We present an analysis and test of an image sampling polarimeter based on the concept of Star Test Polarimetry first introduced by Ramkhalawon. The method makes use of a stress engineered optical element (SEO) placed in the pupil plane of an optical system to induce a polarization dependent point spread function (PSF) at the detector. We describe the calibration requirements of the polarimeter and introduce a new algorithm that can robustly extract the Stokes parameters in a single irradiance measurement. By acquiring statistics on the sampled Stokes parameters of a uniformly illuminated pinhole array, we show that a single frame can provide a root mean square angular error of approximately 10 milliradians on the Poincaré sphere.
© 2016 Optical Society of America
A variety of fields in science, medicine, and engineering employ quantitative polarization measurements in data acquisition. For example, aerosol measurements require accurate polarizations measurements over a wide range of angles; a suitably designed wide angle lens along with a camera of sufficient dynamic range can capture all angles in a single image [1,2]. Particle characterization and atmospheric modeling benefit from polarization information [3,4], as does particle tracking, single molecule imaging and other areas of quantitative microscopy [5–7].
A variety of methods are available for imaging polarimetry [8–10]. Tyo  provided an excellent overview of existing imaging polarimetry methods. Most either require a series of measurements or make use of polarization selective pixels integrated into a sensor array. In this work we extend a concept first introduced by Ramkhalawon, Beckley and Brown [11–13], in which a polarization dependent point spread function produced by a stress-engineered optical (SEO) element can provide quantitative polarization information from the image of a point source. The key to the concept is a natural redundancy in sensor based imaging, in which it is typical for many pixels to be contained within a single optical point spread function. By encoding the polarization into a point spread function, it is possible to recover the polarization state of the source.
So far, two extensions of the method have been reported: Zimmerman  extended the recovery method of references [11,12] to a pinhole array for polarization mapping; Sivankutty et. al.  applied the method to polarization recovery from a multicore fiber using a maximum likelihood estimate. In this paper we report a simplified Stokes recovery algorithm, describe the calibration in detail, and investigate the accuracy of the method using the pinhole array as a spatial ensemble. We show that an angular variance within about 10 milliradians of the nominal Stokes vector can readily be achieved with an inexpensive sensor and no moving parts. In the following section we review how a stressed optical element introduces a polarization dependent aberration that allows an accurate measurement of the Stokes parameters over the image field.
2. System description
Optical stress engineering is the deliberate use of stress to modify an optical element . In this arrangement, stress is symmetrically applied to the periphery of a BK7 glass window (diameter 12.7, thickness 8 mm) using the procedure described in . For threefold (trigonal) symmetry, the birefringence increases from the center in a manner approximately proportional to the radius of the element. In result, the SEO also has a spatially varying fast axis over the cross section. This combination of spatially varying fast axis and radially varying retardance results in a spatially varying wave plate; such that each point over the cross section of a laser beam passing through the SEO experiences a different retardance. An in depth overview of the SEO used in this arrangement, it’s fabrication and extended properties can be found in .
In Fig. 1, a camera lens forms the image of a particular scene or object on the pinhole array. That pinhole array is then the object plane of the relay system; with a stressed engineered optical (SEO) element placed in the pupil plane of the relay system, each object point is imaged to the sensor array with an aberration that is uniquely correlated to the input polarization.
The design, illustrated in Fig. 1, is doubly telecentric. The symmetry of the system reduces field aberrations and allows direct access to the pupil plane. (We require access to the pupil plane for Fourier filtering and any SEO adjustments). With a 2 mm diameter aperture and the use of corrected doublets (Edmund Optics), the system is diffraction limited across the entire field of view. The system has distortion less than 1% to ensure that the point spread functions are uniformly distributed across a CMOS sensor (The Imaging Source DMK 24UP031).
For good accuracy, it is important for the numerical aperture of the system to be low enough that the point spread functions extend over a significant number of pixels. Figure 2 shows PSFs for horizontal (H), vertical (V), + 45 (P) and −45 (M) degree polarizations as well as right hand circular (R) and left hand circular (L) polarizations.
A central assumption in the retrieval of the Stokes parameters using this method is that the PSF of an unknown continuous source can be represented as a linear combination of reference PSFs in the following way:
A major issue in the above formulation is that irradiance measurements are rarely taken in absolute terms, yet the relative scaling of the measured irradiances for the calibration PSFs must be as accurate as possible for accurate recovery of the Stokes parameters.
3. Calibration of the point spread functions
The calibration procedure is similar to a number of other polarimeters: A series of known polarization states are used to illuminate the pinhole array, and a full image is acquired for each polarization. In our case, a polarizer and quarter wave plate were placed in the input plane as a polarization generator for calibration purposes; these states were verified using a calibrated Thorlabs polarimeter. Examples of the calibration PSFs are shown in Fig. 2.
For calibration purposes, it is important to note that the experimental PSF for an incoming state of an unpolarized source must be equal to the sum of the point spread functions of any two orthogonal states. This is illustrated in Fig. 2, showing the unpolarized point spread function on the bottom row, and three corresponding pairs of orthogonal components on the top.
When acquiring each PSF, the polarization optics can result in slightly different power levels for each polarization; the calibration exposures must therefore be scaled such that all PSFs are corrected to a consistent exposure. To solve this problem we introduce a set of scaling variables κ1-6, that multiply the irradiance distributions. We choose Iup as the reference irradiance standard (with a scaling factor of 1). Self-consistency (Fig. 3) requires that:
Each pixel location represents an independent equation, leading to an overdetermined system for the irradiance factors. We solve the system of equations using standard methods of linear algebra, and use the resulting scaling factors to compute the rms error between the unpolarized irradiance distribution and the various fully polarized point spread functions.
It is usualy possible to achieve exposure correction to within 1% of the peak PSF irradiance. Once the calibration images are properly scaled we can solve for the Stokes parameters.
4. Retrieving the Stokes parameters
To retrieve the Stokes polarization parameters, we assume that the PSF of unknown polarization can be written as a combination of the corrected calibration PSFs and the unknown Stokes parameters Sj that must hold at each point x:
5. Polarization mapping
The measurement proceeds as follows: After calibration (the pseudoinverse of the calibration matrix P for each array element is stored in the computer’s memory), images of the pinhole array can be captured in real time and processed. This procedure excerpts the region of interest surrounding each PSF and arranges the data in a column vector. The irradiance vector is then multiplied by the pseudoinverse to retrieve the Stokes parameters for that point.
Because a simple irradiance image is sufficient to recover the Stokes parameters at each point, the system can be readily adapted to real time acquisition. In the cases shown here in Figs. 3 and 4, the measurements take 30-100 ms for processing on a standard (Lenovo) laptop computer.
Figure 3 illustrates a test carried out using the right hand circular calibration element for reference and to test the consistency of the algorithm. It is not a test of absolute accuracy of the measurement, but is carried out in the presence of detector noise and represents that aspect of a realistic measurement. The central image is a superposition of the point spread functions formed in the image of the pinhole array and the reconstructed polarization ellipses. Red coloring represents a right handed ellipse, while green represents a left handed ellipse. Figure 4 shows two examples of screen shots of a real time measurement (clicking on the link will play a movie). The left shows the polarization map of a stressed window such as is used in creating Full Poincaré Beams . The right shows a rotating linear polarizer.
6. Quantifying the method: statistical Stokes analysis
The presence of noise in the sensor (or photon noise in the signal) degrades the fidelity of the measurement. This noise can take the form of photon noise, readout noise, dark noise, or it can be caused by inhomogenieties over the pinhole array. While fixed variations can be corrected in calibration, random noise and/or position errors result in uncertainty. In this study, we present a representative measurement of the spatial statistics measured over the pinhole array as the statistical ensemble.
We quantify the accuracy of the retrieved Stokes parameters by the angular error distribution of the individual Stokes vectors about the mean. To do so, we explore the statistics over the ensemble of pinholes. We define the angular errorfor each PSF as follows:
The evaluation of the spatial statistics is complicated by the fact that, in normal measurement, the illumination pattern over the pinhole array has some nonuniformity. Since the recovery algorithm retrieves S0 (the relative irradiance at the pinhole), we can postselect PSFs that have a value of S0 within 25% of the peak irradiance. Figure 5 shows a histogram of the composite error (over all polarization states measured) for the top 25% irradiance values. The rms variation in angular error for this ensemble is approximately 12 milliradians. For comparison, we also show the error in the DoP. The measurement employed illumination from a white light LED (Radio Shack) passed through polarizers that had been pre-calibrated using the ThorLabs polarimeter.
The nonuniformity of the illumination allowed us to explore the measurement accuracy as a function of signal strength and irradiance signal to noise ratio. Here the ensemble of S0 measurements over the entire pinhole array is sorted into 25 bins and the statistics of each bin separately evaluated for the error in the Stokes parameters. Figure 6 shows the Stokes error distribution as a function of signal to noise ratio η (the ratio of the mean value to the normalized rms fluctuation in S0 over each bin) and as a function of normalized signal power. The curve, intended as a guide to the eye, represents a best fit curve of the form A/ η + B/ η2, where A represents the relative contribution of the photon noise and B the signal independent detector (dark and readout) noise. At SNR values less than 50, the system appears dominated by the detector noise.
7. Discussion and conclusions
It is important to emphasize that the examples shown here are carried out with a room temperature CMOS sensor without any effort at noise reduction beyond that provided by the manufacturer. More sophisticated sensors are available with much lower dark and readout noise. It is therefore expected from the trends in Fig. 6 that a photon noise limited measurement can have accuracy much better than 1 milliradian on the Poincaré sphere. As with other polarimetry methods, the measurement of DoP is especially sensitive to both noise and background signal. We have found that incomplete background subtraction results in an artificially low degree of polarization, while overcompensating the background can yield unphysical values for the DoP. Following the procedure outlined above, in which an average value for dark pixels is subtracted tends to slightly overestimate the background. Therefore to avoid nonphysical values in the DoP histogram, the above measurements subtract 95% of the average computed background signal.
The results shown in this paper were based on calibration and measurement using an LED source; we have also carried out similar tests using the combination of a laser source and rotating diffuser, such that the pinholes are illuminated with spatially incoherent light. For the case of full spatial coherence, there are two issues that can significantly affect the measurement: 1) Neighboring point spread functions will interfere even in the case of a weak overlap, leading to image artifacts that render the analysis more complicated. This is an ongoing area of research. 2) Illumination that strikes the pinhole over a very narrow angular range can cause a significant variation in pupil illumination over the image field, resulting in highly nonuniform point spread functions. While this can, in principle, be accommodated with suitable design optimization, the current system is vulnerable to vignetting for collimated laser illumination.
Because the concept is built around a relay element that can be inserted into an ordinary imaging system, the method can be applied to a wide variety of applications in imaging and metrology. For example, polarization mapping of birefringent material (as is needed for the display industry), polarimetric microscopy, and polarization analysis of telescope images. Some of these applications, including one relevant to aerosol polarimetry, are discussed in reference .
In summary, we have presented the design and implementation of an image sampling polarimeter based on star test polarimetry. The polarimeter allows us to sample the polarization states across a scene of interest with a single irradiance image by observing the polarization dependent point spread functions generated by a stressed element placed in the pupil plane of the system. The statistics of the recovered Stokes parameters of the spatial ensemble show an accuracy of about 10 milliradians on the Poincaré sphere.
National Science Foundation (NSF) (PHY-1507278);NASA Goddard Space Flight Center (GSFC) Graduate Student Research Fellowship Program.
The authors gratefully acknowledge inspiring conversations and helpful advice from Prof. Miguel A. Alonso, Prof. Julie Bentley, Dr. Amber Beckley and Roshita Ramkhalowon. This work was supported in part by the National Science Foundation (PHY-1507278). Dr. Brandon G. Zimmerman gratefully acknowledges financial support received through the NASA (GSFC) GSRP program.
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