Review of passive imaging polarimetry for remote sensing

Review of passive imaging polarimetry for remotesensing applications

J. Scott Tyo, Dennis L. Goldstein, David B. Chenault, and Joseph A. Shaw

Imaging polarimetry has emerged over the past three decades as a powerful tool to enhance the infor-mation available in a variety of remote sensing applications. We discuss the foundations of passiveimaging polarimetry, the phenomenological reasons for designing a polarimetric sensor, and the primaryarchitectures that have been exploited for developing imaging polarimeters. Considerations on imagingpolarimeters such as calibration, optimization, and error performance are also discussed. We reviewmany important sources and examples from the scientific literature. © 2006 Optical Society of America

OCIS codes: 110.0110, 120.0280, 120.5410.

1. Introduction and Background

A. Overview

The primary physical quantities associated with anoptical field are the intensity, wavelength, coherence,and polarization. Conventional panchromatic cam-eras measure the intensity of optical radiation oversome wave band of interest. Spectral imagers mea-sure the intensity in a number of wave bands, whichcan range from one or two (three is common for a colorcamera) through multispectral systems that measureof the order of 10 spectral channels to hyperspectralsystems that may measure 300 spectral channels ormore. Spectral sensors tend to give us informationabout the distribution of material components in ascene. Polarimetry seeks to measure informationabout the vector nature of the optical field across ascene. While the spectral information tells us about

materials, polarization information tells us aboutsurface features, shape, shading, and roughness. Po-larization tends to provide information that is largelyuncorrelated with spectral and intensity images, andthus has the potential to enhance many fields ofoptical metrology. Figure 1 shows one example ofthe ability of polarization to show enhanced contrastwhen there is little contrast in intensity imagery.

Imaging polarimetry is a special case of generalpolarimetry that is dedicated to mapping the stateof polarization across a scene of interest. Applicationsof polarization imagery range from remote sensing tomicroscopy to industrial monitoring. All the concernsof general polarimetry apply; i.e., a measurementmethod still has to be chosen and calibration must beperformed, but now the additional issues associatedwith measuring a 2D region in space exist. Sequen-tial or simultaneous images must be registered, andwe must know that the response of individual detec-tors is linear and, if multiple detectors are used, uni-form in response with respect to all other detectors.

In this paper, we provide what we believe is thefirst in-depth review of the progress that has beenmade specifically in the field of imaging optical pola-rimetry for remote sensing. Most of the work dis-cussed here has been carried out over the past threedecades. Our focus is on imaging, so there are manyimportant references on ellipsometry and other formsof nonimaging polarimetry that are omitted herebecause of scope. Our primary focus is on passiveStokes-vector imagers, though we do discuss some ofthe very recent work that is beginning to emerge inactive Mueller matrix imagers and polarization lidar.Where possible, we refer to the earliest source known

When this research was performed, J. S. Tyo ([email protected]) waswith the Department of Electrical and Computer Engineering,University of New Mexico, Albuquerque, New Mexico 87131. He isnow with the College of Optical Sciences, University of Arizona,Tucson, Arizona 85721. D. L. Goldstein ([email protected]) is with the U.S. Air Force Research Laboratory�MNGI, EglinAirForceBase,Florida32542. D.B.Chenault ([email protected]) is with Polaris Sensor Technologies, Incorporated, 200 WestSide Square, Suite 320, Huntsville, Alabama 35801. J. A. Shaw([email protected]) is with the Department of Electrical andComputer Engineering, Montana State University, Bozeman,Montana 59717.

Received 17 November 2005; revised 4 April 2006; accepted 6April 2006; posted 10 April 2006 (Doc. ID 66019).

0003-6935/06/225453-17$15.00/0© 2006 Optical Society of America

1 August 2006 � Vol. 45, No. 22 � APPLIED OPTICS 5453

to us, preferably from the reviewed scientific litera-ture.

In this introductory section, we offer a set of defi-nition of terms that are used in the paper, as well asa brief historical perspective. Section 2 describesthe phenomenology of imaging polarimetry, and Sec-tion 3 describes types of measurements and datareduction techniques. In Section 4, we give generalmeasurement strategies that have been used, and inSection 5, a discussion of systems engineering issues.Finally, conclusions are presented in Section 6.

B. Definition of Terms

Angle of polarization: the angle of the major axis ofthe polarization ellipse with respect to the x axis.Mathematically in terms of the Stokes-vector ele-ments,

� �12 arctan

s2

s1. (1)

Depolarization: the process of changing polarizedlight into unpolarized light.

Diattenuation: a property of a polarization elementthat describes the intensity contrast ratio betweenorthogonal transmitted polarization states.

Degree of circular polarization (DOCP): the frac-tion of the intensity attributable to circular polarizedlight states. Mathematically in terms of the Stokes-vector elements,

DOCP � s3�s0. (2)

Degree of linear polarization (DOLP): the fractionof the intensity attributable to linear polarized lightstates. Mathematically in terms of the Stokes-vectorelements,

DOLP � �s12 � s2

2�s0. (3)

Degree of polarization (DOP): the fraction ofthe intensity attributable to polarized light states.Mathematically in terms of the Stokes-vector ele-ments,

DOP ��s1

2 � s22 � s3

2

s0. (4)

Division of amplitude polarimeter (DoAmP): a po-larimeter that makes measurements by splitting thelight into different optical paths, each with distinct

Fig. 1. (Color online) Visible picture of two pickup trucks in the shade (top), long-wave IR intensity image (bottom left), and long-waveIR polarization image (bottom right). Strong contrast in the polarization image shows advantages for enhanced target detection usingimaging polarimetry. (Courtesy of Huey Long, U.S. Army Research Laboratory, Adelphi, Maryland.)

5454 APPLIED OPTICS � Vol. 45, No. 22 � 1 August 2006

polarization optics, and using a separate focal-planearray to image each path.

Division of aperture polarimeter (DoAP): a polar-imeter that uses a lens array to focus separate partsof the aperture onto separate focal-plane arrays orsubarrays. Each subarray measures a different po-larization state.

Division of focal-plane polarimeter (DoFP): a polar-imeter that uses a micro-optical array of polarizationelements to make different polarization measure-ments at each pixel on the focal-plane array.1

Imaging polarimetry: the process of measuring po-larization properties of light, an element, or a systemso as to build an extensive 2D description of thepolarization properties, ordinarily recognized as apicture by a human observer.

Jones2 formalism: the mathematical method of de-scribing polarized light in terms of amplitudes andphases. Devised by Jones, light is represented bya two-element (Jones) vector of complex numbers.A polarization element or system is described by a2 � 2 (Jones) matrix of complex numbers. All Jones’matrices represent elements that can be realized inhardware, but not all elements that can be realized inhardware can be represented by a Jones matrix.

Lu–Chipman3 decomposition: a method of inter-preting the Mueller matrix as a factorable product ofa diattenuator matrix, a retarder matrix, and a de-polarizer matrix.

Mueller matrix: the 4 � 4 real matrix representingthe properties of an optical element or system in theMueller–Stokes formalism. The matrix is often nor-malized to the �1, 1�th entry so that the values rangefrom �1 to �1. This normalization is then in terms ofthe unpolarized scattering of the system.

Poincaré sphere: representation of light polariza-tion states as points on a sphere. The coordinates of apoint on the Poincaré sphere corresponds to the threeStokes-vector elements s1, s2, and s3.

Polarizance4: the degree of polarization producedby a polarizer when the incident beam is unpolar-ized. Polarizance is a property of the polarizationelement.

Polarization state analyzer5 (PSA): a collectionof retarders and linear polarizers cascaded to forman elliptical diattenuator used for analyzing an un-known incident Stokes vector.

Polarization state generator5 (PSG): a polarizationstate analyzer used in reverse to create an arbitraryelliptical polarization state.

Retardance: the change of phase introduced by anelement or system between two states of polarizationin a beam of light.

Spectropolarimetry: the process of measuring thepolarization properties of light, an element, or a sys-tem over some defined spectral region.

Stokes6 vector: a four-element real vector describ-ing polarized or partially polarized light, based onintensity measurements. Introduced by Stokes in1852, it can describe partially polarized light. We usethe symbols s0, s1, s2, and s3 for the four Stokes-vectorelements defined as

S � �s0

s1

s2

s3

� � ��Ex�2 � �Ey�2��Ex�2 � �Ey�2�2 Re�ExEy*�

�2 Im�ExEy*��

I0 � I90

I0 � I90

I45 � I135

IL � IR

�. (5)

In Eq. (5), s0 is the total intensity of the light, s1 is thedifference between horizontal and vertical polariza-tion, s2 is the difference between linear �45° and�45° polarization, and s3 is the difference betweenright and left circular polarization. These elementsare often normalized to the value of s0 so that theyhave values between �1 and �1.

C. Historical Perspective

Someone peering through a birefringent crystal andobserving a pair of refracted polarized images probablydid the earliest imaging polarimetry. There are twoimportant early experiments by Arago and Fresnel,7Arago,8 and Millikan9,10 that are often reported as theearliest attempts at quantitative polarimetry. Aragoperformed a number of qualitative experiments involv-ing polarized light, and was the first to observe thephenomena of optical activity and that emitted radia-tion is not always unpolarized. Millikan measured thelinear polarization information from incandescentmolten metals, and there were a number of subsequentstudies that explored polarization of emitted radiation.Sandus11 provides a thorough review of the physicsand these early works.

To discuss imaging polarimetry in the modernquantitative sense, we must leap forward to the ageof solid-state electronics. The earliest work known tous is contained in two originally classified govern-ment reports, the first by Johnson12 in 1974 and thesecond by Chin-Bing13 in 1976. The instrument de-scribed in these reports is a thermal infrared scan-ning camera that was modified by adding a seconddetector and a polarizing prism. A 1976 patent byGarlick et al.,14 described a system that displayed adifferential optical polarization image. The earliestpublications describing imaging polarimetry in thevisible are the papers by Walraven15,16 where a linearpolarizer was rotated in front of a film camera. Thedeveloped film was digitized, and linear Stokes-vector elements calculated. Solomon17 gave an earlyreview of imaging polarimetry in 1981. Polarimetricsensors also have been used on manned and un-manned spacecraft. Pioneer 11 has the Imaging Pho-topolarimeter on board,18 and the space shuttle hascarried dual film cameras19 and later three-color dig-ital cameras with polarization optics20 operated by amission specialist. These systems measured two orthree components of linearly polarized light. Threecameras were used by Prosch et al.21 to obtain the firstthree Stokes-vector elements, and dual piezoelasticmodulators were used by Stenflo and Povel22 to mea-sure the full-Stokes vector. Pezzaniti and Chipman23,24

developed a Mueller matrix imaging polarimeter thathas been used to examine optical elements in trans-mission and reflection. There are many other exam-


ples. The sources cited are each early realizations of aparticular type of imaging instrument.

2. Measurement Considerations

The basic aspects of light that are typically measuredin imaging scenarios are intensity, spectral content,coherence, and polarization. For passive imaging po-larimetry, it is often most convenient to represent thepolarization information in terms of the Stokes vector,which is defined in terms of the time-averaged inten-sity as in Eq. (5). Implied in Eq. (5) is that the intensitymeasurement is made over some spectral range. Therange could be broad or narrow, and the choice of spec-tral bands is discussed in Subsection 2.A.

A. Spectral Considerations

Spectral information usually tells the observer some-thing about the molecular makeup of the materialsthat compose a scene. Multispectral and hyperspec-tral imagers have been developed to exploit this classof information.25 While there are exceptions, polar-ization information is a slowly varying function ofwavelength,26–28 so it provides information that tendsto be uncorrelated with any spectral measurementsthat are made in a system.

When pursuing a particular application of imagingpolarimeters, spectral considerations are among thefirst issues to be addressed. There are advantagesand disadvantages in each spectral band as in inten-sity imaging both from the consideration of detec-tion instrumentation as well as the phenomenologythe user is trying to exploit. Imaging polarimeterstypically are based on silicon in the visible (VIS) tonear-infrared (NIR) spectra, may use InGaAs in theshort-wave infrared (SWIR), InSb in the midwaveinfrared (MWIR), and HgCdTe in the long-wave in-frared (LWIR). The characteristics of these detectortypes that are considered when used in nonimagingsystems apply to imaging polarimeters as well, i.e.,silicon-based imagers are inexpensive relative to IRsystems, IR systems must be cooled but have dayand�or night capability, etc.

In terms of the phenomenology, polarization signa-tures in the visible and NIR parts of the spectrum aredominated by reflection. Thus these signatures dependon an external source for illumination, primarily theSun. The polarization has a wide dynamic range andcan show rapid spatial variation when imaging out-door scenes. The measured polarization informationis dependent on source–scene–sensor geometry, andtherefore can vary significantly depending on the timeof day or sensor location. In the MWIR, polarizationsignatures are a combination of both reflected andemitted radiation, which tend to cancel or reduce theoverall degree of polarization. In the LWIR, the signa-tures are dominated by emission and can be verystable in time when scene temperatures are stable.Unfortunately, in the LWIR, spatial resolution is re-duced and the cost and complexity of building a systemare generally increased.

In outdoor measurements, the most rapid varia-tions of polarization with wavelength result fromatmospheric spectral features.26 In the VIS–NIR–SWIR, there is strong variation with atmosphericaerosol content. The MWIR contains significant emit-ted and reflected terms, and LWIR scenes dependstrongly on atmospheric water vapor. Some of theissues that arise for imaging polarimetry with respectto spectral regions are given in Table 1.

B. One-Dimensional Polarimeters

The simplest possible use of polarimetry in imaging isto put a polarization analyzer in front of a camera andto adjust the polarization state of this polarizer tomaximize the contrast between an object and itsbackground. This is a common technique used in pho-tography, for example, when taking a picture of anobject against linearly polarized skylight. Similartechniques have been used in underwater imagery tomitigate the effect of scattering using both linear29

and circular30 polarization analyzers with both unpo-larized and polarized illumination. The light scatteredby the medium may have a preferred polarization stateowing to the polarization of the source and the illumi-

Table 1. Polarization Phenomenology and Effects from the Visible to the LWIR

Advantages Disadvantages

Visible, NIR, SWIR ● Sun is a strong source ● Strongly dependent on geometryTypical signal: 1%–60% ● High dynamic range of polarization signatures ● High dynamic range of signaturesSensor resolution: �1%–2% ● Sensors cheaper, easier to build and calibrate ● Inconsistent signatures

● Small well size for FPAs limitspolarimetric resolution

● No night operationMWIR ● Good signatures for hot targets ● Signatures combination ofTypical signal: 0.1%–25% ● Night operation emissive and reflectiveSensor resolution: �0.2% ● Large well sizes for FPA for better sensitivity ● Sensors require cooling

● Sensors more expensive anddifficult to build and calibrate

LWIR ● Signatures dominated by emission ● Sensors require coolingTypical signal: 0.1%–20% ● Less dynamic range for polarization signatures ● Sensors most expensive andSensor resolution: �0.1% ● Large well sizes for FPA for better sensitivity difficult to build and calibrate

● Night operation


nation geometry. The general strategy is to select apolarization analyzer that is orthogonal to the polar-ization state of the background or haze.

C. Two-Dimensional Polarimeters

The natural extension of the 1D polarization imageris a polarization difference imager that measures theintensity of light at two polarization states, then addsthem to estimate s0 and subtracts them to estimate s1,s2, or s3, or some linear combination thereof. Simple2D imagers have shown applicability in a numberof scenarios, but are most widely used in clutter re-jection28 and in mitigating the effects of randommedia.31–36 The basic assumption in these cases isthat there is a difference between the polarizationproperties of light coming from the background andthe light coming from a target. In such cases, signif-icant contrast enhancement can be obtained.

Two-dimensional polarimetry has been used withboth unpolarized28,31 and polarized30,32,33,37 illumi-nation. Two-dimensional polarization discriminationhas been widely used in scattering media, and hasbeen shown to increase the range at which targetscan be detected by a factor of 2 to 3.31,32 When usedwith passive or quasi-passive systems, polarizationimaging has been shown to penetrate as much asfive to six photon mean-free paths into random me-dia. For time-gated imagery, polarization can allowpenetration to greater than ten photon paths.33,38

The improved performance of differential polarim-etry over conventional imagery in scattering mediacan be directly attributed to the depolarizing effectof multiple scattering. This results in a spatiallynarrower point spread function for differential po-larization imagery than for intensity imaging.39 Intime-gated imagery, there is a clear temporal de-pendence of the degree of polarization of scatteredlight that can be used to refine the time gate andmitigate the effect of scatterers.37,40

D. Three-Dimensional Polarimeters

The most common class of imaging polarimeter thathas been developed is the linear polarization imagerdesigned to measure s0, s1, and s2. In most passiveimaging scenarios, there is very little expected circu-lar polarization. Since the most complicated Stokesparameter to measure is s3, it is often omitted toreduce the cost of the imaging system. Probably theearliest well-known example of a full linear Stokesimaging polarimeter was reported by Walraven,15,16

who used linear polarizers and photographic film.Other systems have been developed since that per-form full linear polarimetry in all regions of theoptical spectrum.

When a fixed-position retarder of variable retar-dance is combined with a linear polarization ana-lyzer, it is possible to create a 3D Stokes polarimeterthat measures s0, s1, and s3 as discussed in Section3.B. Such a system is sensitive to a linear polarizationdifference and a circular polarization difference, andsystems such as these have been used for imaging inscattering media.30,32,37

E. Full-Stokes Polarimeters

In some applications, it is essential to measure all ofthe available polarization information. For a passiveimaging system, this means that the full-Stokes vec-tor must be measured at every pixel in the scene.Solomon17 provided one of the first early treatmentsthat specifically addressed full-Stokes imaging pola-rimetry in 1981. Since then, numerous systems havebeen built that can perform full-Stokes imaging, andwe review many of these systems in the rest of thepaper organized by the class of spectral imager andthe techniques used to perform the measurement.

F. Active Imaging Polarimeters

The primary focus of this review is passive imagingpolarimeters that measure the state of polarization oflight from an external source. However, it is appro-priate to discuss some of the important recent ad-vances in active systems that measure the Muellermatrix or some subset of the Mueller matrix. Simi-larly, we briefly discuss recent developments in po-larization lidar systems, which record backscatteredlight from a pulsed laser in two or more polarizationstates as a function of range. In all active polarim-eters, the source is known and controlled. The sourcemay generate one or more states of polarization, andthe detection system may sense two or more statesof polarization. Partial or full measurement of theStokes vector of the reflected light may be what thesensor is designed for, but in the most complete formof active imaging, the Mueller matrix for each pixel ofthe illuminated object is obtained. There are two pri-mary forms of active imaging polarimeters. The firstare those that create an entire scene in one imagecollection. The second are lidar systems that scanpixel by pixel to create a scene, and possibly even avolumetric scene with range-gated data.

1. Mueller Matrix and Other ActiveImaging SystemsPezzaniti and Chipman24 and Chipman41 describeMueller matrix imaging polarimeters that are used toexamine samples in transmission or in reflection.Dual rotating retarders are used in these instru-ments according to the scheme devised by Azzam.42

Clémenceau et al.43 operated a Mueller matrix imag-ing polarimeter in a monostatic configuration. Theyalso used a dual-rotating-retarder system, butcollected only 16 images, the minimum number ofmeasurements needed to determine an arbitrary un-known Mueller matrix. All the systems discussedso far use monochromatic sources. Le Hors et al.44

showed a system using a white-light source that wasspectrally filtered prior to entering the CCD camera.A linear polarizer was placed in front of the source,and two linear polarization states were measured.In this way, images at three colors and two polariza-tion states per color were obtained. Breugnot andClémenceau45 have set up a system based on Azzam’sdual-rotating-retarder configuration using a lasersource in a monostatic configuration, but argue that alimited number of Mueller matrix elements are im-


portant and these can be obtained with only two mea-surements. A diagram of such a system is shown inFig. 2. Réfrégier and Goudail46 have also developedcontrast parameters of polarization for active imag-ery and, with others, have looked at the problem ofestimating the degree of polarization in active sys-tems.47 High-speed Mueller matrix imaging systemsfor laboratory samples have been described by Babaet al.48 and by Wolfe and Chipman.49 A different tech-nique was introduced by Mujat et al.,50 that usesinterferometric methods with active imagery. If thedirection within the Poincaré sphere across an imageis uniform and is known or can be assumed, as issometimes the case with active illumination, thenthe degree of polarization and retardance can bemonitored in a single image.

2. Lidar SystemsPolarization is also found to be useful in more tradi-tional lidar remote sensing. For example, the pres-ence of significant cross-polarized light relative to alinearly polarized transmitter can indicate the pres-ence of ice in clouds or nonspherically shaped dustparticles in the atmosphere.51–54 Polarized lidarshave been developed to measure Stokes parametersof backscattered light in studies of forest and Earth-surface properties,55,56 and to enhance contrast in thelidar detection of fish.57

Polarization lidar systems typically employ linearlypolarized laser transmitters that provide ranging fromthe round-trip transit time of a backscattered pulse.The polarization selectivity is typically built into thereceiver, often using polarization beam splitters tosend orthogonally polarized beams to two separate de-tectors for simultaneous detection of copolarized andcross-polarized scattering. Multiple telescopes can alsobe used to provide simultaneous measurement of theStokes parameters of backscattered light.55 Lidarshave been reported recently that use Pockels cells52 or

liquid-crystal variable retarders58 to vary the receiverpolarization state electronically between laser pulses.

G. Spectropolarimetric Imagers

A spectropolarimetric imager allows the measure-ment of polarization as a function of wavelength in animaged scene. When it is not necessary to obtainspectral data rapidly or simultaneously, it is possibleto combine a more traditional imaging polarimeterwith a rotating filter wheel that selects predeter-mined spectral bands.59 One example applicationwhere this kind of system finds use is the study of skypolarization, for which wide angular coverage andrapid polarization measurements are needed, butfor which rapid spectral measurements may not benecessary.60–63 This approach enjoys relatively sim-ple data retrieval and spectral calibration, but is alsoslow (in spectral space) and requires moving parts,making it unsuitable for some applications whererapid spectral data are required. Lemke et al.64 de-scribe a system that uses a combination of rotatingfilters and polarizers to achieve time-sequential po-larization images in an extremely wide wavelengthrange of 2–240 �m.

Loe and Duggin65 described the use of a liquid-crystal tunable spectral filter to electronically tuneacross multiple 10 nm wide wavelength bands in asystem that employed a rotating linear polarizer anda CCD camera to achieve three-Stokes-parameterspectropolarimetric imaging. This was developed as aprototype of a single channel for a four-channel sys-tem. Eventually the full system would employ foursuch systems with a stationary polarization elementoriented to provide a full-Stokes image at each wave-length band.

A faster, but still not simultaneous, method ofachieving electronic spectral tuning in a spectropola-rimeter is to use an acousto-optic tunable filter (AOTF)as a spectral tuning element. The separate ordinary-ray and extraordinary-ray beams from the AOTFcan be used to generate two simultaneous imageswith orthogonal linear polarization. Alternatively, theAOTF can be combined with an external polarizingelement (such as a variable retarder) to obtain time-sequential Stokes-vector images.66–69 AOTF elementsprovide rapid spectral tuning with typical delay timesof 10–20 ms. An active-spectropolarimeter variationof this approach was described by Prasad,70 using asimultaneously tuned AOTF receiver and tunable la-ser source.

Rather than obtaining multiple spatial dimensionssimultaneously and spectral information over time, itis also possible to use one dimension of an imagingarray to capture spectral data while using the otherarray dimension to record 1D spatial data. In thiscase, a full spectropolarimetric image is built up byspatially scanning the sensor’s field of view (FOV)across the scene. For example, Tyo and Turner27 useda polarimeter comprising two liquid-crystal variableretarders and a fixed linear polarizer in combinationwith a monolithic Fourier transform interferometerto achieve line-scanned spectropolarimetric images of

Fig. 2. Experimental setup of the active polarimetric imagingsystem of Breugnot and Clémenceau in Ref. 45.


laboratory test objects. Jensen and Peterson71 used acomplementary strategy of feeding a grating spec-trometer by an infrared liquid crystal for imagingpolarimetry in the SWIR.

Several related schemes exist for obtaining simulta-neous spectropolarimetric images with no movingparts and no temporal delay between spatial, spectral,or polarimetric data. The polarimetric strategy ischanneled spectropolarimetry discussed in Section 3.C.This “snapshot imaging spectropolarimetry” typicallyemploys birefringent crystals72,73 or holographic opti-cal elements74 to record fringe patterns from whichspectropolarimetric images can be retrieved through avariety of numerical inversion techniques. The obviousadvantage is the simultaneous collection of all mea-sured information, but the technique requires inten-sive computation and is not well suited to images withsignificant low-spatial-frequency or high-spectral-frequency content.74

3. Mathematical Basis for Measurement Techniques

The Stokes vector cannot be directly measured. Tocreate an image of a scene, several individual mea-surements must be made and then combined to inferthe Stokes vector. The measurement strategies canbe broadly grouped into three categories: data reduc-tion matrix techniques,5 Fourier-based techniques,75

and channeled spectropolarimeters.76 In this section,we will discuss the general principles of each of thesemethods.

A. Data Reduction Matrix Techniques

The most straightforward method might be to measurefour linearly polarized intensities through a linear an-alyzer oriented at 0°, 45°, 90°, and 135° and through aleft- and right-circular analyzer. The elements couldthen be combined following the definition of the Stokesvector in Eq. (5). However, the Stokes vector has onlyfour degrees of freedom, and this strategy would entailsix measurements. A method has been developedknown as the data reduction matrix method5 that de-scribes the operation of a polarimeter designed to mea-sure the Stokes vector.

A polarimeter is typically composed of a collectionof retarders and polarizers that are cascaded to forma polarization state analyzer (PSA). In general, theremay be one or more retarders placed in front of alinear polarizer. The component Mueller matrices aremultiplied together to form a general elliptical diat-tenuator Mueller matrix as3

MD � Tu1 DhT

Ph �1 � D2I3 � �1 � �1 � D2�aDaD

T. (6)

The three-element column vector Dh

in Eq. (6) is thediattenuation vector3 that gives the location on thePoincaré sphere of the polarization state that passesthe diattenuator with maximum intensity. The unitvector aD points in the direction of D

h, and D is the

diattenuation of the diattenuator, defined as

D ��Tq � Tr�Tq � Tr

, (7)

where q and r are the two orthogonal states that arepassed with maximum and minimum transmission.When we consider ideal polarization optics, we typi-cally have |D

h| � 1, and we can define a diattenua-

tion Stokes vector as

SD � �1 DhT�T

. (8)

When the unknown incident Stokes-vector Sin passesthrough the diattenuator, the resulting output Stokesvector is

Sout � MD · Sin. (9)

Since most photodetector elements are polarizationinsensitive, the output of the detector usually will beproportional to s0,out, which can be written in vectorform as

S0,out � SDT · Sin � mD,00s0,in � mD,01s1,in � mD,02s2,in

� mD,03s3,in. (10)

Equation (10) has four unknowns—the input Stokesparameters—so to solve for these unknowns, we mustbuild up a system of linear equations like Eq. (10)using at least four different realizations of the diat-tenuation Stokes vector in Eq. (8). In matrix form,this system can be written as

X � �s0,out

1

s0,out2

É

s0,outN� � �

�SD1�T

�SD2�T

É

�SDN�T

� · Sin � A · Sin. (11)

The notation �SDi�T represents the ith realization of

the diattenuation Stokes vector. In general, the num-ber of measurements N M, where M is the numberof dimensions that will be reconstructed in the polar-imeter. The matrix A in Eq. (11) is referred to as thesystem matrix, and its inverse is termed the datareduction matrix5 (DRM). We can estimate the un-known input Stokes vector as

Sin � A�1 · X, (12)

where the hat indicates that Eq. (12) is providing onlyan estimate. Sources of error could include noise inthe measurement vector X and calibration measure-ments in determining the DRM. Clearly we need to becareful about the selection of �SD

i�T, as the conditionnumber of the matrix A must be low enough so thatthe inversion process is well behaved. More detailsare provided on this issue in the section on polarim-eter optimization in Subsection 5.B.


The DRM measurement strategy can be inter-preted from a signal processing viewpoint.77 Each ofthe entries in X can be thought of as a projection ofthe unknown input Stokes vector onto an analysisvector �SD

i�T. When N � M, the analysis vectors forma nonorthogonal basis in the conical space that is theallowed space of physically realizable Stokes vectors.When N M, the analysis vectors form an overde-termined basis, or frame.77 As discussed in Section 5,use of a frame can enhance the robustness of themeasurement process.

B. Fourier Modulation Techniques

A common method of polarimetric measurementand data reduction is through the Fourier analysisof polarimetric signals. These methods were devel-oped initially for Mueller and Jones matrix pola-rimeters for nonimaging measurement of polarizedand partially polarized light42,78,79 and for ellipso-metric measurements.80–82 They are readily gener-alized to spectral and imaging instruments.24,34,83,84

In this approach, a series of images are acquired asthe elements of the polarization state analyzer arevaried in a harmonic fashion. The polarization of theincident light is encoded onto the harmonics of thedetected signal. The Stokes-vector elements of the in-cident light are then recovered from a Fourier trans-form of the measured data set. The Stokes vector iscomputed independently for each pixel.

Consider a general polarimeter with incident lightof unknown polarization and a PSA as shown sche-matically in Fig. 3. A series of N intensity measure-ments s0,out

n are made as in Subsection 3.A:

X � �s0,out

1

s0,out2

É

s0,outN� � �

�SD1�T

�SD2�T

É

�SDN�T

� · Sin � A · Sin. (13)

Varying the polarization elements of the analyzermodulates the analyzed polarization states. A typicalmethod of varying the polarization elements is byrotating some or all of the elements in discrete steps.If the angular increments of the polarization ele-ments are constant, only discrete frequencies are gen-erated in the detected intensity X, whose elementsare written as xn. The intensity xn is collected for thenth position of the polarization elements in the PSA.

The detected signal can be written

xn �b0

2 � k�1

�bk cos k�n � ck sin k�n�, (14)

where the largest k is the highest frequency compo-nent in the signal and �n � n� is proportional to theangular frequency of the polarization element. Thepolarization content of the scene being imaged is en-coded onto the various frequencies of the detectedsignal; i.e., the coefficients in the Fourier series ex-pansion are functions of the incident Stokes vector.These relations are inverted to give the Stokes-vectorelements in terms of the Fourier coefficients. Thecoefficients are determined from the set of intensitiesby a discrete Fourier transform,

b0 �1N

n�0

N�1

xn,

bk �2N

n�0

N�1

in cos�2nk�

N �� n�0

N�1

in cos�k�n�,

ck �2N

n�0

N�1

in sin�2nk�

N �� n�0

N�1

in sin�k�n�, (15)

where k is the harmonic, �n � n �, and � is theangular increment of the polarization elements. ForN intensities, the coefficients for the K � N�2 har-monics are found. The step size of the rotation of thepolarization element is determined by the number ofmeasurements � � 2��N.

The highest harmonic K in the polarimetric sig-nal is determined from the analytical expression forthe intensity written as a Fourier series. The min-imum number of measurements Nmin required tocalculate the dc term and all cosine and sine (realand imaginary) terms in the Fourier transform isNmin � 2K � 1. It is often desirable to make moremeasurements than the minimum, or oversample, tohelp reduce the effects of noise. For oversampleddata, the harmonics higher than the frequencies ofthe polarimetric signal are often used as diagnostictools to indicate sources of systematic error.

The Fourier analysis of polarimetric signals pro-vides several significant advantages for data reduc-tion. First, if the analytical form is readily derived viaa system Mueller matrix expression, this data reduc-tion method is straightforward and computationallyfast. Second, the system Mueller matrix may be pa-rameterized such that diattenuation and retardancevalues and orientation of the elements may be deter-

Fig. 3. (Color online) Polarimetric sensor using rotating polarization elements.


mined in calibration. Third, this method often encom-passes instruments where the elements are rotatedcontinuously. In this case, the angular incrementused for discrete steps of the rotated elements is re-placed by the angular increment at which the nextdata acquisition is begun. Any motion of the elementover the integration time of the sensor is compen-sated for in calibration. Fourth, the calculation of thediscrete Fourier transform automatically gives aleast-squares fit to the data. Finally, the discrete Fou-rier transform is a useful analytical tool for investi-gating many types of systematic error such as beamwander and linear drift. The susceptibility to harmfulnoise sources can be reduced through adjusting theparameters of measurements and the correspondingFourier transform. More details of the effect of noiseand errors on the measurements and ways to com-pensate or negate these effects are given in Refs. 85and 86. The chief disadvantage of this approach isthat the system Mueller matrix must be well known.In practice, this requires that the polarizers are purediattenuators and the retarders are pure retarders;i.e., the polarizers contain no retardance and the re-tarders are not diattenuating.

C. Channeled Spectropolarimeters

Most polarization techniques that rely on retardingelements have to go to great lengths to develop awave plate that has uniform retardance across thespectral range of interest. Efforts have been made todevelop achromatic retarders in the visible and theIR spectra. Achromatic retarders are commerciallyavailable at visible wavelengths, but have only re-cently become available for infrared wavelengths.87

When polarimetry and spectrometry are combined,the retardance can be calibrated wavelength bywavelength, reducing the problems associated withthis effect.27

Recent techniques have emerged that couple Fou-rier transform spectrometry with polarimetry to ex-ploit the wavelength dependence of the retardance (inwavelengths) of high-order wave plates.76 We assumethat a wave plate of thickness L can be described byan index of refraction difference n � ne � no, wherene and no are the extraordinary and ordinary indicesof refraction. We ignore dispersion in n over the waveband of interest for the purposes of this development.The phase difference induced between the radiationpolarized parallel and perpendicular to the fast axisis given as

� �2�� n�L

�. (16)

When the thickness of the wave plate is chosen sothat � �� 2�, then the retardance varies rapidly as afunction of the wavelength. The spectrum of the out-put intensity of the PSA is modulated in a manneranalogous to Eq. (14). When the spectrum of thissignal is measured with a Fourier transform spec-trometer, the spectrum is modulated in a mannerthat depends on the polarization state. If the retard-

ers and PSA are designed carefully, and the spectrumof the incident signal is band limited, then spectrallydistinct portions of the signal can be used to deter-mine the Stokes parameters of interest.

It is essential that a Fourier transform spectrom-eter be used to make the spectral measurement. Thisis because the variation in retardance introduced byEq. (16) provides a signal at spatial frequencies thatcorrespond to wavelengths that are typically outsidethe spectral range of the detectors used. The methoddescribed by Oka and Kato76 is not an imagingscheme. Sabatke et al.74 developed a method to couplethe channeled spectropolarimeter with a snapshotFourier-based spectrometer to enable the instanta-neous collection of spectral and polarimetric imageryinformation. This technique has been extended toseveral wave bands of interest.88

Oka and Kaneko89 introduced a novel and com-plimentary strategy for snapshot polarimeter whenusing monochromatic illumination. Whereas thechanneled spectropolarimeter modulates the spec-trum based on the polarization signature, the newmethod uses spatially varying thick retarders tospatially modulate the intensity image. The polari-metric features can be ideally reconstructed using asimilar demodulation technique when the spatialFourier spectrum of the scene is band limited.

4. Imaging Architecture for Integrated Polarimeters

There are several different approaches for polarimet-ric detection. As with spectral imaging where multi-dimensional data are acquired, the data acquisitionprocess is a study in compromises. By the very natureof measuring polarization, multiple images are re-quired to even partially characterize the polarizationstate of a scene. Since polarimetric data reductionmanipulates the same pixel across multiple frames,any motion of the scene in the pixel results in arti-facts that have the potential to mask the true polar-ization signature. Ideally, two spatial dimensions aredesired, but due to this temporal image registrationissue, the images must be acquired simultaneously oracquired as quickly as possible to minimize artifactsfrom platform or scene motion. The best solution forminimizing these artifacts is to acquire multiple im-ages at the same time, but then the issue becomesspatial registration. Spatial registration of multipleimages is complicated by the need to correct for bothmechanical misalignment as well as optical “mis-alignment” arising from aberrations due to separateoptical paths. Conceptually the simplest way to mea-sure the polarization information is to use separatecameras with separate optics that are aligned to thesame portion of the image (coboresighted). Earlyimaging polarimeters did this with both film andelectronic cameras14,19 as well as scanning single-element photodetectors.90 This strategy is difficult toexecute properly, and has largely fallen out of favor.There are a number of integrated techniques that areused now. Trade-offs among these methods, as well asissues of cost and difficulty of fabrication and inte-gration, are listed in Table 2.


A. Division of Time Polarimeter

One commonly used approach is to rotate polariza-tion elements in front of the camera system.16,20

This approach is attractive because it is relativelystraightforward in both system design and data re-duction. However, the obvious drawback is that boththe scene and platform must be stationary to avoidintroducing interframe motion. Figure 2 shows anexample of the common rotating retarder polarime-ter. In this type of polarimeter, the rotation of thepolarization elements causes a modulation of the po-larized light incident on the focal plane from thescene, and the data can be reconstructed using themethods discussed in Section 3. Reducing the data ona pixel-by-pixel basis produces Stokes images thatcan be used to produce images of the degree of linearpolarization, degree of circular polarization, or otherderived quantities such as orientation or ellipticity.

Most often the rotating element has been a polar-izer. Only linear polarization states are detected in thisapproach. In addition, either the rotation rate in pre-vious attempts has been too slow to achieve reasonableframe rates, or the polarizer was moved in steps withthe imagery acquired between movements. Even withrecent successes in continuously rotating the polar-izer,84 artifacts still remain if there is sufficient scenesensor movement during acquisition of if there is beamwander induced by the rotating element. Beam wan-der can result if there is a wedge in the rotatingelement or if the element wobbles in any sense. Nev-ertheless, if proper care is taken, the rotating elementpolarimeter can provide good results with a relativelysmall investment in hardware, design, and integra-tion.

B. Division of Amplitude Polarimeters

DoAmP were first suggested and built by Garlicket al.14 for a two-channel system, then revived laterfor full-Stokes polarimeters,91,92 and have since beenexploited by a number of authors. Figure 3 shows afull Stokes DoAmP polarimeter. This type of polar-

imeter consists of four separate focal-plane arrays.The camera system consists of four separate camerasmounted such that a single objective lens is used incombination with a series of polarizing beam split-ters, retarders, and relay lenses to produce a polari-metric image. Rigid mechanical mounts are used tosupport the cameras in positions facing the four cubeassembly exits. The polarizing beam-splitting cubeassembly is used to balance the linear and circularmeasurements. The cameras simultaneously capturethe four images necessary for computing a completeStokes image, thus eliminating false polarization ef-fects due to scene changes during the collection pro-cess.

In this particular example,93 the polarimetricbeam-splitter assembly is designed to measure thecomplete Stokes vector. The beam-splitting blockincludes three beam splitters, one 80�20 polarizingbeam-splitting cube, two 50�50 polarizing beam-splitting cubes, and a quarter-wave and half-waveretarder. Each path through the beam-splitter blockanalyzes a different aspect of the incident polariza-tion. This makes efficient use of the polarized lightso that none of the light is absorbed or rejected.Furthermore, the analyzed polarization states areas nearly orthogonal as possible, and the analyzedstates evenly span the possible incident polarizationstates.

As described in the first paragraph of Section 4,special care must be taken in alignment, and in prac-tice, mechanical alignment to the required tolerancesjust is not possible. Further, the many degrees offreedom in the relay lens sets the result in differentaberrations in each of the four channels. As a result,postprocessing is required to coregister the four im-ages. One of the chief disadvantages is the size of thesystem with the four focal planes and the breadboardrequired to rigidly mount the focal planes and theiroptics. When full spatial resolution is desired and sizeand cost of components is less of an issue, this ap-proach is suitable.

Table 2. Comparison of Imaging Polarimetry Architectures

Design FeaturesFabrication–Integration

Issues, Cost Misregistration Issues

Rotating element ● Robust ● Easiest to implement ● Scene and platform motion● Relatively small ● Inexpensive ● Beam wander not a problem● Not suitable for dynamic or removed in software

scenes ● Misregistration is linearDivision of amplitude ● Simultaneous acquisition ● High mechanical flexibility ● Must register multiple FPAs

(multiple FPAs) ● Large system size and rigidity required ● Misregistration can be fixed● Expensive ● Can be nonlinear● Large

Division of aperture (single FPA) ● Simultaneous acquisition ● Loss of spatial resolution ● Fixed misregistration● Smaller size ● Expensive ● Can be nonlinear

Division of focal plane ● Simultaneous acquisition ● Fabrication difficult ● IFOVs misregistered● Small and rugged ● Alignment difficult ● Requires interpolation● Loss of spatial resolution ● Very expensive ● Fixed registration

Coboresighted ● Simultaneous acquisition ● Easy integration ● Misregistration not as stable● Best used at long ranges ● Expensive


C. Division of Aperture Polarimeter

Figure 4 (Ref. 94) shows an architecture that can bothacquire all of the polarization data simultaneouslyand ensure that the fields of view (FOVs) of all of thepolarization channels are coboresighted. The archi-tecture uses a single focal-plane array (FPA) and areimaging system to project multiple images onto aFPA that are accurately coaligned. This architecturehas the advantage that once the optics are mechan-ically fixed, the alignment has been shown to be sta-ble in time when compared to DoAmP polarimeters.The improved stability is likely due to the longeroptical paths that are typically necessary in DoAmPsystems, translating small changes into larger devi-ations on the FPA. The architecture can be used bothas a passive sensor (broadband illumination) and asan active monochromatic sensor. The primary disad-vantages of the division of aperture polarimeter arethe loss of spatial resolution (a factor of 2 in eachlinear dimension) and the volume and weight of

the additional reimaging optics. In addition, match-ing transmission, apodization, magnification, anddistortion between the channels is difficult, but canbe accomplished. It should also be noted that thisstrategy is more difficult to employ with coherentillumination due to coherent scattering and interfer-ence.

D. Division of Focal-Plane Array Polarimeters

The recent advances in FPA technology have led tothe integration of micro-optical polarization elementsdirectly onto the FPA.95,96 Most DoFP systems thathave been made to date are only sensitive to linearpolarization, though some discussion of full-StokesDoFP systems has been raised.97 An example systemis shown in Fig. 5. DoFP systems have been manu-factured for imaging in all regions of the optical spec-trum, including visible,98,99 SWIR,1 and LWIR.100

Most DoFP systems have interlaced polarization su-perpixels as shown in Fig. 5, although some systems

Fig. 4. (Color online) Division ofamplitude polarimeter. The fourthcamera is out of the plane of thepage positioned above the PBSblock after the quarter-wave re-tarder.

Fig. 5. (Color online) Division of aperture polarimeter and a raw focal plane image showing the four polarization channels. The fourchannels are reduced to polarization products such as DoLP. For this specific case, the four images are linearly polarized at 0°, 45°, 90°,and 135°.


have been made where the polarization informationis sampled on a line-by-line basis.98 A typical DoFPsystem will compute the Stokes vector at interpola-tion points in the FPA as indicated in Fig. 6. DoFPsystems necessarily trade off spatial resolution forpolarization information, as a 2 � 2 (or larger) con-volution kernel must be applied to the image to esti-mate the Stokes vector at each point.

DoFP systems have the significant advantage thatall polarization measurements are made simulta-neously for every pixel in the scene. The componentmeasurements that go into the Stokes-vector estima-tion are by construction coming from nearby points inthe scene. However, the DoFP system by definitionhas pixel-to-pixel registration error in computingthe Stokes vectors. The instantaneous fields of view(IFOVs) of adjacent pixels are (in principle) nonover-lapping, leading to exactly a 1-pixel registration er-ror. The IFOV error can be partially mitigated byintentionally defocusing the optical point spreadfunction, and recent efforts have been made to min-imize the information loss while simultaneously min-imizing IFOV error through interpolation.100

5. System Considerations

A. Alignment and Calibration of Imaging Polarimeters

Alignment and calibration are important issues in ac-quiring polarimetric imagery, and knowledge of sys-tem characteristics is essential to obtaining good data.The authors have been involved in many measure-ment campaigns that, while qualitatively useful tothose taking the data, cannot be used for quantitativeanalysis because of poor calibration. The credibility ofreported results is dependent upon documented cali-bration procedures.

Each polarimetric system will have its own calibra-tion and alignment issues, and it is not possible tosummarize all possible scenarios here, or offer a pre-scription for handling these issues. We review gen-eral items that must be addressed and give examplesof methods that have been used in the past to acquiredata. System design will determine which methodsmight be relevant, and the designer must be ready toincorporate and justify the appropriate calibrationand alignment procedures.

There are two calibration areas to be addressed:radiometric and polarimetric. The radiometric cali-bration determines the dynamic range of the detec-tors, ensures that the detectors are operated in alinear response region (or if not, guarantees that theresponse is known and consistent), and ensures thatthe detectors are not operated near their saturationpoints. If a detector array is used, and this will morethan likely be the case for modern imaging systems,the response of individual detector elements must beknown. A correction procedure for nonuniformitiesmust be in place (e.g., a nonuniformity correctionNaval Undersea Center look-up table).100 These is-sues are no different from those facing the user of anonpolarimetric system and will not be covered here.However, radiometric calibration is even more impor-tant to the final result because of the potential forinaccurate radiometry among channels to couple intoand invalidate the polarimetric result. Polarimetricmeasurement issues that may need to be addressedinclude polarimetric system calibration, optical ele-ment polarimetric uniformity, optical axis alignmentand optical element rotational alignment, and polar-ization aberrations.

Examples of polarimetric system calibration tech-niques are given in Azzam et al.101 and Goldsteinand Chipman.102 The former contains a description ofthe calibration of the four-detector photopolarimeterwhere a simultaneous measurement of the Stokesvector is made. The latter concerns the calibration ofa dual rotating retarder Mueller matrix polarimeterusing Fourier data reduction techniques where time-sequential measurements are made. While neither ofthese are imaging systems, the approaches to calibra-tion are instructive.

1. Experimental Determination of the DataReduction MatrixThe DRM described in Section 3.A is estimated for anarbitrary polarimeter by measuring at least four lin-early independent Stokes vectors that ideally form amaximum volume polyhedron inscribed inside thePoincaré sphere. When the number of measurementsis exactly four, then the polyhedron is a regular tet-rahedron.101 The equation that describes this mea-surement is similar to Eq. (12) and is given by

X � A · S, (17)

where X is a matrix of the measured intensities, S isa matrix with columns that are input Stokes vectors,and A is the instrument matrix. Each row of theinstrument matrix is the analyzer vector for one mea-surement state of the instrument. Each column in Xis the response of all measurement (analyzer) statesof the instrument to one calibration Stokes vector.The instrument matrix is then empirically calculat-ed as

A � X · S�1. (18)Fig. 6. (Color online) Division of focal plane polarimeter.


Clearly the calibration Stokes vectors that make up Smust be chosen so that the matrix is nonsingular.Measurements of unknown Stokes vectors may bemade with the same measurement states used in thecalibration once the instrument matrix A is estab-lished. This approach is described elsewhere as ageneral formulation.5

2. Calibration of Fourier-Based RotatingRetarder SystemsIn the rotating retarder system referenced in Subsec-tion 2.F.1, the data reduction process is based on aFourier analysis of the modulated signal that occurswhen the retarders are rotated. Fourier coefficientsare obtained as a function of Mueller matrix ele-ments, and the equations are algebraically inverted.The Mueller matrix for the system is

Msys � P2R2MR1P1, (19)

where M is the Mueller matrix of the sample, R1 andR2 are the matrices for the retarders, and the P arethe matrices for the polarizers. The data collectionprocess is allowed to proceed with no sample in placeas a calibration. Since the Mueller matrix in theabsence of a sample is the identity matrix, andexpected and significant errors can be built intothe data reduction equations (e.g., nonideal retardersand element orientation errors), these errors can beevaluated during calibration and used in sample datareduction. Note that for Mueller matrix systems,empty space or a high-quality mirror are good cali-bration samples. For Stokes systems, one has to becareful in the selection of calibration elements suchas polarizers or retarders, and assumptions about thequality or properties of particular polarizers or re-tarders must be made with caution. Proper polari-metric measurement and calibration procedures canbe a process of building up a simultaneous calibrationof the polarimeter and the test signature so that thequality of the measurements made is better thanthe quality of the individual elements used. This isthe case in both of the techniques just described.

3. Polarization Aberrations and ImageMisalignmentAn instantaneous FOV of the individual pixels maybe an issue in polarization measurement, especiallyin imaging systems where the overall FOV of thesystem may be substantial.103 This is an aspect ofpolarization aberrations,104,105 a subject that is inde-pendent of the classical wavefront aberrations morecommonly studied in optics and that result from thedifferences created in amplitudes and phases as po-larized light encounters interfaces.

Image registration, whether for sequential or simul-taneous image collection, is a critical issue. Misregis-tration can result, for example, from separate focalplanes that are not looking at the same region of space,or it can result from beam wander from a rotatingelement. Whatever the cause, Smith et al.106 have sug-gested that images should be registered to 1�20 of a

pixel for acceptable polarimetric results. Ideally, thealignment should be mechanical. In practice, achievingeven a half-pixel alignment mechanically can be diffi-cult and software postprocessing alignment is fre-quently necessary.107

B. Optimization

There have been significant research results recentlyconcerning the optimization of passive and active po-larimeter systems using both Fourier and DRMtechniques. For passive DRM-based polarimeters, theestimate of the unknown incident Stokes vector is asgiven in Eq. (12). However, for real polarimeters, thereare often many sources of uncertainty including, butnot limited to, noise in the measurement process andcalibration errors in determining the DRM. These er-ror sources will be carried through the inversion of theDRM, leading to an error in the reconstructed Stokesvector. Consider first the case of measurement noise. Ifthe DRM for the polarimeter is A, but the measure-ment is

X � A · Sin � n, (20)

where n is a noise vector, then we can define a mea-surement error

�h

� Sin � Sin � A�1 · n. (21)

There are many potential metrics for quantifying thenoise, and a detailed description of the trade-offsis presented by Ambirijan and Look108 and Sabatkeet al.,109 but for the purposes of this discussion we willconcentrate on the 2-norm of the DRM. If we assumethat the noise is independent and identically distrib-uted from pixel to pixel, then the 2-norm predicts themaximum length of the error vector in the recon-structed Stokes images as

�A�2 � �A�1�2 � supn

�A�1 · n�2

�n�2�

sup��h

�2

�N�2, (22)

where �2 is the variance of the elements of the noisevector.

It has been shown that minimization of the 2-normin Eq. (22) means that the �SD

i�T in Eq. (11) should bechosen to be maximally spaced on the surface of thePoincaré sphere101 so as to inscribe the maximumvolume polyhedron within the Poincaré sphere asshown in Fig. 7. Satisfying the maximum volumecondition also guarantees that the signal-to-noise ra-tio is equal in each of the three polarization channelss1, s2, and s3.77,110 When N � 4, this is a tetrahedron,but for larger values of N the geometrical shape hasthe appropriate number of corners. The value ofmaking more than four measurements is that theoverdetermined system in Eqs. (11) and (12) pro-vides redundancy that increases the SNR by a fac-tor of �N,77,111,112 although reduction in noise can bemade with a similar reduction in spatiotemporal


resolution by making four measurements and inte-grating for a longer time.111 This optimization pro-cedure was first carried out for rotating retarderpolarimeters108,111–113 with the result that the com-mon rotating quarter-wave plate polarimeter is sub-optimal to a system composed of a rotating 132° waveplate, and that the optimum rotation angles for thefast axis of the wave plate (independent of retar-dance) are at �15.1° and �51.6° with respect tothe orientation of the linear polarization analyzer.Similar optimization studies have been performed forlinear polarization,114 dual rotating retarder,113 vari-able retardance,77,110 arbitrary-state polarimeters,101

and dual rotating retarder Mueller polarimeters.115 Anumber of experimental validation studies exist thatdemonstrate the value of using an optimized polar-imeter.27,109,111,112,116

In addition to noise considerations, there areusually calibration errors associated with the exper-imental determination of the DRM A. In this case, wecan recast Eq. (11) as

X � �A � � · Sin, (23)

where � is the calibration error matrix. In this case,Eq. (12) becomes

�h

� A�1 · Sin, (24)

which tells us that the error is a function of the cal-ibration error and the input Stokes vector.117 Equa-tion (24) implies that polarimeter optimization is notas straightforward as simply optimizing the conditionnumber of the DRM. The effects of Eq. (24) are ana-lyzed by Tyo77 for a rotating retarder polarimeter,and found that the optimum polarimeter with respectto Eq. (24) is not the optimum polarimeter with re-

spect to Eq. (22). For other classes of polarimeters,Eq. (24) can be used to find the best setting of param-eters to provide minimum error.

6. Conclusions

The sensing of optical polarization information forremote sensing is an historically underused tech-nique. In many applications, polarization phenom-ena are ignored, and the optical field is treated asscalar. While this can be reasonably accurate inmany scenarios, it is clear that the ability to mea-sure polarization information, especially across ascene, provides additional information that can beexploited. Imaging polarimetry is an emerging tech-nique that promises to enhance many fields of op-tical metrology ranging from remote sensing toatmospheric sciences to industrial monitoring.

In this paper, we have discussed many of the im-portant developments in imaging polarimetry. Wehave attempted to survey the literature, and we haveprovided a general overview of most of the strategiesthat can be employed for various tasks. A subjectsuch as imaging polarimetry would be adequatelycovered in a much longer work, but we refer thereader to the extensive reference list that providesmuch greater detail on the topics discussed herein.We have attempted to focus on “first” discussions oftopics where possible, and have necessarily left outmany relevant references throughout. However, aninterested reader can follow through the referencesprovided to get a more complete picture of the state ofthe literature.

Imaging polarimetry is a field rich with potentialfor future work. Advances are needed in sensor tech-nology, data-retrieval and analysis algorithms, andapplications. While much has been accomplished inrecent years, there is still a need for sensor systemswith improved accuracy, precision, and stability. Par-ticularly useful would be better and more completequantification of these characteristics for currentlyexisting and future systems. This implies a need forimproved calibration techniques and more widely ac-cepted and followed characterization procedures.For example, current technology makes use ofMueller matrix images, as promoted by Chipman,118

a practical way of characterizing polarization ele-ments and systems. There is also a significant oppor-tunity for creative ideas in dealing with the temporaland spatial trade-offs that presently exist in imagingpolarimetry. And, finally, as the technology advances,there is a great variety of applications waiting to beaddressed with imaging polarimetry.

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