Polarization calibration in the presence of a varying ... · Polarization calibration in the...

Polarizationcalibration in the

presence of avarying light

source

Schad

IntroductionSources of variability

Key Questions

Polarized Diction

Current MethodsNormalization

X vs O

Remodulation

NormalizedModulationThe Problem

Ideal Case

OLS Solution

Normalizing

Example

Recommendationsfor DKIST PolcalPlanQuestions

References

Polarization calibration in the presence ofa varying light source

Tom Schad

Institute for Astronomy - University of Hawaii

DKIST Optimum Polarization Calibration WorkshopMay 7, 2015



source

Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References

Introduction

I Optical polarimetry differences multiple measurements ofintensity-encoded polarized signals

I Fluctuations in the total intensity during a single modulationcycle introduces anomalous polarization signal.

I Variations in the total intensity throughout a series of polarizationcalibration may introduce large errors in the calibration.

I This talk reviews and extends methods to afford our calibrationsmore resilience to intensity variations.



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Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References

Sources of light variability

I On the best days, the solar radiance varies according tospectral-dependent atmospheric extinction (airmass).

I Total intensity changes are significant over the time it takes tocomplete one polarization calibration (∼15 minutes)

I Early morning and late afternoon irradiance change can be up to∼ 0.01 percent per second.

I ...modulation rates need to be better than ∼4 Hz for ∼ 10−4accuracy .



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Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References


I Beautiful mornings of observing may have unlucky conditions forafternoon calibrations

I Any cloud cover (e.g. cirrus) can impact polarization calibrationsI Do you rely on the repeatability of instrument configurations in

this case, or try to calibrate?



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Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References


I Lamps pose the same, or similar, problems.I If lamp radiance PDF is Gaussian, then lamp variance is a noise

term to be added in quadrature with other sources.I Gaussian noise may be reduced with additional measurementsI If non-Gaussian, then it gets complicated (similarities to seeing

polarization?)



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Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References

Primary questions for discussion:

I How do we setup the calibration problem to be both "optimal"and resilient against intensity fluctuations?

I Can we formulate the problem such that all DKIST polarimeterscan use one method?

I What is the most efficient way to solve the problem, so that fittingpixel-dependent effects is quick?

I How do we take into account known measurement errors (e.g.,photon noise)? What is the criteria for a good fit to thecalibration curves?

I Once solved, how do we compare result with DKIST "errormatrix"?



source

Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References

Polarimetry Diction

Before proceeding, a few notes on the various matricesused/discussed.

II O is the modulation, analysis, or measurement matrixcorresponding to the linear transformation of a Stokes vector Sinto a vector of measured intensities. Each row is a physicallymeaningful Stokes vector.

I D is the demodulation or synthesis matrix reversing themodulation.

I M is a Mueller matrix. A physically realizable transformation froma Stokes vector to a Stokes vector.

I X is a "polarization response" or calibration matrix. Notnecessarily a Mueller matrix, can be a "fudge" matrix.Sraw = (X + δX)Strue, Smeas = X−1Strue

I δX is an error matrix. Smeas = δXStrue



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Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References

First line of defense is normalization

An absolute calibration is not necessary; we can normalize our data tomeasure only changes in relative polarization.

Azzam & Lopez (1989)

Solves for the modulation ("analysis") matrix O using a set of calibrationvectors S :

I = OS (1)

Removes variation in intensity by separate polarization-independentmeasurements of the intensity, ir

In = I/ir (2)

which allows the use of the normalized Stokes vector Sn

In = OSn (3)

This is the ideal method as it can deal with inter-modulation intensitychanges, but requires high frequency spectroradiometricmeasurements. FOV affects can be tricky.Would be nice to have at DKIST (also helps DL-NIRSP and VTFbalance intensities between frames/fields)!



source

Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References

First line of defense is normalization

Solar instruments (ASP, Hinode, FIRS, etc.) often normalize raw"pre-demodulated" Stokes vectors by the demodulated I to removeintensity variations.The X matrix accounts for errors in the demodulation and crosstalk.

ASP Calibration (Skumanich et al. 1997)

|p〉[N]=

XDmθ LnθTΓ1|eu〉

|XDmθ

Lnθ

TΓ1eu〉j=0

Hinode Calibration (Ichimoto et al. 2008)I′kQ′kU′kV ′k

±

= α±Ik

1 x10 x20 x30

x01 x11 x21 x31x02 x12 x22 x32x03 x13 x23 x33

±

1qkukvk

(4)

Noting that

I′±k = α

± Ik (1 + x10qk + x20uk + x30vk ), (5)

Q′k/I′kU′k/I′kV ′k/I′k

=

q′ku′kk′k

=

x01 x11 x21 x31x02 x12 x22 x32x03 x13 x23 x33

1 + x10qk + x20uk + x30vk

1

qkukvk

(6)



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Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References

Is the ASP / Hinode Approach "Optimal"?

The Good....it works!

Hinode X cal is in general repeatable from the ground.



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Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References


The Ugly!I Starts from pre-demodulated measurements. Assumes some

pre-calibration or pre-knowledge of the polarimeter.I If the demodulation is in error, then X does not have to be

Mueller matrixI Example on next slide



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Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References

Example

Imeas = OtrueStrue (7)

O→ modeled polarimeter (8)

Sraw = DImeas = (OT O)−1OT Imeas (9)

Strue = XSraw (10)

For an 8-state (22.5 deg step) simple waveplate modulator of 120degrees followed by a linear polarizer, an error of 5 degrees in themodeled waveplate retardance gives:

X =

1.00 −0.05 0.00 0.000.00 1.05 0.00 0.000.00 0.00 1.05 0.000.00 0.00 0.00 0.96

(11)

X is not a Mueller matrix! Ugly...not necessarily bad! Makes it a bitharder to define bounds on fitting X.



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Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References


The Bad?I Large errors in the "pre-demodulation" can negatively effect the

efficiency with which the calibration Stokes vectors arerecovered.

Polarimetric Efficency (Del Toro Iniesta & Collados (2000))

ξi =

nn∑

j=1

D2ij

−1/2

(12)

S/N of Stokes state i is maximized when the Euclidean distance ofdemodulation row i is minimized

I Example on next slide



source

Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References

Example

For an 8-state (22.5 deg step) simple waveplate modulator of 120degrees retardance followed by a linear polarizer, an error of 10degrees in the modeled waveplate phase angle gives:

X =

1.000 0.058 0.161 0.0000.000 0.766 −0.643 0.0000.000 0.642 0.766 0.0000.000 0.000 0.000 0.940

(13)

The efficiencies in the case of full knowledge of the modulation matrixare:

ξ = {0.905, 0.530, 0.530, 0.612} (14)

After using D and X, the efficencies are:

ξ = {0.905, 0.530, 0.530, 0.575} (15)

Results in a six percent reduction in Stokes V efficiency. Thankfullysmall, but magnitude depends on the polarimeter setup and the typeof error!



source

Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References


Summary:I ASP / Hinode approach to intensity variations works well.

Solution shown for the pre-demodulated Stokes vector.I Nothing wrong with the approach, but its been shown that

pre-demodulation can be risky as it effects efficiencies.I Going forward, we should adapt the approach of ASP / Hinode

to solve for the modulation matrix directly instead of the X matrix.More free parameters to be fit, but this isn’t a problem. This isaddressed below.

I What about for polarimeters than already do on-board / real-timedemodulation for bandwidth reduction? Two differentapproaches?

I No. Pre-demodulated data can be remodulated with theappropriate "remodulation(?)" matrix.



source

Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References

Re-modulating pre-demodulated data

The demodulation matrix (D) is the left-inverse/psuedoinverse of thefull column rank modulation matrix (O). D has no left-inverse!

Similarly, the modulation matrix (O) is the right-inverse/pseudoinverseof the demodulation matrix (D)

Smeas = DImeas (16)

1 = DD+, (17)

where the ′+′ symbol identifies the pseudoinverse (not a †).

Smeas = DD+Smeas = DImeas (18)

Therefore,

Imeas = D+Smeas (19)

Since a properly constructed D has full row rank, its right-inverse isgiven by this analytical expression for the Morse-Penrosepseudoinverse:

D+ = DT (DDT )−1 (20)



source

Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References

Re-modulating pre-demodulated data: Example

The Hinode on-board demodulation scheme for 8-exposures is givenby:

Donboard =

1 1 1 1 1 1 1 11 −1 −1 1 1 −1 −1 11 1 −1 −1 1 1 −1 −1−1 −1 −1 −1 1 1 1 1

(21)

To re-modulate Hinode data, we calculate and apply theright-pseudoinverse of D:

D+onboard = DT (DDT )−1 (22)

D+onboard =

0.125 0.125 0.125 −0.1250.125 −0.125 0.125 −0.1250.125 −0.125 −0.125 −0.1250.125 0.125 −0.125 −0.1250.125 0.125 0.125 0.1250.125 −0.125 0.125 0.1250.125 −0.125 −0.125 0.1250.125 0.125 −0.125 0.125

(23)



source

Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References

Intensity Normalization for the Modulation Matrix Calibration

Goals:I Mitigate intensity variations while solving for the modulation

matrix, not the X matrixI Keep the ability to refine the parameters of the calibration optics

(i.e. self-calibrate)I An efficient solution to solve for the 4xm parameters of the

modulation matrix



source

Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References

The general calibration problem for DKIST

If we take all the optics between GOS and the various detectors, along with themodulating elements, as constituting a single modulation matrix O, we writethe polarimetric measurement equation as

I = OTR(α)Sinput + d (24)

Assuming unpolarized light into the telescope, the calibration problem is writtenas

In = InOMwp(δ, φn + αoffset )Mlp(θn)

1

t10t20t30

, n = 1, 2, 3, . . . ,N (25)

which can be summarized as

In = OScal,n, n = 1, 2, 3, . . . ,N (26)

where n corresponds to the settings of the polarized state generator (i.e. GOS)



source

Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References

Solution in the idealized case

In the case of no intensity variations, and calibration states that areperfectly known, the solution is simple (no use of mpfit, etc.). Startwith...

I1I2I3...

Im

n

=

O11 O12 O13 O14O21 O22 O23 O24O31 O32 O33 O34

.

.

....

.

.

....

OM1 OM2 OM3 OM4

IQUV

n

, n = 1, 2, · · · ,N (27)

A simple solution to this problem can be found by rearranging theterms into a linear regression equation.

Imn =(Om1 Om2 Om3 Om4

)IQUV

n

, n = 1, 2, 3, . . . ,N; m = 1, 2, 3, . . . ,M. (28)

which is also equivalent to

Imn =(I Q U V

)n

Om1Om2Om3Om4

, n = 1, 2, 3, . . . ,N; m = 1, 2, 3, . . . ,M. (29)



source

Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References

This rearrangement results in..Im1Im2

...ImN

=

I1 Q1 U1 V1I2 Q2 U2 V2...

......

...IN QN UN VN

Om1Om2

...OmN

, m = 1, 2, · · · ,M (30)

orIm = ZStokesOT

m, m = 1, 2, · · · ,M (31)

This results in a linear regression model of the form

y = Zβ + ε (32)

where

Z =

zT

1

zT2...

zTn

(33)



source

Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References

Ordinary Least Squares Solution

For the linear regression problem,

y = Zβ + ε (34)

the ordinary least squares parameter estimator can be written using theMorse-Penrose inverse as:

β = (ZT Z)−1ZT y (35)

So, by writing the set of calibration equation as

Im = ZStokesOTm, m = 1, 2, · · · ,M (36)

we can find the solution for the modulation matrix elements row by row asfollows:

OTM = (ZT

StokesZStokes)−1ZTStokes Im, m = 1, 2, . . . ,M (37)



source

Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References

Normalizing the calibration equations

Now, return to the original form of the calibration equations to introduce anormalization to control intensity variations:

I1I2I3...

Im

n

= In

O11 O12 O13 O14O21 O22 O23 O24O31 O32 O33 O34

.

.

....

.

.

....

OM1 OM2 OM3 OM4

1quv

n

, n = 1, 2, · · · ,N (38)

Note that:I1,n = In(O11 + O12q + O13u + O14v). (39)

Dividing each side of Equation 38 by I1, we get

I2/I1I3/I1

.

.

.Im/I1

n

=

O21 O22 O23 O24O31 O32 O33 O34

.

.

....

.

.

....

OM1 OM2 OM3 OM4

1quv

n

(O11 + O12qn + O13un + O14vn), n = 1, 2, · · · ,N (40)



source

Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References

Rearranging the intensity-normalized set of equations

Doing the same rearrangement as before, we get...

ImI1

=—ZStokesOTm, m = 2, · · · ,M (41)

where the compiled Stokes matrix takes a somewhat more complicated form

—ZStokes =

1/ζ1 q1/ζ1 u1/ζ1 v1/ζ11/ζ2 q2/ζ2 u2/ζ2 v2/ζ2

......

......

1/ζN qN/ζN uN/ζN vN/ζN

(42)

whereζn = (1 + O12qn + O13un + O14vn), O11 set to 1 (43)

and 1qnunvn

= Mwp(δ, φn + αoffset )Mlp(θn)

1

t10t20t30

(44)



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Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References

Solving the problem

To solve this problem...

ImI1

=—ZStokesOTm, m = 2, · · · ,M (45)

the non-linear parameters are those of the calibration optics, the telescopeStokes vector, and the three elements of the first row of the modulation matrix.All other parameters found by ordinary least squares estimator.This assumes you have the correct calibrator states to solve for all of this



source

Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References

An example using FIRS data

FIRS Polcal Data from 21 Oct 2013 - 8 states (4 state repeated)



source

Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References


Measurements normalized by the first modulated intensity



source

Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References


Demodulated results compared to calibration vectors



source

Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References

Polcal Fitted Parameters

OFIRS =

1.000 0.638 −0.117 −0.6730.976 0.302 0.245 0.8040.937 −0.519 0.468 −0.4810.911 −0.704 −0.451 0.1880.999 0.620 −0.102 −0.6640.973 0.285 0.246 0.8100.926 −0.479 0.488 −0.4740.913 −0.699 −0.456 0.197

(46)

The offset of the calibration retarder is 1.98 degrees, and its retardance is 88.08 degrees

ξFIRS = {0.94, 0.55, 0.35, 0.58} (47)



source

Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References




source

Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References

Summary

I Solving for the modulation matrix in the presence of intensityvariations is not that difficult

I It may recover the true efficiencies of each polarimeter morereliably

I Assuming uncorrelated errors, the solution is very fast



source

Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References

Recommendations for DKIST Polcal Plan

1. Stay away from the response matrix X, if possible. It has aconfused meaning, and is not always optimal.

2. Solve the problem with the fewest number of assumptions first!

I Do not apply group model to the original measurements!I Solve the Imeas = OStrue problem really well, and understand the

errors!I Solve this problem for the group model:

Omeas(γ, ψ − φ) = Oinst RM56RM34I If DmeasOinst RM56RM34 < δX for all angles, then great! Whew!I If not, we have to improve upon the group model!



source

Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References

Other various questions:

I Can we fit for a time-variable retardance of the calibration linearretarder if we iterate 0 and 45 degree linear polarizationmeasurements?

I Do we need a stabilized light source? Or, can we use asecondary light sensor (e.g. a modular spectrometer) as anormalizor?

I How do we deal with fringes that are in polcal data but not inscience data?

I What is the criteria for a good fit to the calibrationmeasurements?



source

Schad


Key Questions

Polarized Diction


X vs O

Remodulation


Ideal Case

OLS Solution

Normalizing

Example


References

References:1. Azzam, R.M.A. & A.G. Lopez "Accurate calibration of the four-detector photopolarimeter with

imperfect polarizing optical elements" J. Opt. Soc. Am. A, 6, 1513 (1989)

2. Ichimoto, et al. "Polarization Calibration of the Solar Optical Telescope onboard Hinode" SolarPhysics, 249, 233 (2008)

3. Selbing, J. "SST polarization model and polarimeter calibration." (2010)

4. Skumanich et al. "The calibration of the Advanced Stokes Polarimeter A detailed description ofthe telescope model, calibration procedure and results of it for the ASP."Astrophys.J.Suppl.,110,357 (1997)

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