July 15, 2003 / Vol. 28, No. 14 / OPTICS LETTERS 1173

Ortho-Babinet polarization-interrogating filter: aninterferometric approach to polarization measurement

Jay S. Van DeldenOptical Sciences Center, University of Arizona, Tucson, Arizona 85721, and

EigenPhase Technologies, 755 W. Vistoso Highlands Drive, Suite 112, Tucson, Arizona 85737

Received August 13, 2002

A novel, interferometric, polarization-interrogating filter assembly and method for the simultaneous mea-surement of all four Stokes parameters across a partially polarized irradiance image in a no-moving-parts,instantaneous, highly sensitive manner is described. In the reported embodiment of the filter, two spatiallyvarying linear retarders and a linear polarizer comprise an ortho-Babinet, polarization-interrogating (OBPI)filter. The OBPI filter uniquely encodes the incident ensemble of electromagnetic wave fronts comprisinga partially polarized irradiance image in a controlled, deterministic, spatially varying manner to map thecomplete state of polarization across the image to local variations in a superposed interference pattern. Ex-perimental interferograms are reported along with a numerical simulation of the method. © 2003 OpticalSociety of America

OCIS codes: 120.2650, 120.3180, 120.5410, 230.5440, 260.1440, 260.5430.

The vectorial nature of an electromagnetic wavefront, or a plurality of such wave fronts (i.e., a wave-front ensemble), demands that certain informationpertaining to the state of polarization (SOP) beencoded therein. To polarimetrically characterizean arbitrary wave-front ensemble requires four realparameters that may vary, among other things, as afunction of position. These parameters are commonlyreferred to as the spatially varying Stokes parametersor Stokes images, denoted as S0�x, y�, S1�x, y�, S2�x, y�,and S3�x, y�. By decoding the spatially varying SOPacross a partially polarized image, it is possible toacquire additional information, beyond the conven-tional irradiance image, that enables one to potentiallydiscern heretofore indiscernible characteristics of theoriginal conjugate object.

Traditional approaches for measuring the completeSOP of an arbitrary wave-front ensemble include1

(1) discrete time-sequential, (2) continuous time-sequential (i.e., polarization-modulation), (3) division-of-amplitude, and (4) division-of-aperture methods.Whereas all these approaches have been utilized inthe past for measuring the complete SOP in a pointcontext, diff iculties often arise in practice that limitthe general usefulness of such methods for imagingpolarimetry applications. Pixel misregistration andrapidly changing scenes often preclude such methodsfrom being used in an effective manner.

In this Letter an interferometric approach to po-larization measurement is reported that allows all ofthe polarization content within an image (i.e., all fourStokes parameters on a point-by-point basis) to be ac-quired in a single image frame with no moving partsin an instantaneous (i.e., snapshot) manner.

In Fig. 1, four uniaxial, linearly birefringent wedgesand a linear polarizer comprise an ortho-Babinetpolarization-interrogating (OBPI) filter assembly.Operationally, the OBPI filter is characterized by athree-stage optical system consisting of two modifiedBabinet2 compensators [i.e., spatially varying linearretarders (SVLRs)] and a linear polarizer (LP).

In the limit of small wedge angles, such an opticalsystem is amenable to a spatially dependent Mueller

0146-9592/03/141173-03$15.00/0 ©

calculus treatment giving rise to an output Stokes vec-tor of the form

S0�x, y� � MOBPI�x, y�S�x, y� , (1)

where

MOBPI�x, y� � MLP �uTA� ? MSVLR2�uFA2,d2�x,y��

? MSVLR1 �uFA1, d1�x, y�� (2)

so that bipolar retardances d1�x, y� and d2�x, y� varylinearly as a function of transverse �x, y� coordinatesmeasured relative to the mechanical centerline axis(i.e., the z axis) of the system.

The passage of light through the OBPI filter as afunction of the incident SOP is given by the zerothelement of the exiting Stokes vector according to

S00�x, y� � �MOBPI�x, y��11 S0�x, y�

1 �MOBPI�x, y��12 S1�x, y�

1 �MOBPI�x, y��13 S2�x, y�

1 �MOBPI�x, y��14 S3�x, y� . (3)

Fig. 1. OBPI f ilter. Each spatially varying linear re-tarder is composed of a modif ied Babinet compensatorwhose crystal axes are neither parallel nor perpendicularto the coplanar wedge planes.

2003 Optical Society of America

1174 OPTICS LETTERS / Vol. 28, No. 14 / July 15, 2003

The interference patterns produced by the OBPI fil-ter can take on a variety of distinctive forms dependingon (1) the specif ic design of the filter (e.g., crystal axisorientations, wedge magnitude and directions, mate-rials), (2) the spatial and temporal coherence of the in-cident test wave-front ensemble, (3) the spectral SOPdispersion, and (4) the spatially varying SOP acrossthe f ilter’s pupil.

The most important feature of the resulting inter-ferograms is that a unique pattern is generated for anyincident SOP input. Another important feature is thecharacteristic length (i.e., spacing between two adja-cent bright fringes) of the interference pattern. Thislength is typically chosen to minimize the mean-squareerror introduced in the analog-to-digital conversionprocess.

The notion of polarization-dependent interferomet-ric encoding was verified experimentally by construc-tion of an OBPI filter with four yttrium vanadatewedges (Q � 1.429±, uFA1 � 30±, uFA2 � 150±) and anHN-22 dichroic sheet polarizer (uTA � 0±) accordingto the layout depicted in Fig. 1. Figure 2 shows ex-perimental interferograms produced by the assembledOBPI filter under varying conditions of polarizedKoehler illumination. Observations were made withfiltered white light (lc � 535 nm, DlFWHM � 40 nm) ina Leica DM RXA polarized light microscope (0.18-N.A.condenser, 103 objective, and 1.63 tube lens) with aHamamatsu C4742 Orca II digital camera.

Whereas the specific shape of the interferencefringes generated by the OBPI filter gives the ori-entation (azimuth), shape (ellipticity), and chirality(handedness) of the polarization ellipse, the fringevisibility serves to characterize the degree of polar-ization. As such, the complete SOP across the imageis uniquely encoded within the interference pattern.

In practice, an application-specific, front-end opticalsystem will form a partially polarized irradianceimage of some conjugate object onto the OBPI filter.The resulting interferometrically polarization-encodedoutput image is then optically relayed to a digitalcamera, where it is discretized by a detector arrayand digitized by an analog-to-digital converter. Thediscretized–digitized electronic image constitutes aset of raw data within which is encoded the spatiallyvarying SOP across the image.

In the derivation of a mathematical reconstructionalgorithm, let us characterize a pixel in the detectorarray by a set of four indices �i, j ,m,n�, where �i, j �indicates a specif ic unit cell and �m,n� indicates a pixelwithin the selected unit cell. A unit cell is simply asquare 23 2 array of pixels representing the minimumspatially resolvable polarization picture element.

Integrating over the zeroth component of the out-put Stokes vector in Eq. (3), weighted by the functionpixijmn�x, y�, allows one to express the spatially aver-aged optical power that is incident upon a single pixelin the detector array as

�S00�x, y��ijmn �

µ1

dxdy

∂ ZZpixijmn�x, y�S00�x, y�dxdy ,

(4)

where pixijmn�x,y� is a piecewise-continuous, two-dimensional window function that singles out anindividual pixel in the array, dx (dy) is the width of asingle pixel in the x ( y) direction, and �· · ·�ijmn denotesa spatial average over pixel �i, j ,m,n�.

Multiplying Eq. (4) by the responsivity �, relabel-ing the incident Stokes vector components accordingto fp�x, y� � Sp21�x, y�, and employing a summationover four terms results in an expression for the pixelphotocurrent:

gijmn � �

*4X

p�1�MOBPI�1p�x, y�fp�x, y�

+ijmn

. (5)

Equation (5) represents a continuous-to-discretemapping operation from the spatially varying SOPacross the input image (i.e., the polarization signa-ture) to the spatially discretized, polarization-encodedelectronic analog image. This equation characterizeswhat is commonly referred to as the forward problem.Ultimately, what we want to do is solve the reverseproblem, which usually requires some sort of approxi-mation to be made.

In the parlance of the well-known Whittaker–Shannon sampling theorem, the sampling frequency[ fS � 1�dxy , where dxy � �dx

2 1 dy2�1�2] is ordinarily

selected to be at least twice the highest spatial fre-quency contained in the image ( fS $ 2fMAX). Whenthis occurs, the Nyquist criterion ( fMAX # fN � 1/2fS)is satisfied and the spatially discretized image is con-sidered to be a reasonably faithful reproduction of thecontinuous input. However, because each unit cellcomprises a 23 2 array of pixels, the size of the pixelsin the detector array should be selected so that thesampling frequency is at least four times the highestspatial frequency contained in the image ( fS $ 4fMAX).

In short, the price to be paid for instantaneouslysecuring the complete polarization content across theimage (i.e., all four Stokes parameters, for each unitcell) is a requisite oversampling in the detector arrayso that the SOP is slowly varying over any given unitcell. As a result, we can effectively remove fp�x, y�from the argument in Eq. (5), relabel it according to

Fig. 2. Experimental interferograms generated by theOBPI filter of Fig. 1. Alongside each interference patternis the associated incident Stokes vector. A 50-mm scalebar is shown in the lower right-hand corner of each image.

July 15, 2003 / Vol. 28, No. 14 / OPTICS LETTERS 1175

Fig. 3. Numerical simulation of the proposed reconstruc-tion method for a partially polarized test image composedof six overlapping circular regions.

fp�x, y� ) fijp, and interchange the order of integra-tion and summation so that

gijmn �4X

p�1hijmnpfijp , (6)

where hijmnp � ���MOBPI�1p�x, y��ijmn. Unlike thecontinuous-to-discrete mapping case, the integrationcan now be carried out without prior knowledge ofthe incident SOP. The forward problem has beensimplified to a discrete-to-discrete mapping operation.

Letting �m,n� � �1, 1�, �1, 2�, �2, 1�, and �2, 2� inEq. (6) and introducing a slight change in notationso that g � �gij11gij12gij21gij22�t � �g1g2g3g4�t andf � � fij1fij2fij3fij4�t � � f1f2f3f4�t (superscript t de-notes a transpose), with H given by26664hij111 hij112 hij113 hij114

hij121 hij122 hij123 hij124

hij211 hij212 hij213 hij214

hij221 hij222 hij223 hij224

37775 �

26664H11 H12 H13 H14

H21 H22 H23 H24

H31 H32 H33 H34

H41 H42 H43 H44

37775 , (7)

allows the forward problem to be cast in the more fa-miliar form

g � Hf , (8)

where g and f are each 4-element column vectors andH is a 16-element square matrix that characterizesboth the polarization-dependent transmission of theOBPI filter and the pixel discretization process.

Multiplying each side of Eq. (8) by the inverse of ma-trix H and noting that H21H � I, where I is the iden-tity matrix, we find that

f � H21g . (9)

Reverting back to the conventional Stokes parame-ter notation so that fijp � Sij� p21� , where p � 1, 2, 3, or4, and writing out Eq. (9) in the form of a summation,we find that the Stokes parameters of unit cell �i, j�are given by

Sij� p21� �4X

q�1�H21�pqgq . (10)

Let us consider a numerical simulation of theproposed reconstruction method wherein a partiallypolarized test image is interferometrically encodedby the OBPI filter. In Fig. 3(a), the 2.5-mm-squaremonochromatic test image (l � 630.0 nm) is com-posed of six overlapping circular regions (diameter,1.25 mm), for a total of 17 polarimetrically distinctsubregions. (Each subregion has a well-definedStokes vector according to the color-coded table.)The test image is interferometrically encoded by anOBPI filter whose characteristic length is 100 mm(Q � 0.8102± and ne 2 no � 0.2227). Figure 3(b)illustrates the continuous polarization-encoded ana-log output image generated by the OBPI filter. InFig. 3(c), the continuous analog image is spatiallydiscretized by a 100 3 100 array of square-shaped25-mm pixels and further electronically digitized bya 16-bit analog–digital converter. Figures 3(d), 3(e),3(f ), and 3(g) illustrate the reconstructed zeroth, first,second, and third Stokes images, respectively. Witha 16-bit analog–digital converter, the peak and rmsquantization (i.e., digitization) errors in the recon-structed zeroth Stokes image are 0.010 and 0.0006,respectively.

In summary, we have reported on a patent-pending,interferometric, polarization-interrogating f ilter as-sembly and method for determining the completestate of polarization on a point-by-point basis acrossa partially polarized irradiance image from a singleimage frame.

I am indebted to James C. Wyant of the OpticalSciences Center for deepening my appreciation ofinterferometry. Address e-mail correspondence tojayv@u.arizona.edu.

References

1. P. S. Hauge, Proc. SPIE 88, 3 (1976).2. M. Francon and S. Mallick, Polarization Interferometers

(Wiley, London, 1971).