General Polarimeter Temporal and Spatial Modulation Double

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


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


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


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


Combined Spatial and Temporal Modulation


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


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


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


Example of Solar Double Ratio at 1.56 µm (NIM at McMath-Pierce)


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


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


Back-Calibration

use calibration matrix to calibrate calibration observationsas calibration polarization is known⇒ obtain error estimate ofcalibration


Calibration Input Polarization Errorsorientation of calibration optics not know preciselyretardation of calibration retarders not know preciselyconsider as unknown in fitting of calibration data


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


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


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)


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


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


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


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


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General Polarimeter Temporal and Spatial Modulation Double