general polarimeter temporal and spatial modulation double
TRANSCRIPT
Lecture 11: Polarimetry Systems Engineering 2
Outline
1 General Polarimeter2 Temporal and Spatial Modulation3 Double-Ratio Technique4 Calibration5 Polarized Ray Tracing
Christoph U. Keller, [email protected] Lecture 11: Polarimetry Systems Engineering 2 1
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, [email protected] Lecture 11: Polarimetry Systems Engineering 2 2
Spatial Polarization Modulation
DetectorBeamsplitterPolarizing
polarizing beam-splitter polarimetersimple 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, [email protected] Lecture 11: Polarimetry Systems Engineering 2 3
Temporal Polarization Modulation
Detector
rotating waveplate polarimeterrotating retarder, fixed linear polarizermeasured intensity as function of retardance δ, position angle θ
I′ =1
2
„I +
Q
2((1 + cos δ) + (1− cos δ) cos 4θ) +
U
2(1− cos δ) sin 4θ − V sin δ sin 2θ
«
only terms in θ lead to modulated signalequal modulation amplitudes in Q, U, and V for δ=127◦
temporal modulation: sequential measurements of I± one ormore Stokes parameters
Christoph U. Keller, [email protected] Lecture 11: Polarimetry Systems Engineering 2 4
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, [email protected] Lecture 11: Polarimetry Systems Engineering 2 5
Combined Spatial and Temporal Modulation
Christoph U. Keller, [email protected] Lecture 11: Polarimetry Systems Engineering 2 6
Double-Ratio Techniquecombination of spatial and temporal modulationdata reduction minimizes effects of many artifactsrotatable quarter-wave plate, polarizing beam-splitterconsider case of circularly polarized lightquarter-wave plate switches between +45◦ or -45◦ to polarizingbeam-splitterboth beams recorded simultaneouslyfour measurements are combined to obtain estimate ofStokes V/I ratio largely free of effects from seeing and gainvariations between different detector areasexcellent if polarization signal is smallfrequently used in stellar polarimetrycan be applied to any polarized Stokes parameterworks very well for solar applications where the spectrum in thefirst and the second exposures are different
Christoph U. Keller, [email protected] Lecture 11: Polarimetry Systems Engineering 2 7
Double-Ratio Technique (continued)measured intensities in two beams in first exposure
Sl1 = glα1(I1 + V1), Sr
1 = grα1(I1 − V1)
subscript 1: first exposuresubscripts l , r : left, right beamsS: measured signalg: gain in particular beamα: transmission of atmosphere, instrument
second exposure
Sl2 = glα2(I2 − V2), Sr
2 = grα2(I2 + V2)
incoming I and V in second exposure may be completely differentfrom first exposurealso includes beam-wobble induced by rotation of wave plate
Christoph U. Keller, [email protected] Lecture 11: Polarimetry Systems Engineering 2 8
Double-Ratio Technique (continued)combination of 4 measured intensities removes effect oftransmission changes and differential gain variations of differentdetector areas
14
(Sl
1
Sl2
Sr2
Sr1− 1
)=
12
I2V1 + I1V2
I1I2 − I2V1 − I1V2 + V1V2
if V � I12
(V1
I1+
V2
I2
)obtain average V/I signal of two exposuresno spurious polarization signals are introduced
Christoph U. Keller, [email protected] Lecture 11: Polarimetry Systems Engineering 2 9
Example of Solar Double Ratio at 1.56 µm (NIM at McMath-Pierce)
Christoph U. Keller, [email protected] Lecture 11: Polarimetry Systems Engineering 2 10
Calibration
Basic Approachoptimum way to calibrate polarimeter, i.e. measure X
~S = X~I
at least 16 measurements of signal vector elements ~Si , i = 1..mat least 4 different input Stokes vectors~Ic
i , i = 1..mgroup calibration input Stokes vectors, signal vectors into 4 by 4matrices (Azzam et al. 1988)
S = XIc
S =(~S1~S2...~Sm
)Ic =
(~Ic
1~Ic
2 ...~Ic
m
)signal matrix X = SJ with IcJ = 1
Christoph U. Keller, [email protected] Lecture 11: Polarimetry Systems Engineering 2 11
Equivalence with Maximum Polarimetric Performance
choose calibration Stokes vectors~Ici to minimize error in X, given
errors in measurements Sequivalent to optimizing polarimetric efficiencycalibration input Stokes vectors~Ic
i equivalent to rows of the signalmatrix X of maximum performance polarimetervector-polarimeter: corners of a tetrahedron as suggested byAzzam et al. (1988)only true if there are no systematic errors (input polarization isknown completely)better to use many more measurements and also model non-idealcalibration optics
Christoph U. Keller, [email protected] Lecture 11: Polarimetry Systems Engineering 2 12
Back-Calibration
use calibration matrix to calibrate calibration observationsas calibration polarization is known⇒ obtain error estimate ofcalibration
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Calibration Input Polarization Errorsorientation of calibration optics not know preciselyretardation of calibration retarders not know preciselyconsider as unknown in fitting of calibration data
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Measuring Modulation Efficiencypolarization modulator and detector should not depolarizeshould measure fully polarized light on detectorimplies that (X 2
2 + X 23 + X 2
4 )/X 21 = 1
can empirically determine this ratio from calibration data
Christoph U. Keller, [email protected] Lecture 11: Polarimetry Systems Engineering 2 15
Polarized Raytracing
Introduction“non-polarimetric” optical elements influence polarizationcharacteristics of “polarimetric” components depend on angle ofincidence“polarizing” components influence optical performance inunexpected ways (e.g. crystal astigmatism)need methods to predict optical (and polarimetric) performancegeometrical optics: λ� diameter of lenses, apertures etc.locally treat electromagnetic field as plane waveevaluate using rays (plane waves with very limited transverseextensions)trace monochromatic rays through optical systemuse Fresnel equations etc. to determine polarizationcharacteristics of individual rays
Christoph U. Keller, [email protected] Lecture 11: Polarimetry Systems Engineering 2 16
Image Formation
image of a point source: point-spread functionimage of extended object is convolution of object with PSFgeometrical PSF (spot diagram) obtained by tracing many raysfrom point source that uniformly fill entrance aperturegeometrical PSF of perfect optical system is a pointdiffraction at aperture stop makes PSF with finite extensionPSF in case of Frauenhofer diffraction (focal plane far away fromaperture stop) is given by magnitude squared of Fourier transformof electric field over aperture)
Christoph U. Keller, [email protected] Lecture 11: Polarimetry Systems Engineering 2 17
Stokes and Jones Vector Raytracing
highest-end version of commercial software packages (e.g.ZEMAX) can do (limited) calculations with polarized lightcan correctly propagate light in birefringent media and propagateJones vectorsaveraging Stokes parameters in focal plane corresponds toStokes vector of the average over the PSFaverage of Jones vector in focal plane corresponds to Jonesvector at center of PSF
Christoph U. Keller, [email protected] Lecture 11: Polarimetry Systems Engineering 2 18
Example: Ritchie-Chretien telescope with field corrector
SOLIS Vector-Spectromagnetographnot diffraction-limited⇒ Stokes/Mueller calculus appropriatewithout excellent anti-reflection coating on lenses, about 1% linearpolarization towards edge of field
Christoph U. Keller, [email protected] Lecture 11: Polarimetry Systems Engineering 2 19
Polarization Ray Tracing
combines conventional geometric ray tracing and Jones formalismin elegant way (Chipman, 1989)~E0 in plane-wave ansatz
~E = ~E0ei(~k ·~x−ωt)
describes polarization of lightvector has three components Ex , Ey , Ez
in isotropic medium, ~E0 is perpepdnicular to ~kamplitude matrices P operate on incident polarization vectors ~E0to produce emerging polarization vector ~E ′0complex components of P describe influence of interface betweentwo materials on wave vector direction and polarization
Christoph U. Keller, [email protected] Lecture 11: Polarimetry Systems Engineering 2 20
Adding polarization vectors
parallel beams: Jones and Stokes vectors can be addedif beams propagate in different directions, needs to transform intocommon coordinate system before addingpolarization vector approach operates in global coordinate systemwhere ~E0 also contains information about propagation directionpolarization vectors can be added directlycorresponding Jones vector extracted from two components of ~E0that are perpendicular to propagation direction
Christoph U. Keller, [email protected] Lecture 11: Polarimetry Systems Engineering 2 21