Polarization of Light:�from Basics to Instruments�(in less than 100 slides)

N. Manset

CFHT

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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

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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

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Light as an electromagnetic wave

Light is a transverse wave,

an electromagnetic wave

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Part I: Polarization states

Mathematical description of the EM wave

Light wave that propagates in the z direction:

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Part I: Polarization states

Graphical representation of the EM wave (I)

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One can go from:

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to the equation of an ellipse (using trigonometric identities, squaring, adding):

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Part I: Polarization states

Graphical representation of the EM wave (II)

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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...

Part I: Polarization states

Vertically polarized light

If there is no amplitude in x (E_0x = 0), there is only one component, in y (vertical).

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Part I: Polarization states, linear polarization

Polarization at 45º (I)

If there is no phase difference (=0) and

E_0x = E_0y, then E_x = E_y

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Part I: Polarization states, linear polarization

Polarization at 45º (II)

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Part I: Polarization states, linear polarization

Circular polarization (I)

If the phase difference is = 90º and E_0x = E_0y

then: E_x / E_0x = cos Θ , E_y / E_0y = sin Θ

and we get the equation of a circle:

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Part I: Polarization states, circular polarization

Circular polarization (II)

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Part I: Polarization states, circular polarization

Circular polarization (III)

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Part I: Polarization states, circular polarization

Circular polarization (IV)

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

Elliptical polarization

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Part I: Polarization states, elliptical polarization

• Linear + circular polarization = elliptical polarization

Unpolarized light�(natural light)

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Part I: Polarization states, unpolarized light

A cool Applet

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Electromagnetic Wave

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

Part I: Polarization states

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

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Stokes parameters�A tiny itsy-bitsy little bit of history...

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• 1669: Bartholinus discovers double refraction in calcite
• 17^th – 19^th centuries: Huygens, Malus, Brewster, Biot, Fresnel and Arago, Nicol...
• 19^th 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)

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

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

Part II: Stokes parameters

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

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The 4 Stokes parameters are:

Part II: Stokes parameters

Stokes parameters (III)�described in geometrical terms

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Part II: Stokes parameters

Stokes vector

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The Stokes parameters can be arranged in a Stokes vector:

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

Part II: Stokes parameters, Stokes vectors

Pictorial representation of the Stokes parameters

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Part II: Stokes parameters

Stokes vectors for linearly polarized light

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LHP light

LVP light

+45º light

-45º light

Part II: Stokes parameters, examples

Stokes vectors for circularly polarized light

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RCP light

LCP light

Part II: Stokes parameters, examples

(Q,U) to (P,)

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In the case of linear polarization (V=0):

Part II: Stokes parameters

Mueller matrices

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

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

Part II: Stokes parameters, Mueller matrices

Mueller calculus (I)

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Element 1 Element 2 Element 3

I’ = M₃ M₂ M₁ I

Part II: Stokes parameters, Mueller matrices

Mueller calculus (II)

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

M’ = R(- ) M R() with:

Part II: Stokes parameters, Mueller matrices

Jones formalism

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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:

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

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Complex index of refraction

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The index of refraction is actually a complex quantity:

• 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

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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

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Part III: Optical components, polarizers

Wide-grid polarizers (II)� [dichroism]

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Part III: Optical components, polarizers

Dichroic crystals� [dichroism]

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Dichroic crystals absorb one polarization state over the other one.

Example: tourmaline.

Part III: Optical components, polarizers

Polaroids� [dichroism]

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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]

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• 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

Part III: Optical components, polarizers

Crystal polarizers (II)�[birefringence]

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The 2 output beams are polarized (orthogonally).

Part III: Optical components, polarizers

Crystal polarizers (IV)�[birefringence]

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• 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, ...

Part III: Optical components, polarizers

Mueller matrices of polarizers (I)

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• (Ideal) linear polarizer at angle χ:

Part III: Optical components, polarizers

Mueller matrices of polarizers (II)

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Linear (±Q) polarizer at 0º:

Linear (±U) polarizer at 0º :

Part III: Optical components, polarizers

Circular (±V) polarizer at 0º :

Mueller calculus with a polarizer

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Input light: unpolarized --- output light: polarized

Total output intensity: 0.5 I

Part III: Optical components, polarizers

Retarders

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• 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.

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

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Part III: Optical components, retarders

Half-Wave plate (II)

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Part III: Optical components, retarders

Quarter-Wave plate (I)

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

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• Used to convert linear polarization to elliptical.

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Part III: Optical components, retarders

Quarter-Wave plate (II)

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

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• Conversion from linear to circular polarization (vice versa)

Part III: Optical components, retarders

Mueller matrix of retarders (I)

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• Retarder of retardance τ and position angle ψ:

Part III: Optical components, retarders

Mueller matrix of retarders (II)

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• Half-wave oriented at 0º or 90º

• Half-wave oriented at ±45º

Part III: Optical components, retarders

Mueller matrix of retarders (III)

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• Quarter-wave oriented at 0º

• Quarter-wave oriented at ±45º

Part III: Optical components, retarders

Mueller calculus with a retarder

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

Part III: Optical components, retarders

(Back to polarizers, briefly)�Circular polarizers

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• 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.

Part III: Optical components, polarizers

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

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Part III: Optical components, retarders

Achromatic retarders (II)

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Part III: Optical components, retarders

=140-220º

not very achromatic!

= 177-183º

much better!

Retardation on total internal reflection

• Total internal reflection produces retardation (phase shift)

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

Part III: Optical components, retarders

Fresnel rhombs

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

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Part III: Optical components, retarders

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...

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Part III: Optical components, retarders

Part IV: Polarimeters

• Polaroid-type polarimeters
• Dual-beam polarimeters

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Polaroid-type polarimeter�for linear polarimetry (I)

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• 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:

Part IV: Polarimeters, polaroid-type

Polaroid-type polarimeter�for linear polarimetry (II)

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• Advantage: very simple to make
• Disadvantage: half of the light is cut out
• Other disadvantages: non-simultaneous measurements, cross-talk...

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

Part IV: Polarimeters, polaroid-type

Polaroid-type polarimeter�for 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

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Part IV: Polarimeters, polaroid-type

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

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Part IV: Polarimeters, dual-beam type

Dual-beam polarimeters�Switching beams

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Part IV: Polarimeters, dual-beam type

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

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

Dual-beam polarimeters�Switching beams by rotating the prism

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Part IV: Polarimeters, dual-beam type

Dual-beam polarimeters�Switching beams using a ½ wave plate

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Rotated by 45º

Part IV: Polarimeters, dual-beam type

Dual-beam polarimeter for circular polarization - Wollaston and quarter-wave plate

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Part IV: Polarimeters, dual-beam type

• The measurements V/I is:

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• Switch the beams to compensate the gain effects

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

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

Analyser: double calcite crystal

Part IV: Polarimeters, example of circular polarimeter

A real circular polarimeter�free from gain (g) and atmospheric transmission (α) variation effects

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• 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:

Part IV: Polarimeters, example of circular polarimeter

A real circular polarimeter�free from gain and atmospheric transmission variation effects

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

average of the 2 measurements

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

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• 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

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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

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Part V: ESPaDOnS, circular polarimetry mode

ESPaDOnS: circular polarimetry of circular polarization

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• half-wave
• 22.5º positions
• flips polarization
• gain, transmission

• quarter-wave
• fixed
• circular to linear

• analyzer

Part V: ESPaDOnS, circular polarimetry mode

ESPaDOnS: circular polarimetry of (unwanted) linear polarization

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• 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

Part V: ESPaDOnS, circular polarimetry mode

ESPaDOnS: linear polarimetry

• Half-Wave rhombs positioned at 22.5º increments
• Quarter-Wave fixed

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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

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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

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Part V: ESPaDOnS, summary

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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

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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

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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

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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

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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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