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:

for random noise values $n_{\theta }$. In the analysis to follow, we will assume all of the noise terms to have zero mean, i.e., ${⟨I_i+n_i⟩=I}_i$ and $⟨n_i⟩=0$. This causes no difficulties for Poisson-distributed noise, since our definition of the signal $I$ and noise $n$ results in the mean value of the Poisson-distributed variable to be incorporated into $I$ while $n$ retains the zero-mean stochastic portion.

The four equations of (1) have the matrix form

or $\mathbf{I}=\mathbf{Ws}+\mathbf{n}$, where $\mathbf{W}$ is the measurement matrix. By inspection, we can obtain the input linear Stokes vector elements as or i.e., $\mathbf{s}=\mathbf{AI}$, where $\mathbf{A}$ is the analysis matrix. To evaluate the performance of a polarization measurement, we can use the measurement and analysis matrix elements to calculate the Stokes vector covariance matrix $\mathbf{K}$ as^2,3 where $s_k$ is the $k$’th Stokes vector element in units of photoelectrons and $v_{\mathrm{d}}$ is the detector noise variance, also in units of photoelectrons. For an ideal polarizer, this set of four measurements produces variances in the Stokes vector components as³ The equally weighted variance (EWV) is often used to summarize the measurement performance, such that in this case $\mathrm{EWV}=\frac{5}{2}{s}_0+5v_{\mathrm{d}}$.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

for $c=\cos (2\alpha )$ and $s=\sin (2\alpha )$. An ideal polarizer has $D=1$, $\eta =1$, and orientation $\alpha$. The polarizer’s extinction ratio $X$ is derived from the diattenuation as $X=(1+D)/(1-D)$.

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 model⁵

The polarizer efficiency $\eta$ does not appear here because it typically cannot be separated from the pixel’s internal quantum efficiency $q_{\mathrm{int}}$—the two appear together as the single factor $q_{\mathrm{ext}}=\eta q_{\mathrm{int}}$ as the “external” quantum efficiency. Because we do not calibrate the camera’s quantum efficiency here, the equations below assume $q_{\mathrm{ext}}=1$, so that the Stokes parameters are represented in units of photoelectrons rather than photons.

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:

and where ${ϵ}_n$ represents the angular error about the nominal orientation. Likewise, each diattenuation can be represented as $D_i=1-{\mathrm{\Delta }}_i$, where the diattenuation errors ${\mathrm{\Delta }}_i$ are assumed to be small. For this first-order approximation regime, the analysis matrix $\mathbf{A}$ in the Stokes vector estimation: simplifies to Here, ${\widehat{s}}_1^{\prime }$ indicates the calibration-corrected Stokes estimate, while ${\widehat{s}}_i$ without a prime indicates the estimate assuming an ideal polarizer. In Eqs. (9) and (10), note that it is generally not useful to define ${\alpha }_0$ as anything but zero, since the system reference axis is a degree of freedom that we have to choose in most situations. In this case, we can set ${ϵ}_0=0$.

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 $1164\times 874\text{pixels}$ 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 (${ϵ}_i\approx 0$). Thus, in the detector-limited noise regime (uniform Gaussian noise), the variances become

where the variances are the diagonal elements of the Stokes vector covariance matrix, $\mathrm{var}({\widehat{s}}_0^{\prime })=K_{00}$, etc. The approximations in Eq. (11) assume small errors so that only the first-order terms are retained. In this approximation, $\mathrm{var}({\widehat{s}}_0^{\prime })$ is independent of the diattenuation errors ${\mathrm{\Delta }}_i$, while $\mathrm{var}({\widehat{s}}_1^{\prime })$ increases linearly with ${\mathrm{\Delta }}_0$ and ${\mathrm{\Delta }}_2$, and $\mathrm{var}({\widehat{s}}_2^{\prime })$ with ${\mathrm{\Delta }}_1$ and ${\mathrm{\Delta }}_3$, respectively. The Stokes parameters are often expressed in normalized form ${\tilde{s}}_i=s_i/s_0$, for which the expressions in (11) need to be modified. The normalized parameter variances are related to their unnormalized counterparts by⁶ which together with Eq. (11) gives and where we have replaced the corrected parameter means with the true parameter values: $⟨s_0^{\prime }⟩\to s_0$, etc.

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

and calibrated orientation angles of so that the differences from the nominal angles are 0 deg, 4.650 deg, $-1.306\mathrm{deg}$, and $-0.601\mathrm{deg}$. Converting the angular errors to radians for the ${ϵ}_i$ values, we obtain the error parameters: With these parameters, the variances (11) in the detector-limited noise regime (uniform Gaussian noise) become, via Eq. (5), for detector noise variance $v_{\mathrm{d}}$. In comparison to Eq. (6), we see that the corrected nonideal measurement for this pixel group suffers an increase in the variances for measuring $s_1$ and $s_2$ by factors of 1.62 and 1.51, respectively, while the variance for $s_0$ is basically unaffected. These correspond to a loss in signal-to-noise ratio (SNR) by factors of 1.27 in $s_1$ and 1.23 in $s_2$.

Using the same Eqs. (2), (5), and (10), we can obtain similar results for the Poisson noise regime, for which the variance equations become

where only the first-order terms are retained. As we expect, the variances of Eqs. (11) and (16) approach those of the ideal polarizer system³ 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

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