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 = pyEy
px = 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
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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
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 δ
2 00 e−i δ
2
)
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
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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
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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
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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
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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 1
2
(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
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Solar Spectral Line Scattering Polarization
resonance lines exhibit “large” scattering polarization signals
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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
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
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