Polarization of Light - Institute for Astronomy · Polarization of Light: from Basics to...

Polarization of Light:

from Basics to Instruments (in less than 100 slides)

Originally by N. Manset, CFHT,

Modified and expanded by K. Hodapp

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

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


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 parameters (III) described in geometrical terms

2sin

2sin2cos

2cos2cos

V

U

Q

I

2

2

2

2

a

a

a

a

Part II: Stokes parameters


Stokes vector

The Stokes parameters can be arranged in a Stokes vector:

LCPIRCPI

135I45I

90I0I

intensity

εsinEE2

εcosEE2

EE

EE

V

U

Q

I

0y0x

0y0x

2

0y

2

0x

2

0y

2

0x

• 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(



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).

isotropic

crystal

(sodium

chloride)

anisotropic

crystal

(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!


Part IV: Polarimeters

• Polaroid-type polarimeters

• Dual-beam polarimeters


Polaroid-type polarimeter for 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


Dual-beam polarimeters Principle

• 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 polarimeters Switching 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 polarimeters Switching beams by rotating the prism

rotate by

180º



Dual-beam polarimeters Switching beams using a ½ wave plate

Rotated

by 45º


UH DBIP (Masiero, 2007)

Polarization of Light: Basics to Instruments 57


A real circular polarimeter Semel, Donati, Rees (1993)

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

Analyser: double calcite crystal

Part IV: Polarimeters, example of circular polarimeter


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


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 reading Astronomy, 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 reading Astronomy, advanced

• Introduction to Spectropolarimetry, J.C.

del Toro Iniesta – radiative transfer – ouch!

• Astrophysical Spectropolarimetry,

Trujillo-Bueno et al. (eds) – applications to

astronomy

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