mmedia/waves/fix.html

• ESPaDOnS web page and documents

http://www.noao.edu/noao/staff/keller/

http://www.meadowlark.com/

http://www.optics.arizona.edu/jcwyant/JoseDiaz/Polarization-Circular.htm

http://www.colorado.edu/physics/2000/polarization/polarizationII.html


References/Further reading On the Web

• Very short and quick introduction, no equation http://www.cfht.hawaii.edu/~manset/PolarIntro_eng.html

• Easy fun page with Applets, on polarizing filters http://www.colorado.edu/physics/2000/polarization/polarizationI.html

• Polarization short course http://www.glenbrook.k12.il.us/gbssci/phys/Class/light/u12l1e.html

• “Instrumentation for Astrophysical Spectropolarimetry”, a series of 5 lectures given at the IAC Winter School on Astrophysical Spectropolarimetry, November 2000 –http://www.noao.edu/noao/staff/keller/lectures/index.html

http://www.cfht.hawaii.edu/~manset/PolarIntro_eng.html

http://www.colorado.edu/physics/2000/polarization/polarizationI.html

http://www.glenbrook.k12.il.us/gbssci/phys/Class/light/u12l1e.html


References/Further reading Polarization basics

• Polarized Light, D. Goldstein – excellent book, easy read, gives a lot of insight, highly recommended

• Undergraduate textbooks, either will do:– Optics, E. Hecht– Waves, F. S. Crawford, Berkeley Physics Course vol. 3


References/Further readingAstronomy, easy/intermediate

• Astronomical Polarimetry, J. Tinbergen – instrumentation-oriented

• La polarisation de la lumière et l'observation astronomique, J.-L. Leroy – astronomy-oriented

• Planets, Stars and Nebulae Studied With Photopolarimetry, T. Gehrels – old but classic

• 3 papers by K. Serkowski – instrumentation-oriented


References/Further readingAstronomy, advanced

• Introduction to Spectropolarimetry, J.C. del Toro Iniesta – radiative transfer – ouch!

• Astrophysical Spectropolarimetry, Trujillo-Bueno et al. (eds) – applications to astronomy

Polarization Light Intro

Documents

phase difference

linear polarimetry

wave plate

part iv

wave oriented

circular polarimeter

part ii

stokes parameters