Spatially Addressable Polarimetric Calibration of Reflective-Type Spatial Light Modulator Using Mueller–Stokes Polarimetry

Mueller–Stokes polarimetry is emerging as a prominent noninvasive imaging technique to study the structural characteristics of an anisotropic medium. Spatial light modulator (SLM) is a programmable liquid crystal device (LCD), which is used to modulate the amplitude, phase, and polarization of light. The compact design and cumbrous manufacturing process of SLM requires its polarimetric calibration prior to its utilization for various applications. In this study, we experimentally demonstrate Mueller–Stokes imaging of a reflective-type SLM (Holoeye, LCR-720) to calibrate its polarization modulation characteristics with respect to its dynamic gray value range (0–255) at different spatial locations of SLM screen. Mueller matrices at 18 different gray values of SLM at an interval of 15, that is, at gray values 0, 15, 30, up to 255 have been experimentally measured using an improvised Mueller matrix imaging polarimeter (MMIP). Crucial polarimetric characteristics, that is, diattenuation, polarizance, state of polarization (SOP), depolarization, and retardance have been estimated with respect to the gray value range of SLM. Significant polarization modulation characteristics [diattenuation (0.08–0.3), polarizance (0.02–0.2), and retardance (0 to π )] have been determined for the SLM. These results indicate that the SLM exhibits spatially variable depolarizing nature and hence it is not perfectly homogeneous in structure. Therefore, it is expected that the outcomes of this study would be helpful for exploring the applicability of Mueller–Stokes polarimetry in advancement of LC technology.


INTRODUCTION
Spatial light modulator (SLM) is a programmable optical device, which is used to modulate the amplitude and phase of incident light. SLMs are polarization-sensitive devices and these can modulate SOP of light. Having optimized modulation characteristics, SLMs are the most promising dynamic optical elements in modern imaging applications such as beam shaping [1], digital holography [2], phase-shifting interferometry [3], and biophysics [4]. Most of the SLMs are composed of a systematic alignment of liquid crystal (LC) cells in specific patterns [5]. It permits SLM to utilize the ability of LC cells to align themselves with respect to applied voltage, that is, gray values of SLM. Technically, the required modulation characteristics of light originate from the relative rotation of liquid-crystal (LC) cells about their optical axis within the inner structure of SLM. SLMs are homogeneous optical devices but in practice, most of the commercially available SLMs are not perfectly homogeneous in structure [6,7]. In fact, the curvature of the cover glass and silicon backplane polishing of SLM may cause a significant nonuniformity in SLM display. Previously reported studies point out that this nonhomogeneity in SLM display give rise to a higher light modulation capability at the edges than at the center of SLM display [6]. In addition to limited fill factor, the existence of nonactive area, optical efficiency, and refresh rate of SLM display are other crucial parameters that are responsible for the experimentally observable nonuniformity of SLM [7]. It is noteworthy that this spatial nonuniformity may yield a distorted wave front and irregular modulation characteristics at different spatial parts of SLM display. With limited image array portion of SLM display, it is possible that these factors can generate discrepancies in optimized light modulation characteristics of SLM, especially for the applications where a broader laser beam is required. Therefore, an initial calibration at various spatial locations of the SLM display is required before utilizing it for imaging applications.
Numerous techniques on SLM characterization have been reported in the recent past [5,[8][9][10][11][12][13][14]. These techniques can be divided into two main categories, that is, phase characterization techniques [1,3,9,10,15,16] and amplitude and intensity characterization techniques [12,14,15,[17][18][19]. SLMs are most effective for their amplitude and phase modulation characteristics and hence most of the calibration methods revolve around phase characterization of SLM. Jones matrix imaging [8,11,15], fringe shifting interferometry [3], and phase mask-enabled digital holography [10] are well-known techniques to calibrate the phase modulation characteristics of SLM. However, these techniques are not capable to characterize the intensity modulation and other crucial polarimetric characteristics, that is, diattenuation, polarizance, retardance, and SOP of light after passing through SLM. In addition, intensity and polarization modulation are some of the salient features of SLMs. SLMs are capable of optimizing the polarization characteristics of light as a function of its different gray values. However, the existence of spatial nonuniformity is a major barrier against optimized polarization modulation produced due to SLM.
In order to characterize intensity-based polarization characterization of a sample, several techniques are available, that is, Mueller matrix imaging (MMI) [19,20], Stokes polarimetry [21], and Poincare sphere [22,23]. Mueller matrix imaging (MMI) is the most commonly used polarimetric technique in order to characterize the full polarimetric characteristics of a depolarizing medium. MMI is widely used for tissue characterization in biomedical sciences [24,25], material characterization [26], adulteration detection in samples, [20]etc. In context of SLM characterization using MMI, few studies have been also reported in past [14,15,17,18]. In these studies, SLM is considered as a perfectly homogeneous medium. Recently, Dev et al. reported a study for Mueller-Stokes polarimetric characterization of transmissivetype SLM and calibrated the SLM, considering it as a perfectly homogeneous medium [14]. They captured 36 intensity images for each of 18 gray values of SLM (total 648 intensity images) to determine Mueller matrices for SLM. Further, this study does not depict the polarization modulation at different spatial parts of the SLM display. Apart from this, a study has been reported which focuses on the spatial calibration of SLM display while keeping in mind the manufacturing artifacts of SLM display [7]. In this framework, a pixel-wise phase calibration technique for SLM has been introduced [7]. To the best of the author's knowledge, no study has been reported on MMI-based spatial characterization of a reflective-type LC-SLM till date.
In this work, we experimentally demonstrate a spatially addressable polarimetric characterization of a reflective-type LC-SLM (Holoeye, LC-R720) using the Mueller-Stokes polarimetric technique. Conventional MMI techniques require 49 intensity images [27] or 36 intensity images [19] at different SOPs of light. These techniques might be suitable for the characterization of samples where only one set of measurement is sufficient, but these are time-consuming for the cases where one needs to take multiple sets of intensity images for detailed characterization of a sample. A comprehensive polarimetric study of SLM requires multiple sets of Mueller intensity images. Therefore, these conventional techniques are somewhere not time-efficient for SLM characterization. In our study, we are able to retrieve Mueller matrices by capturing only 16 intensity images. Hence, this study provides a better time-efficient approach in order to determine Mueller matrices of a sample (SLM) than earlier studies. Mueller matrices of 17 different gray values of SLM (interval of 15) are retrieved by capturing just 288 intensity images. The obtained Mueller images are then analyzed at different spatial locations of SLM display. Further, Lu-Chipman polar decomposition method [28] is applied to obtain polarimetric properties from obtained Mueller matrices. In addition, SOP modulation produced by SLM has also been evaluated and represented in the Poincare sphere.

Mueller-Stokes Polarimetry
Mueller matrix of a sample is a transformation matrix of order 4 × 4, which transforms the Stokes parameters (SPs) of incident light (S in ) into the SPs of emergent light (S ' out ). Theoretically, Mueller matrix is given as follows: Experimentally, Mueller matrix of a sample is calculated by using eq. 3, that is, by measuring intensity images at various combinations of six SOPs of light, that is, horizontally polarized (H), vertically polarized (V), +45°polarized (P), −45°polarized (M), right circularly polarized (R), and left circularly polarized (L), respectively.
The SOP rule for polarimetry (eq. 4) transforms eq. 3 into the following: Equation 5 enables us to determine Mueller matrix of a sample by capturing just 16 images. The Stokes vectors of emergent light are determined by using eq. 1, that is, multiplying obtained Mueller matrices with Stokes vectors corresponding to six different incident SOPs of light [14,29]. The SOP modulation produced by the SLM can be determined by tracing the exiting normalized Stokes parameters (S 1 , S 2 , and S 3 ) in the Poincare sphere [19].

Polar Decomposition of Mueller Matrices
The polarization properties of sample, that is, diattenuation (D) and polarizance (P) can be directly determined from its Mueller matrix as follows: The difference in these two polarization properties indicates the depolarizing nature of sample. In order to diagnose further polarization properties, that is, depolarization (Δ), retardance (R), and SOP modulation of a sample, its Mueller matrix can be decomposed into three polarization matrices, that is, depolarization matrix (M Δ ), retardation matrix (M R ), and diattenuation matrix (M D ) using Lu-Chipman polar decomposition algorithm [28].  Figure 1 illustrates the schematic of MMIP, which is used for the experimental demonstration of Mueller matrix imaging of a reflective-type LC-SLM (Holoeye, LC-R720). An unpolarized light beam of wavelength 532 nm coming out from a greendiode laser is spatially filtered (SF) and collimated by a convex lens of focal length 20 cm. It is then allowed to pass through an MMIP, which consists of the polarization components, that is, linear polarizers (P 1 and P 2 ), quarter-wave plates (Q 1 and Q 2 ), a half-wave plate (HWP), and a beam splitter (BS). The technical specifications of the sample (SLM) are summarized in Table 1.
Theoretically, MMIP has two arms, that is, the polarization state generator (PSG) and polarization state analyzer (PSA). Four SOPs (H, V, P, and R) are generated in both PSG and PSA arms with the help of polarization components, and corresponding 16 intensity images (different combinations of SOPs) are recorded in the CCD camera (Procilica GX 2750, 2,752 × 2,200 pixels and pixel size of 4.54 μm), placed at the image plane. These images are recorded at a constant frame rate of 10 frames per second (fps) with a fixed exposure time of 0.015 s. A total of 288 intensity images have been recorded for 18 gray values of SLM at the interval of 15 (0, 15, 30, up to 255), respectively, at standard room temperature. Corresponding 18 Mueller matrices have been retrieved from these intensity images using eq. 5.
To allocate spatial dependency of polarization modulation produced by SLM display, we have selected five-pixel values from

RESULTS AND DISCUSSION
Mueller matrix elements as a function of gray values at selected locations of SLM screen are shown in Figure 3.   Figure 4 provides a more heuristic insight of the variation of Mueller matrix images at three gray values (0, 128, and 255) of SLM at the center ( Figures  4A,B,C) and edge ( Figures 4D,E,F). On comparing Mueller matrix images at center with Mueller matrix images at the edge of the SLM screen, a noticeable change in Mueller elements is observed. It implies that the SLM yields different polarization response at its different spatial locations. It validates the existence of spatial nonuniformity of polarization modulation for SLM with respect to its gray values. Mueller matrices work as the footprints for polarization properties of the sample (SLM). Therefore, variation of few crucial polarization properties, that is, diattenuation, polarizance, depolarization, retardance, and degree of polarization has been studied by using polar decomposition of obtained Mueller matrices of SLM and represented in Figure 5.
On comparing Figures 5A,B, we have observed a slight difference in the variation of diattenuation and polarizance of SLM. It implies that the SLM exhibits a significant amount of depolarizing nature. Moreover, Figure 5C exhibits the existence of depolarization of incident light depending on various gray values of SLM. On the other hand, a phase retardance (0 to π) is observed for the SLM. However, minor fluctuations are also observed when considering the spatial configuration of the SLM display. The obtained depolarization and retardance values for the SLM consist of both linear and circular components of light. In addition, SOP modulation characteristics from measured Mueller matrices of SLM have been determined using Stokes polarimetry. Figure 6 represents the Poincare sphere representation of SOP modulation trajectories for SLM within the center part ( Figures 6A-C) and at the left edge ( Figures 6D-F) of SLM display for six SOPs of incident light, respectively. A remarkable SOP modulation is observed with increasing gray values of SLM. On further observing Figures 6A-C, it is evident that SOP modulation trajectories corresponding to incident SOPs "H-polarized," "45°polarized," and right circularly polarized light (trajectories with red dots) are exactly the mirror image of the SOP modulation trajectories corresponding to incident SOPs "vertically polarized," "135°p olarized," and left circularly polarized light (trajectories with blue dots), respectively. On the other hand, Figures 6D-F illustrates the SOP modulation trajectories at edge of SLM

CONCLUSION
We have proposed and experimentally demonstrated spatially addressable polarimetric calibration of a reflective-type twisted nematic LC-SLM (Holoeye, LC-R 720) using Mueller-Stokes polarimetric imaging. Mueller matrices at 18 different gray values of SLM at the interval of 15, that is, at gray values 0, 15, 30, up to 255 has been experimentally measured. The obtained Mueller images are analyzed at different spatial locations of SLM display. Crucial polarimetric characteristics, that is, diattenuation, polarizance, SOP modulation, depolarization, and retardance have been estimated with respect to the dynamic gray value range of the SLM. A significant variation polarization modulation characteristics [diattenuation (0.08-0.3), polarizance (0.02-0.2), and retardance (0 to π)] has been determined for the SLM with respect to different gray values of SLM. It is evident from the results that the SLM is not perfectly homogeneous in structure. Apparently, it shows significant depolarizing characteristics at various spatial locations of its display.
The obtained results indicate that the Mueller matrices of the SLM exhibit the spatial fluctuations especially at the edges of SLM display. The possible reason of this fluctuation would be the existence of dominant diffractive artifacts at the edges than the central part of the SLM display. Although SOP modulation is not altered at various spatial locations of SLM display, it exhibits spatial variation in DOP of incident light. It is due to the fact that LC molecules are arranged between two glass substrates at twisted nematic (TN) configuration in the SLM, which allows it to modulate both characteristics, that is, the amplitude and phase of incident light. Further results imply that SOP modulation characteristics of SLM yield significant changes with respect to its gray values. The change in gray values of SLM constraints the relative rotation of LC molecules between the glass substrates, and it yields the variation of anisotropic properties with respect to gray values of SLM.
In addition, Mueller matrices are intensity-based measurements, whereas SOP modulation includes phase modulation characteristics of SLM as well. The reason for spatial change in Mueller matrices is that the amplitude modulating characteristics varies at different spatial locations of SLM. However, no spatial change in SOP modulation indicates that phase modulation characteristics remains unchanged with respect to spatial locations of SLM display. Therefore, the SLM shows a significant change in Mueller matrices, whereas no spatial fluctuations in SOP are observed. In brief, this study depicts the need for spatial polarimetric calibration of SLM display and point out the discrepancies which may arise due to spatial nonuniformity of SLM display. The obtained results recommend that the light beam should be adjusted over only the central part of SLM display for the majority of applications. In addition, the compensation techniques, that is, computational algorithms and phase mask-enabled techniques can be developed to encounter the nonuniformity of the SLM display in edges to enhance the validity of results, where a large beam size is required. We have calibrated the spatial fluctuations in the polarization modulation (Mueller matrices) characteristics of SLM display for a monochromatic light source. However, a wavelengthdependent Mueller metrics characterization of LC-SLM can be performed in future. On the other hand, the image acquisition time (15 ms) may give rise to minor spatial fluctuations due to flicker effects in CCD acquisition as well. Therefore, it is recommended to minimize the flicker effect, while performing similar studies in future. It is expected that the outcomes of this study would be beneficial for various applications of SLM, especially where intensity modulation is a crucial aspect.

DATA AVAILABILITY STATEMENT
The raw data supporting the conclusion of this article will be made available by the authors, without undue reservation.

AUTHOR CONTRIBUTIONS
VT and NB designed the study. YP has contributed in performing experiment. NB has supervised the study. All authors contributed in article and approved the submitted version.