Department of Physics and Astronomy, Trent University, 1600 West Bank Drive, Peterborough, Ontario K9J 7B8, Canada

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1 alibration method using a single retarder to simultaneously measure polarization and fully characterize a polarimeter over a broad range of wavelengths Matthew J. Romerein, Jeffrey N. Philippson, Robert L. Brooks, and Ralph. Shiell, * Department of Physics and Astronomy, Trent University, 6 West Bank Drive, Peterborough, Ontario K9J 7B8, anada Department of Physics, University of Guelph, 5 Stone Road East, Guelph, Ontario NG W, anada *orresponding author: ralphshiell@trentu.ca Received 9 June ; accepted 7 July ; posted August (Doc. ID 48974); published September A method has been developed to improve the accuracy with which the polarization state of light can be characterized by the rotating quarter-wave plate technique. Through detailed analysis, verified by experiment, we determine the positions of the optic axes of the retarder and linear polarizer, and the wave plate retardance, to better than for typical signal-to-noise ratios. Accurate determination of the Stokes parameters can be achieved using a single wave plate for a wide range of optical wavelengths using this technique to determine the precise retardance at each of the wavelengths of interest. Optical Society of America OIS codes:.54,.453, Introduction Stokes vector polarimetry is a powerful tool able to capture all of the polarization properties of a given light source [,]. ombined with spectral analysis, it forms the basis for Stokes vector spectroscopy, which is employed in diverse applications ranging from industrial sample analysis to solar astronomy [3,4]. Mueller matrix polarimetry uses a source with well-known polarization and applies the techniques of Stokes vector polarimetry to obtain information on the polarization-changing properties of optical elements [5]. When applied to the modes of an optical fiber, this provides a direct measurement of polarization-mode dispersion, which places a limit on data transmission rates [6]. Ellipsometry is a related technique that analyzes the polarization //8538-8$5./ Optical Society of America state of reflected light to measure the dielectric properties of thin films [7]. Imaging polarimetry, in which a Stokes vector is obtained for each pixel in an image, has become a powerful and widely used tool in remote sensing [8]. A number of techniques exist for measuring the polarization state of light [9,] but all are limited both by manufacturing tolerances in the optical elements used and by uncertainties in their experimental application. In this paper we describe a calibrated method that can compensate for a variety of manufacturing and experimental sources of error, and in doing so permits polarization measurements to be obtained at wavelengths far from the specified design wavelength of the optical elements involved. The rotating quarter-wave plate method is a wellknown technique for characterizing the polarization state of light []. In this convenient and widely used method, the intensity of light transmitted through a quarter-wave plate and linear polarizer (LP) is 538 APPLIED OPTIS / Vol. 5, No. 8 / October

2 recorded at a number of azimuthal angles of the retarder s fast axis from a fixed reference axis. The measured intensities depend on these angles, which we denote fβ i g, the precise retardance of the wave plate Δ ( for an ideal quarter-wave plate), and the angle γ of the LP transmission axis from the same fixed reference axis. Accurate determination of the polarization state of incident light, which is often described by the Stokes parameters, therefore requires precise knowledge of fβ i g, Δ, and γ. To give one example of the significance of knowing these accurately, an offset of only þ3 in the retarder alignment and þ% in the retardance introduces a 3% error in the normalized Stokes parameter S for horizontally polarized light, for which one expects unity. In practice these values are not always known within the desired level of uncertainty []. For the first data point (i ), the retarder s fast axis is typically aligned with the reference axis. However, precise azimuthal positioning of the often unlabeled optic axes of the retarder and LP to within is challenging, leading to both an offset, β, in all β i and causing uncertainty in the value of γ [see Fig. (a)]. Further, error in the value of Δ can derive from using a quarter-wave plate at wavelengths away from its design wavelength λ nom, from manufacturing tolerances that are typically :λ nom for a zeroth-order quarter-wave plate [3], or from a slight tilt of the retarder away from normal incidence [4]. A reliable approach to calibration is therefore necessary for accurate characterization of polarization using this method. We have adopted and expanded upon a calibration technique previously developed by one of us (RLB) [5,6]. This technique employed a two-step calibration process using linearly polarized incident light. In the first step, using an assumed value for Δ, the angle of the LP transmission axis from ^x, which we will henceforth call horizontal, and the offset of the retarder fast axis were determined self-consistently from four sets of measurements corresponding to each of the two optics oriented either forward or reversed by rotating a Polarizer z Retarder y x b F' F y F Fig.. (olor online) (a) Illustration of the relevant angles for the calibrated rotating quarter-wave plate method (angles have been exaggerated for clarity). The angle of the LP transmission axis and the offset of the retarder fast axis in its initial position from ^x are denoted by γ and β, respectively. The front-to-back rotation axis of the polarizer defines ^y and the misalignment of the retarder s rotation axis from this is denoted ϕ. (b) Position of the fast axis is shown for the forward (F) and reversed (F ) orientations of the retarder. When reversed, the fast axis is offset by ϕ β from ^x due to the misalignment of the retarder s front-to-back rotation axis from ^y. F' x about a vertical axis. Because a reversal has no effect if an optic axis is aligned either horizontally or vertically for an ideal wave plate, then provided β and γ are correctly accounted for, the Stokes parameters calculated for each of the four cases will be consistent. Subsequently, Δ was determined with a second self-consistent technique requiring an additional two sets of measurements with vertically polarized incident light and the transmission axis of the LP in the polarimeter aligned and then crossed with respect to this polarization. To improve the accuracy of all the determined parameters, fβ ; Δ; γg, the entire process could be repeated iteratively to convergence. Our calibration method accomplishes all this in a single experimental step and also takes into account the deleterious effect of a misalignment between the assumed-vertical rotation axes of the retarder and polarizer, denoted by ϕ, which can occur due to experimental uncertainties. As we show below, a small value of ϕ can introduce errors into the calibration and subsequently into the Stokes parameters if it is not accounted for. Specifically, we present a set of highly consistent Stokes parameters and calibration parameters for an incident laser beam using a set of quarter-wave plates that differ in λ nom by as much as nm and for three distinct experimental misalignments, fϕ nom g. Using this calibration method, one quarter-wave plate could therefore in principle encompass measurements across the whole visible spectrum. Moreover, this calibration method does not require prior knowledge of β, Δ, γ, orϕ.. Theory of the alibrated Rotating Quarter-Wave Plate Method The Stokes parameters completely describe the polarization of fully polarized, partially polarized, or unpolarized light. A Stokes vector S, normally represented as a column vector, can be constructed from the four Stokes parameters S, S, S, S 3. Using the convention that the light travels along þ ^z and h i denotes a time average, S hj ~ E x j iþhj ~ E y j i is the intensity of the light, S hj ~ E x j i hj ~ E y j i is the excess of horizontally over vertically polarized light, S Reh ~ E x ~ Ey i is the excess of þ45 over 45 polarized light, and S 3 Imh ~ E x ~ Ey i is the excess of right over left circularly polarized light for a complex electric field ~E. In all cases, S S þ S þ S 3, where the equality corresponds to fully polarized light. The degree of polarization is expressed by the ratio of the polarized component to the total intensity of the light, P qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S þ S þ S 3 : ðþ S Typically the Stokes vector is normalized such that S, resulting in dimensionless quantities that are independent of the intensity of the light, and we follow this convention herein. October / Vol. 5, No. 8 / APPLIED OPTIS 5383

3 The polarization-changing effect of any optical element can be represented by a 4 4 matrix following the formalism developed by Mueller [7]. The polarization state emerging from any sequence of optics can then be determined by calculating the product of their matrices and the incident Stokes vector. The Stokes vector for light transmitted first through a retarder and then a LP can be calculated as follows: S M pol M ret S: ðþ From this, the intensity of the transmitted light can be found as a function of β i, Δ, and γ: Iðβ i ; Δ; γþ fs þ S ½cos β i cos ðγ β i Þ sin β i sin ðγ β i Þ cos ΔŠ þ S ½sin β i cos ðγ β i Þ þ cos β i sin ðγ β i Þ cos ΔŠ þ S 3 ½sin ðγ β i Þ sin ΔŠg: ð3þ Following [8], this expression can be rearranged into a truncated Fourier series in β i with the values of Δ and γ absorbed within the Fourier coefficients: Iðβ i Þ ða þ a cos β i þ b sin β i þ a 4 cos 4β i þ b 4 sin 4β i Þ: ð4þ The minimum number of intensity measurements required in order to determine a, a, b, a 4, and b 4 is 5. For a set of N intensity measurements taken at equally spaced values of β i ranging from to ðn Þπ=N (effectively spanning to π), the Fourier coefficients can be found from a N a 4 N b 4 N a 4 4 N i i i i I i ; I i cos β i ; I i sin β i ; I i cos 4β i ; ð5þ ð6þ ð7þ ð8þ Using additional measurements with the same spacing of β i between π and π allows Eqs. (5) (9) to still be used but also compensates for systematic errors due to possible inhomogeneities in the retarder. Provided that fβ i g, Δ, and γ are accurately known, the Stokes vector of the incident light can then be derived from the intensity data as follows: S a þ cos Δ cos Δ ða 4 cos 4γ þ b 4 sin 4γÞ; ðþ S S S 3 cos Δ ða 4 cos γ þ b 4 sin γþ; ðþ cos Δ ðb 4 cos γ a 4 sin γþ; ðþ a sin Δ sin γ b sin Δ cos γ : ð3þ In the case where β (there is a finite but unknown offset between the retarder fast axis and ^x), an accurate calibration requires replacing β i with β i þ β in Eqs. (5) (9) to correct for this. However, using the sum difference trigonometric formulas, β can be factored out of the Fourier coefficients and absorbed into Eqs. () (3). Reversing each optic by rotating about a vertical axis is equivalent to reflecting the optic axes in the horizontal reference axis, so β and γ become β and γ. We have chosen the axis orthogonal to both ^z and the vertical rotation axis of the polarizer to be the horizontal reference axis. The offset of the retarder fast axis must then also be measured from the horizontal axis. If its front-to-back rotation axis is offset from the polarizer s vertical rotation axis by ϕ in the plane of the retarder s azimuthal rotation, β must instead be replaced with β þ ϕ when the retarder is reversed as depicted in Fig. (b). The normalized Stokes vectors calculated for the four possible cases with each optic oriented either forward or reversed could be expected to be inconsistent unless the values for β, Δ, γ, and ϕ are correctly accounted for. The formulas for calculating the Stokes vector when both optics are forward (case ) and when only the LP is reversed (case 3) are S ð;3þ ð þ cos ΔÞ a ð cos ΔÞ ½a 4 cos 4ðγ β Þ b 4 sin 4ðγ β ÞŠ; ð4þ b 4 4 N i I i sin 4β i : ð9þ S ð;3þ ð cos ΔÞ ½a 4 cos ðγ β Þ b 4 sin ðγ β ÞŠ; ð5þ 5384 APPLIED OPTIS / Vol. 5, No. 8 / October

4 S ð;3þ ð cos ΔÞ ½b 4 cos ðγ β Þ a 4 sin ðγ β ÞŠ; S ð;3þ a 3 sin Δ sin ðγ β Þ b sin Δ cos ðγ β Þ ; ð6þ ð7þ where the upper sign corresponds to case and the lower sign corresponds to case 3. When only the retarder is reversed (case ) and when both optics are reversed (case 4), S ð;4þ ð þ cos ΔÞ a ð cos ΔÞ ½a 4 cos 4ðγ β ϕþ b 4 sin 4ðγ β ϕþš; S ð;4þ ð cos ΔÞ ½a 4 cos ðγ β 4ϕÞ b 4 sin ðγ β 4ϕÞŠ; S ð;4þ ð cos ΔÞ ½b 4 cos ðγ β 4ϕÞ a 4 sin ðγ β 4ϕÞŠ; S ð;4þ a 3 sin Δ sin ðγ β ϕþ b sin Δ cos ðγ β ϕþ ; ð8þ ð9þ ðþ ðþ where the upper sign corresponds to case and the lower sign to case 4. Here, for each of the four cases the Fourier coefficients a b 4 require calculation only once for any number of values of β considered since it is absorbed into Eqs. (4) (). Although Eqs. (7) and () are valid for all values of β, γ, and ϕ, values for which the denominator approaches zero are more susceptible to experimental uncertainties in a and b. These two definitions for S 3 may be combined to produce a third definition with no explicit dependence on these parameters at the cost of losing any knowledge of the sign of S 3 [8]: S ð;;3;4þ 3 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a þ b sin Δ : ðþ The calibration process is insensitive to the sign of S 3 as discussed below. Subsequent calculation of the Stokes parameters for light with unknown polarization can make use of the appropriate definition from Eq. (7), chosen using the known values of β and γ. If the correct values of β, Δ, γ, and ϕ are used, the normalized Stokes vectors calculated for each of the four cases are expected to be consistent. An additional LP (LP ) placed before the polarimeter and set to provide incident light with comparable S and S components (i.e., set to :5 or67:5 ) ensures that consistency can be determined for values of S =S and S =S well above the experimental uncertainty. We found that coupling between β, Δ, and ϕ leads to multiple possible values of these parameters that produce, within experimental uncertainty, equally consistent results for the calculated Stokes vectors. Fully polarized incident light (due to LP ) adds a supplementary requirement that P ðjþ for all j [9]. Therefore, to determine the set of parameters fβ ; Δ; γ; ϕg closest to the correct values, we calculate the variances of S ðjþ =SðjÞ and S ðjþ =SðjÞ and add them to the sum of the squared differences between P ðjþ and for each j, thus forming a quantity dependent upon both the scatter of S and S and the radial distances of the end points of S ðjþ, anchored at the origin, from the surface of the unit Poincaré sphere [], ξ X4 j " S ðjþ S ðjþ S S! þ SðjÞ S ðjþ S S! þð P ðjþ Þ #: ð3þ Here the summation is over the four cases and h i denotes the arithmetic mean. The value of ξ takes a minimum at the position in fβ ; Δ; γ; ϕg space given by the correct values of these parameters. Note that we have hitherto assumed β refers to the angle of the retarder s fast axis from the reference axis. Since S 3 is squared in Eq. (), this calibration method does not distinguish its fast and slow axes irrespective of the definition of S 3 used. One additional measurement may be made using a circular polarizer to resolve this ambiguity if required. In summary, calibration of the polarimeter involves using linearly polarized incident light and calculating the Fourier coefficients for each of the four cases using Eqs. (5) (9) and subsequently mapping out ξðβ ; Δ; γ; ϕþ, where S S 3 are obtained from Eqs. (4) (). The polarization state of any light with unknown polarization can then be precisely found by removing the additional LP, measuring a set of intensities fi i g, and calculating the Stokes vector for this light (case ) using Eqs. (4) (7) and the parameter values corresponding to the minimum previously found in ξ, as will be illustrated below. 3. alibration Process The polarimeter consists of a retarder and LP in rotational mounts [5]. Light passes first through the retarder and then the LP and the transmitted intensity is measured with a detector as shown in Fig.. An optic axis of the retarder is initially set approximately parallel to the horizontal reference axis ^x by retroreflecting a horizontally polarized October / Vol. 5, No. 8 / APPLIED OPTIS 5385

5 µ PD SM LP λ/4 PD OSA BE ND LP' PBS PBS z Polarimeter x Fig.. (olor online) Schematic of the experimental setup. The beam under test originates from a fiber-coupled external cavity diode laser. A BE increases the beam width to 8 mm. A pair of PBSs produce a horizontally polarized beam with which to test the calibration method. The first s-polarized reflection is sent to an optical spectrum analyzer (OSA), while the weak second reflection serves as a reference for power normalization. The polarimeter itself consists of a quarter-wave plate (λ=4) and LP in rotational mounts. The calibration method employs an additional LP (LP ) set to make S S and is removed for regular beam analysis. The retarder is rotated via a worm gear by the SM and the transmitted light is measured by a PD after attenuation by a neutral density filter (ND). The μ controls the SM and records the PD voltages for transmission to a computer. beam through the retarder and minimizing the vertically polarized component using a polarizing beam splitter (PBS). The LP transmission axis is kept at a fixed angle γ from þ ^x. We have found that the calibration procedure is most robust with γ fixed near 45, which is straight forward to do using a PBS placed after the LP. The mounts of the retarder and LP permit front-to-back (8 ) rotation about an axis perpendicular to the direction of beam propagation (þ ^z), allowing each optic to be oriented either forward or reversed for the four possible cases used in this calibration method. A series of intensity measurements were taken with LP in place and the retarder rotated counterclockwise (as seen looking toward the light source along ^z) to N 5 uniformly spaced angles in 36. The transmission axis of LP was set to provide incident light with comparable S and S components. The retarder s rotational mount was driven by a worm gear for which one full rotation advanced the rotational mount by 3:6. A stepper motor (SM) advanced the worm gear by :8 per step, offering a precision of : in the N angles fβ i þ β g between the horizontal reference axis and the fast axis of the retarder. The SM was controlled by a pair of U377AN SM controllers and a PI8LF3 microcontroller (μ) []. The analog-to-digital converter (AD) capability of the μ read and stored voltages from the photodiodes (PDs) that monitored the light transmitted through the polarimeter and the reference beam as bit values at each β i and transmitted them via RS-3 to a computer. Neutral density filters attenuated the transmitted beam in order to achieve maximum resolution within the μ s 5 V AD range. In cases of significant laser drift, the transmitted intensities can be normalized to the reference intensities to correct for systematic errors. For each of the four cases, a set of intensity data was obtained and the Fourier coefficients were calculated from Eqs. (5) (9). Using Eqs. (4) (), Stokes vectors were calculated for each case followed by a calculation of ξ using Eq. (3) at a range of fβ ; Δ; γ; ϕg values. Our calibration scheme involves an iterative fourdimensional parameter search to find the set of values fβ ; Δ; γ; ϕg that minimizes ξ. An important consideration is the choice of a suitable sampling resolution in each parameter to obtain the true minimum in a reasonable computational time. We found that an initial sampling resolution of no larger than was necessary to locate the region of the global minimum in parameter space in the first iteration. Typical parameter ranges used for the initial search were 5 β 5 ; ð4þ λ nom 85 Δ λ λ nom 95 ; ð5þ λ 4 γ 5 ; ð6þ 5 ϕ 5 : ð7þ This range of Δ is sufficient to find the retardance of a quarter-wave plate within manufacturing tolerances with an approximate value inferred from the design wavelength. The ranges of β, γ, and ϕ allow for typical experimental uncertainties in the alignment of the optics in the polarimeter described above. Experimental results of nine calibrations using this search are presented below. We analyzed the validity of the calibration method by choosing specific values for β, Δ, γ, and ϕ and generating simulated intensity data with a typical signal-to-noise ratio (SNR) of 4 for cases 4. For each case and each value of ðβ ; Δ; γ; ϕþ, ξ was mapped with an initial sampling resolution of in all parameters. In successive iterations, the sampling interval of each parameter was halved while keeping the number of samples fixed and the new range was centered at the position of the minimum value of ξ from the previous iteration. Using this simulated data with random noise added for incident linearly polarized light at 67:5, β, Δ :6 π 93:6, γ 44, ϕ, the following values were found after search iterations: β calc :9, Δ calc 93:56, γ calc 43:93, ϕ calc :9. For illustration, a surface showing the dependence of log ½ξðβ ; ΔÞŠ on β and Δ for the calculated values of γ and ϕ is graphically depicted in Fig. 3. As expected, this shows a global minimum at the position of the correct values. The deleterious effect of a small misalignment of the retarder s front-to-back rotation axis from 5386 APPLIED OPTIS / Vol. 5, No. 8 / October

6 Absolute error deg true Β true Γ true Φ true calc calc calc calc Fig. 3. (olor online) Dependence of ξ on β and Δ using the values γ 43:93 and ϕ :9, which were found after search iterations for data simulated using the values β, Δ :6 π 93:6, γ 44, ϕ, and an incident Stokes vector corresponding to 67:5 polarized light and with noise added. The dashed lines indicate the correct values of β and Δ. þ ^y ðϕ Þ is illustrated in Fig. 4 by two surfaces showing the dependence of ξ on β and Δ for the same simulated data. One surface, with the minimum indicated by dashed lines, assumes ϕ (not including ϕ in the parameter search), and another with the minimum located close to the intersection of the solid lines uses the value obtained for ϕ from the search. The true values of β and Δ are indicated with solid lines. The robustness of the calibration scheme and an indication of its applicability in practice was determined by adding different levels of random noise to the simulated data and calculating the absolute differences between the correct parameter values and those found after eight iterations, averaged over calibration simulations for 4 values of SNR (see Fig. 5). The data were simulated using the same parameter values as were used for Figs. 3 and 4. This indicates that an SNR greater than 7 is sufficient to obtain all parameters to within of the correct values using this calibration method. 4. Experimental Test of the alibration Method The broad applicability of this calibration method was demonstrated using three different quarterwave retarders with 67:7 nm light from a fibercoupled laser diode. A beam expander (BE) increased the beam width to 8 mm to average over inhomogeneities in the polarimeter optics. True zeroth-order Fig. 4. (olor online) Dependences of ξ on β and Δ for data generated using ϕ with noise added. The two surfaces were plotted assuming values of ϕ (minimum indicated by dashed lines, red) and ϕ :9 (minimum located close to the intersection of the solid lines, blue) with a 5 mrad sampling resolution. The solid lines indicate the chosen values of β and Δ. 5 5 SNR Fig. 5. (olor online) Analysis of calibration results for simulated data with added random noise using parameter values fβ ; Δ; γ; ϕg f ; :6 π 93:6 ; 44 ; g. Each datum is the average over sets of results with the same SNR value. For comparison, data used to construct Table had SNR values in the range 4 6 for the three different retarders. mica wave plates were used with design wavelengths λ nom f67 nm; 645 nm; 548 nmg. These design wavelengths correspond to retardances of 89:6, 86:4, and 74:4, respectively, at the optical wavelength used [], taking account of the variation of birefringence with wavelength in mica [3]. Note that inconsistencies in the published data for the dispersion of mica birefringence make these values necessarily approximate [4]. Additionally, to demonstrate the reliability of this calibration technique against experimental misalignment, for each retarder the polarimeter was calibrated for three different values of ϕ by intentionally misaligning the retarder s front-to-back rotation axis in the plane of its azimuthal rotation by ϕ nom f 3:6 ; ; þ3:6 g. The fast axis of each retarder was approximately aligned with ^x (β ). For each of the nine calibrations, the LP transmission axis was kept at a fixed angle from ^x. The results of these nine calibrations are presented in Table. We expect consistency between the three values determined for β and Δ for each retarder, and the results for γ and ϕ ϕ nom should be consistent between all nine calibrations. The observations follow these predictions with the expected similarities clearly visible within each column, and for γ and ϕ ϕ nom throughout the whole table. Further analysis of the 67 nm wave plate using 777:6 nm light resulted in values β 4:46, γ 44:5, and ϕ ϕ nom :88, all within of the values obtained at 67:7 nm, and Δ 76:76, within of the nominally expected value of 67=78 9. These small variations may be attributed to the realignment required to change to a light source at a different wavelength. To verify that each of the nine calibrations could individually provide a normalized Stokes vector consistent with the incident polarization state, LP was removed and an additional set of intensity data was recorded with horizontally polarized incident light provided by the two consecutive PBSs. These results are presented in Table. We expect consistency October / Vol. 5, No. 8 / APPLIED OPTIS 5387

7 Table. Experimental Results for Nine alibrations Using Three Different Quarter-Wave Plates with Design Wavelengths Denoted by λ nom, Each with Three Different Values of ϕnom ϕ nom þ3:6 3:6 λ nom 67 nm λnom between the calculated Stokes vectors, and for these to correspond to that for horizontally polarized light. The nine Stokes vectors are indeed highly consistent with one another, as shown by the low standard errors in S, S,andS 3. The mean values of the normalized Stokes parameters and the degree of polarization with the standard errors derived from these measurements are hsi B :995 : : A ; ð8þ :58 :3 hpi : :: 645 nm λnom 548 nm β 3:34 3:8 :6 Δ 9:47 86:34 75: γ 44: 44:3 44: ϕ ϕ nom :85 :8 :8 β 3:55 3:79 :3 Δ 9:4 86:8 75: γ 44:9 44:9 44: ϕ ϕ nom :94 :8 :85 β 3:9 3:78 :5 Δ 9:47 86:3 75:9 γ 44: 44:3 44:7 ϕ ϕ nom :77 :8 :87 hβ i 3:39ð8Þ 3:787ð5Þ :3ðÞ hδi 9:45ðÞ 86:3ðÞ 75:3ð3Þ hγi 44:5ðÞ hϕ ϕ nom i :84ðÞ ð9þ The average value for S 3 indicates a consistent small ellipticity, possibly due to stress-induced birefringence in a PBS. The values for S and S reflect a small ( 3 ) misalignment between the transmission Table. Stokes Vectors Derived from Measurements of Horizontally Polarized Light Using Three Different Retarders and Three Intentional Misalignments of the Retarder Vertical Rotation Axis Using the Nine alibrations Presented in Table ϕ nom þ3:6 3:6 λ nom 67 nm λnom B : A :6 B :8 A :6 B :7 A :7 645 nm λnom B :6 A :6 B :7 A :6 B :5 A :7 548 nm B :8 A :5 B : A :4 B :8 A :6 axis of the second PBS and the horizontal reference axis defined to be perpendicular to the LP s frontto-back rotation axis. The average value of P is approximately unity, as expected for fully polarized incident light. 5. onclusion We have presented a reliable calibration of the rotating quarter-wave plate method that permits accurate characterization of the polarization of incident light. The transmitted intensities were measured at uniformly spaced angles in one full rotation of the retarder s fast axis with the retarder and LP oriented either forward or reversed. Fourier coefficients were obtained for each of these four cases and Stokes vectors were calculated for a range of values of β, Δ, γ, and ϕ. An iterative search for consistency in calculations of S =S and S =S between the four cases and a degree of polarization equal to unity was employed to determine the set of polarimeter parameters required for accurate determination of the light s polarization state. Our method also accounts for a possible experimental misalignment of the retarder when it is reversed during calibration. The method was experimentally verified by calibrating the polarimeter with three different wave plates with nominal quarter-wave retardance wavelengths between 548 and 67 nm for light at 67:7 nm, each with three different experimental misalignments. Highly consistent normalized Stokes vectors and degrees of polarization corresponding to the expected polarization of a test beam were obtained using the results of the nine calibrations. It was also determined in a simulation of the calibration method that an SNR of 7 is sufficient to obtain accurate results for the experimental parameters. These results demonstrate the broad applicability of this easily implemented method and suggest its potential for enabling accurate determination of the polarization state of incident light over a broad range of wavelengths using a single retarder. This work was supported by the Natural Sciences and Engineering Research ouncil of anada (NSER) and the Research orporation. The authors acknowledge Dr. Keith Donnelly for helpful discussions. References. G. G. Stokes, Mathematical and Physical Papers (ambridge University, 9), Vol. 3.. M. R. Foreman and P. Török, Information and resolution in electromagnetic optical systems, Phys. Rev. A 8, (). 3. F. Meriaudeau, M. Ferraton,. Stolz, O. Morel, and L. Bigue, Polarization imaging for industrial inspection, Proc. SPIE 683, 6838 (8). 4. F. Snik, A. G. de Wijn, K. Ichimoto,. E. Fischer,. U. Keller, and B. W. Lites, Observations of solar scattering polarization at high spatial resolution, Astron. Astrophys. 59, A8 (). 5. D. H. Goldstein, Mueller matrix dual-rotating retarder polarimeter, Appl. Opt. 3, (99) APPLIED OPTIS / Vol. 5, No. 8 / October

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