Optimizing imaging polarimeters constructed with imperfect optics

Transcription

1 Optimizing imaging polarimeters constructed with imperfect optics J. Scott yo and Hua Wei Imaging polarimeters are often designed and optimized by assuming that the polarization properties of the optics are nearly ideal. For example, we often assume that the linear polarizers have infinite extinction ratios. It is also usually assumed that the retarding elements have retardances that do not vary either spatially or with the angle of incidence. We consider the case where the polarization optics used to develop an imaging polarimeter are imperfect. Specifically, we examine the expected performance of a system as the extinction ratio of the diattenuators degrades, as the retardance varies spatially, and as the retardance varies with incidence angle. It is found that the penalty in the signal-to-noise ratio for using diattenuators with low extinction ratios is not severe, as an extinction ratio of 5 causes only a 2.0 db increase in the noise in the reconstructed Stokes parameter images compared with an ideal diattenuator. Likewise, we find that a system can be optimized in the presence of spatially varying retardance, but that angular positioning error is far more important in rotating retarder imaging polarimeters Optical Society of America OCIS codes: , , , Introduction Imaging polarimetry is emerging as a remote sensing tool that can enhance conventional intensity imagery. An imaging polarimeter seeks to measure the Stokes vector (or a portion of the Stokes vector) at every pixel in a scene. 1 Polarization information tends to provide information about the surface features of the object in the scene, such as the orientation of the surface normal, 2 the material parameters, and the surface roughness characteristics. 4 Because a polarimeter generally probes the geometric surface characteristics, it often provides information that is uncorrelated with other classes of sensor such as hyperspectral 5 or multispectral images. 6 Because of the mathematical nature of the Stokes vector as discussed in Section 2, it is impossible to measure the Stokes parameters directly. Instead, they must be inferred from a series of measurements through different polarization analyzers that can be When this research was performed, the authors were with the Department of Electrical and Computer Engineering, University of New Mexico, Albuquerque, New Mexico he authors are now with the College of Optical Sciences, University of Arizona, ucson, Arizona J. S. yo s address is tyo@ieee.org; H. Wei s address is hwei@optics.arizona.edu. Received 17 November 2005; accepted 9 January 2006; posted 21 March 2006 (Doc. ID 66089) /06/ $15.00/ Optical Society of America used to build up a system of linear equations that can be solved in a least-squares sense. 7 Recent research has demonstrated that there is an optimum set or sets of measurements that can maximize the signal-to-noise ratio (SNR) in reconstructed Stokes vector imagery for all classes of polarimeter he results of these studies predict that the output SNR will be dependent on the choice of measurements, not just on the quality of the optics and the SNR of the photodetector array. hese predictions have been verified through experimental measurements. 10,11,1 Most of the analyses that have been conducted to date assume ideal polarization optics. In many cases, these assumptions are nearly correct, especially when working with commercially available polarization optics in the visible. In this regime, extinction ratios of polarization analyzers can exceed several hundred, and achromatic retarders can be obtained with excellent spatial and angular stability. However, as new systems are developed that work in other regions of the spectrum or at high frame rates, the quality of the polarization optics tends to degrade. Liquid-crystal-based optics are widely used to create polarimeters that can be switched at near-video-frame rates One of the most common elements used for such applications is the family of liquid-crystal variable retarders manufactured by Meadowlark Optics. 17 At infrared wavelengths, achromatic wave plates have been developed that use cascaded retarders made of 1 August 2006 Vol. 45, No. 22 APPLIED OPICS 5497

2 infrared crystals such as CdS and CdSe to create retarders that have uniform retardance (in wavelengths) across a broad band. 18 While these retarder elements can have excellent performance characteristics, they tend to have significant variability in their retardance parameters as a function of both spatial location and angle of incidence. 19,20 Likewise, a new class of emerging imaging polarimeters known as division of focal plane (DoFP) or microgrid polarimeters have been developed that integrate a micro-optical array of polarization analyzers directly onto the focal-plane array (FPA). 16,21,22 hese devices provide great flexibility in the design of real-time imaging polarimeters, but the extinction ratio of the polarization elements in these micropolarizer arrays tends to be low, often below 10. Here we investigate the effects of using imperfect polarization optics on the performance of imaging Stokes vector polarimeters. We specifically consider two extremely common sources of error. First, we place a tolerance on the effects of using diattenuators with low extinction ratios. Second, we consider the effects of retarders with spatially varying parameters. he optimization approach that we present here is best used to provide guidance to system designers in their choice of polarization optics and their location within the system. It is important to note that the optimization results presented here consider only specific aspects of a polarimeter, and for a real system, all sources of error should be considered to understand how to optimize the overall performance of the system. In Section 2 we review the data reduction matrix and polarimeter optimization. In Section we discuss the effects of using polarization analyzers with low extinction ratios. In Section 4 we consider retarders that vary both with spatial position and angle of incidence. Discussion is presented in Section 5, and conclusions are drawn in Section Data Reduction Matrix he Stokes vector is a mathematical formalism that describes the time-averaged polarization properties of an electromagnetic field. he details of the theory of Stokes vector are provided elsewhere, 7,2 and only the bare essentials are provided here. he Stokes vector can be defined as 0 s 1 S s I I 0 I 90, (1) s 2 I 45 I 15 s I L I R where I is the intensity of the optical field; I 0, I 45, I 90, and I 15 are the intensities measured through an ideal linear polarization analyzer oriented at the appropriate angle (with respect to an arbitrary reference); and I L and I R are the intensities measured through ideal left- and right-circular polarizers. Because of the nature of S, the Stokes parameters s 0 s cannot be measured directly; they must be computed from a set of measurements through polarization analyzers. he most straightforward way to measure S might be to measure the component intensities in Eq. (1). However, this is usually not convenient, and a number of strategies have been developed that take images through elliptical polarization analyzers and then infer the Stokes parameters indirectly. Such strategies include the rotating retarder, 8 the dual rotating retarder, 24 variable retardance, 5 and multidetector systems operating at arbitrary polarization states he measurement process proceeds by measuring the intensity through a number of discrete polarization analyzers as X 1 SD S D2 É Sin A Sin, (2) S DN where S D i is the diattenuation Stokes vector of the polarization state passed by the ith polarization analyzer, 28 S in is the unknown input Stokes vector, and A is the n 4 data reduction matrix. 7 In general, only four measurements are necessary, but more measurements provide redundancy that can enhance the SNR When there is noise present in the measurement process, Eq. (2) becomes X A S in n, () where n is the noise vector that we will assume to have independent, identically distributed elements with variance 2. he Stokes vector reconstruction error is A 1 n. (4) It has been shown that the SNR is maximized and equalized among the Stokes parameter channels when the 2-norm condition number of the matrix A is minimized. 8,9,11,12 If we write the 4 N inverse matrix (or pseudoinverse matrix if N 4) as b B A b0 1 1 (5) b 2 b, then the noise power in the ith Stokes parameter image is given by 29 n i 2 b i 2 2 2, (6) where b 2 is the Euclidean length of the vector b. For a rotating retarder polarimeter, this occurs when the retardance is 12 and the angular positions of fast 5498 APPLIED OPICS Vol. 45, No August 2006

3 axes with respect to the fixed linear diattenuator are given by 15.1 and ,11 Equation (6) considers the reconstruction error due to noise in an otherwise perfect imaging polarimeter. However, for real systems we can expect there to be other sources of error such as imperfect calibration, nonuniform polarization properties of the optical system, and mechanical variations. he effects of a single error source location of the measurement angles were studied previously, and it was found that, when position accuracy is the dominant source of error, the optimum retardance shifts to 120 with the same measurement angles. 12 In Sections and 4, we consider two additional sources of error that are common in imaging Stokes polarimeters. hese are linear diattenuators with low extinction ratios and retarder elements with retardance that is nonuniform either spatially or with respect to the incident angle.. Analyzers with Low Extinction Ratio he diattenuation Stokes vector used to construct the data reduction matrix (DRM) in Eq. (2) takes the form 28 S D 1 D, (7) where D isa 1 diattenuation vector that gives the location within the Poincaré sphere of the polarization state that passes the diattenuator with maximum intensity. he diattenuation Stokes vector for a linear polarizer with copolarized transmission q and cross-polarized transmission r oriented at an angle is 7 S D 1 q r cos 2 q q r r sin 2 q r 0. (8) he extinction ratio of a linear polarization analyzer is defined as the ratio of the maximum and minimum transmission through the analyzer R q r. (9) Commercially available polarizers, especially in the visible, will often have extinction ratios in excess of 100, in which case we can say that D 2 1, (10) and the condition number of A can approach the theoretical minimum of. 10,12 When Eq. (10) holds and the polarimeter is optimized, we have 12 b i 0 2 b 0 2. (11) Fig. 1. (Color online) Condition number of the DRM as a function of the extinction ratio of the linear polarization analyzer used in an optimized rotating retarder polarimeter. However, when the extinction ratio of the polarization analyzers is less than approximately 10, Eq. (10) does not hold. In Fig. 1, the condition number of A is plotted as a function of the extinction ratio for an optimal four-measurement rotating retarder polarimeter with retardance 0.67 and angles { 15.1, 51.5 }. 11 We see that the condition number does increase as the extinction ratio drops below 10. However, reasonable condition numbers can be obtained for extremely low extinction ratios. Furthermore, the condition number of a matrix is equal to the ratio given in Eq. (11), which can increase either by increasing b i 0 2 or by decreasing b 0 2. he lengths of the second to fourth rows of B, which are all equal, are plotted in Fig. 1, as is the length of the first row of B. We see that while the noise power in the measurement of s 1 s increases, the noise power in the measurement of s 0 decreases as the extinction ratio increases. Furthermore, there is not a significant penalty to be paid, as there is only a 2 db decrease in SNR in the s 1 s images for R 5 and a.5 db decrease in SNR for R. At the same time, the SNR in the s 0 images is expected to increase by 1.5 db at R 5 and by 2.5 db at R. he reason for this increase in SNR for the s 0 image should be expected, as the lower extinction ratio leads to greater redundancy in the measurement of the intensity. hese changes in SNR are independent of the input polarization state, as they depend only on the DRM and the random noise vector. he penalty associated with the decrease in SNR in the s 1 s channels may be small enough to tolerate in many applications. 4. Nonuniform Retarder Elements Retarders designed to perform difficult tasks, such as liquid-crystal retarders for temporally rapid variation of the retardance in the visible portion of the spectrum or achromatic retarders in the infrared, often pay a price in uniformity of the retardance parameters. Liquid-crystal retarders have been found 1 August 2006 Vol. 45, No. 22 APPLIED OPICS 5499

4 Fig. 2. (Color online) he retarder element is typically located at either (a) an aperture plane or (b) an intermediate field plane in the optical system. to have fast-axis orientation and retardance values that can vary appreciably both spatially and as a function of the angle of incidence. 19,20 hese variations lead to errors that may or may not be able to be calibrated, depending on the nature of the error and the location of the retarding element in the optical path. Retarders are typically located either at an aperture plane (in front of the telescope, for example) or at an intermediate field plane as indicated in Fig. 2. he choice of the plane for the retarding elements depends on many factors. For example, in rotating retarder systems, any wedge that exists in the retarding element will lead to beam wander. his will result in a shifting of the image on the FPA as the retarder is rotated. Image registration is an extremely important feature in imaging polarimetry, and it is widely held that image registration of of a pixel is necessary to minimize motion artifacts. 0 If the rotating element is located at a field plane, then beam wander can be minimized, reducing the image registration problem. 1 Consider first the case of a retarder placed at an aperture plane as indicated in Fig. 2(a). In the geometrical optics approximation, all rays entering the retarder at a given angle of incidence will go on to contribute to a specific pixel in the final image. If the retardance parameters vary as a function of the angle of incidence, this can, in principle, be calibrated on a pixel-by-pixel basis. he effects of this retardance variation can be greatly mitigated with only postcalibration error due to factors such as thermal variations and calibration accuracy limits. In contrast, if the retardance varies as a function of position, then the diattenuation vector must be averaged over the entire aperture when computing the Stokes vector at each pixel as x i S i D x, ỹ S in, (12) where x and ỹ represent the Cartesian coordinates in the aperture plane. If the unknown input Stokes vector S in is truly constant over the aperture, then Eq. (12) can be accounted for in a fully empirical calibration procedure. However, in imaging scenarios it is expected that S in will vary across the aperture. In this case, calibration of the variation in Eq. (12) is greatly complicated and may not be possible even using fully empirical methods. Next we consider the case of a retarder placed at a field plane as shown in Fig. 2(b). In this case, if there is spatial variation in the retardance parameters, these variations can be calibrated on a pixel-by-pixel basis. However, if there are variations in the retardance parameters as a function of the angle of incidence, these errors must be averaged over the angles of incidence that contribute to the final image at a pixel in a manner analogous to Eq. (12). he sources of the variations in these two cases are different, but the net effect on the final imagery is the same. he first type of error in each case leads to a small residual error that is determined by how well the device can be calibrated. he second type of error in each case cannot be calibrated and must be considered as an error source along with other sources of system errors such as the angular positioning error in operating the polarimeter. 12 he choice of the location for the rotating retarder will now be influenced by the dominant source of retardance variation. If the retardance varies more rapidly as a function of angle of incidence, the retarder might be best placed at an aperture plane. If the retardance varies more rapidly as a function of position, it might better to place it at a field plane. We will explicitly consider the case of randomly spatially varying retardance for the case in Fig. 2(a). Following from the general theory proposed by yo, 12 we can consider the error matrix experienced at each location in the aperture as A A d A d 0, (1) where A is the DRM for a retarder with nominal redardance 0 at the angles i, and A is the perturbed DRM, assuming that the angles are the 5500 APPLIED OPICS Vol. 45, No August 2006

5 same but 0 d. For a rotating retarder system with an ideal linear polarizer at 0 and an ideal linear retarder with retardance 0 with its fast axis oriented at angle i, the ith row of the DRM is given by the diattenuation Stokes vector product of the Frobhenius norms of these two matrices, which is given as B F 2 i 0 j 0 B ij 2, (18) S Di 1 cos 4 i sin cos2 sin 4 i sin sin 2 i sin 0. (14) he corresponding row of the matrix is i d S i D sin 0 cos 4 i sin 0 sin 4 i 1 2 sin 2 i cos 0. (15) If the noise in the measurement process is small compared with the effect of the retardance variation, we have a mean squared reconstruction error E 22 E Ŝ S in 22 E B S in 22, (16) where E[x] is the expected value averaged over the ensemble of input polarization states and over the entire aperture. If we assume that the input states are uniformly distributed over the Poincaré sphere and the retardance varies randomly with standard deviation d, then we have E 22 2 d i 0 Bij2 j 0 k 0 jk 2. (17) predicts the performance of the system. he optimization discussed in Section 2 minimizes B F, but the matrix, which gives the rate of change of A with respect to the variable parameters, also plays a key role. he retarder optimization presented here complements earlier results, and tells us that we need to understand what the dominant noise and error sources are in the operation of our polarimeter. he settings of the polarimeter that we choose to operate will depend on the sources of our error. his points to the importance of a detailed and reliable calibration of the polarimeter, and an understanding of the quality of each compent in the system. Figure 4 shows the Optimizing the polarimeter involves finding the combination of nominal retardance 0 and angular position settings i that minimize the average square error in Eq. (17). he optimization was performed using the MALAB (version 7.0.4) optimization toolbox (version.0.2). he angular positions i were chosen to be the values of { 15.1, 51.6 }, which are the optimum angles for minimizing SNR in the system. 11,12 Figure shows the evaluation of Eq. (17) for these optimum angles as a function of the nominal retardance 0 for different values of d. We see that the optimum retardance for this configuration is 0.40, with a broad minimum. 5. Discussion he optimization study in Section 4 provides another calibration accuracy issue that must be considered in the design of imaging polarimeters. We can interpret the optimum setting as that which minimizes the error caused by both the inverse DRM and the error matrix. 12 It was found in that reference that the Fig.. (Color online) Mean squared error from Eq. (17) normalized to the length of the input Stokes vector for a rotating retarder polarimeter. he retarder is assumed to be located at a field plane, but with spatially varying retardance with standard deviation d with respect to the nominal retardance value. he optimum value of 0 is determined to be August 2006 Vol. 45, No. 22 APPLIED OPICS 5501

6 where  is the estimate of the true DRM, and 1 is the inverse or pseudoinverse as the case may be. On the other hand, the variability of the matrix A can only be determined through careful characterization and calibration. Fig. 4. (Color online) Average error in Stokes vector reconstruction for a system with both angular position error and retardance variation as a function of 0. he angular positioning error a is assumed to be fixed with a standard deviation of 0.5, and the angular position error is varied as shown in the figure. optimization results when there is both angular positioning error and retardance variation in the system as a function of nominal retardance. he optimal angles for maximizing SNR are used in these calculations. We see from Fig. 4 that the angular positioning error dominates the retardance error. When the two error standard deviations are comparable, the optimum retardance is nearly 0.0, which is the optimum for angular positioning error alone. 12 Not until the retardance error exceeds the positioning error by a factor of 5 does the optimum retardance 0 start to shift towards 0.4. he result in Section is somewhat surprising in that reliable polarimetry can be conducted using diattenuators with extinction ratios as low as. It is conventional wisdom that extinction ratios in excess of 20 are necessary to get reliable polarimetry. It turns out that it is less important that the diattenuator has a high extinction ratio than it is to simply know the extinction ratio accurately. If properly calibrated, the imperfect diattenuators will have no effect on optimum settings for the angular position and retardance; rather, it will only affect the minimum value of the mean squared error. he results presented above point to the need for a detailed and accurate understanding of the calibration of the polarimeter. he calibration problem has attracted recent interest with the advent of imaging polarimeters. 7,22 hese results indicate that we need to know the accuracy of the DRM as well as its variability in the system. he former can be obtained using standard methods, whereby the polarimeter is used to analyze a number of known polarization states as 7 X A S 1 S 2,..., S M A, (19)  1 X, (20) 6. Conclusions Stokes vector imaging polarimeters are designed to simultaneously measure the polarization information at as many as 10 6 pixels across a scene. In many applications the desired accuracy of the polarimeter is of the order of 1% degree of polarization or lower. In these demanding situations, a detailed understanding of the sources of error and the accuracy of the calibration is essential. hese points are relevant for all classes of polarization imagers, but the rotating retarder polarimeter is the most common class of full Stokes polarimeter, and is a good surrogate for understanding calibration artifacts. Past studies have considered the effect of noise 8,11,12 and angular positioning error 12 on the performance of rotating retarder polarimeters. In this paper, we consider two additional sources of error in imaging polarimeters that are very common and often ignored: the extinction ratio of the diattenuators, and spatial and angular variability of the retarding elements. We find here that extinction ratios as low as can be tolerated in four-channel rotating retarder polarimeters with only a.5 db decrease in SNR in the resulting imagery. As a side effect, the use of diattenuators with low extinction ratios provides redundancy in the measurement of s 0, allowing for the SNR in the intensity image to increase by 1.5 db over the same polarimeter with an infinite extinction ratio. he case of spatial and angular variation in retardance is a bit more complicated. When the spatial variation of retardance dominates over angular variation, then the retarder should be placed at a field plane to enable calibration of the retarder error. Likewise, when angular variation in retardance dominates, then the retarder should be placed at an aperture plane to allow calibration. However, when both types of error are present, only one can be calibrated out, and the other will result in residual error. We find here that optimizing the rotating retarder polarimeter in the presence of these types of error leads to a different set of optimum positions and retardance than when optimizing for either noise 11 or position error. 12 he issues considered in this paper are especially important for imaging polarimeters, but are broadly applicable. he same principles apply for active Mueller matrix polarimeters; but they must be applied for both the polarization state generator and polarization state analyzer separately. Similar optimization studies could be performed for specific results pertaining to other classes or polarimeters, including linear polarimeters and variable retardance systems. his work was supported by the National Science Foundation under CAREER award 2809 and in 5502 APPLIED OPICS Vol. 45, No August 2006

7 part by the Air Force Office of Scientific Research under award FA he authors are grateful to Peng Li for discussions related to this work. References 1. R. Walraven, Polarization imagery, Opt. Eng. 20, (1981). 2. L. B. Wolff, Surface orientation from polarization images, in Optics, Illumination, and Image Sensing for Machine Vision II, D. J. Svetkoff, ed., Proc. SPIE 850, (1987).. L. B. Wolff and. E. Boult, Constraining object features using a polarization reflectance model, IEEE rans. Pattern Anal. Mach. Intell. 1, (1991). 4. G. D. Lewis, D. L. Jordan, and E. Jakeman, Backscatter linear and circular polarization analysis of roughened aluminum, Appl. Opt. 7, (1998). 5. J. S. yo and. S. urner, Variable retardance, Fourier transform imaging spectropolarimeters for visible spectrum remote sensing, Appl. Opt. 40, (2001). 6. W. G. Egan, W. R. Johnson, and V. S. Whitehead, errestrial polarization imagery obtained from the space shuttle: characterization and interpretation, Appl. Opt. 0, (1991). 7. R. A. Chipman, Polarimetry, in Handbook of Optics, M. Bass, ed. (McGraw-Hill, 1995), Vol. 2, Chap A. Ambirajan and D. C. Look, Optimum angles for a polarimeter: part I, Opt. Eng. 4, (1995). 9. A. Ambirajan and D. C. Look, Optimum angles for a polarimeter: part II, Opt. Eng. 4, (1995). 10. V. A. Dlugunovich, V. N. Snopko, and A. V. saruk, Minimizing the measurement error of the stokes parameters when the setting angles of a phase plate are varied, J. Opt. echnol. 67, (2000). 11. D. S. Sabatke, M. R. Descour, E. Dereniak, W. C. Sweatt, S. A. Kemme, and G. S. Phipps, Optimization of retardance for a complete Stokes polarimeter, Opt. Lett. 25, (2000). 12. J. S. yo, Design of optimal polarimers: maximization of signal-to-noise ratio and minimization of systematic error, Appl. Opt. 41, (2002). 1. P. Li and J. S. yo, Experimental measurement of optimal polarimeter systems, in Polarization Science and Remote Sensing, J. A. Shaw and J. S. yo, eds., Proc. SPIE 5158, (200). 14. L. B. Wolff, Polarization camera for computer vision with a beam splitter, J. Opt. Soc. Am. A 11, (1994). 15. D. C. Dayton, B. G. Hoover, and J. D. Gonglewski, Full-order Mueller matrix polarimeter using liquid-crystal variable retarders, in Optics in Atmospheric Propagation and Adaptive Systems V, A. Kohnle and J. D. Gonglewski, eds., Proc. SPIE 4884, (200). 16. C. K. Harnett and H. G. Craighead, Liquid-crystal micropolarizer for polarization-difference imaging, Appl. Opt. 41, (2002). 17. Meadowlark Optics, Boulder, Colo., Liquid Crystal Variable Retarder, D. B. Chenault and R. A. Chipman, Infrared birefringence spectra for cadmium sulfide and cadmium selenide, Appl. Opt. 2, (199). 19. N. J. Pust and J. A. Shaw, Dual-field imaging polarimeter for studying the effect of clouds on sky and target polarization, in Polarization Science and Remote Sensing II, J. A. Shaw and J. S. yo, eds., Proc. SPIE 5888, (2005). 20. N. A. Beaudry, Y. Zhao, and R. Chipman, Multi-angle generalized ellipsometry of anisotropic optical structures, in Polarizaiton Science and Remote Sensing II, J. A. Shaw and J. S. yo, eds., Proc. SPIE 5888, (2005). 21. G. P. Nordin, J.. Meier, P. C. Deguzman, and M. Jones, Diffractive optical element for Stokes vector measurement with a focal plane array, in Polarization Measurement, Analysis, and Remote Sensing II, D. H. Goldstein and D. B. Chenault, eds., Proc. SPIE 754, (1999). 22. J. K. Boger, J. S. yo, B. M. Ratliff, M. P. Fetrow, W. Black, and R. Kumar, Modeling precision and acuracy of a LWIR microgrid array imaging polarimeter, in Polarization Science and Remote Sensing II, J. A. Shaw and J. S. yo, eds., Proc. SPIE 5888, (2005). 2. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977). 24. R. M. A. Azzam, Photopolarimetric measurement of the Mueller matrix by Fourier analysis of a single detected signal, Opt. Lett. 2, (1977). 25. R. M. A. Azzam, I. M. Elminyawi, and A. M. El-Saba, General analysis and optimization of the four-detector photopolarimeter, J. Opt. Soc. Am. A 5, (1988). 26. C. A. Farlow, D. B. Chenault, K. D. Spradley, M. G. Gulley, M. W. Jones, and C. M. Persons, Automated registration of polarimetric imagery using fourier transform techniques, in Polarization Measurement, Analysis, and Remote Sensing IV, D. B. Chenault and D. H. Goldstein, eds., Proc. SPIE 4819, (2002). 27. V. L. Gamiz and J. F. Belsher, Performance limitations of a four-channel polarimeter in the presence of detection noise, Opt. Eng. 41, (2002). 28. S.-Y. Lu and R. A. Chipman, Interpretation of Mueller matrices based on the polar decomposition, J. Opt. Soc. Am. A 1, (1996). 29. J. S. yo, Noise equalization in Stokes parameter images obtained by use of variable retardance polarimeters, Opt. Lett. 25, (2000). 0. M. H. Smith, J. B. Woodruff, and J. D. Howe, Beam wander considerations in imaging polarimetry, in Polarization Measurement, Analysis, and Remote Sensing II, D. H. Goldstein and D. B. Chenault, eds., Proc. SPIE 754, (1999). 1. S. H. Sposato, M. P. Fetrow, K. P. Bishop, and. R. Caudill, wo long-wave infrared spectral polarimeters for use in remote sensing applications, Opt. Eng. 41, (2002). 1 August 2006 Vol. 45, No. 22 APPLIED OPICS 550