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Abstract
A spatially adaptive Mueller matrix imaging polarimeter is described, simulated, and demonstrated with preliminary experiments. The system uses a spatial light modulator (SLM) in the polarization state generator (PSG) to create spatial carriers that controlled by the pattern written to the SLM. The polarization state analyzer (PSA) is a commercial division of focal plane imaging polarimeter. The PSG/PSA pair form a 9-channeled partial Mueller matrix polarimeter that measures a 3 × 3 sub-matrix of the Mueller matrix. We demonstrate that adapting the PSG modulation to the spatial frequency structure of the scene can reduce channel crosstalk and improve reconstruction accuracy. Initial experiments are performed that demonstrate the SLM’s ability to produce sufficient modulation diversity to create the desired channel structure. Though there are several experimental challenges to obtain accurate Mueller matrix imagery, we demonstrate a system that adapts to the particular scene spatial frequency structure.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical detectors do not directly measure the polarization state of light, so an optical polarimeter must infer the polarization state. The first primary polarimeter strategy divides the light into multiple paths using beamsplitters [1] or lens arrays [2] and analyzes each path using separate polarization optics. These are then measured by independent detectors or detector arrays (or sections of a single FPA). The polarization is determined by combining the outputs of these detectors using a data reduction matrix [3]. We term these wavefront division instruments. The second strategy uses a single detector or detector array to measure a distribution of information that includes a set of polarization carriers that have been modulated as a function of time [4], space [5], wavelength [6], angle of incidence [7], or some combination of these domains [8,9]. The modulation is usually accomplished using periodic carriers that are polarization dependent, and this modulation results in a set of sidebands in the corresponding frequency space that carry combinations of the polarization parameters [10]. We term these modulated or channeled polarimeters. Both strategies are equally applicable for passive polarimeters that measure the Stokes parameters and active polarimeters that measure the Mueller matrix [11]. It should be noted that there are classic polarimeter designs that are not obviously one type or the other, but we categorize the division of time (DoT) and division of focal plane (DoFP) polarimeters as channeled polarimeters and multiple camera systems as wavefront division polarimeters.
We limit our discussion in this paper to channeled polarimeters. Since the advent of the spectrally channeled snapshot polarimeter independently by Oka and Kato [12] and Ianarelli, et al. [13], numerous strategies have been developed for imaging and non-imaging designs that could be either passive or active [14]. We include temporally modulated instruments in this category even though they are not snapshot instruments because the Fourier-based processing methods are identical [10]. It should be noted that channeled polarimeters can be reconstructed using non-Fourier methods, as demonstrated by Diner, et al., who used a Taylor series expansion method in the data reduction matrix of their high-speed temporally modulated polarimeter [15,16].
While channeled polarimeters have numerous benefits, they also have two primary drawbacks. The first is that they sacrifice resolution. This is because the native bandwidth of the detector is used to measure multiple pieces of information. This limitation can be mitigated by modulating in multiple domains simultaneously [9,17–19], allowing the sidebands to be placed advantageously in the Fourier domain so as to minimize channel crosstalk. Our group has demonstrated a spatiotemporally modulated system that can achieve performance that is never worse than spatial or temporal modulation alone, and often better [8]. Additional research on channeled systems has further demonstrated that using artificial intelligence/machine learning (AI/ML) methods to predict adaptive interpolation/filtering strategies can have significant benefits [20–24]. The second challenge is that systems never have perfect, periodic modulation. Imperfections in the system violate the channeled assumption, resulting in spurious channels and other effects that fundamentally limit performance [25]. Calibration errors in channeled polarimeters can be mitigated by starting with a nominal calibration and allowing the AI/ML reconstruction scheme to “solve for” the correct calibration parameters during data reduction [24,26].
In this paper we present a concept for an adaptive, spatially channeled Mueller matrix polarimeter. The system presented here uses a spatial light modulator (SLM) in the illumination arm (the polarization state generator, or PSG, see below) to generate spatial carriers that can be adapted to the scene. This structured illumination is then used on a scene, and the scattered light is analyzed by a conventional DoFP camera. The camera produces spatial sidebands that interact with those of the PSG to create a set of nine channels used for polarimetry. Choosing the PSG modulation in a way that adapts to the spatial frequency structure of the scene allows for significant improvement in image quality. We discuss the concept using canonical images in simulation, and we present some very preliminary experimental measurements that demonstrate that the technique is viable, if complicated to implement currently. The concept in this paper was originally presented with much less detail and analysis in an earlier report [27].
2. SLM-based spatially channeled polarimeter
2.1 Stokes and Mueller notation
The Stokes parameters arranged as a vector are defined as
We prefer the definition of $\textbf {S}$ in Eq. (1) because it expresses the Stokes parameters in terms of a set of directly measurable intensities through linear and circular polarization analyzers. The vector $\textbf {x}$ represents the independent variables such as space, time, wavelength, etc., that the sensor measures over. In order to estimate $\textbf {S}$, the light is passed through a polarization state analyzer (PSA) described by the analyzer Stokes vector,
The intensity measured by a detector is given by the inner product between $\textbf {S}_A$ and $\textbf {S}$
The job of the designer is then to choose functional forms for the elements of $\textbf {S}_A$ that allow the multiplexed information in Eq. (3) to be separated and reconstructed.
The above analysis considers a passive system that measures the light’s polarization field, but in many applications the interest is in the polarization scattering properties of objects in a scene, which are fully described by the $4\times 4$ real Mueller matrix $\underline {\underline {\mathbf {M}}}{=}(\textbf {x})$. We use the convention that $m_{00}$ is the top left element and $m_{33}$ is the bottom right consistent with the Stokes indices in Eq. (1). If the system illuminates the scene using PSG that creates illuminating Stokes parameters
The collection of scalar measurements in Eq. (5) make up the data set that is used to infer the desired polarization quantities.
2.2 System channel structure
In typical channeled polarimeters, the elements $\textbf {S}_A$ and $\textbf {S}_G$ are chosen to be periodic functions of one or more of the independent variables [28]. In this paper we consider the two spatial variables, so $\textbf {x}=(x,y)$. The SLM allows the system to also be modulated in time $(t)$ to make this a hybrid modulated instrument, but we do not consider that here. The PSA in our system is assumed to be a conventional $2\times 2$ DoFP array that measures linear polarization only [29] with analyzer vector
The PSG of our system includes a SLM where each individual pixel is a voltage-controlled retarder with redardance $\delta (m,n)$ and a common fast axis oriented at nominal angle $\theta _0$ that is taken as ${0}^{\circ }$ and used as the reference for computing the Stokes parameters in Eq. (1). The SLM is illuminated by linear polarization at ${45}^{\circ }$. The SLM pixels therefore modulate the polarization in the $s_1$ and $s_3$ Stokes parameters. We add a quarter waveplate after the SLM with fast axis oriented at ${0}^{\circ }$ so that the $s_1$ parameter is unchanged and the $s_3$ parameter is converted into $s_2$. The system setup is shown in Fig. 1(a).
The polarization state generated by the $(m,n)^{\mathrm {th}}$ pixel of the PSG is
The parameters of the pixels can be chosen so that the distribution of $\textbf {S}_G$ is arbitrarily complicated. Initially we choose one dimensional sinusoids of arbitrary orientation and frequency:
Both $\textbf {S}_A$ and $\textbf {S}_G$ contain information only in the $s_0$ – $s_2$ components, so the polarimeter is only able to reconstruct the upper $3\times 3$ block of the Mueller matrix. Hence the device is a channeled partial Mueller matrix polarimeter (c-pMMP). While not providing the full details of the Mueller matrix, a pMMP is still useful for many tasks [30,31]. In general, channeled Mueller polarimeters have many more channels than reconstructed Mueller elements. However, this particular system is a special class of c-pMMP known as a nine-channeled pMMP (9-c-pMMP) that has symmetry properties that makes it optimal from a signal to noise ratio (SNR) perspective and also make it an ideal building block for multi-snapshot systems [32].
Alenin and Tyo [28] provide a general theory of channeled polarimeters that allows us to relate the channel structure created by $\textbf {S}_A$ and $\textbf {S}_G$ to the Mueller matrix elements through the ${Q}$-matrix formalism. The matrix $\underline {\underline {\textbf {Q}}}$ is a bookkeeping device that contains information about how each Mueller element is mapped into the channels of the system, and the Mueller matrix elements can be estimated by inverting the system of equations
The vector $\textbf {C}$ includes the channels created by the modulation strategy (after being demodulated back to base band), and the vector $\textbf {M}$ is formed by taking a row-by-row reorganization of the Mueller matrix. The various channels in $\textbf {C}$ must also be isolated from one another by use of an appropriate channel filter as discussed in more detail below, and Eq. (9) inherently assumes that the individual channels are demodulated before recombination.
In order to illustrate $\underline {\underline {\textbf {Q}}}$ for this pMMP, consider the canonical sample image shown in Fig. 1(b). The scene consists of an ideal depolarizing background with Mueller matrix $\underline {\underline {\textbf {M}}}_B$ and two target regions ($\underline {\underline {\textbf {M}}}_1$ and $\underline {\underline {\textbf {M}}}_2$) taken to be ideal linear polarizers oriented at $0^\circ$ and $45^\circ$, respectively:
The analyzer and generator vectors in Eq. (6) and Eq. (7) allow us to predict the location of the system channels. The DoFP system creates spatial sidebands at $(\xi _0,\eta _0) = (\pm 0.5,0)$ and $(\xi _0,\eta _0) = (0,\pm 0.5)$ in addition to the base-band channel at (0,0), where $\xi$ and $\eta$ are the spatial frequency variables corresponding to $x$ and $y$ in units of samples per pixel [33]. Each of these channels is then split into three channels by $\textbf {S}_G$, one at the original channel center location $(\xi _0,\eta _0)$ and two offset channels at $(\xi _0\pm \rho _0\cos \theta _g,\eta _0\pm \rho _0\sin \theta _g)$. Figure 2 shows the simulated image measured by the DoFP sensor and the Fourier transform that shows the nine channels.
Fig. 2. Image measured by the PSA (left) and its Fourier transform (right); the generator was set to $\rho _g = \sqrt {2}/6$ cycles per pixel and $\theta _g = 45^\circ$.
2.3 Pixel size issues
There are two very important simplifications that we have used up until this point. The first is that there is a one-to-one mapping of the PSG and PSA pixels. While this is possible, it is highly unlikely. Modern DoFP cameras have five megapixel (MP) arrays, whereas SLMs are typically on the order of $512\times 512$ or $1024 \times 1024$. When there are many PSA pixels for each PSG pixel, the spatial frequency of the PSG ($\rho _g$) needs to be as high as possible. Even when the pixel ratio is intended to be 1:1, achieving that exactly in both size and alignment is unlikely and unnecessarily difficult.
Using higher PSG frequencies and having many PSA pixels per PSG pixel brings us to the second issue, which is that the PSG carriers are not purely sinusoidal as implied by Eq. (7). Rather, the modulation is piecewise constant at the resolution of the PSA. The resulting waveform has many harmonics when measured by the PSA. Figure 3 shows the channel structure for the case when the ratio of the PSG pixel size to the PSA pixel size is 9:4 (81:16 in area), so there are 5.0625 PSA pixels for each PSG pixel. This produces additional channels at the odd harmonics of the PSG fundamental. Fortunately, this issue is closely linked to the diffraction behavior of the optical system, and can be partially mitigated through spatial filtering in the optical path as discussed in section 3 below.
Fig. 3. Channels structure when the ratio of the area of the PSG:PSA pixel size is 5.0625 (9:4 on a side) with $\rho _g=\sqrt {2}/4$ and $\theta _g=45^\circ$. The larger PSG pixels mean that the modulation is piecewise constant instead of sinusoidal at the resolution of the PSA, which produces extra channels at the odd harmonics of the PSG fundamental.
2.4 Channel structure adaptation
Figure 2 shows a specific example of a scene whose spatial frequency structure spreads along the $\xi$ and $\eta$ axes. For this example, $\theta _g$ was chosen as ${45}^{\circ }$ so that the cross talk among the various side bands was minimized. However, when the object rotates by ${45}^{\circ }$, this modulation direction is no longer optimal as is shown in Fig. 4. For this image, the width of the PSG pixels is four times that of the PSA pixels (pixel area ratio of $4^2:1$), and the frequency of the PSG sinusoid is $\rho _g=\sqrt {2}/4$. The channels are extracted before reconstruction using simple, ideal low pass filters (LPFs) that are circular with a cutoff frequency of $\rho _g/2$ converted into the spatial frequency of the PSA as indicated in the figure. These filters are not ideal because they give rise to ringing near object edges, and it is known that more sophisticated filters can produce significantly better reconstruction [8,22,34]. However, we choose these particularly simple filters so as to focus on the overall system concept. The optimal choice of filter is complicated and depends on the specific scene capture configuration. We previously developed a scene-adaptive filter selection method for use with passive DoFP systems like the PSA [22], and we are currently working to extend that technique for use with this pMMP. Also, the reconstruction is only using the nine channels highlighted in Fig. 4, but we have recently developed methods for use with multi-harmonic instruments (such as MMPs based on photoelastic modulators) that are directly relevant to this problem [26].
Figure 4 demonstrates that the interaction between the channel structure and the object spectrum is important. In order to illustrate this, we have created an example that highlights the crosstalk issues when the scene severely violates the bandwidth limitation of the system. The choice of PSG modulation can mitigate this by placing the channels in portions of the frequency space that minimize channel crosstalk. Figure 5 shows the reconstructed Mueller matrix elements and the corresponding error for the object of Fig. 4 at $\theta _g = 0^\circ$ and ${45}^{\circ }$ with $\rho _g=\sqrt {2}/4$. There is significant error in both reconstructions, but the cumulative mean squared error across the Mueller matrix elements is 53% lower for $\theta _g=0^\circ$ than for $\theta _g=45^\circ$.
Fig. 4. Top row: Measured image and channel structure for object shown in Fig. 2 rotated by ${45}^{\circ }$ with $\theta _g=0^\circ$. Bottom row: Same scene, but with $\theta _g=45^\circ$. In both cases the ratio of PSA:PSG pixels is $42:1$ and $\rho g = \sqrt {2}/4$ cycles per PSG pixel. The circles show the ideal, circular LPFs with cutoff frequency $\sqrt {2}/8$ that are used to extract the channels.
Fig. 5. Top three rows: $\theta _g = 0^\circ$. Bottom three rows: $\theta _g=45^\circ$. Left three columns: Reconstructed Mueller matrix images. Right three columns: Absolute reconstruction error. These images correspond to the case of Fig. 4 and were reconstructed assuming $4^2$ PSA pixels per PSA pixel, $\rho _g = \sqrt {2}/4$ cycles per PSG pixel, and a reconstruction filter cutoff frequency of 0.0706 cycles per PSA pixel.
In this paper, we have only adapted the angle of modulation $\theta _g$ to the scene’s structure, but the spatial frequency $\rho _0$ in Eq. (8) is also a degree of freedom. In general, for this simple one-dimensional modulation, it is advantageous to choose $\rho _0$ to be as large as possible for a given $\theta _g$. In the experimental work presented below, we are severely limited in $\rho _g$ of our SLM.
2.5 Channel filtering and polarization reconstruction
In order to reconstruct the Mueller matrix images, each of the nine channels must be extracted, demodulated, and then added together using the weights in the pseudo-inverse $\underline {\underline {\textbf {Q}}}^\dagger$. The matrix $\underline {\underline {\textbf {Q}}}$ is depicted graphically in Fig. 6. The interpretation of the matrix is as follows. The base band channel ($c_1$) relates to $m_{00}$. The original DoFP channels ($c_2$ and $c_3$) relate to $m_{10}$ and $m_{20}$ with real coefficients. The channels offset from baseband ($c_4$ and $c_5$) carry $m_{01}$ information in the real part and $m_{02}$ in the imaginary part, and the channels offset from the DoFP channels ($c_6$ – $c_9$) carry the Mueller matrix elements related to row/column 1 in their real part and that related to row/column 2 in their imaginary part. Note that the amplitudes of the channels reduces like $2^{-N}$, where $N$ is the number of sinusoids that create the channel (0, 1, or 2 in this case) [28].
Fig. 6. Graphical depiction of $\underline {\underline {\textbf {Q}}}$. The nine channel locations are labeled in Fig. 2. The amplitude of the channels is indicated at right, and the phase is given by the hue used. For this simple example, the phase is ${0}^{\circ }$, ${90}^{\circ }$, ${180}^{\circ }$, or ${270}^{\circ }$, but in general the coefficients can be complex. This is a pMMP, so seven of the Mueller matrix elements are not reconstructable [31].
For each channel, a filter must be applied that isolates that channel’s information from the other eight channels. Conventionally these filters are some form of low-pass filter; in our work we have generally used Planck-taper filters for this task [8,27]. More recently, we have demonstrated the use of deep learning methods to predict ideal ratio filters (IRFs) that adapt to the particular scenes being imaged for a Stokes polarimeter [22]. These IRFs can provide significant accuracy improvements by increasing the bandwidth while simultaneously suppressing channel cross talk. The specific choice of filter is important for the reconstruction accuracy of any particular collection. However, the choice of filter does not affect interpretation, and it is not treated in this study. Here we use ideal circular low pass filters as depicted graphically in Fig. 4.
In this paper, we have chosen canonical images with a particular spatial structure that makes the adaptivity of the PSG very useful. The strong, orthogonally oriented edges create broad spreading in the frequency domain that can be mitigated by adapting the modulation direction. While strong horizontal and vertical edges are not uncommon, real-world images will have a range of spatial frequency structures. Some will benefit greatly from adaptivity of the angle of modulation while others will not; we presented some simple examples in our preliminary publication on this system [27]. Our group and others have also shown that adapting the modulation and/or the reconstruction filter to the scene’s spatial structure in a passive system can yield great benefits [22,35]. Ultimately, our goal is to allow many more degrees of freedom in the PSG than the one or two that are evident in Eq. (8) and exploit modern machine learning methods to adapt the PSG in a complicated fashion in near-real-time. That is the subject of future work.
3. Experimental study
To examine the predictions made here, we constructed a prototype system based on the design in Fig. 1(a). The illumination is a Thorlabs 0.5 mW polarized HeNe laser operating at 632 nm. The system uses a $512\times 512$ Meadowlark Optics SLM with a 16-bit PCIe Controller. The camera used for this demonstration is a Lucid Laboratories Triton 5 megapixel DoFP camera with an extinction ratio of 145.89 at 630 nm [36], and a measured accuracy of 98.8%. Our system also includes a $10\times$ beam expander before the SLM. The initial results below show that the behavior predicted in simulation can be realized in a real system. However, our current setup has a number of limitations that will need to be overcome to make this strategy competitive with other spatially modulated methods [5]. These include speckle from the laser source, spatially varying illumination, diffraction issues, multiple reflections, and spatially varying calibration of the SLM. As the purpose of the present manuscript is to show a proof of concept, we leave most of these issues for future work focused on instrument development.
3.1 Mitigating diffraction effects
In section 2.3 above we discussed the signal processing issues associated with the different size pixels in the PSG and PSA. The fact that the PSG pixels are piecewise-constant creates a second issue related to the diffraction of the fields that can be used to partially compensate for this effect. The sharp discontinuities at the edges of the PSG pixels creates multiple harmonics of the fundamental PSG frequency. The laboratory system includes a relay lens between the SLM and the scene that re-images the PSG pattern onto the scene. Figure 7 shows a picture taken at an iris located at the back focal plane of the relay lens, where we see an array of diffraction orders created by the diffraction from the SLM. We can use this iris to control the number of diffraction orders that are allowed to pass the system, which in turn helps address the pixel size issues mentioned above.
Fig. 7. Photograph of the diffraction orders at the back focal plane of the relay lens. An iris is inserted at this location to filter out undesired diffraction orders.
In addition to diffraction effects from the piecewise constant PSG pixels, the particular SLM we are using includes masking between pixels to prevent crosstalk at the device level. Our SLM has only 80% of the pixel area open, and this creates many more high-frequency diffraction orders that are not desired. The iris discussed above eliminates these as well. More modern SLMs have as much as 95% of the pixel area open, greatly reducing this effect. We are working to incorporate such a device into future experiments.
3.2 Simple scene analysis
Figure 8 shows the example scene used to illustrate the concept. The setup of Fig. 1(a) is used in reflective mode, and the scene is composed of two linear polarizers placed in front of a black cardboard diffusing background (Fig. 8(a)). The SLM illuminated by the expanded laser is imaged onto the scene, which is then re-imaged onto the PSA (Fig. 8(b)). The resulting PSA image and its Fourier transform are shown in Fig. 9.
Fig. 9. (a) Image corresponding to Fig. 8(b) taken with the system in Fig. 1(a). The squares mark regions of interest either completely contained in the polarizers (B and C) or at the boundary between polarizer and the background (A and D). (b) DFT of panel (a) in logarithmic scale.
Figure 9 shows an example of the versatility of the proposed method, where different parts of the scene can be illuminated with patterns modulated in different directions. In this case, the illumination is modulated on the left- and right-hand sides of the image in the horizontal direction and at $+45^{\circ }$, respectively, with a spatial frequency of 1/32 pixels$^{-1}$ on the SLM and a magnification of about 3 at the scene. Ideally, we would be able to modulate at a much higher spatial frequency on the PSG, but our particular device loses modulation contrast as the spatial frequency increases. As mentioned above, we intend to replace the SLM in future work.
Figure 9(b) shows the amplitude of the DFT of the full scene in Fig. 9(a). When the spatial modulation is variable across the scene, a full-image DFT would not be used, but it is shown here for reference. The four regions of interest (ROIs) marked in Fig. 9(a) with red squares are either completely contained in the area covered by the polarizers (ROIs B and C) or at the boundary between the polarizers and the background (ROIs A and D). As an example, we show the intensity modulation and DFT of ROI C in Fig. 10. Similar distributions are obtained for the DFT of the other ROIs. However, since the frequency content of the DFT of all those ROIs is concentrated in specific zones of the corresponding Fourier domains, marked with rectangles in Fig. 10 for ROI C, we focus our attention in those zones. Figure 11 shows zoom-in images of the three zones marked in Fig. 10 for the four ROIs in Fig. 9(a). Note that zones 2 and 3 are at the edges of the Nyquist domain at spatial frequencies of $\pm 1/2$, but they are shown in the figure after demodulation to base band. We can see that the frequency content of the ROIs completely contained within the polarizers (B and C) has minimal spread, whereas that of ROIs at the boundary polarizer-background (A and D) spreads in the direction orthogonal to the edge. Hence, the amplitude of the DFTs in Fig. 11 shows that the distribution of the frequency content of the ROIs depends on the modulation of the illumination and the structure of the scene.
Fig. 10. (a) Intensity modulation of ROI C in Fig. 9(a) and (b) amplitude of its DFT with rectangles marking the zones where the frequency content is concentrated. Zones 2 and 3 are split at opposite edges of the Fourier domain but are considered as consolidated zones in the rest of this section.
Fig. 11. Amplitude of the DFT of the four ROIs in Fig. 9(a) at the zones marked in Fig. 10. In each figure, the label indicates the ROI (capital letter) and zone (number) on the corresponding Fourier domain.
For scenes with similar polarization properties, the amplitude of the DFT may remain the same for the nine channels. For instance, if the scene is a rotating linear polarizer it would not be possible to distinguish between different orientations of the polarizer by looking only at the amplitude of the channels. In that case, the information about its orientation will be encoded on the phase, as can be inferred by inspection of Fig. 6. The expected phase shift can be seen in Fig. 12, where we show experimental results for a ROI completely contained within a linear polarizer, illuminated with the same polarization modulation along the horizontal direction ($\theta _{g} = 0^{\circ }$), when its transmission axis is at $\theta _{p} = 0^{\circ }$ and $\theta _{p} = 45^{\circ }$ with respect to the horizontal.
Fig. 12. Intensity modulation for a linear polarizer at (a) $\theta _{p} = 0^{\circ }$ and (b) $\theta _{p} = 45^{\circ }$ illuminated with the same polarization modulation along the horizontal direction. The displacement of the fringes indicates a phase shift that reveals the direction of the transmission axis of the polarizer.
According to Fig. 6, rotating a linear polarizer from $0^{\circ }$ to $45^{\circ }$ yields no phase shift in channels $C_1$ and $C_2$, a phase shift of $\pi$ radians in channel $C_3$, and a phase shift of $\pm \pi /2$ radians in channels $C_4$ – $C_9$. Table 1 shows the amplitude, phase, and phase shift for the nine channels of the DFT of ROI C in Fig. 9 for a linear polarizer with its transmission axis at the two angles considered above. The results in Table 1 show that, although there are differences from the actual values expected from the Q-matrix in Fig. 6 due to the limitations of our current experimental setup, the proposed method can distinguish between different orientation of the transmission axis of a linear polarizer. There are numerous potential causes for the discrepancies between theory and observation, including alignment & calibration issues, residual circular polarization, and spatial nonuniformity in the SLM. However, there is still room for improvement of the performance of our experimental setup, but our preliminary experimental results constitute a proof of concept of the capabilities of the proposed method.
Table 1. Amplitude, phase, and phase shift at the nine channels for a linear polarizer oriented at $\theta _p = 45^{\circ }$ (P45) and $\theta _p = 0^{\circ }$ (P0). In both cases the illumination was modulated at $\theta _g = 0^{\circ }$ with a spatial frequency $\rho _g = 1/32$ pixels$^{-1}$, and the amplitudes are normalized with respect to the amplitude of C1.
We can see a number of experimental challenges in Fig. 9(a) including speckle from the laser illumination and spatially nonuniform illumination. Because of the Fourier-nature of the processing these issues affect the SNR of the reconstruction, but not generally the accuracy. However, there are areas where the bright pixels are saturated. In these regions the linear systems assumptions of the $Q$-matrix formalism break down. However, the analysis that we have presented shows that the information content in the various channels is consistent with the underlying theory. We aim to improve the imaging system through several upgrades including updating the SLM to reduce the pixel masking and increase the number of PSG pixels and replacing the laser source with an LED to reduce speckle effects and provide a less collimated beam. In spite of these challenges, we can see that the basic concept is illustrated. Examining the Fourier transform of the data shows that information is present in all nine channels, as predicted in the simulations above. Furthermore, because the two polarizers are rotated by ${45}^{\circ }$ with respect to each other, we are able to see the spreading of the scene’s frequency data in multiple directions.
The experimental challenges are significant, but the proposed geometry provides a path to have an adaptive, snapshot Mueller matrix measurement capability. In our recent work [22], we demonstrated that artificial intelligence/machine learning (AI/ML) methods can be used to improve the filters used to extract the channels in a passive Stokes polarimeter. We are currently working to extend those methods to the present architecture, and we hope to also exploit AI/ML to predict optimal PSG illumination patterns for any particular scene. Ultimately, we intend to see if the combination of AI/ML-based PSG selection and image reconstruction can be used to step away from the Fourier paradigm completely, allowing for a fully spatially adaptive channeled system that can change in time as the scene being observed evolves.
4. Discussion and conclusions
Channeled polarimeters create a number of options for using the available resolution of the imaging system in order to place the sidebands that carry the polarization information. In recent years, several groups have been exploring ways of locating the sidebands that minimize channel crosstalk in many different polarimeter configurations. Recent results have demonstrated how the particular spatial frequency distribution of an image can interact with the channel structure to produce better or worse results [35]. In this paper, we propose a system that uses a spatially adaptive PSG in order to tune the channel structure to the particular scene being imaged. We have demonstrated in simulation that the system can produce excellent results, and we have showed experimentally that the concept can be implemented. There remain many experimental challenges to achieve high quality imaging results, and we are working on overcoming these in our ongoing work.
The examples considered in this paper are particularly simple, as the PSG illumination is limited to one-dimensional sinusoids whose orientation can be changed to minimize channel crosstalk caused by the object’s spatial frequency structure. A simple next step in complexity would be to use different orientations of $\theta _g$ in different portions of the scene in order to optimally measure particular features. Flannery, et al., [27] demonstrated in simulation that this concept works. Extending that concept to arbitrary complexity would allow for completely adaptable PSG illumination patters to within the allowed complexity of the SLM.
An additional improvement in this system would result from the incorporation of more advanced filter design for channel extraction. It would be straightforward to replace the simple low-pass filters used here with ideal ratio filters using the method proposed by Song, et al., [22], since the PSA used here is identical to the DoFP polarimeter that they studied. Ultimately we are hopeful of combining these techniques by considering the PSG spatial modulation and the channel isolation filters as a single system that can be tuned to any particular polarization scene using machine learning or other methods. This is a main line of development that we are pursuing for future work.
The experimental system presented here is severely limited by practical performance parameters of the specific hardware we have used. Of particular note, the SLM has large guard bands that increase diffraction effects and the maximum modulation frequencies we were able to achieve were on the order of 1/32 cycles per pixel. We will be upgrading the SLM in future experiments to one that has much smaller guard bands and that hopefully will be able to be addressed at frequencies closer to the theoretical maximum of 1/2 cycle per pixel. A second major limitation of this experiment is the laser illumination, which has issues with speckle and intensity uniformity. In our future experiments we will replace the laser with a narrowband LED. We expect the introduction of an LED with spectral bandwidth will improve intensity uniformity and eliminate speckle at the cost of a reduction in fringe contrast [5].
Funding
Universidad Nacional Autónoma de México (IG100121); USAF Asian Office of Aerospace Research and Development (FA2396-15-1-4098).
Acknowledgements
Oscar G. Rodríguez-Herrera is grateful to UNAM for support through programs DGAPA-PAPIIT (IG100121) and DGAPA-PASPA. The hardware used for the experimental demonstration was purchased through US Air Force Asian Office of Aerospace Research & Development Project FA2386-15-1-4098. Joe Kurtz would like to thank Lowell Lamb of San Ramon, CA, USA, for his help at the start of what has been a long and varied career in science that comes to an end with this paper.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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