An efficient modulation scheme for dual beam polarimetry
K. Nagaraju, K. B. Ramesh, K. Sankarasubramanian, K. E. Rangarajan
aa r X i v : . [ a s t r o - ph ] O c t Bull. Astr. Soc. India (0000) , 000–000 An Efficient Modulation Scheme for Dual BeamPolarimetry
K. Nagaraju ∗ , K. B. Ramesh, K. Sankarasubramanian † and K. E. Rangarajan Indian Institute Astrophysics, Bangalore 560 034, India † ISRO Satellite center, Bangalore 560094, India
Abstract.
An eight stage balanced modulation scheme for dual beam po-larimetry is presented in this paper. The four Stokes parameters are weightedequally in all the eight stages of modulation resulting in total polarimetric effi-ciency of unity. The gain table error inherent in dual beam system is reduced byusing the well known beam swapping technique. The wavelength dependent po-larimetric efficiencies of Stokes parameters due to the chromatic nature of thewaveplates are presented. The proposed modulation scheme produces betterStokes Q and V efficiencies for wavelengths larger than the design wavelengthwhereas Stokes U has better efficiency in the shorter wavelength region. Cali-bration of the polarimeter installed as a backend instrument of the KodaikanalTower Telescope is presented. It is found through computer simulation thata 14% sky transparency variation during calibration of the polarimeter canintroduce ≈ .
8% uncertainty in the determination of its response matrix.
Keywords : Instrumentation : polarimeter
1. Introduction
Polarimetric accuracy is one of the most important goals in modern astronomy. It islimited since most optical elements encountered by the light on its path from the sourceto the detector, can alter its state of polarization (for eg. telescope optics, imagingsystem, grating, etc). Apart from these, variation in sky transparency, image motion andblurring due to the atmosphere are a major concern in high precision ground based solar ∗ e-mail:[email protected] K. Nagaraju, K. B. Ramesh, K. Sankarasubramanian and K. E. Rangarajan polarimetry. The effect of atmosphere, which is commonly known as seeing induced effect,can be reduced by fast modulation schemes(Stenflo and Povel, 1985). The modulationfrequencies in these schemes are generally higher than seeing fluctuations, which is ≈ kHz (Stenflo and Povel, 1985 and Lites, 1987). Large format CCDs, which are requiredto cover reasonable spectral and spatial range, will pose difficulty in reading out the dataat kHz speed. Stenflo and Povel(1985) proposed a scheme whereby rapidly modulatedsignal is demodulated by optical means, thereby avoiding the need to read the detectorsat a rapid rate. Lites(1987) has proposed a rotating waveplate modulation scheme as analternative to minimize the seeing induced cross-talk among Stokes parameters. Therehe has shown that the faster the rotation rate of the modulator, the lower the cross-talkamong Stokes parameters. And the seeing induced cross-talk levels of a dual beam systemare factors 3-5 smaller than those of a single beam system. However, in dual beam system,the error introduced due to flat field residual is a matter of concern in high precisionpolarimetry. A possible solution to the above mentioned problems can be found by usinga mixed scheme in which spatial and temporal modulations are performed(Elmore et al.1992, Martinez Pillett et al. 1999 and Sankarasubramanian et al. 2003). The gain tableuncertainties are avoided using the beam swapping technique(Donati et al. 1990, Semelet al. 1993 and Bianda et al. 1998)A low cost dual beam polarimeter has been installed as a backend instrument for theKodaikanal Tower Telescope (KTT). Different modulation schemes were studied and anoptimum scheme is identified. The proposed scheme requires eight stages of modulationof input light in order to obtain the maximum polarimetric efficiency. Laboratory ex-periments have been performed to verify the theoretical understanding of the proposedscheme. The studies are extended to other wavelengths apart from the design wavelengthof λ
2. Proposed Modulation Scheme
A zero-order quarter wave (R1) and a zero-order half wave (R2) retarders at λ > with respect to a polarizingGlan-Thomson prism (GTP). A simplest way of measuring Stokes parameters is to use ahalf waveplate (HWP) along with PBS for linear polarization measurement and a quarterwaveplate (QWP) along with PBS for circular polarization measurement (Bianda et al.1998). However, this way of modulation will introduce a possible differential optical n efficient modulation scheme for dual beam polarimetry The input polarization is modulated on to intensity by using the waveplate orientationsgiven in Table.1. Modulation Orientation of Orientation ofstage QWP(R1) HWP(R2)1 22.5 02 22.5 453 67.5 454 67.5 905 112.5 906 112.5 1357 157.5 1358 157.5 180
Table 1.
Orientation of Waveplates for different stages of modulation expressed in degrees.
The modulated intensities ~I ± = ( I ± , I ± , I ± , I ± , I ± , I ± , I ± , I ± ) T , where T representstranspose operator, can be written in terms of input Stokes parameters as ~I ± = g ± O ± ~S in . (1)Where ± indicate the two orthogonally polarized beams emerging out of the polarimeterrespectively. ~S in = [ I, Q, U, V ] T is the input Stokes vector to the polarimeter. Here, thestandard definition of the Stokes vector is used with I representing the total intensity, Q and U representing the linear polarization state and V representing the circular polariza-tion state. The multiplication factor of the two orthogonally polarized beams g ± , knownas the gain factor, is a product of transparency of the corresponding optical path and thedetector gain factor. The analyser Mueller matrices of respective beams can be obtainedby multiplying the Mueller matrices of retarders( M R and M R ) and PBS( M ± P ) in theorder M ± P M R M R (del Toro Iniesta, 2003 and Stenflo, 1994). The modulation matrices O ± are constructed by arranging the first row of the analyser matrix of the respectivebeam for each of the measurement steps(see del Toro Iniesta, 2003 for details).The theoretical modulation matrices O ± at the design wavelength are given below. K. Nagaraju, K. B. Ramesh, K. Sankarasubramanian and K. E. Rangarajan O ± = 0 . . ± . ∓ . ± . . ∓ . ± . ∓ . . ∓ . ∓ . ∓ . . ± . ± . ± . . ± . ∓ . ∓ . . ∓ . ± . ± . . ∓ . ∓ . ± . . ± . ± . ∓ . . It is to be noted here that the four Stokes parameters are modulated on to intensityin all the eight stages of measurements. Also that in all the eight stages of measure-ments, each Stokes parameter is weighted equally. Matrices O ± show that the alternatemeasurements are obtained by swapping the orthogonally polarized beams (seen as signchange in the corresponding Stokes parameters).The maximum efficiencies of the modulation scheme (see del Toro Iniesta and Col-lados, 2000 and del Toro Iniesta, 2003 for details) in measuring Stokes I , Q , U , V are1.0, 0.5, 0.5, 0.707 respectively, at the design wavelength. The total polarimetric effi-ciency is √ . + 0 . + 0 . = 0 . O ± are not the same at different wavelengths and hencethe maximum efficiencies of modulation scheme in measuring Stokes QUV are different.However, the total polarimetric efficiency will remain close to unity.For comparison, the maximum efficiencies of some of the well known polarimetersare given below: ASP-(1.0, 0.546, 0.41, 0.659), ZIMPOL-(1.0, 0.474, 0.467, 0.534), TIP-(1.0, 0.617, 0.41, 0.659), POLIS-(1.0, 0.494, 0.464, 0.496). In the above examples, onlyTIP has a total polarimetric efficiency close to unity. The demodulation of the input Stokes parameters from the modulated intensities involvethe following steps. As a first step, the signal vectors ~S ± (Gandorfer, 1999 and Stenflo,1984) of the orthogonally polarised beams are constructed from the modulated intensities(Eq. 1) using the equation, ~S ± = D ~I ± / . (2)Where the matrix D is defined as, D = . . . . . . . . . − . − . . . − . − . . − . . − . . − . . − . . . − . − . . − . . . − . . n efficient modulation scheme for dual beam polarimetry D that, to derive signal vectors all the eight stage intensitymeasurements are considered. Hence the final derived input Stokes parameters will bewell balanced with respect to changes, if there are any, during measurements in the inputStokes parameters. Also, the final derived Stokes parameters will be an average over thetime taken for the eight stages of modulation.The second step involves combining the signal vectors of the orthogonally polarizedbeams after correcting for the gain factors g ± . Gain table corrections can be done eitherby regular flat field procedure or normalizing the elements of signal vectors to theirrespective first element (i.e. ~S ± /S ± (0)). Since the regular flat field procedure limits thepolarimetric precision and to make use of the advantage of the beam swapping techniqueincorporated in the modulation scheme, second method is used to derive the combinedsignal vector. The signal vector( ~S ′ ) of the combined beam can be written as S ′ (0) = S + (0) + S − (0) (3) S ′ (1) = S + (1) /S + (0) − S − (1) /S − (0) S ′ (2) = S + (2) /S + (0) − S − (2) /S − (0) S ′ (3) = S + (3) /S + (0) − S − (3) /S − (0)where S ′ ( i ), i = 0 , , ,
3, are the elements of ~S ′ . Similarly S ± ( i ) are defined. The indices i = 0 , , , ~S ′ = M ~S ′ in . (4)Where ~S ′ in = [ I, Q/I, U/I, V /I ] T is the input Stokes vector and M is a 4 × M = . . . . The third and final step involves obtaining the input Stokes vector( ~S ′ in ) from Eq.(4).During the observations, the response matrix is obtained using a polarimetric calibrationprocedure which will be detailed in Section 4. K. Nagaraju, K. B. Ramesh, K. Sankarasubramanian and K. E. Rangarajan
The efficiency of the polarimeter in measuring respective Stokes parameter is defined as(Beck et al. 2005) ǫ i = s X j =1 , M ji . (5)where i = 0, 1, 2, 3 corresponds to I, Q, U, V respectively. Since the response matrix M in Eq.(4) is diagonal, the efficiency in the Eq. (5) will be simplified (using Eq. 4) to ǫ i = | S ′ ( i ) /S ′ in ( i ) | . (6)We would like to note here that the response matrix M in Eq.(4) is diagonal at all thewavelengths considered here. However, at the wavelengths away from the design wave-length, the signals S ± (0) in Eq.(2) are not just proportional to input Stokes I but witha small contribution from the input Stokes Q. If the signal vectors( ~S ± ) of orthogonallypolarised beams are combined without normalising to their respective Stokes I signal thenthis cross-talk term will not appear in the signal vector( ~S ′ ) of the combined beam. But,the flat fielding is essential to remove the gain factors g ± . In this paper the signal vectorsare combined in such a way that the Stokes QUV signals are normalised to Stokes I signalin order to remove the gain factors. This results in an over estimation of efficiency ǫ Q .This over estimation is about 1.8% at λ λ λ λ λ λ
3. Laboratory Experiment
The efficiency of the polarimeter is different for different wavelengths due to the chromaticnature of the retarders used in the polarimeter. If the polarimeter is used at differentwavelengths then it is important to understand its performance at the desired wavelength.The variation of the efficiency factor for different input Stokes parameters is studied bycarrying out a few laboratory experiments.The experimental setup is shown in Fig.1. The light from the monochromator wasset at the desired wavelength and then passed through a 1 mm rectangular aperture.This rectangular aperture was imaged on to a CCD detector using a two lens systemwith an effective focal length of 12 . cm . The polarimeter optics were placed betweenthe lens and the detector. The first in the light beam is the QWP (R1), followed by theHWP (R2) and the PBS. The retarders (R1 and R2) of the polarimeter were mountedon two different rotating stages whose rotational accuracy is 0 . o . A known state of n efficient modulation scheme for dual beam polarimetry R2R1GTP RAS CCDPBSL2L1 W (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)
Figure 1.
Block diagram of an experimental setup used to study the polarimetric efficiency.The symbols in the figure are S-monochromator, A-mount holding a 1 mm square aperture, L1and L2-lenses(f=25cm), GTP-Glan Thomson prism used to produce linear polarization, R-QWPused along with GTP to produce circular polarization, R1 and R2- Half and Quarter waveplateswhich forms a part of the polarimeter, PBS-polarizing beam splitter, W- CCD window. polarization was produced using the GTP and calibration retarder(CU) R. Stokes Q and U were produced by using only the GTP where as both GTP and R were used toproduce Stokes V . (The CU retarder is a zero order chromatic waveplate which acts asa quarter waveplate at λ D using Eq.(2). The corresponding Stokes QUV signals werenormalised to the respective Stokes I signal as in Eq.(3).We noted in the section 2.3 that the theoretical response matrix of the polarimeterpresented in this paper is diagonal at all the wavelengths considered. However, in practicethere will be off-diagonal elements which are nothing but the cross-talk among Stokesparameters. But, in the experiment performed to study the polarimetric efficiency, themeasured cross-talk terms are small. Since the cross-talk terms are small, the simplifieddefinition of Eq.(6) is used to calculate the efficiency.Plots of polarimetric efficiency in measuring Stokes Q , U and V as a function ofwavelength are shown in Fig.2. The diamond symbols shown in the plots are the experi-mental values and the solid lines are theoretical curves. It is clear from the plot that theexperimental values closely resemble the theoretical predictions. From this figure it canbe concluded that Q and V are measured with better efficiencies in longer wavelength K. Nagaraju, K. B. Ramesh, K. Sankarasubramanian and K. E. Rangarajan region compared to the design wavelength, where as U is measured better at shorterwavelengths.
4. Calibration of the Polarimeter Installed at KTT
The calibration of the polarimeter, which uses the modulation scheme presented in section2, installed as a backend instrument at KTT is discussed in this section. KTT is athree mirror coelostat system for solar observations(Bappu, 1967). It is equipped with aLittrow mount spectrograph using a grating of 600 lines per mm. The polarimeter setupis placed in the converging beam (f/90) of the telescope before the spectrograph slit. Theretarders are mounted on rotating stages which can be rotated from 0 to 360 o in steps of22 . o . PBS is fixed in position with one of its optic axes parallel to the slit direction. Alinear polarizer, called as a compensator, is placed behind the PBS to compensate for thedifferential grating efficiency of the two orthogonally polarized beams. The compensatoris oriented at 45 o to the grating grooves to make the efficiency of the orthogonal beamsnearly equal. Ideally a QWP or a HWP will be preferred as the compensator. But, linearpolarizer is used to make use of the polarimeter at other wavelengths of interest.To calibrate the polarimeter, a calibration unit consisting of a linear polarizer followedby a quarter waveplate at λ o to 180 o in steps of 15 o and hence producing 13 inputstates of polarisation. Out of 13 states of polarisation, only 11 are different because thepolarisation states correspond to the retarder orientation 0 o , 90 o and 180 o are essentiallythe same. The corresponding input Stokes parameters and measured Stokes signals (Eq.3) are arranged in a 13 × S cin represents the 13 × S cop represents the measured 13 × S cop = S cin M T . (7)The response matrix of the polarimeter setup ( M ) is solved by defining ( S cin ) T S cop =( S cin ) T S cin M T = SM T as (see Beck et al. 2005 for details) M T = S − ( S cin ) T S cop . (8)Where, S = ( S cin ) T S cin . The structures of S cin and S cop are given in the appendix for thesake of clarity. n efficient modulation scheme for dual beam polarimetry Figure 2.
Plots of polarimetric efficiency of Stokes Q , U and V parameters as a function ofwavelength. The solid curves are theoretical values where as the diamond symbols correspondto the measured values. The error bars shown in these plots are ten times the obtained noiserms values. K. Nagaraju, K. B. Ramesh, K. Sankarasubramanian and K. E. Rangarajan
The calibration data were obtained using the procedure as explained in the beginning ofthis section. Data was subjected to the standard dark and flat corrections(for eg. seeBeck et al. 2005). The response matrix of the polarimeter setup has been derived usingthe 13 × S cin ) constructed out of input Stokes parameters and 13 × S cop ) constructed out of measured Stokes signals using Eq.(8). A typicalderived response matrix( M ) of the polarimeter setup is, − . . . . . . − . . − . . − . − . − . . . . The wavelength of observation is in the continuum of λ QU V , are 0.0044, 0.0048 and 0.0022respectively. The corresponding noise rms of the measurements are 0.0017, 0.0018 and0.0023 respectively.The observed efficiency(Eq.5) of the polarimeter in measuring Stokes
QU V are 0.5644,0.4228 and 0.685086 respectively, which are close to theoretically expected values of 0 . .
423 and 0 .
698 at this wavelength. In an ideal case, off-diagonal elements of the responsematrix M are expected to be zero. However, in practice telescope induced cross-talk andthe variation in the sky transparency can influence the calibration of the polarimeter.Sky transparency variation means that the variation in the input intensity to the tele-scope. To take into account the telescope induced cross-talk, telescope model of KTToriginally developed by Balasubramaniam et al.(1985) and later modified by Sankara-subramanian(2000) is used. A computer simulation has been performed to understandthe effect of sky transparency variation on the calibration of the polarimeter. From thissimulation it is found that the maximum cross-talk produced among Stokes parameters is ≈ .
8% for a sky transparency variation of 14% within the eight stages of measurement.In the actual measurements for this calibration, the intensity variation is of the orderof 14%. The origin of most of the off-diagonal elements M , can be explained based onthe sky transparency variation. However, there are off-diagonal elements which are largerthan the 1 .
8% expected due to the sky transparency variation. An off set angle of ≈ − . o in the HWP is required to produce the observed cross-talk from U to Q . However, theorigin of cross-talk from U to V has not been traced out. n efficient modulation scheme for dual beam polarimetry
5. Conclusions
An eight stage modulation scheme to measure the general state of polarization is pre-sented here. Beam swapping technique is incorporated in this scheme, which helps inalleviating the gain correction errors. The total polarimetric efficiency is close to unityas the Stokes parameters are weighted equally in all the stages of modulation. The fi-nal Stokes parameters are demodulated using all the stages of intensity measurements.Hence, the derived input Stokes parameters are equally weighted time averaged quantitiesover the time of measurement.Since the retarders used in the polarimeter are chromatic, the efficiency of the po-larimeter in measuring Stokes
QU V is wavelength dependent. The laboratory exper-iments performed to study the wavelength dependence of efficiency of the polarimeterconfirms the theoretical expectations.It is found through computer simulation that a 14% sky transparency variation cancause ≈ .
8% uncertainty in the elements of the polarimetric response matrix duringits calibration, for the modulation/demodulation scheme presented here. The non-zerovalues of the off-diagonal elements are not a serious concern if those values do not changedrastically in short time scales. During any solar polarimetric measurements, data for thecalibration of the polarimeter are taken at least once a day. Calibration of the polarimeterare carried out on a few days over a period of 10-day and the response matrix derived overthis period did not show any appreciable variations. The variations in the off-diagonalelements are less than the fit errors ( < ≈ .
986 at λ Acknowledgments
We would like to thank the anonymous referee for the useful suggestions which madethe contents of the paper clearer. We thank B. R. Prasad and Ravinder Kumar Baynalfor providing some of the laboratory equipments necessary for the experiment and P. K.Mahesh for his help in procuring the laboratory equipments. We thank P. U. Kamathfor his help in mechanical design and fabrication of the polarimeter. The help of P.Devendran and P. Hariharan during observations are thankfully acknowledged.
References
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13 correspond to 13 orientations of the calibrationretarder and i = 0 ,,
13 correspond to 13 orientations of the calibrationretarder and i = 0 ,, ,,
13 correspond to 13 orientations of the calibrationretarder and i = 0 ,, ,, ,,