Optimizing imaging polarimeters constructed withimperfect optics

J. Scott Tyo and Hua Wei

Imaging polarimeters are often designed and optimized by assuming that the polarization properties ofthe optics are nearly ideal. For example, we often assume that the linear polarizers have infiniteextinction ratios. It is also usually assumed that the retarding elements have retardances that do not varyeither spatially or with the angle of incidence. We consider the case where the polarization optics usedto develop an imaging polarimeter are imperfect. Specifically, we examine the expected performance ofa system as the extinction ratio of the diattenuators degrades, as the retardance varies spatially, and asthe retardance varies with incidence angle. It is found that the penalty in the signal-to-noise ratio forusing diattenuators with low extinction ratios is not severe, as an extinction ratio of 5 causes only a 2.0 dBincrease 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 thatangular positioning error is far more important in rotating retarder imaging polarimeters. © 2006Optical Society of America

OCIS codes: 120.2130, 120.5410, 260.2130, 260.5430.

1. Introduction

Imaging polarimetry is emerging as a remote sensingtool that can enhance conventional intensity imag-ery. An imaging polarimeter seeks to measure theStokes vector (or a portion of the Stokes vector) atevery pixel in a scene.1 Polarization informationtends to provide information about the surface fea-tures of the object in the scene, such as the orienta-tion of the surface normal,2 the material parameters,3and the surface roughness characteristics.4 Because apolarimeter generally probes the geometric surfacecharacteristics, it often provides information that isuncorrelated with other classes of sensor such as hy-perspectral5 or multispectral images.6

Because of the mathematical nature of the Stokesvector as discussed in Section 2, it is impossible tomeasure the Stokes parameters directly. Instead,they must be inferred from a series of measurementsthrough different polarization analyzers that can be

used to build up a system of linear equations that canbe solved in a least-squares sense.7

Recent research has demonstrated that there isan optimum set or sets of measurements that canmaximize the signal-to-noise ratio (SNR) in recon-structed Stokes vector imagery for all classes ofpolarimeter.8–12 The results of these studies predictthat the output SNR will be dependent on the choiceof measurements, not just on the quality of theoptics and the SNR of the photodetector array.These predictions have been verified through exper-imental measurements.10,11,13

Most of the analyses that have been conducted todate assume ideal polarization optics. In many cases,these assumptions are nearly correct, especiallywhen working with commercially available polariza-tion optics in the visible. In this regime, extinctionratios of polarization analyzers can exceed severalhundred, and achromatic retarders can be obtainedwith excellent spatial and angular stability. How-ever, as new systems are developed that work inother 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 createpolarimeters that can be switched at near-video-framerates.14–16 One of the most common elements used forsuch applications is the family of liquid-crystal vari-able retarders manufactured by Meadowlark Optics.17

At infrared wavelengths, achromatic wave plates havebeen developed that use cascaded retarders made of

When this research was performed, the authors were with theDepartment of Electrical and Computer Engineering, University ofNew Mexico, Albuquerque, New Mexico 87131. The authors arenow with the College of Optical Sciences, University of Arizona,Tucson, Arizona 85721. J. S. Tyo’s e-mail address is tyo@ieee.org;H. Wei’s e-mail address is hwei@optics.arizona.edu.

Received 17 November 2005; accepted 9 January 2006; posted 21March 2006 (Doc. ID 66089).

0003-6935/06/225497-07$15.00/0© 2006 Optical Society of America

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infrared crystals such as CdS and CdSe to create re-tarders that have uniform retardance (in wavelengths)across a broad band.18 While these retarder elementscan have excellent performance characteristics, theytend to have significant variability in their retardanceparameters as a function of both spatial location andangle of incidence.19,20

Likewise, a new class of emerging imaging pola-rimeters known as division of focal plane (DoFP) ormicrogrid polarimeters have been developed that in-tegrate a micro-optical array of polarization analyz-ers directly onto the focal-plane array (FPA).16,21,22

These devices provide great flexibility in the design ofreal-time imaging polarimeters, but the extinctionratio of the polarization elements in these micropo-larizer arrays tends to be low, often below 10.

Here we investigate the effects of using imperfectpolarization optics on the performance of imagingStokes vector polarimeters. We specifically considertwo extremely common sources of error. First, weplace a tolerance on the effects of using diattenuatorswith low extinction ratios. Second, we consider theeffects of retarders with spatially varying parame-ters. The optimization approach that we present hereis best used to provide guidance to system designersin their choice of polarization optics and their locationwithin the system. It is important to note that theoptimization results presented here consider onlyspecific aspects of a polarimeter, and for a real sys-tem, all sources of error should be considered to un-derstand how to optimize the overall performance ofthe system. In Section 2 we review the data reductionmatrix and polarimeter optimization. In Section 3 wediscuss the effects of using polarization analyzerswith low extinction ratios. In Section 4 we considerretarders that vary both with spatial position andangle of incidence. Discussion is presented in Section5, and conclusions are drawn in Section 6.

2. Data Reduction Matrix

The Stokes vector is a mathematical formalism thatdescribes the time-averaged polarization propertiesof an electromagnetic field. The details of the theoryof Stokes vector are provided elsewhere,7,23 and onlythe bare essentials are provided here.

The Stokes vector can be defined as

S � �s0

s1

s2

s3

� � �I

I0 � I90

I45 � I135

IL � IR

�, (1)

where I is the intensity of the optical field; I0, I45, I90,and I135 are the intensities measured through an ideallinear polarization analyzer oriented at the appropri-ate angle (with respect to an arbitrary reference); andIL and IR are the intensities measured through idealleft- and right-circular polarizers.

Because of the nature of S, the Stokes parameterss0–s3 cannot be measured directly; they must be com-puted from a set of measurements through polar-

ization analyzers. The most straightforward way tomeasure S might be to measure the component in-tensities in Eq. (1). However, this is usually notconvenient, and a number of strategies have beendeveloped that take images through elliptical polar-ization analyzers and then infer the Stokes parame-ters indirectly. Such strategies include the rotatingretarder,8 the dual rotating retarder,24 variable re-tardance,5 and multidetector systems operating atarbitrary polarization states.25–27

The measurement process proceeds by measuringthe intensity through a number of discrete polariza-tion analyzers as

X � ��SD

1�T

�SD2�T

É

�SDN�T

� · Sin � A · Sin, (2)

where �SDi�T is the diattenuation Stokes vector of the

polarization state passed by the ith polarization an-alyzer,28 Sin is the unknown input Stokes vector, andA is the n � 4 data reduction matrix.7 In general, onlyfour measurements are necessary, but more mea-surements provide redundancy that can enhance theSNR.10–12

When there is noise present in the measurementprocess, Eq. (2) becomes

X � A · Sin � n, (3)

where n is the noise vector that we will assume tohave independent, identically distributed elementswith variance �2. The Stokes vector reconstructionerror is

�→

� A�1 · n. (4)

It has been shown that the SNR is maximized andequalized among the Stokes parameter channelswhen the 2-norm condition number of the matrix A isminimized.8,9,11,12 If we write the 4 � N inverse ma-trix (or pseudoinverse matrix if N � 4) as

B � A�1 � �b0

T

b1T

b2T

b3T�, (5)

then the noise power in the ith Stokes parameterimage is given by29

ni2 � �bi�2

2�2, (6)

where �b�2 is the Euclidean length of the vector b. Fora rotating retarder polarimeter, this occurs when theretardance is 132° and the angular positions of fast

5498 APPLIED OPTICS � Vol. 45, No. 22 � 1 August 2006

axes with respect to the fixed linear diattenuator aregiven by �15.1° and �51.6°.8,11

Equation (6) considers the reconstruction error dueto noise in an otherwise perfect imaging polarimeter.However, for real systems we can expect there to beother sources of error such as imperfect calibration,nonuniform polarization properties of the opticalsystem, and mechanical variations. The effects of asingle error source—location of the measurementangles—were studied previously, and it was foundthat, when position accuracy is the dominant sourceof error, the optimum retardance shifts to 120° withthe same measurement angles.12

In Sections 3 and 4, we consider two additionalsources of error that are common in imaging Stokespolarimeters. These are linear diattenuators with lowextinction ratios and retarder elements with retar-dance that is nonuniform either spatially or with re-spect to the incident angle.

3. Analyzers with Low Extinction Ratio

The diattenuation Stokes vector used to construct thedata reduction matrix (DRM) in Eq. (2) takes theform28

SD � �1 D→T�T

, (7)

where D→

is a 3 � 1 diattenuation vector that givesthe location within the Poincaré sphere of the po-larization state that passes the diattenuator withmaximum intensity. The diattenuation Stokes vec-tor for a linear polarizer with copolarized transmis-sion q and cross-polarized transmission r orientedat an angle � is7

SD � �1 q � rq � r cos 2

q � rq � r sin 2 0T

. (8)

The extinction ratio of a linear polarization ana-lyzer is defined as the ratio of the maximum andminimum transmission through the analyzer

R �qr . (9)

Commercially available polarizers, especially in thevisible, will often have extinction ratios in excess of100, in which case we can say that

�D→

�2 1, (10)

and the condition number of A can approach the the-oretical minimum of �3.10,12 When Eq. (10) holds andthe polarimeter is optimized, we have12

�bi0�2

�b0�2� �3. (11)

However, when the extinction ratio of the polariza-tion analyzers is less than approximately 10, Eq. (10)does not hold. In Fig. 1, the condition number of A isplotted as a function of the extinction ratio for anoptimal four-measurement rotating retarder polar-imeter with retardance � � 0.367� and angles{�15.1°, �51.5°}.11 We see that the condition numberdoes increase as the extinction ratio drops below 10.However, reasonable condition numbers can be ob-tained for extremely low extinction ratios. Further-more, the condition number of a matrix is equal to theratio given in Eq. (11), which can increase either byincreasing �bi0�2 or by decreasing �b0�2.

The lengths of the second to fourth rows of B, whichare all equal, are plotted in Fig. 1, as is the length ofthe first row of B. We see that while the noise powerin the measurement of s1–s3 increases, the noisepower in the measurement of s0 decreases as theextinction ratio increases. Furthermore, there is not asignificant penalty to be paid, as there is only a 2 dBdecrease in SNR in the s1–s3 images for R � 5 and a3.5 dB decrease in SNR for R � 3. At the same time,the SNR in the s0 images is expected to increase by1.5 dB at R � 5 and by 2.5 dB at R � 3. The reasonfor this increase in SNR for the s0 image should beexpected, as the lower extinction ratio leads to greaterredundancy in the measurement of the intensity.These changes in SNR are independent of the inputpolarization state, as they depend only on the DRMand the random noise vector. The penalty associatedwith the decrease in SNR in the s1–s3 channels maybe small enough to tolerate in many applications.

4. Nonuniform Retarder Elements

Retarders designed to perform difficult tasks, such asliquid-crystal retarders for temporally rapid varia-tion of the retardance in the visible portion of thespectrum or achromatic retarders in the infrared, of-ten pay a price in uniformity of the retardance pa-rameters. Liquid-crystal retarders have been found

Fig. 1. (Color online) Condition number of the DRM as a functionof the extinction ratio of the linear polarization analyzer used in anoptimized rotating retarder polarimeter.

1 August 2006 � Vol. 45, No. 22 � APPLIED OPTICS 5499

to have fast-axis orientation and retardance valuesthat can vary appreciably both spatially and as afunction of the angle of incidence.19,20 These varia-tions lead to errors that may or may not be able to becalibrated, depending on the nature of the error andthe location of the retarding element in the opticalpath.

Retarders are typically located either at an aper-ture plane (in front of the telescope, for example) orat an intermediate field plane as indicated in Fig. 2.The choice of the plane for the retarding elementsdepends on many factors. For example, in rotatingretarder systems, any wedge that exists in the re-tarding element will lead to beam wander. This willresult in a shifting of the image on the FPA as theretarder is rotated. Image registration is an ex-tremely important feature in imaging polarimetry,and it is widely held that image registration of1�10–1�20 of a pixel is necessary to minimize motionartifacts.30 If the rotating element is located at a fieldplane, then beam wander can be minimized, reducingthe image registration problem.31

Consider first the case of a retarder placed at anaperture plane as indicated in Fig. 2(a). In the geo-metrical optics approximation, all rays entering theretarder at a given angle of incidence will go on tocontribute to a specific pixel in the final image. If theretardance parameters vary as a function of the angleof incidence, this can, in principle, be calibrated on apixel-by-pixel basis. The effects of this retardancevariation can be greatly mitigated with only postcali-bration error due to factors such as thermal varia-tions and calibration accuracy limits. In contrast, ifthe retardance varies as a function of position, thenthe diattenuation vector must be averaged over theentire aperture when computing the Stokes vector ateach pixel as

xi ��� �SDi�x, y��T · Sin, (12)

where x and y represent the Cartesian coordinatesin the aperture plane. If the unknown input Stokesvector Sin is truly constant over the aperture, then

Eq. (12) can be accounted for in a fully empiricalcalibration procedure. However, in imaging scenar-ios it is expected that Sin will vary across the aper-ture. In this case, calibration of the variation in Eq.(12) is greatly complicated and may not be possibleeven using fully empirical methods.

Next we consider the case of a retarder placed at afield plane as shown in Fig. 2(b). In this case, if thereis spatial variation in the retardance parameters,these variations can be calibrated on a pixel-by-pixelbasis. However, if there are variations in the retar-dance parameters as a function of the angle of inci-dence, these errors must be averaged over the anglesof incidence that contribute to the final image at apixel in a manner analogous to Eq. (12).

The sources of the variations in these two cases aredifferent, but the net effect on the final imagery is thesame. The first type of error in each case leads to asmall residual error that is determined by how wellthe device can be calibrated. The second type of errorin each case cannot be calibrated and must be con-sidered as an error source along with other sources ofsystem errors such as the angular positioning error inoperating the polarimeter.12 The choice of the loca-tion for the rotating retarder will now be influencedby the dominant source of retardance variation. If theretardance varies more rapidly as a function of angleof incidence, the retarder might be best placed at anaperture plane. If the retardance varies more rapidlyas a function of position, it might better to place it ata field plane. We will explicitly consider the case ofrandomly spatially varying retardance for the case inFig. 2(a).

Following from the general theory proposed byTyo,12 we can consider the error matrix experiencedat each location in the aperture as

� A� � A d�A�d

���0

, (13)

where A is the DRM for a retarder with nominalredardance � � �0 at the angles ��i�, and A� is theperturbed DRM, assuming that the angles are the

Fig. 2. (Color online) The retarder element is typically located at either (a) an aperture plane or (b) an intermediate field plane in theoptical system.

5500 APPLIED OPTICS � Vol. 45, No. 22 � 1 August 2006

same but � � �0 � d. For a rotating retarder systemwith an ideal linear polarizer at 0° and an ideal linearretarder with retardance �0 with its fast axis orientedat angle �i, the ith row of the DRM is given by thediattenuation Stokes vector

The corresponding row of the matrix � is

If the noise in the measurement process is smallcompared with the effect of the retardance variation,we have a mean squared reconstruction error

E����22� � E��S � Sin�2

2� � E��B · Sin�22�, (16)

where E[x] is the expected value averaged over theensemble of input polarization states and over theentire aperture. If we assume that the input statesare uniformly distributed over the Poincaré sphereand the retardance varies randomly with standarddeviation �d, then we have

E����22� �

�d2

3 �i�0

3

�j�0

3

Bij2 �

k�0

3

jk2. (17)

Optimizing the polarimeter involves finding the com-bination of nominal retardance �0 and angular posi-tion settings ��i� that minimize the average squareerror in Eq. (17).

The optimization was performed using theMATLAB (version 7.0.4) optimization toolbox (ver-sion 3.0.2). The angular positions ��i� were chosen tobe the values of {�15.1°, �51.6°}, which are the op-timum angles for minimizing SNR in the system.11,12

Figure 3 shows the evaluation of Eq. (17) for theseoptimum angles as a function of the nominal retar-dance �0 for different values of �d. We see that theoptimum retardance for this configuration is 0.40�,with a broad minimum.

5. Discussion

The optimization study in Section 4 provides anothercalibration accuracy issue that must be considered inthe design of imaging polarimeters. We can interpretthe optimum setting as that which minimizes theerror caused by both the inverse DRM and the errormatrix �.12 It was found in that reference that the

product of the Frobhenius norms of these two matri-ces, which is given as

�B�F2 � �

i�0

3

�j�0

3

�Bij�2, (18)

predicts the performance of the system. The optimi-zation discussed in Section 2 minimizes �B�F, but thematrix �, which gives the rate of change of A withrespect to the variable parameters, also plays a keyrole.

The retarder optimization presented here comple-ments earlier results, and tells us that we need tounderstand what the dominant noise and errorsources are in the operation of our polarimeter. Thesettings of the polarimeter that we choose to operatewill depend on the sources of our error. This points tothe importance of a detailed and reliable calibrationof the polarimeter, and an understanding of the qual-ity of each compent in the system. Figure 4 shows the

�SDi�T � �1 �cos 4�i sin2

�0

2 � cos2�0

2 � sin 4�i sin2�0

2 �12 sin 2�i sin �0. (14)

i � d�SD

i

�� �0

�0 12 sin �0�cos 4�i � 1�

12 sin �0 sin 4�i �

12 sin 2�i cos �0. (15)

Fig. 3. (Color online) Mean squared error from Eq. (17) normal-ized to the length of the input Stokes vector for a rotating retarderpolarimeter. The 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. The optimum valueof �0 is determined to be 0.40�.

1 August 2006 � Vol. 45, No. 22 � APPLIED OPTICS 5501

optimization results when there is both angular po-sitioning error and retardance variation in the sys-tem as a function of nominal retardance. The optimalangles for maximizing SNR are used in these cal-culations. We see from Fig. 4 that the angular po-sitioning error dominates the retardance error. Whenthe two error standard deviations are comparable, theoptimum retardance is nearly 0.30�, which is the op-timum for angular positioning error alone.12 Not un-til the retardance error exceeds the positioning errorby a factor of 5 does the optimum retardance �0 startto shift towards 0.4�.

The result in Section 3 is somewhat surprising inthat reliable polarimetry can be conducted using di-attenuators with extinction ratios as low as 3. It isconventional wisdom that extinction ratios in excessof 20 are necessary to get reliable polarimetry. Itturns out that it is less important that the diattenu-ator has a high extinction ratio than it is to simplyknow the extinction ratio accurately. If properly cal-ibrated, the imperfect diattenuators will have no ef-fect on optimum settings for the angular position andretardance; rather, it will only affect the minimumvalue of the mean squared error.

The results presented above point to the need for adetailed and accurate understanding of the calibra-tion of the polarimeter. The calibration problem hasattracted recent interest with the advent of imagingpolarimeters.7,22 These results indicate that we needto know the accuracy of the DRM as well as its vari-ability in the system. The former can be obtainedusing standard methods, whereby the polarimeter isused to analyze a number of known polarizationstates as7

X � A · �S1 S2, . . . , SM� � A · �, (19)

A � ��1 · X, (20)

where A is the estimate of the true DRM, and ��1 isthe inverse or pseudoinverse as the case may be. Onthe other hand, the variability of the matrix A canonly be determined through careful characterizationand calibration.

6. Conclusions

Stokes vector imaging polarimeters are designed tosimultaneously measure the polarization informa-tion at as many as 106 pixels across a scene. In manyapplications the desired accuracy of the polarimeteris of the order of 1% degree of polarization or lower. Inthese demanding situations, a detailed understand-ing of the sources of error and the accuracy of thecalibration is essential. These points are relevant forall classes of polarization imagers, but the rotatingretarder polarimeter is the most common class of fullStokes polarimeter, and is a good surrogate for un-derstanding calibration artifacts.

Past studies have considered the effect ofnoise8,11,12 and angular positioning error12 on the per-formance of rotating retarder polarimeters. In thispaper, we consider two additional sources of error inimaging polarimeters that are very common and of-ten ignored: the extinction ratio of the diattenuators,and spatial and angular variability of the retardingelements.

We find here that extinction ratios as low as 3 canbe tolerated in four-channel rotating retarder pola-rimeters with only a 3.5 dB decrease in SNR in theresulting imagery. As a side effect, the use of diat-tenuators with low extinction ratios provides redun-dancy in the measurement of s0, allowing for the SNRin the intensity image to increase by 1.5 dB over thesame polarimeter with an infinite extinction ratio.

The case of spatial and angular variation in retar-dance is a bit more complicated. When the spatial vari-ation of retardance dominates over angular variation,then the retarder should be placed at a field plane toenable calibration of the retarder error. Likewise,when angular variation in retardance dominates, thenthe retarder should be placed at an aperture plane toallow calibration. However, when both types of errorare present, only one can be calibrated out, and theother will result in residual error. We find here thatoptimizing the rotating retarder polarimeter in thepresence of these types of error leads to a different setof optimum positions and retardance than when opti-mizing for either noise11 or position error.12

The issues considered in this paper are especiallyimportant for imaging polarimeters, but are broadlyapplicable. The same principles apply for activeMueller matrix polarimeters; but they must be ap-plied for both the polarization state generator andpolarization state analyzer separately. Similar opti-mization studies could be performed for specific re-sults pertaining to other classes or polarimeters,including linear polarimeters and variable retar-dance systems.

This work was supported by the National ScienceFoundation under CAREER award 238309 and in

Fig. 4. (Color online) Average error in Stokes vector reconstruc-tion for a system with both angular position error and retardancevariation as a function of �0. The angular positioning error �a isassumed to be fixed with a standard deviation of 0.5°, and theangular position error is varied as shown in the figure.

5502 APPLIED OPTICS � Vol. 45, No. 22 � 1 August 2006

part by the Air Force Office of Scientific Researchunder award FA9550-05-1-0090. The authors aregrateful to Peng Li for discussions related to thiswork.

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1 August 2006 � Vol. 45, No. 22 � APPLIED OPTICS 5503