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 / CFHTPolarization of Light: Basics to Instruments2 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

N. Manset / CFHTPolarization of Light: Basics to Instruments3 Light as an electromagnetic wave Light is a transverse wave, an electromagnetic wave Part I: Polarization states ?!?

N. Manset / CFHTPolarization of Light: Basics to Instruments4 Mathematical description of the EM wave Light wave that propagates in the z direction: Part I: Polarization states

N. Manset / CFHTPolarization of Light: Basics to Instruments5 Graphical representation of the EM wave (I) One can go from: to the equation of an ellipse (using trigonometric identities, squaring, adding): Part I: Polarization states

N. Manset / CFHTPolarization of Light: Basics to Instruments6 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... Part I: Polarization states

N. Manset / CFHTPolarization of Light: Basics to Instruments7 Vertically polarized light If there is no amplitude in x (E 0x = 0), there is only one component, in y (vertical). Part I: Polarization states, linear polarization

N. Manset / CFHTPolarization of Light: Basics to Instruments8 Polarization at 45º (I) If there is no phase difference (  =0) and E 0x = E 0y, then E x = E y Part I: Polarization states, linear polarization

N. Manset / CFHTPolarization of Light: Basics to Instruments9 Polarization at 45º (II) Part I: Polarization states, linear polarization

N. Manset / CFHTPolarization of Light: Basics to Instruments10 Circular polarization (I) If the phase difference is  = 90º and E 0x = E 0y then: E x / E 0x = cos , E y / E 0y = sin  and we get the equation of a circle: Part I: Polarization states, circular polarization

N. Manset / CFHTPolarization of Light: Basics to Instruments11 Circular polarization (II) Part I: Polarization states, circular polarization

N. Manset / CFHTPolarization of Light: Basics to Instruments12 Circular polarization (III) Part I: Polarization states, circular polarization

N. Manset / CFHTPolarization of Light: Basics to Instruments13 Circular polarization (IV) Part I: Polarization states, circular polarization... see it now?

N. Manset / CFHTPolarization of Light: Basics to Instruments14 Elliptical polarization Part I: Polarization states, elliptical polarization Linear + circular polarization = elliptical polarization

N. Manset / CFHTPolarization of Light: Basics to Instruments15 Unpolarized light (natural light) Part I: Polarization states, unpolarized light

N. Manset / CFHTPolarization of Light: Basics to Instruments16 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

N. Manset / CFHTPolarization of Light: Basics to Instruments17 Stokes parameters A tiny itsy-bitsy little bit of history : Bartholinus discovers double refraction in calcite 17 th – 19 th centuries: Huygens, Malus, Brewster, Biot, Fresnel and Arago, Nicol 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

N. Manset / CFHTPolarization of Light: Basics to Instruments18 Stokes parameters (I) 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

N. Manset / CFHTPolarization of Light: Basics to Instruments19 Stokes parameters (II) described in terms of the electric field The 4 Stokes parameters are: Part II: Stokes parameters

N. Manset / CFHTPolarization of Light: Basics to Instruments20 Stokes parameters (III) described in geometrical terms Part II: Stokes parameters

N. Manset / CFHTPolarization of Light: Basics to Instruments21 Stokes vector 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

N. Manset / CFHTPolarization of Light: Basics to Instruments22 Pictorial representation of the Stokes parameters Part II: Stokes parameters

N. Manset / CFHTPolarization of Light: Basics to Instruments23 Stokes vectors for linearly polarized light LHP lightLVP light+45º light-45º light Part II: Stokes parameters, examples

N. Manset / CFHTPolarization of Light: Basics to Instruments24 Stokes vectors for circularly polarized light RCP lightLCP light Part II: Stokes parameters, examples

N. Manset / CFHTPolarization of Light: Basics to Instruments25 (Q,U) to (P,  ) In the case of linear polarization (V=0): Part II: Stokes parameters

N. Manset / CFHTPolarization of Light: Basics to Instruments26 Mueller matrices 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

N. Manset / CFHTPolarization of Light: Basics to Instruments27 Mueller calculus (I) Element 1 Element 2 Element 3 I’ = M 3 M 2 M 1 I Part II: Stokes parameters, Mueller matrices

N. Manset / CFHTPolarization of Light: Basics to Instruments28 Mueller calculus (II) 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

N. Manset / CFHTPolarization of Light: Basics to Instruments29 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: BUT: Jones formalism can only deal with 100% polarization... Part II: Stokes parameters, Jones formalism, not that important here...

N. Manset / CFHTPolarization of Light: Basics to Instruments30 Part III: Optical components for polarimetry Complex index of refraction Polarizers Retarders

N. Manset / CFHTPolarization of Light: Basics to Instruments31 Complex index of refraction 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

N. Manset / CFHTPolarization of Light: Basics to Instruments32 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

N. Manset / CFHTPolarization of Light: Basics to Instruments33 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 Part III: Optical components, polarizers

N. Manset / CFHTPolarization of Light: Basics to Instruments34 Wide-grid polarizers (II) [dichroism] Part III: Optical components, polarizers

N. Manset / CFHTPolarization of Light: Basics to Instruments35 Dichroic crystals [dichroism] Dichroic crystals absorb one polarization state over the other one. Example: tourmaline. Part III: Optical components, polarizers

N. Manset / CFHTPolarization of Light: Basics to Instruments36 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 Part III: Optical components, polarizers – Polaroids, like in sunglasses!

N. Manset / CFHTPolarization of Light: Basics to Instruments37 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 Part III: Optical components, polarizers

N. Manset / CFHTPolarization of Light: Basics to Instruments38 Crystal polarizers (II) [birefringence] The 2 output beams are polarized (orthogonally). isotropic crystal (sodium chloride) anisotropic crystal (calcite) Part III: Optical components, polarizers

N. Manset / CFHTPolarization of Light: Basics to Instruments39 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,... Part III: Optical components, polarizers

N. Manset / CFHTPolarization of Light: Basics to Instruments40 Mueller matrices of polarizers (I) (Ideal) linear polarizer at angle  : Part III: Optical components, polarizers

N. Manset / CFHTPolarization of Light: Basics to Instruments41 Mueller matrices of polarizers (II) Linear (±Q) polarizer at 0º: Linear (±U) polarizer at 0º : Part III: Optical components, polarizers Circular (±V) polarizer at 0º :

N. Manset / CFHTPolarization of Light: Basics to Instruments42 Mueller calculus with a polarizer Input light: unpolarized --- output light: polarized Total output intensity: 0.5 I Part III: Optical components, polarizers

N. Manset / CFHTPolarization of Light: Basics to Instruments43 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

N. Manset / CFHTPolarization of Light: Basics to Instruments44 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. Part III: Optical components, retarders

N. Manset / CFHTPolarization of Light: Basics to Instruments45 Half-Wave plate (II) Part III: Optical components, retarders

N. Manset / CFHTPolarization of Light: Basics to Instruments46 Quarter-Wave plate (I) Retardation of ¼ wave or 90º for one of the polarizations Used to convert linear polarization to elliptical. Part III: Optical components, retarders

N. Manset / CFHTPolarization of Light: Basics to Instruments47 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) Part III: Optical components, retarders

N. Manset / CFHTPolarization of Light: Basics to Instruments48 Mueller matrix of retarders (I) Retarder of retardance  and position angle  : Part III: Optical components, retarders

N. Manset / CFHTPolarization of Light: Basics to Instruments49 Mueller matrix of retarders (II) Half-wave oriented at 0º or 90º Half-wave oriented at ±45º Part III: Optical components, retarders

N. Manset / CFHTPolarization of Light: Basics to Instruments50 Mueller matrix of retarders (III) Quarter-wave oriented at 0º Quarter-wave oriented at ±45º Part III: Optical components, retarders

N. Manset / CFHTPolarization of Light: Basics to Instruments51 Mueller calculus with a retarder Input light linear polarized (Q=1) Quarter-wave at +45º Output light circularly polarized (V=1) Part III: Optical components, retarders

N. Manset / CFHTPolarization of Light: Basics to Instruments52 (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. Part III: Optical components, polarizers

N. Manset / CFHTPolarization of Light: Basics to Instruments53 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 MgF 2 plates Part III: Optical components, retarders

N. Manset / CFHTPolarization of Light: Basics to Instruments54 Achromatic retarders (II) Part III: Optical components, retarders  = º not very achromatic!  = º much better!

N. Manset / CFHTPolarization of Light: Basics to Instruments55 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 Part III: Optical components, retarders

N. Manset / CFHTPolarization of Light: Basics to Instruments56 Fresnel rhombs Quarter-wave and half-wave rhombs are achieved with 2 or 4 reflections Part III: Optical components, retarders

N. Manset / CFHTPolarization of Light: Basics to Instruments57 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 III: Optical components, retarders

N. Manset / CFHTPolarization of Light: Basics to Instruments58 Part IV: Polarimeters Polaroid-type polarimeters Dual-beam polarimeters

N. Manset / CFHTPolarization of Light: Basics to Instruments59 Polaroid-type polarimeter for linear polarimetry (I) Use a linear polarizer (polaroid) to measure linear polarization... [another cool applet] Location: cool applet] Polarization percentage and position angle: Part IV: Polarimeters, polaroid-type

N. Manset / CFHTPolarization of Light: Basics to Instruments60 Polaroid-type polarimeter for 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) Part IV: Polarimeters, polaroid-type

N. Manset / CFHTPolarization of Light: Basics to Instruments61 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 Part IV: Polarimeters, polaroid-type

N. Manset / CFHTPolarization of Light: Basics to Instruments62 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

N. Manset / CFHTPolarization of Light: Basics to Instruments63 Dual-beam polarimeters Switching beams 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 To compensate different gains, switch the 2 beams and average the 2 measurements

N. Manset / CFHTPolarization of Light: Basics to Instruments64 Dual-beam polarimeters Switching beams by rotating the prism rotate by 180º Part IV: Polarimeters, dual-beam type

N. Manset / CFHTPolarization of Light: Basics to Instruments65 Dual-beam polarimeters Switching beams using a ½ wave plate Rotated by 45º Part IV: Polarimeters, dual-beam type

UH DBIP (Masiero, 2007) Polarization of Light: Basics to Instruments66

N. Manset / CFHTPolarization of Light: Basics to Instruments67 Dual-beam polarimeter for circular polarization - Wollaston and quarter-wave plate Part IV: Polarimeters, dual-beam type The measurements V/I is: Switch the beams to compensate the gain effects

N. Manset / CFHTPolarization of Light: Basics to Instruments68 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

N. Manset / CFHTPolarization of Light: Basics to Instruments69 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

N. Manset / CFHTPolarization of Light: Basics to Instruments70 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

N. Manset / CFHTPolarization of Light: Basics to Instruments71 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

N. Manset / CFHTPolarization of Light: Basics to Instruments72 ESPaDOnS: circular polarimetry of circular polarization half-wave 22.5º positions flips polarization gain, transmission quarter- wave fixed circular to linear analyzer Part V: ESPaDOnS, circular polarimetry mode

N. Manset / CFHTPolarization of Light: Basics to Instruments73 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 Part V: ESPaDOnS, circular polarimetry mode

N. Manset / CFHTPolarization of Light: Basics to Instruments74 ESPaDOnS: linear polarimetry Half-Wave rhombs positioned at 22.5º increments Quarter-Wave fixed Part V: ESPaDOnS, linear polarimetry

N. Manset / CFHTPolarization of Light: Basics to Instruments75 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

N. Manset / CFHTPolarization of Light: Basics to Instruments76 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

N. Manset / CFHTPolarization of Light: Basics to Instruments77 Credits for pictures and movies Christoph Keller’s home page – his 5 lectures “Basic Polarisation techniques and devices”, Meadowlark Optics Inc. Optics, E. Hecht and Astronomical Polarimetry, J. Tinbergen Planets, Stars and Nebulae Studied With Photopolarimetry, T. Gehrels Circular polarization movie Unpolarized light movie Reflection of wave ESPaDOnS web page and documents

N. Manset / CFHTPolarization of Light: Basics to Instruments78 References/Further reading On the Web Very short and quick introduction, no equation Easy fun page with Applets, on polarizing filters Polarization short course “Instrumentation for Astrophysical Spectropolarimetry”, a series of 5 lectures given at the IAC Winter School on Astrophysical Spectropolarimetry, November 2000 –

N. Manset / CFHTPolarization of Light: Basics to Instruments79 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

N. Manset / CFHTPolarization of Light: Basics to Instruments80 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

N. Manset / CFHTPolarization of Light: Basics to Instruments81 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