Microgrid polarization cameras have historically experienced lower performance than is typical for monolithic polarization systems. Specifically, their polarizer elements have had a lower extinction ratio and a larger orientation error. We show how to use the calibrated parameters of a nonideal polarizer to modify the polarization measurement model, effectively allowing one to generate high-performance measurements from low-performance elements. We demonstrate the effectiveness of this approach on a commercial polarization camera and estimate the signal-to-noise ratio penalty for using nonideal polarizers in this camera as being 1.25 × versus a system using ideal polarizers.

A polarizer can be used to estimate the orientation and degree of linear polarization of light by rotating the polarizer to the four angles: 0 deg, 45 deg, 90 deg, and 135 deg and measuring the light intensity transmitted at each angle. In a polarization camera, this can be done in a single snapshot by combining the measurements of four adjacent pixels, each of which have micropolarizers oriented at each of the four orientation angles (Fig. 1).^{1} For incoherent light, we can use the Stokes vector $\mathbf{s}={({s}_{0},{s}_{1},{s}_{2},{s}_{3})}^{T}$ to represent the input polarization state and the Mueller calculus to model the four intensities ${I}_{\theta}$ measured at the detector:

## Eq. (1)

$${I}_{0}=\frac{1}{2}({s}_{0}+{s}_{1})+{n}_{0},\phantom{\rule{0ex}{0ex}}{I}_{45}=\frac{1}{2}({s}_{0}+{s}_{2})+{n}_{45},\phantom{\rule{0ex}{0ex}}{I}_{90}=\frac{1}{2}({s}_{0}-{s}_{1})+{n}_{90},\phantom{\rule{0ex}{0ex}}{I}_{135}=\frac{1}{2}({s}_{0}-{s}_{2})+{n}_{135},$$The four equations of (1) have the matrix form

## Eq. (2)

$$\underset{\mathbf{I}}{\underset{\u23df}{\left(\begin{array}{c}{I}_{0}\\ {I}_{45}\\ {I}_{90}\\ {I}_{135}\end{array}\right)}}=\underset{\mathbf{W}}{\underset{\u23df}{\frac{1}{2}\left(\begin{array}{cccc}1& 1& 0& 0\\ 1& 0& 1& 0\\ 1& -1& 0& 0\\ 1& 0& -1& 0\end{array}\right)}}\underset{\mathbf{s}}{\underset{\u23df}{\left(\begin{array}{c}{s}_{0}\\ {s}_{1}\\ {s}_{2}\\ {s}_{3}\end{array}\right)}}+\underset{\mathbf{n}}{\underset{\u23df}{\left(\begin{array}{c}{n}_{0}\\ {n}_{45}\\ {n}_{90}\\ {n}_{135}\end{array}\right)}}$$## Eq. (3)

$${\widehat{s}}_{0}=\frac{1}{2}({I}_{0}+{I}_{45}+{I}_{90}+{I}_{135}),\phantom{\rule{0ex}{0ex}}{\widehat{s}}_{1}={I}_{0}-{I}_{90},\phantom{\rule{0ex}{0ex}}{\widehat{s}}_{2}={I}_{45}-{I}_{135},$$## Eq. (4)

$$\underset{\widehat{\mathbf{s}}}{\underset{\u23df}{\left(\begin{array}{c}{\widehat{s}}_{0}\\ {\widehat{s}}_{1}\\ {\widehat{s}}_{2}\end{array}\right)}}=\underset{\mathbf{A}}{\underset{\u23df}{\frac{1}{2}\left(\begin{array}{cccc}1& 1& 1& 1\\ 2& 0& -2& 0\\ 0& 2& 0& -2\end{array}\right)}}\underset{\mathbf{I}}{\underset{\u23df}{\left(\begin{array}{c}{I}_{0}\\ {I}_{45}\\ {I}_{90}\\ {I}_{135}\end{array}\right)}},$$^{2}

^{,}

^{3}

## Eq. (5)

$${K}_{ij}=\underset{\text{Poisson noise part}}{\underset{\u23df}{\left[\sum _{k=0}^{3}{s}_{k}\sum _{\ell =0}^{3}{A}_{i\ell}{A}_{j\ell}{W}_{\ell k}\right]}}+\underset{\text{Gaussian noise part}}{\underset{\u23df}{\left[{v}_{\mathrm{d}}\sum _{\ell =0}^{3}{A}_{i\ell}{A}_{j\ell}\right]}},$$^{3}

## Eq. (6)

$$\mathrm{var}({\widehat{s}}_{0})=\frac{1}{2}{s}_{0}+{v}_{\mathrm{d}},\phantom{\rule{0ex}{0ex}}\mathrm{var}({\widehat{s}}_{1})={s}_{0}+2{v}_{\mathrm{d}},\phantom{\rule{0ex}{0ex}}\mathrm{var}({\widehat{s}}_{2})={s}_{0}+2{v}_{\mathrm{d}}.$$^{3}

A nonideal linear polarizer is a diattenuator defined by its diattenuation $D$, its orientation $\alpha $, and its efficiency $\eta $ (the average transmission of the diattenuator for all input polarization angles):^{4}

## Eq. (7)

$${\mathbf{M}}_{\mathrm{ld}}=\eta \left(\begin{array}{cccc}1& Dc& Ds& 0\\ Dc& 1-D{s}^{2}& \frac{1}{2\eta}Dcs& 0\\ Ds& \frac{1}{2\eta}Dcs& 1-D{c}^{2}& 0\\ 0& 0& 0& 1-D\end{array}\right),$$Modeling each pixel of a polarization camera as a nonideal polarizer, we can calibrate the diattenuation ${D}_{n}$ and orientation ${\alpha}_{n}$ parameters at each pixel $n$, then use the Mueller calculus to estimate the polarization state by combining measurements ${I}_{i}$ from each set of four neighboring pixels. This produces the measurement model^{5}

## Eq. (8)

$$\underset{\mathbf{I}}{\underset{\u23df}{\left(\begin{array}{c}{I}_{0}\\ {I}_{2}\\ {I}_{3}\\ {I}_{4}\end{array}\right)}}=\underset{\mathbf{W}}{\underset{\u23df}{\frac{1}{2}\left(\begin{array}{ccc}1& {D}_{0}\text{\hspace{0.17em}}\mathrm{cos}(2{\alpha}_{0})& {D}_{0}\text{\hspace{0.17em}}\mathrm{sin}(2{\alpha}_{0})\\ 1& {D}_{1}\text{\hspace{0.17em}}\mathrm{cos}(2{\alpha}_{1})& {D}_{1}\text{\hspace{0.17em}}\mathrm{sin}(2{\alpha}_{1})\\ 1& {D}_{2}\text{\hspace{0.17em}}\mathrm{cos}(2{\alpha}_{2})& {D}_{2}\text{\hspace{0.17em}}\mathrm{sin}(2{\alpha}_{2})\\ 1& {D}_{3}\text{\hspace{0.17em}}\mathrm{cos}(2{\alpha}_{3})& {D}_{3}\text{\hspace{0.17em}}\mathrm{sin}(2{\alpha}_{3})\end{array}\right)}}\underset{\mathbf{s}}{\underset{\u23df}{\left(\begin{array}{c}{s}_{0}\\ {s}_{1}\\ {s}_{2}\end{array}\right)}}.$$The general analytical form of $\mathbf{A}$ is quite complex, but if we assume that the orientation errors are small then we can use a Taylor expansion about their nominal values:

## Eq. (9)

$$\underset{{\widehat{\mathbf{s}}}^{\prime}}{\underset{\u23df}{\left(\begin{array}{c}\widehat{{s}_{0}^{\prime}}\\ {\widehat{s}}_{1}^{\prime}\\ {\widehat{s}}_{2}^{\prime}\end{array}\right)}}=\underset{\mathbf{A}}{\underset{\u23df}{\frac{1}{4}\left(\begin{array}{cccc}{A}_{00}& {A}_{01}& {A}_{02}& {A}_{03}\\ {A}_{10}& {A}_{11}& {A}_{12}& {A}_{13}\\ {A}_{20}& {A}_{21}& {A}_{22}& {A}_{23}\end{array}\right)}}\underset{\mathbf{I}}{\underset{\u23df}{\left(\begin{array}{c}{I}_{0}\\ {I}_{1}\\ {I}_{2}\\ {I}_{3}\end{array}\right)}}$$## Eq. (10)

$${A}_{00}=2+{\mathrm{\Delta}}_{0}-{\mathrm{\Delta}}_{2}+2{\u03f5}_{1}-2{\u03f5}_{3},\phantom{\rule{0ex}{0ex}}{A}_{01}=2+{\mathrm{\Delta}}_{1}-{\mathrm{\Delta}}_{3}-2{\u03f5}_{0}+2{\u03f5}_{2},\phantom{\rule{0ex}{0ex}}{A}_{02}=2-{\mathrm{\Delta}}_{0}+{\mathrm{\Delta}}_{2}-2{\u03f5}_{1}+2{\u03f5}_{3},\phantom{\rule{0ex}{0ex}}{A}_{03}=2-{\mathrm{\Delta}}_{1}+{\mathrm{\Delta}}_{3}+2{\u03f5}_{0}-2{\u03f5}_{2},\phantom{\rule{0ex}{0ex}}{A}_{10}=4+{\mathrm{\Delta}}_{0}+3{\mathrm{\Delta}}_{2}+2{\u03f5}_{1}-2{\u03f5}_{3},\phantom{\rule{0ex}{0ex}}{A}_{11}={\mathrm{\Delta}}_{0}-{\mathrm{\Delta}}_{2}-4{\u03f5}_{0}+2{\u03f5}_{1}+4{\u03f5}_{2}-2{\u03f5}_{3},\phantom{\rule{0ex}{0ex}}{A}_{12}=-4-3{\mathrm{\Delta}}_{0}-{\mathrm{\Delta}}_{2}+2{\u03f5}_{1}-2{\u03f5}_{3},\phantom{\rule{0ex}{0ex}}{A}_{13}={\mathrm{\Delta}}_{0}-{\mathrm{\Delta}}_{2}+4{\u03f5}_{0}-2{\u03f5}_{1}+4{\u03f5}_{2}+2{\u03f5}_{3},\phantom{\rule{0ex}{0ex}}{A}_{20}={\mathrm{\Delta}}_{1}-{\mathrm{\Delta}}_{3}+2{\u03f5}_{0}+4{\u03f5}_{1}-2{\u03f5}_{2}+4{\u03f5}_{3},\phantom{\rule{0ex}{0ex}}{A}_{21}=4+{\mathrm{\Delta}}_{1}+3{\mathrm{\Delta}}_{3}-2{\u03f5}_{0}+2{\u03f5}_{2}\phantom{\rule{0ex}{0ex}}{A}_{22}={\mathrm{\Delta}}_{1}-{\mathrm{\Delta}}_{3}+2{\u03f5}_{0}-4{\u03f5}_{1}-2{\u03f5}_{2}-4{\u03f5}_{3},\phantom{\rule{0ex}{0ex}}{A}_{23}=-4-3{\mathrm{\Delta}}_{1}-{\mathrm{\Delta}}_{3}-2{\u03f5}_{0}+2{\u03f5}_{2}.$$To demonstrate this technique, we use a commercially available polarization camera to measure the Stokes vector elements of the light transmitted through a high-extinction-ratio Glan–Thompson polarizer as the polarizer is rotated through a set of angles from 0 deg to 180 deg. The polarization camera is a photonic lattice PI-110, having $1164\times 874\text{\hspace{0.17em}\hspace{0.17em}}\text{pixels}$ of $4.65\text{-}\mu \mathrm{m}$ size, 20-Hz frame rate, 12-bit depth, and $520\pm 20\text{-}\mathrm{nm}$ wavelength range. Figure 2 shows the resulting Stokes parameter estimates before (${\widehat{s}}_{1}$ and ${\widehat{s}}_{2}$) and after (${\widehat{s}}_{1}^{\prime}$ and ${\widehat{s}}_{2}^{\prime}$) applying the correction, corresponding to using the ideal polarization analysis matrix [Eq. (4)] or a corrected analysis matrix [Eq. (10)]. We can see that the corrected curves display a higher contrast than the uncorrected curves, so that the mean values of the Stokes parameter (averaged across all camera pixels) at each angle of the input polarizer are closer to the correct values. Looking closely at the curves, we can also see that the standard deviation of the corrected result is slightly larger than for the uncorrected measurement.

To see why the corrected result has higher noise, we can use Eqs. (5), (8), and (10) in their analytical form to calculate the general equation for the variances. To keep the expressions as simple as possible, and because our experimental data show that the diattenuation errors are much larger than the angular errors in our system, we set the angular errors to zero (${\u03f5}_{i}\approx 0$). Thus, in the detector-limited noise regime (uniform Gaussian noise), the variances become

## Eq. (11)

$$\mathrm{var}({\widehat{s}}_{0}^{\prime})=\frac{{v}_{\mathrm{d}}}{8}[8+{\mathrm{\Delta}}_{0}^{2}+{\mathrm{\Delta}}_{1}^{2}+{\mathrm{\Delta}}_{2}^{2}+{\mathrm{\Delta}}_{3}^{2}-2{\mathrm{\Delta}}_{0}{\mathrm{\Delta}}_{2}-2{\mathrm{\Delta}}_{1}{\mathrm{\Delta}}_{3}]\approx {v}_{\mathrm{d}},\phantom{\rule{0ex}{0ex}}\mathrm{var}({\widehat{s}}_{1}^{\prime})=\frac{{v}_{\mathrm{d}}}{4}[8+3{\mathrm{\Delta}}_{0}^{2}+3{\mathrm{\Delta}}_{2}^{2}+2{\mathrm{\Delta}}_{0}({\mathrm{\Delta}}_{2}+4)+8{\mathrm{\Delta}}_{2}]\phantom{\rule{0ex}{0ex}}\approx 2{v}_{\mathrm{d}}[1+{\mathrm{\Delta}}_{0}+{\mathrm{\Delta}}_{2}],\phantom{\rule{0ex}{0ex}}\mathrm{var}({\widehat{s}}_{2}^{\prime})=\frac{{v}_{\mathrm{d}}}{4}[8+3{\mathrm{\Delta}}_{1}^{2}+3{\mathrm{\Delta}}_{3}^{2}+2{\mathrm{\Delta}}_{1}({\mathrm{\Delta}}_{3}+4)+8{\mathrm{\Delta}}_{3}]\phantom{\rule{0ex}{0ex}}\approx 2{v}_{\mathrm{d}}[1+{\mathrm{\Delta}}_{1}+{\mathrm{\Delta}}_{3}],$$^{6}

## Eq. (12)

$$\mathrm{var}({\tilde{s}}_{i}^{\prime})=\frac{1}{{\u27e8{s}_{0}^{\prime}\u27e9}^{2}}\text{\hspace{0.17em}}\mathrm{var}({s}_{i}^{\prime})+\frac{{\u27e8{s}_{i}^{\prime}\u27e9}^{2}}{{\u27e8{s}_{0}^{\prime}\u27e9}^{4}}\mathrm{var}({s}_{0}^{\prime}),$$## Eq. (13)

$$\mathrm{var}({\tilde{s}}_{1}^{\prime})=\frac{{v}_{\mathrm{d}}}{{s}_{0}^{2}}(2[1+{\mathrm{\Delta}}_{0}+{\mathrm{\Delta}}_{2}]+{\tilde{s}}_{1}^{2}),\phantom{\rule{0ex}{0ex}}\mathrm{var}({\tilde{s}}_{2}^{\prime})=\frac{{v}_{\mathrm{d}}}{{s}_{0}^{2}}(2[1+{\mathrm{\Delta}}_{1}+{\mathrm{\Delta}}_{3}]+{\tilde{s}}_{2}^{2}),$$To see how these variance equations behave quantitatively, we can look at an example group of 4 pixels on the polarization camera. For this group, we obtain calibrated diattenuations of

## Eq. (14)

$${\mathrm{\Delta}}_{0}=0.1723,\phantom{\rule[-0.0ex]{1em}{0.0ex}}{\mathrm{\Delta}}_{1}=0.2565,\phantom{\rule[-0.0ex]{1em}{0.0ex}}{\mathrm{\Delta}}_{2}=0.1867,\phantom{\rule[-0.0ex]{1em}{0.0ex}}{\mathrm{\Delta}}_{3}=0.2662,\phantom{\rule{0ex}{0ex}}{\u03f5}_{0}=0,\phantom{\rule[-0.0ex]{1em}{0.0ex}}{\u03f5}_{1}=0.0811,\phantom{\rule[-0.0ex]{1em}{0.0ex}}{\u03f5}_{2}=-0.0228,\phantom{\rule[-0.0ex]{1em}{0.0ex}}{\u03f5}_{3}=-0.0105.$$## Eq. (15)

$$\mathrm{var}({\widehat{s}}_{0}^{\prime})=1.00{v}_{\mathrm{d}},\phantom{\rule{0ex}{0ex}}\mathrm{var}({\widehat{s}}_{1}^{\prime})=3.23{v}_{\mathrm{d}},\phantom{\rule{0ex}{0ex}}\mathrm{var}({\widehat{s}}_{2}^{\prime})=3.01{v}_{\mathrm{d}},$$Using the same Eqs. (2), (5), and (10), we can obtain similar results for the Poisson noise regime, for which the variance equations become

## Eq. (16)

$$\mathrm{var}({\widehat{s}}_{0}^{\prime})\approx \frac{1}{2}{s}_{0}+\frac{1}{8}{s}_{1}({\mathrm{\Delta}}_{0}-{\mathrm{\Delta}}_{2})+\frac{1}{8}{s}_{2}({\mathrm{\Delta}}_{1}-{\mathrm{\Delta}}_{3}),\phantom{\rule{0ex}{0ex}}\mathrm{var}({\widehat{s}}_{1}^{\prime})\approx {s}_{0}(1+{\mathrm{\Delta}}_{0}+{\mathrm{\Delta}}_{2})+{s}_{1}({\mathrm{\Delta}}_{2}-{\mathrm{\Delta}}_{0}),\phantom{\rule{0ex}{0ex}}\mathrm{var}({\widehat{s}}_{2}^{\prime})\approx {s}_{0}(1+{\mathrm{\Delta}}_{1}+{\mathrm{\Delta}}_{3})+{s}_{2}({\mathrm{\Delta}}_{3}-{\mathrm{\Delta}}_{1}),$$^{3}as the diattenuation errors approach zero (${\mathrm{\Delta}}_{i}\to 0$).

For mixed Gaussian–Poisson noise, the Gaussian and Poisson expressions (11) and (16) are simply added together to get the total variance. Using the example diattenuation errors of Eq. (14), the Stokes parameter variances under Poisson noise become

## Eq. (17)

$$\mathrm{var}({\widehat{s}}_{0}^{\prime})=0.500{s}_{0}-0.001{s}_{1}-0.002{s}_{2},\phantom{\rule{0ex}{0ex}}\mathrm{var}({\widehat{s}}_{1}^{\prime})=1.542{s}_{0}+0.092{s}_{1}+0{s}_{2},\phantom{\rule{0ex}{0ex}}\mathrm{var}({\widehat{s}}_{2}^{\prime})=1.381{s}_{0}+0{s}_{1}+0.014{s}_{2}.$$The results of Eqs. (15) and (17) and Fig. 3 demonstrate an SNR penalty due to reduced diattenuation that is similar to that calculated by Tyo and Wei,^{7} where the authors concluded via numerical simulation that polarizer extinction ratios as low as 3 give a 3.5 dB (i.e., a factor of 2.2) penalty in SNR. Although the measurement system analyzed by Tyo and Wei is not the same as the one we consider, here, the SNR penalty is similar in magnitude. Our results also agree with those of Roussel et al.,^{8} who analyzed microgrid polarization cameras and concluded that diattenuations of 0.75 to 0.85 will increase the variance in the estimated degree of polarization by 1.6 to 3. What is different from Roussel et al. is that the analytic dependence of the Stokes parameter variance is expressed here in terms of the calibration parameters directly, allowing us to see more clearly the effect of misalignment or low diattenuation on the measurements.

In Fig. 2, we can see that the mean values of the corrected curves come close to the ideal behavior of a cosine curve oscillating between $+1$ and $-1$. The shaded regions surrounding each curve describe the values within one standard deviation about the mean, where we can expect to find the data most of the time. The noise fluctuations result primarily from the shot noise of the measurements, but are also influenced by any error in the calibration parameters. As we can expect for any linear estimation procedure, as the estimate approaches the ideal cosine curve, noise fluctuations cause some of the data at the peaks and valleys to extend beyond the physically valid region between $+1$ and $-1$.

From these results, we can see that most polarimeters can tolerate a surprisingly low diattenuation if calibrated well. As often is the case for indirect measurements, it is the accuracy with which one can calibrate the system measurement model that limits the measurement performance more than the inherent accuracy of the components themselves. If we work with low-extinction-ratio polarizers, we can still achieve high-accuracy measurements, but only if we take extreme care in the calibration and in ensuring the accuracy of our reconstruction.^{9} For example, we should keep in mind that the diattenuation value of a pixel is a spectrally dependent quantity, so that the measurement spectrum must be similar to the spectrum used during calibration in order for the results to be accurate.^{10}