Near-field Fourier transform polarimetry by use of a discrete space-variant subwavelength grating

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1 1940 J. Opt. Soc. Am. A/ Vol. 20, o. 10/ October 2003 Biener et al. ear-fiel Fourier transform polarimetry by use of a iscrete space-variant subwavelength grating Gabriel Biener, Avi iv, Vlaimir Kleiner, an Erez Hasman Optical Engineering Laboratory, Faculty of Mechanical Engineering, Technion Israel Institute of Technology, Haifa, Israel Receive February 18, 2003; revise manuscript receive May 15, 2003; accepte May 28, 2003 We present a unique metho for real-time polarization measurement by use of a iscrete space-variant subwavelength grating. The formation of the grating is one by iscrete orientation of the local subwavelength grooves. The complete polarization analysis of the incient beam is etermine by spatial Fourier transform of the near-fiel intensity istribution transmitte through the iscrete subwavelength ielectric grating followe by a subwavelength metal polarizer. We iscuss a theoretical analysis base on Stokes Mueller formalism, as well as on Jones calculus, an experimentally emonstrate our approach with polarization measurements of infrare raiation at a wavelength of 10.6 m Optical Society of America OCIS coes: , , , ITRODUCTIO Optical polarimetry measurement has been wiely use for a large range of applications such as ellipsometry, 1 bioimaging, 2 imaging polarimetry, 3 an optical communications. 4 A commonly use metho is the measuring of the time-epenent signal once the beam is transmitte through a photoelastic moulator 5 or a rotating quarter-wave plate (QWP) followe by an analyzer. 6 The polarization state of the beam can be erive by Fourier analysis of the etecte signal. This metho, however, requires a sequence of consecutive measurements, thus making it impractical for real-time polarization measurement in an application such as aaptive polarizationmoe ispersion compensation in optical communications. Moreover, it involves either mechanically or electronically active elements, resulting in a complicate an cumbersome evice. An increasing eman for faster an simpler methos has le to the evelopment of the simultaneously fourchannel ellipsometer. 7 In this metho the beam is split into four channels; each is analyze by using ifferent polarization optics, while the real-time polarization state is calculate from the measure intensities. The main rawback of this metho is its high sensitivity to statistical errors as a result of the low number of measurements. Recently, Gori 8 propose real-time polarimetry by use of a space-variant polarizer, in what is basically a manifestation of the four-channel technique. This metho relies on measuring the far-fiel intensity an therefore is not suitable for on-chip integrate applications. In our recent letter, 9 we presente a space-omain analogy to the rotating QWP metho. In this metho spatial intensity istribution analysis is applie for realtime near-fiel polarimetry. The intensity moulation is achieve by a space-variant wave plate, realize as a computer-generate space-variant continuous subwavelength grating, followe by an analyzer. However, continuity of the subwavelength grating ha le to a spaceepenent local perio. Therefore, in orer for the perio not to excee the Woo anomaly, 9,10 the elements were restricte in their physical imensions. Moreover, the varying perioicity ha complicate the optimization of the photolithographic process an ha le to spatial variations in the retaration of the element. In this paper we propose real-time near-fiel polarimetry by spatial iscrete rotating of the groove orientation of a subwavelength ielectric grating. The grating of this type of element is ivie into equal-size zones. The subwavelength grooves are of uniform orientation an perio at each zone an are rotate at iscrete angles respective to each zone. The resulting elements are unlimite in their imensions an have uniform optical parameters. We name such elements iscrete spacevariant subwavelength ielectric gratings (DSGs). Unlike the case for other methos base on Fourier analysis, no active elements are require to etermine the polarization state of an incient beam. Our metho is less sensitive to statistical errors because of the increase number of measurements, an it is suitable for real-time applications an can be use in compact configurations. It is possible to integrate our polarimeter on a twoimensional etector array for laboratory on-chip applications to achieve a high-throughput an low-cost commercial polarimeter for biosensing. In Section 2 we iscuss a theoretical analysis of the near-fiel intensity istributions of the beams transmitte through our computer-generate space-variant subwavelength gratings as a function of the polarization state of the incient beams by use of Stokes Mueller formalism. In Appenix A we use Jones calculus to gain physical insight into the main results of Section 2. In Section 3 we escribe the realization proceure an ex /2003/ $ Optical Society of America

2 Biener et al. Vol. 20, o. 10/October 2003/J. Opt. Soc. Am. A 1941 perimentally emonstrate the ability of our metho to measure the polarization state as well as the egree of polarization for fully an partially polarize light. Section 4 is evote to concluing remarks. 2. THEORETICAL AALYSIS BY USE OF STOKES MUELLER FORMALISM The concept of near-fiel polarimetry base on subwavelength gratings is presente in Fig. 1. A uniformly polarize light is incient upon a polarization-sensitive meium (e.g., biological tissue, optical fiber, wave plate, etc.) an then transmitte through a DSG, which acts as a space-variant wave plate, followe by a polarizer. The resulting intensity istribution is image onto a camera an capture for further analysis. It will show further that the emerging intensity istribution is uniquely relate to the polarization state of the incoming beam. This epenence is given by a spatial Fourier series analysis, whereby the resulting Fourier coefficients completely etermine the polarization state of the incoming beam. The DSGs are consiere wave plates with constant retaration an space-variant fast axes, the orientation of which is enote by (x, y). It is convenient to form such space-variant wave plates by use of a subwavelength grating. When the perio of a subwavelength perioic structure is smaller than the incient wavelength, only the zero orer is a propagating orer, an all other orers are evanescent. The subwavelength perioic structure behaves as a uniaxial crystal with the optical axes parallel an perpenicular to the subwavelength grooves. 11 Therefore, by fabricating quasi-perioic subwavelength structures, for which the orientation of the subwavelength grooves is change along the length of the element, one can form space-variant wave plates. The creation of a DSG is one by iscrete orientation of the local subwavelength grating, as illustrate in Fig. 2(a). The DSG is obtaine by iviing the element into equal-size zones along the x axis, where each zone consists of a uniform orientation as well as a uniform subwavelength grating perio. The orientation of the grooves is efine by the angle between the grating vector K g of the subwavelength grating (perpenicular to the grooves) an the x axis; therefore is a function of the x coorinate (x). The grating perio is efine as the istance between the nearest zones having ientical orientations. We consier the perio of grating as larger than the incient wavelength, whereby the local subwavelength perio of the grooves,, is smaller than the incient wavelength. Figure 2(a) presents a DSG with a perio that consists of 4 zones of uniform orientation of the subwavelength grating. We enote as the number of iscrete levels. The polarization state within the Stokes representation is escribe by a Stokes vector S (S 0, S 1, S 2, S 3 ) T, where S 0 is the intensity of the beam, whereas S 1, S 2, an S 3 represent the polarization state. In general, S 2 0 S 2 1 S 2 2 S 2 3, where the equality hols for fully polarize beams. 6 The polarization state emerging from an optical system (i.e., wave plates, polarizers, etc.) is linearly relate to the incoming polarization state through S MS, where M isa4 4 real Mueller matrix of the system an S an S are the Stokes vectors of the incoming an outgoing polarization states, respectively. The optical system uner consieration consists of a DSG followe by a polarizer. This composite element can be escribe, in Cartesian coorinates, by the Mueller matrix where M PRWR, (1) cos2 sin2 0 R 0 sin2 cos is the Mueller matrix that represents rotation of the axis frame by angle an Fig. 1. Schematic presentation of near-fiel Fourier transform polarimetry base on a iscrete space-variant subwavelength ielectric grating followe by a subwavelength metal polarizer.

3 1942 J. Opt. Soc. Am. A/ Vol. 20, o. 10/ October 2003 Biener et al. Fig. 2. (a) Magnifie geometry of the iscrete space-variant ielectric grating with a number of iscrete levels 4. (b) Discrete rotation angle of the subwavelength grating as a function of x coorinate; the local groove orientations are inicate. (c) Scanning electron microscopy image of a region on the subwavelength structure. () Scanning electron microscopy image of a cross section of the subwavelength grooves. W 2tx 2 2 t y t 2 x t y t 2 2 x t y t 2 2 x t y t x t y cos 2t x t y sin 0 0 2t x t y sin 2t x t y cos is the Mueller matrix of an ieal horizontal polarizer. The outgoing intensity can be relate to the incoming polarization state of the beam by calculating the Mueller matrix given by Eq. (1) an using the linear relation between the incoming an outgoing Stokes vectors, yieling is the Mueller matrix of a transversally uniform retarer 6 with retaration an real transmission coefficients for two eigenpolarizations t x an t y. Finally, P S 0 x 1 4 (AS A CS 1 BS 1 S 0 cos2x BS 2 DS 3 sin2x 1 2 A C S 1 cos4x S 2 sin4x), (2) where A t x 2 t y 2, B t x 2 t y 2, C 2t x t y cos, an D 2t x t y sin. Equation (2) escribes the intensity of the outgoing beam as a truncate Fourier series with coefficients that epen on the Stokes parameters of the incient beam. In our case represents the iscrete rotation angle of the

4 Biener et al. Vol. 20, o. 10/October 2003/J. Opt. Soc. Am. A 1943 retarer (subwavelength grating) as a function of the location along the x axis. A single perio of can be written explicitly as 0, 0 x, 2 x ] ] x m 1 m 1, x m, (3) ] ] 1 1, x where m is an integer number, with m 0. Figure 2(b) illustrates (x) for a number of iscrete levels 4. ote that (x) is perioic in. Inserting the iscrete angle escribe by Eq. (3) into Eq. (2) an then expaning the trigonometric expressions into a Fourier series yiels the resulting intensity istribution 4S 0 x AS A CS 1 BS 1 S 0 n1 a n cos2nx/ b n sin2nx/ BS 2 DS 3 n c n sin2nx/ 1 n cos2nx/ 1 A C 2 S 1 n1 S 2 n1 e n cos4nx/ f n sin4nx/ g n sin4nx/ h n cos4nx/. (4) The corresponing Fourier coefficients in Eq. (4) are given by a n c n 2n sin 2, b n n n sin2, e n g n 4n sin 4, sin2 f n h n 2 (5) 2n for n k 1(k 0, 1, 2, 3,...) an are zero otherwise. One can see from Eqs. (5) that increasing the number of iscrete levels increases a 1, c 1, e 1, an g 1 towar unity an ecreases b 1, 1, f 1, an h 1 towar zero. Moreover, as the number of iscrete levels increases, the higher-orer terms ten to reach zero as well; thus, at the limit of an infinite number of iscrete levels, Eq. (4) egenerates to the case of a continuous space-variant subwavelength grating. 9 With the use of Fourier analysis, the first coefficients of Eq. (4) yiel AS A CS 1 4 S 0 xx, 0 BS 1 S 0 a 1 BS 2 DS 3 b BS 2 DS 3 a 1 BS 1 S 0 b 1 S 0 xcos 2x (6a) x, (6b) 8 S 0 xsin 2x x, (6c) 0 A CS 1 e 1 S 2 f 1 16 S 0 xcos 4x x, 0 (6) A CS 1 f 1 S 2 e 1 16 S 0 xsin 4x x. 0 (6e) These equations are a linear combination of the Stokes parameters of the incient beam. To extract S 0 S 3, Eqs. (6) shoul represent four inepenent equations, which are obtaine for 5. However, a larger number of iscrete levels is esirable, in which case a larger portion of the intensity is represente by the first harmonics of every series of Eq. (4) an therefore a larger signal-to-noise ratio can be obtaine. We note that the grating coefficients A, B, C, an D shoul be etermine by irect measurement of the subwavelength grating parameters t x, t y, an or by performing a suitable calibration process. The full synthesis of the incoming polarization state is given in Appenix B. Moreover, an interesting physical insight into our polarimeter approach is given in Appenix A by use of Jones calculus. 3. REALIZATIO PROCEDURE AD EXPERIMETAL RESULTS The DSG element for CO 2 laser raiation of 10.6-m wavelength was fabricate upon a 500-m-thick GaAs wafer with 2 m, 2.5 mm, an 16. The imensions of the element were 30 mm 3 mm an consiste of 12 perios of. First, a binary chrome mask of the grating was fabricate by using high-resolution laser lithography. The pattern was then transferre onto the GaAs wafer by use of photolithography, after which we etche the grating by electron cyclotron resonance with

5 1944 J. Opt. Soc. Am. A/ Vol. 20, o. 10/ October 2003 Biener et al. t y 0.81, an ra that were achieve by using a rigorous couple-wave analysis 12 for a groove profile epicte in Fig. 2() an to the irect measurement of the optical parameters using ellipsometric techniques, 6 which resulte in t x 0.940, t y 0.833, an ra with a stanar eviation of approximately 0.02 for t x an t y an of ra for. We foun that in our etching process, the errors in the etching epth were approximately 5% of the nominal epth (3.3 m). Therefore we can relate the eviations of t x, t y, an to the spatial errors in the etching process. ote that the calibration values also take into account other imperfections of the optical setup. To test the ability of our evice in conucting polarization measurements of fully polarize light, we use a CO 2 laser that emitte linearly polarize light at a wavelength of 10.6 m an replace the polarization-sensitive meium with a QWP. The images were capture by a Spiricon Pyrocam III two-imensional pyroelectric etection array at a rate of 24 Hz. Figure 3 shows the Fig. 3. Measure (circles) an preicte (soli curves) values of the normalize transmitte intensity as a function of the x coorinate along the DSG when the fast axis of the rotating QWP was at angles (a) 0, (b) 20, an (c) 45 ; the insets show the experimental images of the near-fiel transmitte intensities. BCl 3 for 35 min, resulting in an approximately 2.5-m groove epth. Finally, an antireflection coating was applie to the back sie of the wafer. Figure 2(c) shows a scanning electron microscopy image of a region on the subwavelength structure that we fabricate, whereas Fig. 2() epicts an image of a cross section of the subwavelength grooves. Following the measurements, we use the setup epicte in Fig. 1 to test our concept for polarization measurements, whereby a subwavelength metal wire grating was use as a polarizer. 10 First, a calibration process was performe by illuminating the DSG followe by a polarizer with horizontal, vertical, an right-han circularly polarize light. We etermine the experimental optical parameters by fitting the curve of Eq. (4) to the measure intensity istributions, using a least-mean-squares algorithm, with t x, t y, an of the DSG as free parameters. The calibration process yiele a DSG having t x 0.9, t y 0.8, an retaration 0.3 ra. These values are close both to the theoretical preictions of t x 0.87, Fig. 4. Measure (circles) an preicte (soli curves) values of the normalize Stokes parameters (a) S 1 /S 0, (b) S 2 /S 0, an (c) S 3 /S 0 as a function of the orientation of the QWP.

6 Biener et al. Vol. 20, o. 10/October 2003/J. Opt. Soc. Am. A 1945 Fig. 5. Measure (circles) an preicte (soli curves) values for (a) azimuthal angle an (b) ellipticity angle as a function of orientation of the QWP. Fig. 6. Calculate (soli curve) an measure (circles) DOP as a function of intensity ratio of the two inepenent lasers having orthogonal linear polarization states, as use in a setup epicte in the top inset. The bottom inset shows calculate (soli curves) an measure (circles) intensity cross sections for the two extremes, I 1 I 2 (DOP 0.059) an I 2 0 (DOP 0.975). measure intensity istributions capture in a single camera frame, when the fast axis of the QWP was set at angles 0, 20, an 45, as well as the preicte results. The preiction was obtaine by inserting the expecte values for the incoming Stokes parameters into Eq. (4) for A 1.45, B 0.17, C , an D Consequently, Fig. 4 shows the measure an preicte Stokes parameters of a resulting beam as a function of the orientation of the QWP. We etermine the experimental values of S 1, S 2, an S 3 by using Eqs. (B1a) (B1). The integrals,,, an were numerically evaluate from the acquire ata by using the Simpson algorithm. There is a goo agreement between the preictions an the experimental results. Moreover, Fig. 5 shows the experimental an theoretical azimuthal angle an the ellipticity angle, calculate from the ata in Fig. 4, by use of the relations tan(2) S 2 /S 1 an sin(2) S 3 /S 0. 6 The measurements yiel a stanar eviation error with respect to the theoretical preictions of 2.6 an 0.6 for the azimuthal angle an the ellipticity angle, respectively. The errors of the polarization measurements result mainly from systematic errors such as the nature of the algorithm, imprecision in the rotation of the QWP, spatial inhomogeneity in the QWP retaration, which is approximately 2, spatial inhomogeneity in the retaration of the subwavelength grating, which is approximately 1.3 (8% of the nominal phase), an low resolution of the IR camera an ynamic range, as well as statistical noise ue to spatial an temporal fluctuation of the light emitte from the laser, shot noise an amplifier noise of the IR camera, pixel response nonuniformity, an quantization noise. The stanar eviations in the azimuthal angle an the ellipticity angle for a series of successive measurements were 0.2 an 0.07, respectively. To emonstrate the use of a DSG for polarimetry of partially polarize beams, we combine two inepenent CO 2 lasers of orthogonal linear polarization states by use of the setup epicte in the inset at the top of Fig. 6. The 2 egree of polarization (DOP) is efine by DOP (S 1 S 2 2 S 2 3 ) 1/2 /S 0. For incoherent beam summation, the Stokes vector of the resulting beam is the sum of the Stokes vectors of the combine beams. 6 In the case of two orthogonal linearly polarize beams, the DOP is given by DOP (I 1 I 2 )/(I 1 I 2 ), where I 1 an I 2 are the intensities of the horizontally an vertically polarize beams, respectively. Figure 6 shows the measure an preicte DOP as a function of the intensity ratio I 1 /I 2. The inset shows the experimental intensity istributions for two extreme cases. The first is for equal intensities (I 1 I 2 ), in which the measure DOP is , inicating unpolarize light. The secon is for illumination of a single laser only (i.e., I 2 0), in which the measure DOP is 0.975, inicating fully polarize light. This experiment shows the ability to obtain all four Stokes parameters simultaneously, thereby emphasizing the goo agreement between preiction an measurement for partially polarize light. 4. COCLUSIOS We have theoretically analyze an experimentally emonstrate the use of a computer-generate space-variant iscrete subwavelength ielectric grating for real-time polarization measurement. The iscrete subwavelength gratings are unlimite in their imensions, have uniform optical properties (t x, t y, an ), an, in general, are lightweight, compact, an highly efficient. Both the space-variant subwavelength ielectric grating an the polarizer were realize by using photolithographic techniques commonly use in the prouction of microelectric evices. Therefore the camera an the gratings coul be combine into a single chip, resulting in a very small evice for polarization measurements.

7 1946 J. Opt. Soc. Am. A/ Vol. 20, o. 10/ October 2003 Biener et al. APPEDIX A: THEORETICAL AALYSIS FROM THE ITERFERECE POIT OF VIEW Subwavelength quasi-perioic structures are conveniently escribe as epicte in Fig. 2 by using Jones calculus. The analysis in this case is limite to fully polarize coherent beams. The DSG, which is a birefringent element with optical axes (parallel an perpenicular to the grating grooves) that rotate perioically in the x irection, can be represente as a polarization iffraction grating. When a plane wave with a uniform polarization is incient upon such a perioic subwavelength structure, the transmitte fiel will be perioic in both the polarization state an the phase front; therefore we can expect this fiel to yiel iscrete iffraction orers. The interference between the iffraction orers in the near fiel yiels spatial intensity moulation. Eviently, the resulting interferogram pattern is irectly relate to the incient polarization state. In this representation a uniform perioic subwavelength structure, the grooves of which are oriente along the y axis, can be escribe by the Jones matrix J t x 0 (A1) 0 t y expi, where t x an t y are the real amplitue transmission coefficients for light polarize perpenicular an parallel to the optical axes an is the retaration of the grating. Consequently, the space-epenent transmission matrix of the DSG can be escribe by T C x M(x)JM 1 (x), (A2) where (x) is the iscrete local orientation of the optical axis given by Eq. (3) an Fig. 2(b), an M() cos sin sin cos is the two-imensional rotation matrix. ote that, meanwhile, the polarizer is omitte from the optical system, an this will be referre to below. For convenience, we convert T C (x) to the helicity basis; therefore the space-variant polarization grating can be escribe by the matrix T(x) UT C U 1, where U 1 1/2 i i 1 is a unitary conversion matrix. Explicit calculation of T(x) yiels Tx 1 2 t x t y expi t x t y expi 0 expi2x. (A3) expi2x 0 We further aopt the Dirac bra ket notation, in which R (1, 0) T an L (0, 1) T are the two-imensional unit vectors for right-han an left-han circularly polarize light, respectively. Thus the resulting fiel is the prouct of an incient plane wave with arbitrary polarization E in an the space-epenent transmission matrix T(x) given by Eq. (A3), yieling where E out E E in R expi2x, yr L expi2x, yl, (A4) E 1 2 t x t y expi, R 1 2 t x t y expi E in L, L 1 2 t x t y expi E in R are the complex fiel coefficients an enotes the inner prouct. From Eq. (A4) it is evient that the emerging beam from the DSG, which is enote by E out, comprises three polarization orers. The first maintains the original polarization state an phase of the incient beam, the secon is right-han circularly polarize with a phase moification of 2(x), an the thir has polarization irection an phase moification opposite to those of the former polarization orer. ote that the phase moification of the R an L polarization orers results solely from local changes in the polarization state an therefore is geometrical in nature. 13 Since (x) is a perioic function [Eq. (3), Fig. 2(b)], the functions expi2 (x) an expi2 (x) in Eq. (A4) can be expane into a Fourier series. Taking into account the connection between the Fourier series of the conjugate functions expi2 (x) an expi2 (x) leas to the equation E out E E in R L m m m expi2mx/r m * expi2mx/l, (A5) where m (2/) 0 expi2 (x) mx/x. Base on Eq. (A5), we fin that the iffraction efficiency into the R mth iffracte orer of the R polarization orer ( m m 2 ) is equal to the iffraction efficiency into the L mth iffracte orer of L ( m m * 2 ); thus R an L iffract in opposite manners. Therefore, as epicte in Fig. 7, once a uniformly polarize beam is incient upon the DSG, the resulting beam comprises three polarization states escribe by Eq. (A4), whereas the polarization orers of R an L states are split into multiple iffraction orers as a result of the iscontinuity of the phase, 2(x). In our near-fiel polarimetry concept, the wave front emerging from the polarization grating is incient upon a linear polarizer, escribe in the helicity basis by a Jones matrix of the form P The linearly polarize fiel emerging from the polarizer is given by the prouct of Eq. (A5) an P, yieling Ẽ out 1 2 E E in R E in L R L m m m expi2mx/ m * expi2mx/. (A6)

8 Biener et al. Vol. 20, o. 10/October 2003/J. Opt. Soc. Am. A S 0 xsin4x/x, 0 whereby the coefficient matrix is A A C/2 0 0 Bb 1 Bb 1 Ba 1 Da 1 H. 0 A Ce 1 A Cf A Cf 1 A Ce 1 0 We note that the coefficients A,B,C, an D coul be etermine from a suitable calibration process. In that case H is fully known by use of Eqs. (5). Consequently, the incoming polarization state can be calculate, by using Kramer s metho, to yiel S A C e 1 f 1 A 2 A C e f 1, (B1a) Fig. 7. Diffraction orers emerging from the DSG: zero orer (ashe), first orer for R an L polarize beams (soli), an higher orer for R an L polarize beams (otte ashe). Consequently, the intensity istribution is given by Ẽ out 2, which results, after some algebraic manipulations, in an expression ientical to that given in Eq. (4). ote that if we use a 4-f telescope configuration for imaging the emerging beam from our polarimeter an insert a spatial filter in the Fourier plane, the higher iffracte orers can be eliminate. Therefore this can result in an intensity istribution resembling the pattern obtaine by the continuous-grating-base polarimeter. 9 APPEDIX B: SYTHESIS OF THE ICOMIG POLARIZATIO STATE AS A FUCTIO OF THE MEASURED ITESITY In this appenix we show the full synthesis of the incoming polarization state as a function of the measure intensity by solving Eqs. (6a) (6e). Only four equations are require to fully etermine the polarization state, whereas the possible choice of equations epens on the number of iscrete levels. However, for 8 any four equations out of the given five is suitable. From our evaluation the best results coul be obtaine by omitting Eq. (6b). The remaining equations can be written in matrix form as HS R, where S (S 0, S 1, S 2, S 3 ) T is the incoming polarization state, R (,,, ) T, an H is the coefficient matrix. Explicitly,,,, an are given by 4 S 0 xx, 0 16 S 0 xcos4x/x, 0 8 S 0 xsin2x/x, 0 S 1 S 2 1 A C 1 A C e 1 f 1 e 1 2 f 1 2, e 1 f 1 e 1 2 f 1 2, S 3 1 Da 1 B A b 1 B e 1 2 f 1 2 (B1b) (B1c) b 1e 1 2A a 1f 1 A C b 1f 1 2A a 1e 1 A C. REFERECES (B1) 1. J. Lee, J. Koh, an R. W. Collins, Multichannel Mueller matrix ellipsometer for real-time spectroscopy of anisotropic surfaces an films, Opt. Lett. 25, (2000). 2. V. Sankaran, M. J. Everett, D. J. Maitlan, an J. T. Walsh, Jr., Comparison of polarize light propagation in biological tissue an phantoms, Opt. Lett. 24, (1999). 3. G. P. orin, J. T. Meier, P. C. Deguzman, an M. W. Jones, Micropolarizer array for infrare imaging polarimetry, J. Opt. Soc. Am. A 16, (1999). 4. P. C. Chou, J. M. Fini, an H. A. Haus, Real-time principle state characterization for use in PMD compensators, IEEE Photon. Technol. Lett. 13, (2001). 5. G. E. Jellison, Jr., Four channel polarimeter for timeresolve ellipsometry, Opt. Lett. 12, (1987). 6. E. Collet, Polarize Light (Marcel Dekker, ew York, 1993). 7. R. M. A. Azzam, Integrate polarimeters base on anisotropic photoetectors, Opt. Lett. 12, (1987). 8. F. Gori, Measuring Stokes parameters by means of a polarization grating, Opt. Lett. 24, (1999). 9. Z. Bomzon, G. Biener, V. Kleiner, an E. Hasman, Realtime analysis of partially polarize light with a spacevariant subwavelength ielectric grating, Opt. Lett. 27, (2002). 10. Z. Bomzon, V. Kleiner, an E. Hasman, Space-variant polarization state manipulation with computer-generate

9 1948 J. Opt. Soc. Am. A/ Vol. 20, o. 10/ October 2003 Biener et al. subwavelength metal stripe gratings, Opt. Commun. 192, (2001). 11. P. C. Deguzman an G. P. orin, Stacke subwavelength gratings as circular polarization filters, Appl. Opt. 40, (2001). 12. P. Lalanne an G. M. Morris, Highly improve convergence of the couple-wave metho for TM polarization, J. Opt. Soc. Am. A 13, (1996). 13. E. Hasman, V. Kleiner, G. Biener, an A. iv, Polarization epenent focusing lens by use of quantize Pancharatnam Berry phase iffractive optics, Appl. Phys. Lett. 82, (2003).