Seeing-Induced Polarization Cross-Talk and Modulation Scheme Performance

Transcription

1 Seeing-Induced Polarization Cross-Talk and Modulation Scheme Performance R. Casini, A. G. de Wijn, & P. G. Judge High Altitude Observatory, National Center for Atmospheric Research, P. O. Box 000, Boulder, CO , U.S.A. We consider the effect of atmospheric seeing on the generation of polarization cross-talk in Stokes polarimeters. After a general introduction to the argument, where we derive the essential formalism, we generalize this to the case of dual-beam polarimetry, to discuss two popular polarization modulation schemes, and investigate their performance in the presence of seeing. We introduce explicitly the seeing power spectrum in our formalism and compare our results with previous work on the subject. Finally, we derive a simple condition to establish the maximum calibration error for a given polarimetric sensitivity target. c 2011 Optical Society of America 1. Introduction Many important solar phenomena are driven by the interaction of turbulent plasma with magnetic fields. Measurements of the full state of polarization of solar spectral lines are routinely used to infer properties of vector solar magnetic fields from their imprints in the emergent spectra through the Zeeman and Hanle effects. With the exception of instruments on the Hinode (Solar-B) and Solar Dynamics Observatory spacecrafts, which by necessity are fed by relatively small telescopes, all measurements of the solar vector magnetic field must be done from the ground. All such measurements are detrimentally affected by image distortions introduced by atmospheric seeing, which has significant power at frequencies of at least 100 Hz, significantly higher than the frame rates of typical CCD cameras. The seeinginduced errors are expected to increase as telescope apertures increase in size: the telescope can resolve more structure(and hence steeper gradients) in images; the number of isoplanatic patches to be corrected through adaptive optical techniques increases with the square of the telescope aperture. The Advanced Technology Solar Telescope (ATST), a 4m aperture, off-axis telescope, will soon enter construction phase. The European Solar Telescope (EST) is a 4m class telescope 1

2 currently under a study of conceptual design. Spectropolarimetry is the prime mode of operation of this new generation of solar telescopes, but there is an urgent need to identify potential pitfalls before committing to a particular design for the modulation and detection of polarized light using such telescopes. The purpose of the present paper is to re-examine and extend earlier theoretical studies of errors introduced by the effects of atmospheric seeing, and to offer pointers for the design of polarization systems for these large, multi-national projects. 2. Statistical Effects of Atmospheric Seeing Let M (m ij ) be the n 4 matrix for a given polarization modulation scheme. We will assume that this modulation matrix is perfectly known (either by design or calibration) and constant in time. This assumption corresponds to the case of a modulator that is stepped through the different states of the modulation cycle. In the case of continuously modulating devices (such as a rotating waveplate), M is instead a function of time. In such systems, the above assumption must be interpreted in the sense that M(t+t i ) = M(t+t j ) when t j t i is a multiple of the modulation period. For the sake of simplicity, in this section we focus on the case of stepped modulators, although the extension of the formalism to the case of continuously modulating devices is straightforward, and will be considered in Sect. 5. Let S be the incoming Stokes vector averaged over the spatial resolution element. We assume that the physical conditions of the light emitting region on the Sun do not vary over the integration time, T, needed to perform a spectro-polarimetric observation of a given element of spatial or spectral sampling with the required sensitivity, 1 and therefore that the corresponding S is constant over this characteristic time. In the presence of atmospheric seeing, a given pixel on the detector collects photons from different elements on the Sun, with a characteristic time scale t 0 ( 0.01s). The lowest order, and largest effect (tip-tilt distortion) can be modeled by assuming that, at any given time within the duration T of a particular observation, the average [why average?] Stokes vector on the Sun that is mapped to a given pixel on the detector is given by [1,2] S(t) = S + S x(t), (1) where x(t) is the displacement vector on the plane of the sky (e.g., expressed in fraction of the pixel unit) due to atmospheric seeing, whereas by assumption S and S are constant during the observation. The modulated intensity recorded by an ideal detector (i.e., without bias or shot noise), for the i-th stage of the modulation cycle, centered around the time t i, and integrated over 1 In what follows, the term observation will always refer to such elemental operation, for instance, the observation of one slit position of a spatial scan, in the case of a grating-based spectro-polarimeter, or of one wavelength position of a spectral scan, in the case of a tunable, imaging spectro-polarimeter. 2

3 the exposure time t, is given by I i = κ = κ t t i + t/2 t i t/2 dt m ij [S j + S j x(t)] m ij (S j + S j x i ), i = 1,...,n, (2) where κ is a dimensional scaling constant for the detector, and x i is the time average of the seeing-displacement vector, x(t), over the interval (t i t/2,t i + t/2). Under the reasonable assumption that seeing can be regarded as an ergodic random process, it makes sense to derive the expressions for the expectation value and variance of I i respectively, E(I i ) and σ 2 (I i ) as these provide essential information about the quality of the observations. Once these quantities are known, they can be used in principle to assess the quality of the Stokes data inversion. Two cases are of particular interest. Slit-based spectro-polarimeters often use a long integration time T that includes a large number of modulation cycles N (typically of the order of a few tens). Tunable imaging spectro-polarimeters typically use only one or a few measurements of I i. Because of the random nature of atmospheric seeing, E( x i ) = 0, hence, using eq. (2), I i E(I i ) = κ t = κ t m ij [S j + S j E( x i )] m ij S j, i = 1,...,n. () This is a fundamental result, telling us that the expectation value of the i-th intensity signal is determined exclusively by the physical Stokes vector S, regardless of the modulation scheme adopted. It is important to note that, in eq. (), the subscript i no longer represents a specific instant t i, like in eq. (2), but strictly only the corresponding step position of the n-state modulation cycle. Similarly, the variance of I i is given by [ σ 2 (I i ) E([I i I i ] 2 ) = E κ t ] 2 m ij S j x i = κ 2 t 2 m ij m ik E([ S j x i ][ S k x i ]), i = 1,...,n. (4) j,k=1 The above expression has no direct applicability to Stokes data analysis, since it relies on physical parameters, such as S i and x i, which are not directly measurable. Instead, the

4 quantities on the l.h.s. of eq. (4) are determined in practice from the statistics of repeated intensity measurements over many modulation cycles. On the other hand, eq. (4) clarifies the physical origin of the noise in the measurement of the modulated intensities, and therefore can provide insights on the optimal choices of modulation schemes and frequencies for reducing the final error. Following[], the optimal demodulation matrix, which allows the inference of the incoming Stokes vector from the modulated intensity signals, is given by D (d ij ) = (M t M) 1 M t. If we indicate the measured Stokes vector with S, we then have S i = n d ij I j, i = 1,...,4. (5) It is useful to derive the expectation value and variance also in the case of S i. Using eq. (), we find S i E(S i) = n = κ t d ij I j ( n d ij m jk )S k = κ ts i, i = 1,...,4, (6) k=1 where in the last line we used j d ijm jk = δ ik. Similarly, for the variance of S i we find, using eqs. (5) and (6), [ n ] 2 σ 2 (S i) E([S i S i] 2 ) = E d ij (I j I j ) = n j,k=1 d ij d ik E([I j I j ][I k I k ]), i = 1,...,4. (7) When the exposure time t is sufficiently large compared to the coherence time of the atmospheric seeing, like in the case of slow modulation cycles, then we can assume that the covariance terms E([I j I j ][I k I k ]) in eq. (7) are negligible for j k, and the variance of S i can in such case be expressed directly as a diagonal quadratic form of the variances of the modulated intensity signals, σ 2 (S i) = n d 2 ijσ 2 (I j ), i = 1,...,4. (8) In Sect. 5 we provide a rigorous demonstration of the above statement. However, in many cases certainly when fast cameras with frames rates 10Hz are employed the time interval t = t i+1 t i (whose inverse conventionally defines the modulation frequency) is 4

5 of the same order or less than the time scale t 0 of the atmospheric seeing. As a result, the intensity variations induced by the average seeing displacement x i, at different sampling times t i during a modulation cycle, are in general statistically correlated, and then one must use the more general expression of eq. (7). Finally, we note that eq. (8) also holds in the absence of seeing, in which case σ 2 (I j ) represents the photon noise on the measurement of the modulated intensity at the step j of the modulation cycle. Because the signals from different camera exposures are always statistically independent, no covariance terms arise in this case.. Interlude: Comparison with earlier work Equations () (4) and (6) (7) form the basis of our following discussion, so it is important to compare these results with previous work. In [1, 2], the variances there defined depend explicitly on the total observation time and the modulation frequency through the integral of the product of the seeing power spectrum with a real function of frequency that depends implicitly on both (cf. eq. [15] of [1], and points 1 5 of Sect. 2 of [2]). In our formalism, the total integration time does not appear explicitly because eq. (4) represents the variance on a single measurement of I i. However, if a number N of such measurements are made, and those measurements can be considered statistically uncorrelated, then the variance on the average signal I i isreducedbyafactorn.insuchcase,forafixedmodulationfrequency,but increasing the total observation time, we expect the same qualitative scaling of variance with the integration time as derived in previous work. Secondly, the dependence on the modulation frequency and the power spectrum of the seeing is apparent within our formalism already in the case of a single measurement when we consider eq. (2). This is formally presented in Sect. 5. The importance of the covariances E([I j I j ][I k I k ]) in typical cases has fundamental implications for the concept of polarimetric measurements. It must be expected that seeing-induced correlations between different modulation states will extend largely beyond the time interval of one modulation cycle. In other words, the Stokes vector measurements corresponding to different modulation cycles during an elemental observation will in general be statistically correlated. Under this condition, we must expect that the variance of the average signal I i will obey the ordinary scaling law by the total number N of cycles of the elemental observation only approximatively. Formally, one should consider instead such elemental observation as a single measurement that is realized through the totality of the N modulation cycles, and thus characterized by a corresponding nn 4 modulation matrix. The variance of this measurement can be determined through the usual equations, adopting such extended definition of the modulation and demodulation matrices. This is also derived in Sect. 5. 5

6 Our formalism also reveals that the earlier work [1, 2] have neglected covariance terms corresponding to contributions proportional to S i S j, with i j (see eq. [22]). In fact, in [1] the variances were written down as eq. (12) of that paper directly from eq. (11), which are explicitly of diagonal form. The additional off-diagonal components arise when properly evaluating the expectation value of (O r Ōr) 2 in the notation of [1]. At the end of Sect. 5, we use our formalism to show how the results of [1] can be generalized to include these off-diagonal components. Finally, we observe that the diagonality of σ 2 (S i), as expressed by eq. (8), is a fundamental assumption in the derivation of the optimal form of demodulation matrices presented in []. In other words, the formal definition of the efficiency of polarization modulation schemes must rely on the fact that seeing-induced noise be negligible with respect to photon noise, or at least on the vanishing of seeing-induced covariances of the type appearing in eq. (7). 4. Dual-Beam Polarimetry In the case of dual-beam polarimetry, one has two independent sets of n measurements that can be combined into 2n new intensity signals of the form I ± i Ii a ± Ii b, where a and b refer to the two beams. For an ideal polarimeter, the modulation matrices for the two beams satisfy the simple relations m a i1 = m b i1 = 1, and m a ij = m b ij for j = 2,,4. However, in practical cases the two beams are never perfectly balanced. We can take into account beam unbalance through the gain factor in front of eq. (). To this purpose, we introduce new modulation matrices for the dual-beam system, M + and M, according to κm ± ij = κa m a ij ±κ b m b ij, κ κ a +κ b. (9) Through these matrices, eqs. () and (4) can be directly extended to take into account the dual-beam redundancy. For this purpose, it is convenient to introduce a new 2n intensity vector with the corresponding modulation matrix, I ± = (I +, I ) T, M ± = (M +,M ) T. We then find, and I ± i E(I ± i ) = κ t m ± ij S j, i = 1,...,2n, (10) σ 2 (I ± i ) E([I ± i I ± i ] 2 ) = κ 2 t 2 m ± ij m± ik E([ S j x i ][ S k x i ]), i = 1,...,2n. (11) j,k=1 6

7 It must be noted that the index i of x i in the last equation must be intended modulus n, since it is related to the actual number of steps in the modulation cycle. In the usual way, we can associate to M ± an optimal, dual-beam demodulation matrix, D ±. Then, the extensions of eqs. (6) and (7) to the case of dual-beam polarimetry become, respectively, S i = σ 2 (S i) = 2n d ± iji ± 2n j,k=1 j, i = 1,...,4, (12) d ± ij d± ik E([I ± j I ± j ][I ± k I ± k ]), i = 1,...,4. (1) Equations(9) (10) and(12) (1) provide the basic formulas through which we can evaluate the performance of different modulation schemes in the presence of atmospheric seeing. For eq.(10) the same conclusions drawn after eq.() apply. As before, the off-diagonal covariances in eq. (1) can be neglected only in the limit of exposure times much larger than the seeing coherence time ( t t 0 ). In order to illustrate the effect of seeing on polarization measurements, we will consider two specific modulation schemes: a Stokes definition scheme and a balanced scheme, with modulation matrices given respectively by M Sdef , M bal (14) Both schemes provide maximum, balanced modulation efficiencies for all Stokes parameters (1 for S 1, and 1/ for S i, with i 2). If we introduce the degree of beam unbalance, ρ = (κ a κ b )/(κ a + κ b ), the modulation matrices M ± appearing in eqs. (10) and (11) are accordingly given by ρ ρ ρ ρ ρ ρ M ± Sdef +ρ ρ ρ ρ ρ ρ ρ ρ ρ ρ M ± bal + ρ + ρ ρ ρ ρ ρ ρ + ρ ρ ρ + ρ ρ T T,, (15)

8 where the top and bottom halves correspond respectively to the + and linear combinations of the n measurements (with n Sdef = 6 and n bal = 4) from the two beams. The respective optimal modulation matrices are ρ ρ ρ ρ ρ ρ D ± Sdef K +ρ ρ ρ ρ , ρ ρ ρ ρ ρ ρ D ± bal K + ρ + ρ ρ ρ ρ ρ ρ + ρ + +, + ρ ρ + ρ ρ + + where we also introduced the dimensionless constant K = (1/2)(κ a +κ b ) 2 /(κ 2 a+κ 2 b ). In case of perfectly balanced beams, we obviously have ρ = 0 and K = 1. From eqs. (11) and (15) we see immediately that the seeing variations on the intensity and those on the polarization parameters of the incoming Stokes vector are decoupled when ρ = 0. Such removal of cross-talk between intensity and polarization is in fact a characteristic advantage of dual-beam polarimetry. However, this advantage is attained only if the two intensity signals of the two beams can be balanced (either by camera gain adjustment, or by rescaling of the signals during data reduction) with an error which must be better than the target polarimetric sensitivity. Since the two beams can always be rescaled post facto, we will assume in the following that we are always dealing with perfectly balanced beams. It is understood that the possible rescaling of the two beams has to be taken into account for the proper determination of the noise on the the combined beams, according to the usual formula σ 2 (ax + by) = a 2 σ 2 (X) + b 2 σ 2 (Y), where a and b are real numbers. This has an effect on the derivation of the quantities σ 2 (I ± j ) in eq. (11) from the noise statistics of the individual beams. In the appendix we illustrate this problem for the particular case where the beam unbalance is produced by the differential efficiency of a diffraction grating in the p and s polarizations. We are now ready to compare the performance of the two modulation schemes introduced earlier. In the case of the Stokes-definition scheme, we see that only one particular Stokes parameter contributes at any time to any given intensity signal combined from the two beams, so that all cross-talk terms between different Stokes parameters are eliminated in this scheme. In addition, the seeing-induced Stokes variations enter the Stokes-definition scheme only through diagonal terms of the form E([ S j x i ] 2 ). In the case of a balanced modulation scheme, instead, one must take into account also the general covariance terms E([ S j x i ][ S k x i ]), for j,k = 2,,4, which slightly complicate the expressions of the seeing-induced polarization cross-talk (cf. eqs. [28] and [29] in Sect. 5). 8

9 Let us now consider a particular case, in which E([ S 1 x i ] 2 ) = α 2, S 2 = S = 0, and E([ S 4 x i ] 2 ) = γ 2. Then, eq. (11) reduces to σ 2 (I ± i ) = κ 2 t 2 [ (m ± i1 α)2 +(m ± i4 γ)2 ], (16) for both modulation schemes. Here we assume that the exposure time, t, is sufficiently large compared to the characteristic time of the seeing, t 0, such that we can neglect the off-diagonal terms in eq. (1). Substitution of eq. (16) into eq. (1), then yields {σ 2 (S i) Sdef } 4 i=1 = 1 6 κ2 t 2 {α 2,0,0,γ 2 }, (17) {σ 2 (S i) bal } 4 i=1 = 1 4 κ2 t 2 {α 2,γ 2,γ 2,γ 2 }, (18) respectively for the Stokes-definition scheme and the balanced scheme. This result shows that the two schemes are affected by the same total error on the inferred Stokes vector, under identical conditions of camera exposure and total integration time. This is because there are 6/4 = 1.5 more modulation cycles for the balanced scheme than for the Stokes-definition scheme, during the same time interval. The above result also shows that the seeing-induced error in the Stokes-definition scheme only affects the Stokes parameters that have non-vanishing gradients at the entrance of the modulator. In the example given above, we assumed that the two linear-polarization Stokes parameters were constant over the displacement x(t). This assumption can be justified in terms of characteristic distributions of magnetic fields at the solar surface (e.g., emerging flux tubes near disk center). It is important to remark that the result expressed by eq. (17) strictly applies when the observation is performed with a polarization-free telescope. Otherwise, the scrambling of the solarpolarizationdeterminedbythetelescope smuellermatrixmakessothatthegradient S 4 attheentranceofthetelescopeismappedontoanewgradientvector( S 1, S 2, S, S 4) at the entrance of the modulator, so that all inferred Stokes parameters will be affected by seeing-induced errors. In such case, there is no evident advantage in adopting the Stokesdefinition scheme over a balanced modulation scheme. In the next section we consider the case in which the off-diagonal terms of eq. (1) cannot be neglected. 5. The power spectrum of the seeing We want to evaluate the covariance term E([I j I j ][I k I k ]) in eq. (7) (or eq. [1]), and to study its behavior as a function of the modulation frequency. We use the definition (2), generalizing it to the case of a continuously modulating device. The covariance term is then 9

10 given by E([I j I j ][I k I k ]) (19) t/2 t/2 = κ 2 E dt m ji (t) S i x(t+t j ) dt m kl (t ) S l x(t +t k ) t/2 = κ 2 t 2 i=1 l=1 i=1 α S i β S l 1 t 2 t/2 t/2 dt t/2 t/2 t/2 l=1 dt m ji (t)m kl (t )E(x α (t+t j )x β (t +t k )), where in the last line a summation over the indexes α,β = 1,2 is implied. Because seeing can be considered a stationary random process, the expectation value in the last line of eq. (19) can be written as the two-time correlation matrix, Γ αβ (t t) E(x α (t)x β (t )), α,β = 1,2, (20) and because of the isotropy of the seeing motion, we also have 2 Γ 11 (t) = Γ 22 (t) Γ(t), Γ 12 (t) = Γ 21 (t), (21a) (21b) since the two components of such a motion cannot be distinguishable. In addition, the two component motions x 1 (t) and x 2 (t) are orthogonal, and therefore they can be assumed to be independent random processes. Hence, E(x α (t)x β (t )) = E(x α (t))e(x β (t )) = 0. α β, since x 1 (t) and x 2 (t) are random processes with zero average. Thus the correlation matrix is diagonal, and proportional to the unit matrix. Another way to state this result is to consider that since the correlation matrix (21) is symmetric, it can be diagonalized via a similarity transformation involving standard rotation matrices in O(2). On the other hand, because of the isotropy of the seeing motion, there cannot be any preferential direction to attain such a diagonal form, and so the correlation matrix for the seeing displacement must always be diagonal, Equation (19) then becomes E([I j I j ][I k I k ]) = κ 2 t 2 i=1 l=1 S i S l 1 t 2 Γ αβ (t) = δ αβ Γ(t). t/2 t/2 dt t/2 t/2 dt m ji (t)m kl (t )Γ(t t+t k t j ). (22) 2 Thecross-correlationfunctionofarealstationaryrandomprocesssatisfiestherelationΓ αβ (t) = Γ βα ( t). The additional symmetry constraint provided by eq.(21b) can be viewed as a consequence of the time-reversal symmetry of the seeing motion. 10

11 Fig. 1. Two-time correlations of the seeing-displacement normalized amplitudes, T jk ( t), plotted against the normalized modulation time, χ t, for various values of k j : 0 (variance; thick continuous line); 1 (first neighbor; dotted line); 2 (second neighbor; dashed line); (third neighbor; dot-dashed line). The expression (24) for the auto-correlation function Γ(t) was assumed. For a n-state modulation scheme in dual-beam configuration, j and k vary from 1 to 2n. However, the indexes j and k in the argument of Γ(t) must be taken modulus n, because these refer to the actual steps of the modulation cycle. No further simplifications can be made at this point in the case of a continuously modulating device. In the following, we will restrict ourselves to the case of stepped modulators, so that the product of the modulation matrix elements m ji (t)m kl (t ) is independent of the time. In such case, using standard manipulations [5], the double integral in the above expression can be transformed into a single integral, T jk ( t) = 1 t 2 = 1 t t/2 t/2 t t dτ dt t/2 t/2 dt Γ(t t+t k t j ) ( 1 τ ) Γ(τ +t k t j ). (2) t 11

12 For the sake of demonstration, in the following we assume for Γ(t) a functional dependence typical of a Gauss-Markov random process, Γ(t) = Γ(0)e χ t, χ > 0. (24) It is known [6] that the Kolmogorov description of atmospheric turbulence leads instead to an auto-correlation function of the form e χ t 5/. However this does not change qualitatively the conclusions of this section. Using the auto-correlation function (24), T jk ( t) can be integrated analytically. Noting that t k t j = (k j) t in our case, we find T jk ( t) = Γ(0) (eχ t 1) 2 χ 2 t 2 e ( k j +1)χ t, k j, T jk ( t) = Γ(0) 2(χ t+e χ t 1) χ 2 t 2, k = j. We then see that T jk ( t) 0 when t χ 1 as expected, because of the random nature of atmospheric seeing. In particular, for very long modulation times, T jk ( t) tends to zero at least as (χ t) 2 for k j, while T jk ( t) (χ t) 1 for k = j. For typical atmospheric conditions, χ 1 t s, which is of the same order of magnitude of typical exposure times. Hence, the term E([I j I j ][I k I k ]) must in general be taken into account for a proper determination of the measurement errors on the Stokes vector, even for k j. Figure 1 shows that these off-diagonal terms drop rapidly as t k t j increases. This opens to the possibility of reducing the importance of the contribution from these terms to the noise in the measured Stokes vector, by means of an optimal choice of the sequence of modulation states. For example, by implementing the Stokes definition scheme via a modulation matrix so modified (cf. eq. [14]), M Sdef , (25) all off-diagonal terms in eq. (1) are of the third-neighbor type, and become practically negligible already with modulation rates of the order of 50 Hz. Use of more realistic autocorrelation functions [6] results in off-diagonal terms falling off significantly faster than in the case of the choice (24), so that the above scheme should remain applicable up to about 100 Hz modulation rates. In order to illustrate the effects of the covariances (22) on the seeing noise on the measured Stokes parameters, we introduce the Stokes gradient matrix, G ij = S i S j, which allows 12

13 Fig. 2. Power spectra of a seeing-like random process. The thin curve represents the normalized power spectrum corresponding to the auto-correlation function (24). The thick curve represents a typical modification of this spectrum produced by adaptive optics. us to recast eq. (22) in matrix form, Cov jk ( t) E([I j I j ][I k I k ]) = κ 2 t 2( MGM T) jk T j k ( t). (26) Here we also used eq. (2), with the identification T j k ( t) T jk ( t), which is suggested by the explicit functional form that those integrals assume in practical cases. Similarly, eq.(7) can be written in matrix form as σ 2 (S i) = ( DCov( t)d T) ii. (27) We then consider again the example treated at the end of Sect. 4, with S 2 = S = 0, and assume for simplicity that S 1 is parallel to S 4 [e.g., magnetic bright point?]. Introducing then the quantities g 1 = S 1 and g 4 = S 4, we find, for the Stokes definition 1

14 scheme with dual-beam polarimetry, [ σ 2 (S 1) = 1 6 κ2 t 2 g1 2 T0 ( t)+ 1 5 n=1 (6 n)t n( t) ], σ 2 (S 2) = σ 2 (S ) = 0, σ 2 (S 4) = 1 2 κ2 t 2 g4[t 2 0 ( t)+t p ( t)], (28a) (28b) (28c) where in the last line p = 1 for the traditional Stokes definition scheme (see eq. [14]), whereas p = for the modified scheme (25). We note that these errors coincide with those estimated at the end of Sect. 4 (cf. eq. [17]) when we neglect all the covariances, i.e., we retain only the terms proportional to T 0 ( t). For the balanced modulation scheme of eq. (14), with dual-beam polarimetry, we find instead [ σ 2 (S 1) = 1 4 κ2 t 2 g1 2 T0 ( t)+ 1 2 n=1 (4 n)t n( t) ], (29a) σ 2 (S 2) = 1 4 κ2 t 2 g4{ 2 T0 ( t) T 2 ( t) 1[T 2 1( t) T ( t)] }, (29b) σ 2 (S ) = 1 4 κ2 t 2 g4{ 2 T0 ( t) T 2 ( t)+ 1[T 2 1( t) T ( t)] }, (29c) [ σ 2 (S 4) = 1 4 κ2 t 2 g4 2 T0 ( t)+ 1 2 n=1 (4 n)t n( t) ], (29d) which again coincide with the results found previously (cf. eq. [18]), in the limit of vanishing coherences. [Should we mention/treat what happens in the single-beam case?] We want to conclude this section by studying the modification of the covariances (22) as a result of the typical seeing correction provided by a system of adaptive optics (AO). The general effect is an important reduction of the low-frequency part of the power spectrum. Figure 2 shows an example based on the model of seeing correlations described by eq. (24). Here the power spectrum S(ω) is defined as S(ω) = + dt Γ(t)e iωt = Γ(0) + dt γ(t)e iωt, (0) where γ(t) = Γ(t)/Γ(0). The thin curve in Fig. 2 represents the power spectrum of the uncorrected seeing motion, whereas the thick curve shows a typical modification of this spectrum produced by the action of adaptive optics. Note how the high-frequency part of the spectrum is not modified by the seeing correction. In practical cases, the seeing will be described by an observed power spectrum, S(ω), rather than by a model auto-correlation function, Γ(t). From eq. (0), we then have Γ(t) = 1 2π + dωs(ω)e iωt, (1) 14

15 Fig.. Normalized auto-correlation function corresponding to the two power spectra of Fig. 2. The thin curve corresponds to eq. (24) [after normalization by Γ(0)], whereas the thick curve corresponds to the AO-corrected spectrum. Note the significant reduction of the time interval within which an efficient suppression of the auto-correlation of the seeing is attained in the presence of AO correction. through which eq. (2) becomes T jk ( t) = 1 2π t = 1 π + + dω S(ω) t t dτ ( 1 τ ) e iω(τ+[k j] t) t iω(k j) t 1 cosω t dωs(ω)e. (2) ω 2 t 2 Application of eq. (1) to observed power spectra allows the determination of realistic autocorrelation functions of the seeing motion, even in the case of AO-corrected systems. As an example, Figure shows the normalized auto-correlation functions corresponding to the two power spectra of Fig. 2. Obviously, the thin curve corresponds to the auto-correlation function (24). Once the true auto-correlation function for an AO-corrected system is known, one can use eqs. (22) and (2) to determine how the modulated intensity covariances are modified by the AO correction. Figure 4 shows this effect in the case of the AO-corrected, 15

16 Fig. 4. Same as Fig. 1, but assuming the AO-corrected auto-correlation function shown in Fig.. Note the significant damping of the covariances for a given value of χ t, compared to the case of uncorrected seeing. auto-correlation function shown in Fig. (thick curve). Comparing these results to those of Fig. 1, which correspond to the case of uncorrected seeing motion, we see that the variances drop faster in the presence of AO correction. On the other hand, the covariances do not converge to zero any faster than in the absence of AO correction. These covariance however change sign, so one can choose modulation schemes and modulation rates such that some of the covariances are either vanishing, or even contributing a negative term to the expression of σ 2 (S i), thus possibly reducing the final error. 5.A. Generalization of previous work We rewrite eq. (22) in the following way, through the substitutions ξ = t+t j and ξ = t +t k : E([I j I j ][I k I k ]) = κ 2 S p S q p,q=1 t j + t/2 t j t/2 dξ t k + t/2 t k t/2 dξ m ji (ξ t j )m kl (ξ t k )Γ(ξ ξ). We then consider the expression of the Stokes variances, eq. (7), and extend it to the case of a series N of modulation cycles. As commented in section, this corresponds to regarding 16

17 the entire elemental observation as a single measurement. We obtain σ 2 (S i) = nn j,k=1 d ij d ik E([I j I j ][I k I k ]) nn = κ 2 S p S q p,q=1 t j + t/2 j,k=1 t j t/2 dξ t k + t/2 t k t/2 dξ d ij d ik m jp (ξ t j )m kq (ξ t k )Γ(ξ ξ). By introducing the box function 1, ξ a/2 Π a (ξ) = 0, ξ > a/2 and eq. (1), we arrive at σ 2 (S i) = κ2 2π S p S q p,q=1 + dω S(ω) + + nn dξ e iωξ d ij Π t (ξ t j )m jp (ξ t j ) nn dξ e iωξ d ik Π t (ξ t k )m kq (ξ t k ). k=1 To make this formula applicable to specific modulation schemes and seeing realizations, we must evaluate the Fourier transform of the functions H ij (ξ) = 1 t nn k=1 d ik Π t (ξ t k )m kj (ξ t k ). First we notice that the elements d ik and m kj are periodic in k with period n. So we can rewrite H ij (ξ) = 1 t = 1 t n k=1 N 1 d ik r=0 Π t ( ξ t1 [k 1] t rn t ) m kj ( ξ t1 [k 1] t rn t ) ( n N 1 d ik δ ( ξ t 1 [k 1] t rn t ) [ Π t (ξ)m kj (ξ) ]), () k=1 r=0 where we used explicitly the fact that t k t j = (k j) t. We then have the following Fourier transform pairs, for the unit box and the windowed comb functions, N 1 r=0 Π a (ξ ξ 0 ) ae iωξ 0 sinc(ωa/2), δ(ξ ξ 0 ra) N e iω(ξ 0+(N 1)a/2) sinc(nωa/2) sinc(ωa/2), 17

18 through which the Fourier transform of H ij (ξ) can easily be evaluated. Let us consider first the case of a stepped modulator. Then, from the convolution theorem, we find H ij (ω) + dξe iωξ H ij (ξ) = N sinc(ω t/2)sinc(nnω t/2) sinc(nω t/2) e iω[t 1+(N 1)n t/2] n d ik m kj e i(k 1)ω t Evidently, H ij(ω) = H ij ( ω).wealsonotethatd ik isanelementofthedemodulationmatrix corresponding to the extended measurement of N cycles. Such matrix contains a factor 1/N with respect to the analogous matrix for one cycle. Therefore, we can replace d ik with the standard (one-cycle) demodulation matrix, and drop the factor N in front of eq. (). In conclusion, the Stokes variances in the case of a stepped modulator are given by σ 2 (S i) = κ2 t 2 2π = κ2 t 2 2π S p S q p,q=1 + + k=1 dωs(ω) H ip (ω) H iq(ω) dωs(ω) ( H(ω)G H (ω) ) ii. (4) Because the matrix H(ω) always appears in a product with its Hermitian conjugate H (ω), the common phase factors can be dropped from its definition, so that we can rewrite H ij (ω) = sinc(ω t/2)sinc(nnω t/2) n d ik m kj e i(k 1)ω t. (5) sinc(nω t/2) Using the symmetry properties of H(ω) and G, it is simple to demonstrate that the integrand in eq. (4) is an even function of ω. This allows to restrict the integration domain to [0,+ ), which is where the observed power spectrum is naturally defined. We also note that H ip (ω) H iq(ω) δ ip δ iq for vanishing t. Therefore, only G ii = S i 2 contributes to σ 2 (S i) in eq. (4) for t 0. This is in complete agreement with the result that can be derived from eqs. (26) and (27) under the same limit conditions. The results presented in [1,2] correspond evidently to the diagonal case p = q. Finally, we observe that, in the case of dual-beam polarimetry, only the summation in eq. (5) extends to 2n, while at the same time (k 1) in the exponential must be taken modulus n. Instead, the factors n appearing in the arguments of the sinc functions remain the same since they define the time interval for one modulation cycle. [We need to add the continuously rotating modulator, at least to the extent k=1 of providing the Fourier transform of Π t (ξ)m kj (ξ).] 18

19 Fig. 5. Plots of the cross-talk errors of eq. (4) normalized to the incoming intensity, as a function of the modulation rate, and in the case of the balanced scheme of eq. (14) in dual-beam configuration (perfectly balanced beams): continuous curve, S 4 S 4 ; dashed curve, S 4 S 2 ; dash-dotted curve, S 4 S. The thin curves are for an observed power spectrum of the seeing, while the thick curves show the effects of AO correction (power spectra, courtesy of T. Rimmele). The case shown corresponds to a non-zero gradient of S 4 such that g 4 = S 0 arcsec 1. These plots are for a total integration time T = 10s. The total number of modulation cycles N spans between 1250 and 1 over the t domain. 6. The effects of non-ideal systems In practical applications, the modulator will be characterized by calibration to finite accuracy. We may write the modulation matrix inferred from the calibration as M M+δM. 19

20 The calibrated demodulation matrix D is determined from M in the usual way. We then find, for the measured Stokes vector, S = D MS = D M S D δms = S D δms, and the error on the true Stokes vector due to calibration errors is then given by δs = D δms D δms, (6) where in the second equivalence we neglected errors of order higher than the first. The error matrix δm can be estimated from the error statistics of the calibration procedure itself. Equation (6) is useful to estimate the calibration accuracy needed to meet a given target for the polarimetric sensitivity of the observations. If we indicate with ǫi 0 the polarimetric sensitivity,wherei 0 isareferenceintensitylevel(e.g.,theneighboringcontinuumofaspectral line), we must require (apart perhaps from some factor of order one) δs D δms ǫi 0, where, for vectors, is the usual Euclidean norm, whereas, for matrices, we choose the absolute maximum row sum norm, [4]. The above inequality is certainly satisfied if we impose that D δm 2I0 ǫi 0, or δm ǫ 2 D. (7) [The following needs conclusions or should be dropped altogether.] In the absence of systematic errors, we may assume that E(S ) = S (after a possible rescaling; cf. eq. [6]). We then have σ 2 (S i) = E ( [S i E(S i)] 2) E ( ) δsi 2 ( n E D ijδm jm S m = E m=1 7. Comments and Conclusions n=1 k=1 ( D δm [ S S T] δm T D T ) ii ) n D ikδm kn S n, i = 1,...,4. (8) In this paper, we have approached the determination of seeing-induced cross-talk noise from a statistical point of view, in contrast to previous investigations [1, 2]. 20

21 In our approach the power spectrum of the seeing displacement appears most naturally as the Fourier transform of the two-time correlation function of the displacement vector. In contrast, the implementation of the seeing power spectrum in previous work should, in our opinion, be revised so to make it fully mathematically consistent (e.g., by a proper use of the generalized Fourier transform of the seeing displacement). Another problem of previous work is that it only considers the diagonal contributions to the seeing-induced noise on a given Stokes parameter. In other words, looking at our eqs. (26) (27), previous work has computed the seeing-induced noise always under the assumption that the gradient matrix G were diagonal. Through our analysis we were able to quantify the importance of seeing-induced correlations between measurements corresponding to different modulation steps, and how these correlations decay for decreasing modulation frequencies. These correlations are captured also in previous work, and they are made manifest by performing a spectral analysis of the demodulated signal over the entire time interval of an elemental observation (e.g., one slit position), which may include many modulation cycles. In the Appendix A, we present a full analysis of this approach based on our formalism. Such study leads to a natural generalization of the results of [1,2] where both diagonal and off-diagonal elements of the gradient matrix are taken into account. Equations (26) (27) also clarify the vanishing of the cross-talk terms in the limit of large modulation rates (χ t 0). Because all the integrals T jk ( t) tend to the same value in such limit, the vector of the Stokes variances becomes simply proportional to the diagonal of the gradient matrix, G. Appendix We consider a beam of unpolarized light incident on a diffraction grating. Typically the diffractionefficiencyofthegratingisdifferentinthepandsdirections,sothebeamdiffracted by the grating consists of a mixture of two orthogonally polarized beams with generally different intensities S p and S s. Let us consider the case in which these two beams get separated by a perfectly polarizing beam-splitter placed after a grating. The emerging beams will then have intensity S p and S s, respectively. Let us consider the combined signal of intensity S so defined, κs = κ p S p +κ s S s, κ = κ p +κ s. where κ p and κ s are the two gain factors applied to the detector to balance the two beams. Theratioκ p /κ s thencorrespondstotheratioofthetwoorthogonalefficienciesofthegrating, ǫ p and ǫ s, according to κ p /κ s = ǫ s /ǫ p. (9) 21

22 The photon noise associated with the combined signal S is given by ( ) 2 ( ) 2 S S σ 2 (S) = σ 2 (S p )+ σ 2 (S s ) S p S s ( κp ) 2σ ( = 2 κs ) 2σ (S p )+ 2 (S s ). κ κ Correspondingly, the relative error is given by σ 2 (S) S 2 = ( ) 2 κp S p σ 2 (S p ) + κs Sp 2 ( ) 2 κs S s σ 2 (S s ) κs Ss 2 Evidently, for balanced beams κ p S p = κ s S s = 1 2 κs, (40) and therefore σ 2 (S) = 1 [ ] σ 2 (S p ) + σ2 (S s ) S 2 4 Sp 2 Ss 2. (41) If we express the signals in terms of photon counts, assuming Poisson s statistics for the photon flux, and indicating with r N the read-out noise of the camera, we can rewrite eq. (41) as σ 2 (S)S 2 = 1 ( Np +rn 2 + N ) s +rn 2 4 Np 2 Ns 2 = 1 ( ) ( + r2 N 1 4 N p N s 4 N 2 p + 1 ) Ns 2. (42) If we indicate with N in the photon count before the grating, evidently N p = 1 2 ǫ pn in and N s = 1 2 ǫ sn in, so we find σ 2 (S) S 2 = 1 2N in ǫ p +ǫ s ǫ p ǫ s + r2 N N 2 in = 1 N in ǫ ǫ p ǫ s + r2 N N 2 in ǫ 2 p +ǫ 2 s ǫ 2 pǫ 2 s ǫ 2 p +ǫ 2 s ǫ 2 pǫ 2 s where we also introduced the grating s average efficiency ǫ = 1 2 (ǫ p +ǫ s )., (4) If the incident beam is weakly polarized (as it is typically the case for solar radiation), then eq. (9) is still valid, but eqs. (40) (4) are only approximately verified. References 1. B. W. Lites, Appl. Opt. 26, 88 (1987) 2. P. G. Judge, D. F. Elmore, B. W. Lites, C. U. Keller, T. Rimmele, Appl. Opt. 4, 817 (2004) 22

23 . J. C. del Toro Iniesta and M. Collados, Appl. Opt. 9, 167 (2000) 4. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, Academic Press, New York (1980) 5. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, Cambridge (1995) 6. V. I. Tatarski, Wave Propagation in a Turbulent Medium, Dover, New York (1961) 2