1 SOT Polarization Calibration -- method and results for FG -- K.Ichimoto and SOT Team SOT#

2 0. Descriptions of the SOT polarimeter Schematics of the SOT polarimeter Pupil image HDM Polarization modulator (PMU) CTM-TM OTA M1 M2 FG/NFI SP FG-CCD SP-CCD left/right SP- Polarization analyzer (polarizing beam splitter) NFI- Polarization analyzer Non- polarizing beam splitter Collimato r lens unit (CLU) Tunable filter Slit scan mirror Mask wheel Mech. shutter Astigmatism collector lens (ACL) Reimaging lens Slit

3 Modulation and sampling scheme The polarization modulator is a continuously rotating waveplate at the rate of 1rev./1.6sec. Retardation is optimized for equally modulating circular and linear polarization at 6302A and 5172A. Both SP and FG take multiple images in synchronous with PMU and appropriate demodulation is applied onboard to reduce the amount of telemetry data and to improve S/N. SP takes 16 frames in every PMU revolution in both orthogonal states of polarization. FG has a variety of sampling scheme. In ‘shutter mode’, the mechanical shutter is used to take large area of CCD. In ‘shutterless mode’, continuous readout is performed for central area of CCD with masked outer parts of the CCD. Typical sampling schemes are as follow: Example of demodulation matrix for IQUV 16 frame continuous sampling Shutter mode: -8 exposures by 22.5deg step for IQUV -4 exposures for IQUV -2 exposures for IV -etc. (exposure time is flexible) Shutterless mode: -16 frames/rev. as SP for IQUV -4 frames/half rev. for IV -2 frames/half rev. for IV -etc. (accumulation number is flexible)

4 1. Definition of polarimeter response matrix and its tolerance S’ = XS, X: SOT ‘polarimeter response matrix’ Incident Stokes vector is obtained by S” = X -1 S’ ‘Polarization calibration’ is to determine the X for each SOT product. Calibration error :  S ” = S ” - S = { X r -1 X- E } S Statistical noise :  S ” = X r -1  S ’ = X r -1  where X : true matrix (unknown) X r : matrix used in calibration Polarization modulation Onboard demodulation SOT product Incident Stokes vector Modulated intensity on CCD IiIi S S’

5 Requirement on the calibration accuracy Solar-B, SOT  = a = 0.05 P l = 0.15 （ max of Q,U ） P v = 0.2 （ max of V ） Tolerance of X 1) for crosstalk among different elements of S (off diagonal of X)  S ” <  S ”  { X - X r } S =  X S <  2) for scale error (diagonal of X and QUV  I crosstalk )  Q”/I” ) < a Q/I

6 Sheet polarizer window (I,Q,U,V) mask FPP Heliostat 2. Polarization calibration test method Test configuration -Entire SOT is located under a heliostat in a clean room. -Sunlight fed by the heliostat -Sheet polarizers (linear, L/R circular) on OTA -Room T=20C, CLU T>25C

7 FPP +Q +U UU View from the top of SOT QQ VV +V View towards the sun S/C +Y S/C +X W N S E +Q QQ UU +U VV +V Definition of SOT polarization coordinate

8 FPP 0゜0゜ 45 ゜ 90 ゜ 135 ゜ HNCP37R HN38 +Y +X View from top Created Stokes vectors HNCP37L (only for ) Configuration of sheet polarizer in SOT suntest / ゜0゜ P l ~ 1

9 Test cases

10 3. Derivation of X matrix: (FG/NFI) # of unknowns: x ij 15 = 15 (with x 00 = 1) P lR, P lL,  R,  L (linear polarization and offset angle of RCP,LCP) are determined from average over the CCD and then fixed in fitting for each pixel assume P cR 2 + P lR 2 = 1, P cL 2 + P lL 2 = 1 # of equations : 3x12 = 36 S k ’ : polarimeter product s k : incident Stokes vector with I=1. k stands for polarizer config. 0,1,~,11 Relation between FG products and incident Stokes vector Fitting equation; normalized by I’ k to eliminate the sky fluctuation Incident Stokes vectors determined by sheet polarizers SOT products

11 4. Fitting results for polari. cal. data (an example) NFI shutterless: 630nm, CCD center Q V U I Symbols: observed Lines: fitting

12 5. SP X matrix Be presented by B.Lites

13 6. FG X matrices 6303, Shutter 2048x1024 (2x2sum, OBS_ID=3) right: theta= 3.555deg left: theta= 3.648deg Mean X matrices for left and right halves of CCD Horizontal lines show the tolerance of each element.

14 right: theta= deg left: theta= deg Horizontal position All points of CCD plotted Dot lines are tolerance Mean X matrices for left and right halves of CCD Rotation of Q-U frame between left and right halves of CCD caused by the delay of exposure. Cause a rotation of B azimuth by about 3 deg. 6302, Shutterless 80x1024 (2x2sum) spxmat_0506p.pro

15 Left: d  = Repeatability, shutterless 6302 Difference 6/14 – 6/13 Repeatability is good enough compared with the tolerance matrix. right : d  = right: theta= deg left: theta= deg /6/ /6/14 right: theta= deg left: theta= deg

16 7. Summary of representative X matrices Delay between left and right CCD in PMU angle (deg.)

17 Delay between left and right CCD in PMU angle (deg.)

18 8. Modeling of SOT polarization (for NFI) FG has a variety of observing sequence with different exposure, on-chip summing and polarization sampling scheme, and we do not have the experimental X matrix for all of them. To extend our knowledge of the X matrix of the tested cases, a simple SOT polarization model is created with which one can obtain the X matrix for arbitrary observing scheme. Assumptions in the model: -Ideal PMU retarder and polarization analyzer, -Exposure length and mutual separation of exposure are as specified by the command, while a constant delay of exposure is incorporated, -Residual deviations of X from the theoretical matrix are attributed to the ‘telescope’ matrix. S SOT = D W T S in, X =D W T D : demodulation matrix W (k,*) = (1,1,0,0) P(  ,  k,  t, dt) : polarization modulation matrix T : ‘telescope’ matrix   : retardation of the waveplate  k : phase angles of PMU at each exposure  t : exposure time dt : delay of exposure timing

19 Example of DWT matrix: OBS_ID=3, FGIQUV (shutter mode) 8 exposures at  k = *[0,1,2,3,4,5,6,7] D W T Demodulation matrix Modulation matrix‘Telescope’ matrix

20 Least square fitting to the experimental X matrices X experiment  X fit = D W(  k,  t, dt,   ) Least square fitting  dt,  T ( ) = X fit, -1 X ex Wavelength , …. modeshutterlessshuttered Data setData-1Data-2….Data-1Data-2,,,…. X ex left X ex right X ex left X ex right X ex left X ex right X ex left X ex right X ex Fitting parameters   ….   dt ….dt …. TTTT TTTT Average for SOT model  dt_left, dt_rightdt T  k and  t are specified for each data set X ex are averaged over the each CCD-left and right Standard deviation of the fitting residual  X = X ex (i)  X mode is compared with the tolerance matrix. SOT polarization model is given by ,  dt, T( ) are determined for each data set, and then averaged over the wavelength or mdoes

21 More about the ‘exposure’ in FG shutterless mode x p  mask Geometrical sketch In shutterless mode of FG, each pixel experiences ‘smearing periods’ during the frame transfer. xmxm t0t0 t2t2  t =  t 1 +  t 2 +  t 3 ‘exposure’ cycle (typ. =100ms) time Time sketch illuminated period t3t3 Start of exposure: t s Start at CCD center: t 0 End of exposure Start of transfer: t e  t 2 +  t 3 = 1024  2048 t1t1 CCD CCD center x=0 pixel position x p  t 1 : exposure at the pixel position  t 0,  t 2 : smearing period  t 3 : transfer time under mask  t eff =  t 0 +  t 1 +  t 2 : illuminated period

22 In polarization calibration test: S in0 = S in2 = S in1 S SOT = D [ W(  t 0 ) + W(  t 1 ) + W(  t 2 )] T S in1  X ex = X(  t 0 ) + X(  t 1 ) + X(  t 2 ) In real observation: S in0 = S in2 = (I,0,0,0) t (assume that smear regions have mixed polarity to give Q,U,V =0) S SOT = (  I, 0, 0, 0) t + X(  t 1 ) S in1 (assume T ~ 1)  I = I (  t 0 +  t 2 ) / (  t 0 +  t 1 +  t 2 ) -- bias intensity due to smearing  X(  t 1 ) is what we need for polarization calibration for NFI/shutterless mode. - X ex depends on both mask size and pixel position on the CCD. - X(  t 1 ) is independent on the mask size nor pixel position. The SOT polarization model takes this point into account and can provide ether X(  t 1 ) or X ex. SOT product is summation of the contributions from three periods  t 0,  t 1,  t 2 S SOT = D [ W(  t 0 ) T S in0 + W(  t 1 ) T S in1 + W(  t 2 ) T S in2 ] Modification of X matrix due to smearing

23 9. Results of model fitting Average , dt left, dt right Wavelength (nm) Retardation (wave) Modulation amplitude ( Diagonal element of X) Time delay of t c (ms) shutterlessshuttered design measured QUVleftrightleftright Averaged retardation of the waveplate (  ) and exposure delay (dt left, dt right ) obtained by the model fitting are given below for each wavelength and for shuttered or shutterless modes.

24 Data points refer to t c, independant on mask nor pix.pos. Exposure timing wrt PMU phase, t c : center of readout cycle ~10ms Shutterless mode Tolerance of exposure timing ~ 2ms

Q U V SOT modulation profiles with obtained PMU retardance Wavelength (nm) Retardation (wave)

26 Red elements are larger than tolerance T-matrix is sometimes unphysical. This may be due to incomplete modeling of the SOT polarization.  X = X obs  X model SOT polarization model well reproduces the experimental X matrix except the first column. Fitting residual

Other observing schemes NFI IV observation (shutter mode) Shuttered IV (Obs_ID = 2) exposure = 100, 150, 200, 300, 400ms demodulation NFI can takes IV information with only 2 exposures centered at the PMU phases of +45 deg. (see figure below). The exposure time is selectable. tt 2 intensities are given by. I + = I + c Q Q + c V V I  = I + c Q Q  c V V where c Q : Q  I crosstalk c V : Efficiency of V measurement I ’ = x 00 x 10 x 20 x 30 I V ’ x 03 x 13 x 23 x 33 Q U V In this case the X matrix is a 4x2 matrix. The X matrices for this mode with different exposures were not measured with real SOT and the verification test was performed with FPP+PMU(backup) on For details, ‘polarization t-cal.ppt’

28 dash: Q  I crosstalk, c Q ms 304ms 380ms Theoretical c Q, c V vs. exposure time solid: sensitivity to V, c V

Critical components – CLU – In the development of SOT, the polarization properties of all optical elements were verified by theoretical prediction and experiments. Critical components in polarizational point of view were identified as PMU, CLU and astigmatism corrector lens (ACL) since they are located in upstream of the optics and their thermal environment is not well controled as in the FPP. Special attention was paid on their opto-thermal characteristics. It was turned out that the PMU and ACL are stable enough against the possible temperature excursion in orbit, while the CLU is quite sensitive to temperature; especially in the cold case, the mechanical stress on the glasses induced from the metal housing cause a significant retardation, and this drove us to set the lowest ‘operational temperature range’ of CLU as 25C. Extensive tests was made by using the ‘Component Polarization Analyzer’ of HAO for the CLU flight model mounted in a thermal shroud.

30 T=15C (from 20C) T=30C (from 40C) CLU Mueller matrix image at different temperatures (example) Rectangular shows the SOT field of view. Interval of contours indicates the tolerance of each Mueller matrix element.

31 Hysteresis of (3,4) element (=linear retardation) of the CLU Mueller matrix against temperature after vibration after 2 nd /3 rd cold cycle after 1 st cold cycle initial after 4th cold cycle torelance

32 -The only significant polarization property of CLU is the linear retardation. -The CLU retardation can be regarded as uniform over the SOT field of view and constant against T if temperature is higher than 25C (=lower limit of operational temp.). -The experimental X matrices of SOT include the CLU retardation, but the CLU may have a small retardation offset after the launch vibration and the initial low-T cycle. -Signature of circular to linear crosstalk needs to be checked after launch using sunspots. Summary of the CLU polarization-thermal properties :

33 Summary -Polarimeter response matrices (X) were obtained experimentally using entire SOT for representative products of NFI as functions of position in FOV. -The accuracy of measurements inferred from the repeatability meets the required accuracy of X except for the first column. -The X-matrices can be regarded as uniform over the field of view except the NFI shutterless mode, in which the left and right halves of CCD have a non negligible difference due to the relative delay of exposures. -The ‘SOT polarization model’ reproduces experimental X matrices of NFI with the required accuracy, and can be used to get the X matrices of other observing sequences for which the experimental X matrix was not obtained. IDL procedure ‘nfi_modelx’ is prepared (Appendix). -The first columns of X matrices will be determined more exactly after launch using the continuum of the sun light. -The SOT polarization characteristics is expected to be fairly stable in orbit, while the linear retardation of CLU might have a small offset after experiencing the launch environment. This will be checked in real sun data.

34 X = nfi_modelx(wav, obs_id=obs_id, expo=expo) or X = nfi_modelx(wav, pmupos=pmupos, Dmat=Dmat, expo=expo, delay=delay,, $ Tmat=Tmat,,mask=mask, ix=ix ) ; wav- wavelength, [nm] ;obs_id- if set, pmupos and Dmat are taken from fpp_obsid.pro ; expo - exposure, [ms] ; pmupos(*) - PMU angles at the center of exposure, [deg]. ; Dmat(*,4) - demodulation matrix ; delay(1 or 2)- delay of exposure for (left/right) CCD, [ms], if not set, use cal.data ; Tmat(4,4) - Telescope matrix, if not set, use cal.data, Tmat =1 for unit matrix ;mask- mask# for shutterless mode ;ix- pixel position from CCD center ; if mask and ix are set, return experimental X IDL procedure to obtain NFI X If ‘delay’ and ‘Tmat’ are not specified, experimental data are used, thus X is the most probable X(  t 1 ) for use of real sun data.