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  • Lecture 8: Polarimetry 2

    Outline

    1 Polarizers and Retarders2 Polarimeters3 Scattering Polarization4 Zeeman Effect

    Christoph U. Keller, Utrecht University, [email protected] Observational Astrophysics 2, Lecture 8: Polarimetry 2 1

  • Polarization Summarypolarization is an intrinsic property of lightpolarization properties and intensity of light can be described by 4parameters:

    coherent Jones calculusincoherent Stokes/Mueller calculus

    degree of polarization is the fraction of the intensity that is fullypolarizedtypical values for degree of polarization:

    45 degree reflection off aluminum mirror: 5%clear blue sky: up to 75%45 degree reflection off glass: 90%LCD screen: 100%solar scattering polarization: 1% to 0.001%exoplanet signal: 0.001%

    Christoph U. Keller, Utrecht University, [email protected] Observational Astrophysics 2, Lecture 8: Polarimetry 2 2

  • Polarizerspolarizer: optical element that produces polarized light fromunpolarized input lightlinear, circular, or in general elliptical polarizer, depending on typeof transmitted polarizationlinear polarizers by far the most commonlarge variety of polarizers

    Christoph U. Keller, Utrecht University, [email protected] Observational Astrophysics 2, Lecture 8: Polarimetry 2 3

  • Jones Matrix for Linear PolarizersJones matrix for linear polarizer:

    Jp =(

    px 00 py

    )0 ≤ px ≤ 1 and 0 ≤ py ≤ 1, real: transmission factors for x ,y -components of electric field: E ′x = pxEx , E ′y = pyEypx = 1, py = 0: linear polarizer in +Q directionpx = 0, py = 1: linear polarizer in −Q directionpx = py : neutral density filter

    Mueller Matrix for Linear Polarizers

    Mp =12

    1 1 0 01 1 0 00 0 0 00 0 0 0

    Christoph U. Keller, Utrecht University, [email protected] Observational Astrophysics 2, Lecture 8: Polarimetry 2 4

  • Mueller Matrix for Ideal Linear Polarizer at Angle θ

    Mpol (θ) =12

    1 cos 2θ sin 2θ 0

    cos 2θ cos2 2θ sin 2θ cos 2θ 0sin 2θ sin 2θ cos 2θ sin2 2θ 0

    0 0 0 0

    Poincare Sphere

    polarizer is a point on the Poincaré spheretransmitted intensity: cos2(l/2), l is arch length of great circlebetween incoming polarization and polarizer on Poincaré sphere

    Christoph U. Keller, Utrecht University, [email protected] Observational Astrophysics 2, Lecture 8: Polarimetry 2 5

  • Wire Grid Polarizers

    parallel conducting wires, spacing d . λ act as polarizerelectric field parallel to wires induces electrical currents in wiresinduced electrical current reflects polarization parallel to wirespolarization perpendicular to wires is transmittedrule of thumb:

    d < λ/2⇒ strong polarizationd � λ⇒ high transmission of both polarization states (weakpolarization)

    mostly used in infrared

    Christoph U. Keller, Utrecht University, [email protected] Observational Astrophysics 2, Lecture 8: Polarimetry 2 6

  • Polaroid-type Polarizers

    developed by Edwin Land in 1938⇒ Polaroidsheet polarizers: stretched polyvynil alcohol (PVA) sheet,laminated to sheet of cellulose acetate butyrate, treated withiodinePVA-iodine complex analogous to short, conducting wirecheap, can be manufactured in large sizes

    Christoph U. Keller, Utrecht University, [email protected] Observational Astrophysics 2, Lecture 8: Polarimetry 2 7

  • Crystal-Based Polarizers

    crystals are basis of highest-quality polarizersprecise arrangement of atom/molecules and anisotropy of index ofrefraction separate incoming beam into two beams with preciselyorthogonal linear polarization stateswork well over large wavelength rangemany different configurationscalcite most often used in crystal-based polarizers because oflarge birefringence, low absorption in visiblemany other suitable materials

    Christoph U. Keller, Utrecht University, [email protected] Observational Astrophysics 2, Lecture 8: Polarimetry 2 8

  • Retarders

    en.wikipedia.org/wiki/Wave_plate

    General Retarders or Wave Platesretarder: retards (delays) phase of one electric field componentwith respect to the orthogonal component

    Christoph U. Keller, Utrecht University, [email protected] Observational Astrophysics 2, Lecture 8: Polarimetry 2 9

    http://en.wikipedia.org/wiki/Wave_plate

  • Retarder Propertiesdoes not change intensity or degree of polarizationcharacterized by two (not identical, not trivial) Stokes vectors ofincoming light that are not changed by retarder⇒ eigenvectors ofretarderdepending on polarization described by eigenvectors, retarder is

    linear retardercircular retarderelliptical retarder

    linear, circular retarders are special cases of elliptical retarderscircular retarders sometimes called rotators since they rotate theorientation of linearly polarized lightlinear retarders by far the most common type of retarder

    Christoph U. Keller, Utrecht University, [email protected] Observational Astrophysics 2, Lecture 8: Polarimetry 2 10

  • Jones Matrix for Linear Retarderslinear retarder with fast axis at 0◦ characterized by Jones matrix

    Jr (δ) =(

    eiδ 00 1

    ), Jr (δ) =

    (ei

    δ2 0

    0 e−iδ2

    )

    δ: phase shift between two linear polarization components (inradians)absolute phase does not matter

    Mueller Matrix for Linear Retarder

    Mr =

    1 0 0 00 1 0 00 0 cos δ − sin δ0 0 sin δ cos δ

    Christoph U. Keller, Utrecht University, [email protected] Observational Astrophysics 2, Lecture 8: Polarimetry 2 11

  • Retarders on the Poincaré Sphere

    retarder eigenvector (fast axis) in Poincaré spherepoints on sphere are rotated around retarder axis by amount ofretardation

    Christoph U. Keller, Utrecht University, [email protected] Observational Astrophysics 2, Lecture 8: Polarimetry 2 12

  • Variable Retarders

    Introductionsensitive polarimeters requires retarders whose properties(retardance, fast axis orientation) can be varied quickly(modulated)retardance changes (change of birefringence):

    liquid crystalsFaraday, Kerr, Pockels cellspiezo-elastic modulators (PEM)

    fast axis orientation changes (change of c-axis direction):rotating fixed retarderferro-electric liquid crystals (FLC)

    Christoph U. Keller, Utrecht University, [email protected] Observational Astrophysics 2, Lecture 8: Polarimetry 2 13

  • Liquid Crystals

    liquid crystals: fluids with elongated moleculesat high temperatures: liquid crystal is isotropicat lower temperature: molecules become ordered in orientationand sometimes also space in one or more dimensionsliquid crystals can line up parallel or perpendicular to externalelectrical field

    Christoph U. Keller, Utrecht University, [email protected] Observational Astrophysics 2, Lecture 8: Polarimetry 2 14

  • Liquid Crystal Retarders

    dielectric constant anisotropy often large⇒ very responsive tochanges in applied electric fieldbirefringence δn can be very large (larger than typical crystalbirefringence)liquid crystal layer only a few µm thickbirefringence shows strong temperature dependence

    Christoph U. Keller, Utrecht University, [email protected] Observational Astrophysics 2, Lecture 8: Polarimetry 2 15

  • Rotating Mueller Matricesoptical element with Mueller matrix MMueller matrix of the same element rotated by θ around the beamgiven by

    M(θ) = R(−θ)MR(θ)

    with

    R (θ) =

    1 0 0 00 cos 2θ sin 2θ 00 − sin 2θ cos 2θ 00 0 0 1

    Christoph U. Keller, Utrecht University, [email protected] Observational Astrophysics 2, Lecture 8: Polarimetry 2 16

  • Temporal and Spatial Modulation

    General Polarimeters

    Source Polarization

    OpticsCalibrationPolarization

    DetectorModulatorSpectro-meterTelescope

    polarimeters: optical elements (e.g. retarders, polarizers) thatchange polarization state of incoming light in controlled waydetectors always measure only intensitiesintensity measurements combined to retrieve polarization state ofincoming lightpolarimeters vary by polarization modulation schemepolarimeter should also include polarization calibration optics

    Christoph U. Keller, Utrecht University, [email protected] Observational Astrophysics 2, Lecture 8: Polarimetry 2 17

  • Polarizing Beam-Splitter Polarimeter

    DetectorBeamsplitterPolarizing

    simple linear polarimeter: polarizing beam-splitter producing 2beams corresponding to 2 orthogonal linear polarization statesfull linear polarization information from rotating assemblyspatial modulation: simultaneous measurements of two (or more)Stokes parameters

    Christoph U. Keller, Utrecht University, [email protected] Observational Astrophysics 2, Lecture 8: Polarimetry 2 18

  • Rotating Waveplate Polarimeter

    Detector

    rotating retarder, fixed linear polarizermeasured intensity is function of retardance δ, position angle θonly terms in θ lead to modulated signaltemporal modulation: sequential measurements of I± one ormore Stokes parameters

    Christoph U. Keller, Utrecht University, [email protected] Observational Astrophysics 2, Lecture 8: Polarimetry 2 19

  • Comparison of Temporal and Spatial Modulation Schemes

    Modulation Advantages Disadvantagestemporal negligible effects of flat

    field and optical aber-rations

    influence of seeing ifmodulation is slow

    potentially high polari-metric sensitivity

    limited read-out rate ofarray detectors

    spatial off-the-shelf array de-tectors

    requires up to fourtimes larger sensor

    high photon collectionefficiency

    influence of flat field

    allows post-facto re-construction

    influence of differentialaberrations

    schemes rather complementary⇒ modern, sensitive polarimeters useboth to combine advantages and minimize disadvantages

    Christoph U. Keller, Utrecht University, [email protected] Observational Astrophysics 2, Lecture 8: Polarimetry 2 20

  • Example: ExPo (Extreme Polarimeter)

    Christoph U. Keller, Utrecht University, [email protected] Observational Astrophysics 2, Lecture 8: Polarimetry 2 21

  • Scattering Polarization

    Single Particle Scatteringlight is absorbed and re-emittedif light has low enough energy, no energy transfered to electron,but photon changes direction⇒ elastic scatteringfor high enough energy, photon transfers energy onto electron⇒inelastic (Compton) scatteringThomson scattering on free electronsRayleigh scattering on bound electronsbased on very basic physics, scattered light is linearly polarized

    Christoph U. Keller, Utrecht University, [email protected] Observational Astrophysics 2, Lecture 8: Polarimetry 2 22

  • Polarization as a Function of Scattering Angle

    same variation of polarization with scattering angle applies toThomson and Rayleigh scatteringscattering angle θprojection of amplitudes:

    1 for polarization direction perpendicular to scattering planecos θ for linear polarization in scattering plane

    intensities = amplitudes squaredratio of +Q to −Q is cos2 θ (to 1)total scattered intensity (unpolarized = averaged over allpolarization states) proportional to 12

    (1 + cos2 θ

    )

    Christoph U. Keller, Utrecht University, [email protected] Observational Astrophysics 2, Lecture 8: Polarimetry 2 25

  • Extrasolar Planetary Systems in Polarized LightPast and Present: Detecting planetary systems with indirectmethodsThe Future: Understanding formation and evolution of planetarysystems by direct characterizationThe Problem: Scattered light from central star dominates

    Disks: about 104 times fainter than central starJupiter: 109 times fainter than Sun

    Hubble Space Telescope image of M4 Star Cluster

    Christoph U. Keller, Utrecht University, [email protected] Observational Astrophysics 2, Lecture 8: Polarimetry 2 26

  • Limb Darkening

    Christoph U. Keller, Utrecht University, [email protected] Observational Astrophysics 2, Lecture 8: Polarimetry 2 29

  • Solar Spectral Line Scattering Polarization

    resonance lines exhibit “large” scattering polarization signals

    Christoph U. Keller, Utrecht University, [email protected] Observational Astrophysics 2, Lecture 8: Polarimetry 2 30

  • Zeeman Effect

    photos.aip.org/

    Splitting/Polarization of Spectral Linesdiscovered in 1896 by Dutch physicist PieterZeemandifferent spectral lines show different splittingpatternssplitting proportional to magnetic fieldsplit components are polarizednormal Zeeman effect with 3 componentsexplained by H.A.Lorentz using classicalphysicssplitting of sodium D doublet could not beexplained by classical physics (anomalousZeeman effect)quantum theory and electron’s intrinsic spinled to satisfactory explanation

    Christoph U. Keller, Utrecht University, [email protected] Observational Astrophysics 2, Lecture 8: Polarimetry 2 31

    http://photos.aip.org/

  • Bernasconi et al. 1998

    Zeeman Effect in Solar Physicsdiscovered in sunspots byG.E.Hale in 1908splitting small except for insunspotsmuch of intensity profile due tonon-magnetic area⇒filling factora lot of strong fields outside ofsunspotsfull Stokes polarizationmeasurements are key todetermine solar magneticfields180 degree ambiguity

    Christoph U. Keller, Utrecht University, [email protected] Observational Astrophysics 2, Lecture 8: Polarimetry 2 35

    http://aa.springer.de/bibs/8329002/2300704/small.htm