An analysis of the temporal noise in the Stokes parameters computed by division of focal plane polarimeters is presented. Theoretical estimations of the Stokes parameter signal-to-noise ratios for CCD polarization imaging sensors with both 4-polarizer and 2-polarizer micropolarization filter arrays are derived. The theoretical derivation is verified with measurements from an integrated polarization imaging sensor composed of a CCD imaging array and aluminum nanowire polarization filters. The measured data obtained from the CCD polarimeters matches the theoretical derivations of the temporal noise model of the Stokes parameters.
©2010 Optical Society of America
Recovering polarization information in the visible spectrum has typically been implemented using CMOS or CCD imaging sensors coupled with linear polarization filters . Various imaging systems have been designed and tested in the last two decades with a goal of recovering polarization information with high spatial and temporal resolution. The current state-of-the-art polarization imaging sensors can be divided into division of time [2,3], division of amplitude [4,5], division of aperture [6,7] and division of focal plane polarimeters [8–13].
Division of focal plane (DoFP) polarimeters include imaging elements and micropolarization filters on the same substrate [8–13]. The sampling of the imaged environment is achieved with spatially distributed polarization filters pixel-pitch-matched over a neighborhood of pixels. One of the first DoFP sensors was reported by Andreou et al. where birefringent materials and thin film polarization filters were monolithically integrated with a custom CMOS imaging chip . The neuromoropic approach toward realizing polarization sensors [10,11] lead to sampling the light wave with two orthogonal pixel-pitch matched linear polarization filters. The large pixel pitch, limited extinction ratios as well as computing only polarization contrast information are the shortcomings of these sensors.
Research in the area of division of focal plane sensors has focused not just on the creation of these sensors but also in better fabrication techniques for micropolarization filters. Recent improvements have been made in nano- and micro-fabrication techniques [13,14] and have yielded high-resolution polarization imaging sensors [8,9]. These sensors are capable of extracting the complete set of polarimetric properties for partially linearly polarized light. In  a pixilated filter array is presented for designing a DoFP polarimeter for extracting all four Stokes parameters by incorporating liquid crystals structures in the micropolarizer array.
Monolithic integration of pixilated linear polarization filters with an array of CCD imaging elements has been reported in . This high-resolution polarimetric sensor extracts the first three Stokes parameters of the incident light for every acquired frame on a neighborhood of pixels. The first Stokes parameter, S 0, is a measure of the total intensity of the light incident on the sensor. The second and third Stokes parameters, S 1 and S 2, provide information about the linear polarization state of the incident light. The fourth Stokes parameter, S 3, provides information about the circular polarization state of the incident light and cannot be determined using only linear polarization filters, thus making it not measureable by the type of sensor described in this paper.
There are several different ways of computing the first three Stokes parameters depending on the number of linear polarization filters that are used to sample a scene. For sampling with N linear polarization filters, Tyo et al. have described the optimal configuration for these filters to minimize correlation of information recorded . The estimated precision in the Stokes equations, when sampling the imaged scene with N linear polarization filters, has been addressed in . Two configurations of linear polarization filters are presented next. The first approach involves filtering the incident light wave with four linear polarization filters offset by 45° and recording the filtered light intensity with four different photodiodes. The Stokes parameters are computed based on the intensity measurements from the four photodiodes and are presented by Eqs. (1) through (3):Eqs. (1) through (3), I 0 is the intensity of the 0° filtered light wave, I 90 is the intensity of the 90° filtered light wave, and so on. An imaging sensor capable of characterizing partially polarized light based on Eqs. (1) through (3) must employ four linear polarization filters offset by 45° together with an array of imaging elements.
The second approach for computing the Stokes parameters encompasses a direct measurement of the incident light intensity in conjunction with 2 measurements taken with linear polarization filters oriented 45° apart. Based on this approach, the Stokes parameters can be computed as presented by Eqs. (4) through (6):Eqs. (4) through (6) must employ two linear polarization filters offset by 45° together with an array of imaging elements and record the light intensity of the incoming light wave without any polarization filters.
Due to inherent noise in the intensity values recorded by the pixels in the imaging array, there will always be some uncertainty in the Stokes parameters computed using a polarization imaging sensor. The amount of uncertainty in the Stokes parameters depends on which of the two previously described set of calculations is used. For sensors capable of extracting polarization information in real time, knowledge of these uncertainties allows the placement of boundaries on the accuracy of acquired polarization information as well as the ability to distinguish between sensor noise and a true change in a dynamic polarization scene.
In this paper, we derive analytical expressions for the signal-to-noise ratios for the first three Stokes parameters computed by degree of focal plane CCD polarization imaging sensors. This noise analysis is performed both for sensors using Eqs. (1) through (3) to compute the Stokes parameters as well as for those using Eqs. (4) through (6). The validity of these analytical expressions is then evaluated using data taken with an actual CCD polarization imaging sensor.
2. Pixilated polarization imaging sensor architecture
A pixilated filter array of differently oriented pixilated micropolarization filters can be deposited onto an imaging array to allow real-time polarization information extraction from a scene. The pixilated filter array has the same pixel-pitch as the underlying imaging sensor pixels in order to accurately filter the incoming light wave. These types of imaging sensors are known as division of focal plane polarimeters.
The DoFP architecture capable of computing polarization information based on Eqs. (1) through (3) is shown in Fig. 1-a . This imaging sensor monolithically integrates a pixilated micropolarizaiton filter array composed of filter-pixels with 4 linear polarization filters oriented at 0°, 45°, 90° and 135° with an array of imaging elements. An alternative array architecture capable of extracting polarization information based on Eqs. (4) through (6) is presented in Fig. 1-b. The second imaging architecture is composed of a pattern of 2 linear polarization filters oriented at 0° and 45° and a third, intensity-only pixel without a polarization filter monolithically integrated with an array of imaging elements. In both imaging architectures, polarization information is computed on each 2 by 2 neighborhood of pixels in the imaging array using either Eqs. (1) through (3) or Eqs. (4) through (6) depending on the filter array architecture implemented on the imager.
We have designed, fabricated and successfully tested a 1 megapixel CCD imaging array with integrated nanowire polarization filters. The CCD sensor is composed of 1000 by 1000 imaging elements with 7.4μm pixel pitch covered with an array of pixel-pitch matched aluminum nanowire optical filters. The 4-polarizer and 2-polarizer architectures are implemented on separate halves of the same imaging chip for simultaneous testing of the two array types. The aluminum nanowire filter array is fabricated by an interference lithography procedure  and is deposited directly on top of the imaging sensor. The nanowires of each micropolarizer are 70nm wide, 70nm high and 130 nm in pitch. The CCD sensor has a conversion gain of 2.9 e-/ADU and a total readout noise of 18 e-.
The sensor collects raw polarization images at 40 frames per second. The raw polarization image information is digitized and sent to a digital processing unit. Finally, for each 2 by 2 neighborhood of pixels on the sensor, the Stokes parameters are computed using either Eq. (1) through Eq. (3) or Eq. (4) through Eq. (6) depending on the filter array architecture implemented in that neighborhood on the imager.
3. Theoretical signal-to-noise analysis for the polarization imaging sensor
The most significant contributions to temporal noise of the CCD polarization imaging sensor are photon shot noise and the readout electronics thermal and 1/f noise. Only these three noise sources will be considered in the following analysis. While dark current contributes to the sensor noise, for the current CCD polarization sensor in our possession it contributes less than 1 e- to the noise at operating temperature (40°C). Therefore, for simplicity, dark current will be ignored in this analysis.
For a single pixel in a CCD array, the number of photons collected during an integration period is described by a Poisson distribution, consequently giving a standard deviation in the signal due to photon shot noise as the square root of the mean signal in units of electrons. The photon shot noise in units of electrons is presented by Eq. (7):
The readout electronics consist of two stages of source follower transistors. The readout noise in the CCD imaging sensor, R, is independent of the number of incident photons integrated at the photodiode node. The thermal noise associated with the source follower and the biasing transistors for the source follower  are described by Eq. (8):18] is signal independent as well and can be computed as shown in Eq. (9):Eq. (9), Ibias is the bias current for the source follower which is constant during the various levels of readout signal, a is a constant dependent on the semiconductor fabrication process and f is operating frequency. The noise contributions due to the thermal and 1/f noise of the readout electronics can be combined together (i.e. the noise power of all noise sources can be added together assuming uncorrelated noise sources) and will be represented as R with units of electrons in the rest of the paper.
Summing the photon shot noise and readout noise in quadrature gives an estimation of the total temporal noise for a single pixel in the CCD array. That is, for a pixel in the array with a mean signal Ii, the standard deviation of that pixel's signal is given by Eq. (10):
3.1 Signal-to-noise analysis for a sensor with a 4-polarizer filter array
Using the Stokes parameter calculations in Eq. (1) through Eq. (3) and standard error propagation techniques , the standard deviations of the computed Stokes parameters by a sensor with a 4 pixilated polarization filter array may be estimated by Eq. (11) through (13):Eq. (1) through Eq. (3) and obtaining the expressions using Eq. (10) with the appropriate pixel signals gives the uncertainties in the calculated Stokes parameters and are presented by Eq. (14) through Eq. (16):
Because S 0 = I 0 + I 90 = I 45 + I 135, the uncertainties in the calculated Stokes parameters all reduce to the same formula, expressed in terms of the mean calculated value of S0 and are presented by Eq. (17):Eq. (18) through Eq. (20):
3.2 Signal-to-noise analysis for a sensor with a 2-polarizer filter array
A similar noise treatment to the 4-polarizer filter array is derived for the 2-polarizer filter array. Using the Stokes parameter calculations presented by Eq. (4) through Eq. (6) and again applying error propagation techniques, the uncertainty estimations for the calculated Stokes parameters using a sensor with a 2-polarizer filter array are presented by Eq. (21) through Eq. (23):Eq. (4) through Eq. (6) and obtaining the expressions using Eq. (10) as before, the uncertainties in the calculated Stokes parameters are described by Eq. (24) through Eq. (26):Eq. (4) through Eq. (6) gives the estimated uncertainties in the calculated Stokes parameters using the 2-polarizer array sensor in terms of the mean calculated values for the parameters. The uncertainties for the three Stokes parameters are presented by Eq. (27) through Eq. (29):Eq. (30) through Eq. (32):Eq. (18) through Eq. (20) with Eq. (30) through Eq. (32) respectively shows that the signal-to-noise ratio for S 0 is always slightly better for the 2-polarizer architecture, though the two converge as S 0 gets larger. This is due to the fact that the dominant noise source for high light intensities is the shot noise and the read noise term contributions are insignificant compared to the shot noise contributions. The signal-to-noise ratios for S 1 and S 2 are functions of S 1 and S 2 respectively for the 2-polarizer architecture but independent of S 1 and S 2 for the 4-polarizer architecture. Furthermore, even when the signal-to-noise ratios for S 1 and S 2 for the 2-polarizer architecture are at their maximum (i.e. S 1 or S 2 equal to -S 0) the signal-to-noise for the 4-polarizer architecture is larger. Thus, the signal-to-noise for S 1 and S 2 for the 4-polarizer architecture are always better than those for the 2-polarizer architecture. Another observation can be made that the 4-polarizer architecture allows for uniform sampling of the incoming light wave on the Poincare sphere. In other words, the 4 linear polarization filters are equally spaced i.e. equally offset by 45° and allow for an optimal SNR of the Stokes parameters [20–22].
Another interesting observation can be made regarding the computation of the first Stokes parameter in both polarization sensors. The first Stokes parameter in the 4-polarier filter array which is presented by Eq. (1), can also be computed as presented by Eq. (33):Equation (33) takes into account the intensity values of all four pixels in a neighborhood of 2 by 2 pixels. It can easily be shown that the SNR for the S0 parameter is described by Eq. (34):Eq. (4), can also be computed as presented by Eq. (35):Eq. (35), Itot,1 and Itot,2 are the intensity values of both pixels without polarization filters in a neighborhood of 2 by 2 pixels. The SNR for the S0 parameter in this sensor is described by Eq. (36).Equations (34) and (36) allow for a comparison of the SNR of the first Stokes parameter when computed on a neighborhood of 2 by 2 pixels. The SNR of the 2-polarizer filter array is better when compared to 4-polarizer filter array as previously concluded.
4. Experimental methods and results
In order to evaluate the noise of the CCD polarization imaging sensor, the sensor was illuminated with a collimated, uniform, linearly polarized light source at 625 ± 10 nm. Two arrays of narrow-band, high intensity LED were used as inputs to a 3-port 4” integrating sphere to produce uniform illumination. A pinhole at the integrating sphere’s output followed by a condensing lens with a 4” diameter were used to collimate the light. The uniform, collimated light was then passed through a commercial linear polarization filter mounted on a motor-controlled rotating stage. By rotating the linear polarization filter, S 1 and S 2 could be varied while maintaining a constant S 0.
The rotating stage was swept through 180° in 1° increments, and the light intensity was swept through 60 different values at each angle. For each angle and intensity, 1000 raw images were collected from the sensor. The Stokes parameters were calculated (as described in Section 2) for each image for both the 4-polarizer and 2-polarizer filter array architectures. The mean values and standard deviations for S 0, S 1 and S 2 for both array architectures were calculated for each set of 1000 images. All measurements were taken with the sensor at an operating temperature of 40°C.
The computed and theoretical signal-to-noise ratios for the S 0 parameter in a 2 by 2 neighborhood of pixels for both filter array architectures are presented in Fig. 2 . Both data sets closely follow their respective theoretical curves, with average deviations from the modeled curve of 2.46% and 1.83% for the 4-polarizer and 2-polarizer array architectures respectively. As predicted by the theoretical model, the signal-to-noise for the 2-polarizer filter array is slightly greater than that of the 4-polarizer filter array at low values of S 0 and converges with that of the 4-polarizer array at higher values of S 0.
The computed and theoretical signal-to-noise ratios for the S 1 and S 2 parameters in a 2 by 2 neighborhood of pixels for both filter array architectures at 3 different incident intensities are presented in Fig. 3 and Fig. 4 respectively. For both the S 1 and S 2 signal-to-noise ratios, the data follows the theoretical curves very closely for all three S 0 values. For the S 1 signal-to-noise, the average deviations from the theoretical model for the lowest, middle and highest intensity values are 2.04%, 2.86% and 2.98% for the 4-polarizer architecture and 1.93%, 2.16% and 2.06% for the 2-polarizer architecture. For the S 2 signal-to-noise, the average deviations from the theoretical model for the lowest, middle and highest intensity values are 1.89%, 2.63% and 2.60% for the 4-polarizer architecture and 2.06%, 2.10% and 2.09% for the 2-polarizer architecture.
Ideally, a 180° angle sweep of the linearly polarized incident light should result in S 1 and S 2 values that range from -S 0 to + S 0, however likely due to the non-ideal response of the micropolarizers this full range is not achieved, particularly at low incident intensities. While this complication does not change the theoretical model for the sensor noise, this problem does affect the sensor accuracy. A calibration routine would be required to increase the accuracy of measurements taken with the sensor; however implementation of such a routine is beyond the scope of this paper. A similar problem with division of focal plane polarimeters in the long wave infrared spectrum (LWIR) was observed by Bowers et al.  who used a routine developed by Tyo and Wei  to compensate for the non-ideal optics in the sensor.
A theoretical analysis for the noise in the Stokes parameters computed by a CCD polarization imaging sensor has been presented. These theoretical noise derivations for both 4-polarizer and 2-polarizer micropolarizer filter array sensors are general to any polarization state of the incident light wave. Experimental results confirm that this analysis is a reasonable estimate of the real noise behavior of such imaging sensors. Both the theoretical model and experiment show a slightly larger signal-to-noise ratio for S 0 for the 2-polarizer filter array at low light intensities and better signal-to-noise ratios for S 1 and S 2 for the 4-polarizer filter array. Thus, the choice of filter array architecture for a polarization imaging sensor is dependent on the relative importance of the measurements of S 0 versus S 1 and S 2 for a particular application.
The work described in this paper was supported by National Science Foundation (NSF) grant number 0905368, Air Force of Scientific Research grant number FA9550-10-1-0121 and the Center for Material Innovation at Washington University in St. Louis.
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