Spatial Light Modulators and Their Applications in Polarization Holography

Abstract

Liquid crystal spatial light modulators (LC-SLMs) have gained substantial interest of the research fraternity due to their remarkable light modulation characteristics in modern imaging applications. Replacing the conventional optical elements from the SLM-based computer-generated holograms (CGHs) is a trending approach in modern digital holographic applications due to the optimized phase shift depending on the phase modulation features of SLMs. Apparently; SLMs serve a crucial role in the experimental implementation of digital holographic techniques. However, the resolution of the CGHs are sometimes limited by the structural discrepancies (fill factor, spatial anomalies, refresh rate, etc.) of SLM. Therefore, it is recommended to calibrate the modulation characteristics of SLMs prior to their implementation for imaging applications. This chapter provides comprehensive literature (review) of the LC-SLMs along with their major calibration methods. In addition, recent interesting applications of LC-SLMs have been discussed thoroughly within the framework of polarization holography.

1. Introduction

Spatial light modulators (SLMs) are electro-optical devices, pertaining to manipulating the fundamental characteristics, viz., amplitude, phase, and polarization state of light. SLMs have gained significant attention of the research fraternity due to their versatile applicability in various optical imaging applications, including digital holography [1, 2], adaptive optics [3], computational imaging [4, 5, 6], holographic display [7]. In fact, SLM driven CGHs have the potential to eliminate the undesired artifacts (low resolution, noise, etc.) in holographic applications resulting in dynamic modulation response and high resolution.

The structure of SLM is primarily based on either the optical micro-electro-mechanical system (MEMS) [8] or liquid crystal on silicon (LCOS) technology [9, 10]. Digital micrometer devices (DMDs) are commonly used MEMS-based SLMs, which consist of pixel-dependent micro-mirrors. The rapid switching between fixed states of micro-mirrors makes DMDs suitable for amplitude modulation of light with a high frame rate (up to 40 kHz) [9]. However, corresponding amplitude modulation is binary (depends on two fixed states of micro-mirrors) and therefore DMDs are not sufficient for customized phase and polarization modulation of light. On the other hand, LCOS SLMs have the potential to modulate the amplitude, phase, and polarization of light due to the electrically controlled birefringence (ECB) properties of liquid crystals (LCs). The desired phase shift can be obtained by addressing different grayscales (pixel voltages) of the SLM. In practice, the phase response of the SLM can vary from the ideal grayscale-phase response [provided in look-up-table (LUT) of SLM model] due to various exogenous factors, viz., manufacturing defects [11, 12], non-linear optical response of LC, spatial anomalies [10, 13, 14], etc. Therefore, calibration of LC-SLM for grayscale-phase is recommended for its precise utilization in various holographic applications.

Digital holography is a conventional optical technique that enables the simultaneous recording of amplitude and phase of light (encoded in hologram) using the complex electric field modulation of light [15]. However, polarization is another crucial degree of freedom of light along with its amplitude and phase. The inclusion of polarization encoding in conventional digital holograms paves the route for the development of polarization holography [16]. Conventional polarization holography requires polarization-sensitive holograms (PSH) for reconstructing polarization multiplexed holographic images. In general, these PSHs are encoded with the help of polarization-dependent birefringent materials [17]. However, recent developments in polarization holography use LC-SLM for encoding the polarization-sensitive holograms [18, 19, 20, 21]. In addition, LC-SLM serves key role in the experimental demonstration of advanced digital holographic techniques, viz. coded aperture correlation holography (COACH) [22, 23, 24], Fresnel incoherent correlation holography (FINCH) [2, 24], Stokes correlation holography [21, 25, 26], polarization holography [18, 19, 27, 28, 29], etc. This chapter presents comprehensive literature (review) on the LC-SLM and its applicability in digital holography. Subsequent sections of the chapter illustrate the construction, working principle, and the major calibration methods for LC-SLM, followed by some crucial applications of LC-SLMs in the context of polarization holography.

2. Liquid crystal on silicon (LCOS) spatial light modulator

2.1 Structure and working principle of LCOS SLM

LCOS-based SLMs are programmed such that the orientation of LC molecules can be changed depending on their applied pixel voltage (gray scales). The use of LC materials in SLMs depends on their optical and electrical anisotropic properties. Typically, a thin layer of LC material can be described as a birefringent material with two refractive indices. The orientation of the index ellipsoid depends on the direction of the director of LC molecules. Ferroelectric and nematic alignments are the most commonly deployed LC mesophases for SLM fabrication [30]. The ferroelectric LC SLMs exhibit a faster refresh rate but suffer from the zeroth order diffraction artifact and are limited up to binary phase modulation only [31]. However, the nematic LC SLMs can provide customized multi-grayscale phase modulation. Further, the commonly used nematic LC alignments in SLMs are parallel aligned nematic (PAN) or homogeneous, vertically aligned nematic (VAN) or homeotropic cells. Figure 1 represents the schematic of the LC alignments. In homogeneous and homeotropic configuration, the alignment layers are parallel to each other, so the LC molecules have the same orientation and are generally used to obtain amplitude modulation of light. On the other hand, in the twisted nematic liquid crystal (TNLC) alignment, the orientation of the molecules can be adjusted such that the uppermost and bottom layers are differed by a fixed angle (0°, 45°, 90°).

Figure 1.

Schematic of liquid crystal alignments in LC-SLM (a) Homogeneous, (b) homeotropic, and (c) tilted.

SLMs can be used either in the transmission or reflection arm of an optical system, depending upon their structure and specifications. A schematic of the fundamental structure of the LC-SLM is shown in Figure 2. The inner structure of SLMs is comprised of a typical alignment of LC molecules (nematic phase), sandwiched between transparent electrodes and glass substrates. The active matrix circuit is formed on the silicon substrate to control each pixel electrode by the applied electrical potential individually using semiconductor technology. The pixel electrodes are typically made of reflective aluminum mirrors with pixel circuitry, which contains gate lines and buried transistors for a maximum fill factor with high reflectivity. Two alignment layers with rubbing direction could be used to initialize the orientation of the modulated LC molecules. The transparent electrode is deployed to work with the pixel electrode for generating the electric field across the LC cell.

Figure 2.

Schematic of the structure of LC-SLM.

The typical alignment of LC molecules of SLMs allows the change only along their director axis while the other orthogonal direction remains invariant. It results in optical anisotropy (phase retardation) due to SLM and is responsible for the optimized phase shift at any location of the SLM display. The optimized phase shift for each pixel of the SLM is programmed as a function of 256 gray scales ranging from 0 to 255. Therefore the typical anisotropic properties of SLM (phase shift and birefringence) can also be interpreted as a function of its gray scales.

2.2 Electro-optic effect in LC-SLM

LC-SLM can exhibit anisotropic nature due to the relative phase delay (birefringence) for the orthogonal polarization components of its LC array. Let us consider the SLM is designed to modulate the phase of light along the vertically polarized (y-polarized) light. The phase delay experienced by the SLM is given as [32]

$${\phi }_x\left(x,y\right)=\frac{4\pi n_o}{\lambda }.d$$E1

$${\phi }_y\left(x,y\right)=\frac{4\pi n_e}{\lambda }.d$$E2

Here, $n_o\mathit{\text{and}}{n}_e$ are the refractive indices for the slow and fast axis of the LC-SLM respectively and d is the thickness of the LC cell. The relative rotation of LC molecules of SLM for applied pixel voltage (grayscale) is shown in Figure 3. The LC molecules of SLM exhibit the tendency to align themselves along the direction of an external applied electric field. This effect is known as electrically controlled birefringence (ECB).

Figure 3.

Electo-optic effect in TNLC-SLM.

When pixel voltage (grayscale) is applied to the SLM, it provides an additional phase to the y-polarized light, whereas x-component of light remains unaffected.

The resultant phase of the x-polarized component of light is

$${\delta }_x\left(x,y\right)={\phi }_x\left(x,y\right)$$E3

The resultant phase of the y-polarized component of light is

$${\delta }_y\left(x,y\right)={\phi }_y\left(x,y\right)+\gamma \left(x,y\right)$$E4

Here, $\gamma \left(x,y\right)$ is the additional phase due to the applied grayscale of the SLM.

The retardance of the SLM can be defined as

$$R\left(x,y\right)={\delta }_y\left(x,y\right)-{\delta }_x\left(x,y\right)=\left(1-\frac{n_o}{n_e}\right){\phi }_y\left(x,y\right)+\gamma \left(x,y\right)$$E5

2.3 Light modulation characteristics of LC-SLM

It is already discussed that LC-SLM is known for customized light modulation characteristics in terms of its amplitude, phase, and polarization with respect to their grayscale values. The amplitude modulation can be obtained by illuminating twisted PAN or VAN-based SLMs with linearly polarized light. It should be transmitted through a polarizer with orthogonal direction with respect to the initial polarization of light. The level of attenuation by the second polarizer can be tuned by applying an electric field to the cell, which leads to a change of the birefringence β [9]. The phase-only modulation of light can be conveniently achieved using PAN LC cell-based SLMs. The reason is that PAN LC cells are most responsive to the ECB effect, i.e., light passing through the extraordinary axis of PAN cells exhibits phase retardation as a function of voltage-controlled birefringence (gray scales). On the other hand, TNLC SLMs are very complicated to handle for phase-only modulation as there exists amplitude modulation as well in that case [33, 34]. Therefore, TNLC-SLMs exhibit complex field (amplitude and phase) modulation or phase mostly modulation of light.

Polarization modulation is another crucial feature of LC-SLMs, which lead to very interesting applications of LC-SLMs in polarization holography. It is evident that when SLM is placed between two quarter-wave plates (QWPs) with their fast axis rotated by +45° and −45° with respect to the incident light beam. The Jones matrix of this combined optical configuration is typically a rotation matrix, which represents the phase shift of SLM. Therefore, it is possible to convert a phase-only SLM into a pixel-dependent phase-shifting device by inserting it between two QWPs. For example, a linearly polarized light can be converted into azimuthally polarized light by exposing SLM with a vortex structure under polarization modulation configuration.

3. Characterization of LC-SLM

SLMs are pixelated programmable devices. Exogenous parameters like pixel size, and fill factor are too crucial to investigate in order to obtain the desired phase response of LC-SLM. For instance, smaller pixel size leads to better resolution and larger diffraction angles whereas a higher fill factor is preferred as it can eliminate the zeroth order diffraction issue in SLM projection. However, the said issue cannot be perfectly eliminated due to pixel crosstalk in SLM. Moreover, other factors (frame rate, power supply, incident angle, etc) can also perturb the ideal modulation characteristics of LC-SLM.

The effect of LC cells on a coherent light source with a fixed polarization state can be described by using Jones calculus [35]. In particular, the modulation characteristics of light due to TNLC cells can be represented in terms of its Jones matrix as [36]

$$J_{\mathit{\text{TNLC}}}=exp\left(\mathit{i\phi }\right)\left(\begin{array}{cc}a-\mathit{ib}& c-\mathit{id}\\ -c-\mathit{id}& a+\mathit{ib}\end{array}\right)$$E6

Where, a, b, c, and d are known as anisotropic coefficients, which typically depend on the LC physical parameters (twist angle, director orientation angle, etc.). These physical parameters are usually not provided by manufacturers of LC-SLM and therefore it is very challenging to determine precise modulation features of TNLC-SLM.in a straightforward manner. However, the Jones matrix of LC-SLM can be measured using optical characterization techniques (interferometry, digital holography, cryptography, etc).

3.1 LC-SLM calibration methods

3.1.1 Interferometric methods

The phase modulation characteristics of LC-SLM can be calibrated using interferometric measurements [14, 27, 37, 38, 39, 40, 41, 42]. The interferometry enables to record the intensity and phase of light simultaneously in the form of fringes (interference pattern). Interferometry relies on the principle of the reference beam (unperturbed light beam) and object beam (beam with phase shifting), where the phase difference between these two beams results in the fringe-shift depending upon the provided phase to the object beam. In the context of phase calibration of LC-SLM, the interference pattern is obtained from the superposition of the SLM modulated beam (object beam) and the reference beam. The fringe shift is obtained by changing the gray scales on SLM and the phase modulation of SLM can be retrieved from the relative displacement of fringes. However, a good contrast is required between the reference beam and object beam in order to obtain sharp fringes. Therefore, generating the reference arm of the interferometer is a critical factor to be optimized in these calibration methods. In particular, the reference beam can be generated either from the LC-SLM itself (self-reference) or by splitting the wavefront into two parts before interacting with SLM (out-reference). Figure 4 depicts the schematic of the self-referenced interferometric method and reference calibration using a splitted SLM screen scheme [43]. In this technique, two different phase masks were imported into LC-SLM [PLUTO-VIS, Holoeye] such that half of the screen was addressed with uniform varying gray values while the other half portion was exposed with vertical binary diffraction grating (Figure 4b).

Figure 4.

(a) Schematic of self-referenced interferometric technique for phase calibration of reflective LC-SLM [PLUTO-VIS, Holoeye] calibration, and (b) imported pattern on LC-SLM (left) and recorded fringe pattern (right). [OBJ: Microscopic objective, L: Lens, P: Polarizer] (adapted from [43]).

In a self-referenced method, the active area of SLM is divided into two parts. In one part different grayscales are provided (modulating part) meanwhile a fixed gray scale ‘0’ is provided to the other half portion of the SLM screen (reference part) [27]. The beam passing through the modulating part undergoes a phase-shift depending on the addressed gray values on the SLM pixels. Hence, the phase-shift due to SLM modulation can be calibrated by investigating the fringe displacement (deformation) of the recorded interference pattern. Although the self-referenced interferometric calibration methods are compact and stable but these methods are not compatible to determine the spatially varying (pixel addressable) modulation characteristics of LC-SLM.

On the other hand, an external reference beam (independent of LC-SLM) is used in the off-axis-referenced method. The simplest experimental configuration of these types of methods is the Twyman-Green interferometer, where the light beam is splitted into two orthogonal directions with the help of a beam-splitter [41]. In one arm, a plane-mirror is inserted whereas the SLM is placed on the other arm of the interferometer. The interference pattern formed due to the superposition of these two light beams is recorded at the camera plane (Figure 5). Although off-axis-referenced calibration methods require complicated experimental set-ups but these methods are very useful for precise measurements of spatial uniformity in SLM.

Figure 5.

Schematic of off-axis-referenced Twyman-green interferometer for phase calibration of LC-SLM [HWP: Half wave plate, MO1: Micro-objective, L1: Lens, P: Polarizer, BS: Beam splitter, M: Mirror, A: Analyzer, CCD: Charge-coupled device] (adapted from [44]).

3.1.2 Polarization-sensitive digital holography

SLMs are polarization-sensitive devices and therefore the polarization properties of light are expected to be modulated after interacting with an LC-SLM. It is evident that the polarization properties [diattenuation, degree of polarization (DOP), retardance, etc.] cannot be determined using conventional interferometric methods. Therefore, polarization-sensitive characterization methods are developed to account for the polarimetric calibration of SLM. In this context, spatially resolved Jones matrix imaging has the potential to extract the phase information of LC-SLM from the complex electric field of light.

The modern SLM calibration techniques are typically based on Polarimetric methods accompanied by digital holography [45, 46]. Moreover, the consolidation of digital holography with polarimetry has paved the convenient approach to SLM calibration. Figure 6 illustrates the polarization-sensitive digital holography scheme for Jones matrix imaging of a reflective type LC-SLM (Holoeye, LC-R 720). The proposed experimental technique is equipped with a combination of Mach-Zehnder interferometer and Sagnac interferometer, which provide angular multiplexing to recover orthogonal polarization components. Moreover, the triangular Sagnac interferometer in the reference arm of the experimental setup is capable to tune the carrier frequency of reference orthogonal components according to object beam requirements.

Figure 6.

(a) Schematic of polarization-sensitive digital holography for reflective LC-SLM (LC-R 720, holoeye), and (b) recorded fringe pattern and corresponding Fourier spectrum. [HWP: Half-wave plate, L: Lens, BS: Beam splitter, PBS: Polarization beam splitter, CCD: Charge coupled device, BD: Beam dump, M: Mirrors] (adapted from [32]).

Using the proposed experimental set-up, different interference patterns have been recorded at various grayscales of LC-SLM in two shots for input states of polarization (SOPs) +45° and −45° respectively. The phase shift can be measured by observing fringe-shift for the recorded interference patterns. For instance, Figure 7 illustrates the recorded fringe shifts for gray values 0, 180, and 255 for input SOPs +45° (a-c) and −45° (d–f) respectively. The observed fringe shift with respect to the gray values indicates the SOP modulating characteristics of the LC-SLM.

Figure 7.

Recorded interference patterns for +45° polarized incident light (a-c), for −45° polarized incident light (d–f) at grayscales 5 (a, d), 180 (b, e), and 255 (c, f) of LC-SLM respectively (adapted from [32]).

Mathematically, the interference pattern for input +45° polarized ($I_p\left(x,y\right)$) and −45° polarized ($I_m\left(x,y\right)$) input light beam can be given as

$$I_p\left(x,y\right)={\left|R_{\mathit{px}}\left(x,y\right)+R_{\mathit{py}}\left(x,y\right)+J{E}_p\left(x,y\right)\right|}^2$$E7

$$I_m\left(x,y\right)={\left|R_{\mathit{mx}}\left(x,y\right)+R_{\mathit{my}}\left(x,y\right)+J{E}_m\left(x,y\right)\right|}^2$$E8

Here, $E_p\left(x,y\right),E_m\left(x,y\right)$ denotes the modulated electric field, $R_{\mathit{px}}\left(x,y\right),R_{\mathit{py}}\left(x,y\right)$ and $R_{\mathit{mx}}\left(x,y\right),R_{\mathit{my}}\left(x,y\right)$ denotes the reference beams for x and y components for input light field corresponding to +45° and −45° polarized input light beam respectively. J is the Jones matrix of the LC-SLM.

Jones matrix calculus establishes a connection between the electric field components of emerging light from the object ($E_x^{\prime },E_y^{\prime }$) and electric field components of input light beam ($E_x,E_y$) as

$$\left(\begin{array}{c}{E}_x^{\prime }\\ E_y^{\prime }\end{array}\right)=\left(\begin{array}{cc}{J}_{\mathit{xx}}& J_{\mathit{xy}}\\ J_{\mathit{yx}}& J_{\mathit{yy}}\end{array}\right)\left(\begin{array}{c}{E}_x\\ E_y\end{array}\right)$$E9

Here, $J=\left(\begin{array}{cc}{J}_{\mathit{xx}}& J_{\mathit{xy}}\\ J_{\mathit{yx}}& J_{\mathit{yy}}\end{array}\right)$ is the Jones matrix of the object.

Taking reference calibration of SLM display into account, the electric field components of the modulated light can be defined as [27]

$$E_{\mathit{\text{calibrated}}}^{\prime }=\frac{E^{\prime }\mathit{\text{modulated}}}{E^{\prime }\mathit{\text{reference}}}$$E10

Here,$E_{\mathit{\text{modulated}}}^{\prime }$ and $E_{\mathit{\text{reference}}}^{\prime }$ is the electric field component corresponding to modulating part and reference part (grayscale ‘0’) of the SLM display respectively.

For an illuminating object with +45° and −45° polarized input light, Eq. (9) turns into

$$\left(\begin{array}{c}{E}_{\mathit{px}}^{\prime }\\ E_{\mathit{py}}^{\prime }\end{array}\right)=\left(\begin{array}{cc}{J}_{\mathit{xx}}& J_{\mathit{xy}}\\ J_{\mathit{yx}}& J_{\mathit{yy}}\end{array}\right)\left(\begin{array}{c}1\\ 1\end{array}\right)$$E11

$$\left(\begin{array}{c}{E}_{\mathit{mx}}^{\prime }\\ E_{\mathit{my}}^{\prime }\end{array}\right)=\left(\begin{array}{cc}{J}_{\mathit{xx}}& J_{\mathit{xy}}\\ J_{\mathit{yx}}& J_{\mathit{yy}}\end{array}\right)\left(\begin{array}{c}1\\ -1\end{array}\right)$$E12

The Jones matrix of the object from the modulated field components can be retrieved as

$$\left.\begin{array}{c}{J}_{\mathit{xx}}=\frac{1}{2}\left[E_{\mathit{px}}+E_{\mathit{mx}}\right]\\ J_{\mathit{xy}}=\frac{1}{2}\left[E_{\mathit{px}}-E_{\mathit{mx}}\right]\\ J_{\mathit{yx}}=\frac{1}{2}\left[E_{\mathit{py}}+E_{\mathit{my}}\right]\\ J_{\mathit{yy}}=\frac{1}{2}\left[E_{\mathit{py}}-E_{\mathit{my}}\right]\\ \end{array}\right\}$$E13

The measured Jones matrix of the LC-SLM with respect to the gray values of the LC-SLM is depicted in Figure 8. The Jones matrix elements contain the vital information of the SLM, including the amplitude, phase, and anisotropic properties (birefringence, dichroism, retardance, etc.), which can be easily decomposed using the Jones decomposition methods [32, 47]. However, Jones matrix cannot provide the polarization information of the medium, containing depolarization. In this case, other polarimetric methods (Mueller matrix imaging, etc.) can be used for polarimetric calibration of LC-SLM [48, 49].

Figure 8.

Jones matrix components as a function of gray scales of the SLM. Amplitude (a–d) and phase (e–h) of Jones matrix components respectively (adapted from [32]).

3.1.3 Determination of spatial anomalies in LC-SLM using Mueller-Stokes polarimetry

The inner structure of SLMs is comprised of a systematic alignment of Liquid crystal (LC) cells in specific patterns. It permits SLM to utilize the ability of LC cells to align themselves with respect to applied voltage i.e. 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. In general, SLMs are considered as homogenous in structure but they are not perfectly homogeneous in practice due to manufacturing discrepancies [10, 50]. The curvature of cover glass and silicon backplane polishing of SLM may also cause a significant non-uniformity in SLM display. This non-homogeneity in the SLM display gives rise to a higher light modulation capability at the edges than at the centre of the SLM display [51]. In addition, limited fill factor, the existence of non-active area, optical efficiency, and refresh rate of SLM display are other crucial parameters that are responsible for the experimentally observable non-uniformity of SLM [52]. It is noteworthy that this spatial non-uniformity may yield distorted wavefront and irregular modulation characteristics at different spatial parts of the SLM display. With a limited image array portion of the SLM display, it can be 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.

The spatial anomalies of LC-SLM can be calibrated using Mueller matrix imaging. Figure 9 illustrates the schematic of the improvised Mueller matrix imaging polarimeter (MMIP), which is used for rapid Mueller matrix measurement of a reflective type LC-SLM (Holoeye, LC-R720) with respect to its grayscales. A partially polarized light beam of wavelength 532 nm is emerging from a green-diode laser is spatially filtered (SF) and collimated by a collimating lens of the focal length of 20 cm. It is then allowed to pass through an MMIP, which consists of the polarization components i.e. linear polarizers (P₁ and P₂), quarter-wave plates (Q₁ and Q₂), a Half Wave Plate (HWP), and a Beam Splitter (BS). Theoretically, MMIP has two arms i.e. Polarization State Generator (PSG) and Polarization State Analyzer (PSA). Four SOPs (H, V, P, 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 a CCD camera (Procilica GT 2750, 2752 × 2200 pixels and pixel size of 4.54 μm), placed at the image plane. 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. Polarization properties (diattenuation, retardance, polarizance, and depolarization) have been retrieved from the measured Mueller matrices using the polar decomposition method of Mueller matrices [53].

Figure 9.

(a) Schematic of Mueller matrix imaging Polarimeter (MMIP) for Mueller matrix imaging of reflective LC-SLM (LC-R 720, holoeye), and (b) selection of spatial locations of SLM display for measurements of spatial anomalies (adapted from [12]).

To allocate spatial anomalies in polarization modulation produced due to SLM display, we have selected five spatial coordinates (pixel values) from different parts i.e. central part, right edge, left edge, upper part, and lower part of the light beam over SLM display (Figure 9b). Spatially addressable polarization properties are plotted for different grayscales of the LC-SLM and are represented in Figure 10. Interestingly, the determined polarization properties exhibit spatial fluctuations (particularly at the edges of the SLM display) and thus indicate the spatial non-uniformity in the LC-SLM.

Figure 10.

Polarization characteristics [diattenuation (a), polarizance (b), depolarization (c), and retardance (d)] of the LC-SLM as function of gray values of SLM at different spatial locations (adapted from [12]).

4. Applications of LC-SLMs in digital holography

4.1 Phase-shifting digital holography

SLMs are widely employed in phase-shifting digital holography (PSDH), where multiple digital holograms are imported with known phase shifts in the reference beam. In such cases, LC-SLM offers a convenient means to import CGHs with optimized phase shifts for the experimental implementation of PSDH.

The reference beams with four pre-determined phase-shifts $\left(0,\frac{\pi }{2},\pi ,\mathit{\text{and}}\frac{3\pi }{2}\right)$ can be mathematically represented as

$$H_0={\left|R\right|}^2+{\left|O\right|}^2+2\left|R\right|\left|O\right|cos\left({\phi }_O-{\phi }_R\right)$$E14

$$H_{\pi /2}={\left|R\right|}^2+{\left|O\right|}^2-2\left|R\right|\left|O\right|sin\left({\phi }_O-{\phi }_R\right)$$E15

$$H_{\pi }={\left|R\right|}^2+{\left|O\right|}^2-2\left|R\right|\left|O\right|cos\left({\phi }_O-{\phi }_R\right)$$E16

$$H_{3\pi /2}={\left|R\right|}^2+{\left|O\right|}^2+2\left|R\right|\left|O\right|sin\left({\phi }_O-{\phi }_R\right)$$E17

Here, ${\phi }_O$ and ${\phi }_R$ denote the 2-D phase functions corresponding to the object and reference beams respectively.

For a known reference phase [${\phi }_R\left(x,y\right)$], the object beam phase can be given as

$${\phi }_O\left(x,y\right)={\phi }_R\left(x,y\right)+\mathit{arctan}\left(\frac{H_{3\pi /2}-H_{\pi /2}}{H_0-H_{\pi }}\right)$$E18

Phase retrieval of a hologram (vortex of topological charge 1) using the PSDH technique is represented in Figure 11. Four digital phase shifts have been provided to a hologram computationally and the corresponding phase is retrieved from the phase-shifted holograms.

Figure 11.

Phase reconstruction of hologram (vortex, topological charge 1) using PSDH technique. (a–d) Holograms with phase shift 0, π/2, π, and 3π/2 respectively, and (e) reconstructed phase of a vortex (charge 1). [PSDH: Polarization sensitive digital holography].

4.2 Polarization holography

Polarization holography can be interpreted as an advanced version of conventional digital holography, which enables to simultaneous recording of the SOPs of light beam along with its amplitude, and phase. Polarization holography is an emerging domain in digital holography leveraging several interesting applications such as holographic storage technology [16], multichannel polarization multiplexing [54], vector beam generation [19, 27, 55, 56], and optical functional devices [57], etc.

In polarization holography, a periodic change in the polarization state of light is required in terms of polarization-sensitive holograms. These polarization-sensitive holograms can be obtained with the help of SLMs using their pixel-dependent phase-shift. In this context, SLMs can be employed as key components for the experimental implementation of polarization holography. In a recent work, Nabadda et al. have demonstrated a phase-shifting interferometry (PSI) to evaluate the reconstruction of complex-valued holograms displayed onto a phase-only SLM (Hamamatsu X10,468-01, with 800 × 600 square pixels, 20 μ pixel pitch and 98% fill factor) and corresponding experimental scheme is presented in Figure 12a. Figure 12b shows the experimental observations for the encoded Hermite-Gauss (HG) and a Laguerre-Gauss (LG) beam. It was observed that the theoretical intensity and phase distributions exhibit its characteristic double ring and spiral pattern. In this case, the experimental interferograms show three bright and three dark lobes inside each of the two intensity rings. The bright and dark lobes are opposite from one ring to the other, as corresponds to the π-phase shift between them, and they progressively rotate as phase-shifts are added in successive interferograms [58] .

Figure 12.

(a) Scheme of the optical system. SF: Spatial filter; QWP: Quarter-wave plate;P: Linear polarizer; CGH: Displayed phase-only computer-generated hologram, and (b) realization of a HG₃₃ (A–D) and LG₁₃ (E–H) beam. Theoretical intensity (A, E) and phase (B, F) distributions. Experimental intensity (C, G) and phase (D, H) distributions. The phase distribution of each beam is retrieved from the interferograms shown below each case with eight consecutive phase shifts of step Δφ = π/4 (adapted from [58]).

In another development under polarization holography, vectorial wavefront holography is proposed using the wavefront shaping of a structured vector beam [59]. The study shows that the phase hologram can be used to tailor the polarization interference of a vector beam in momentum space, resulting in arbitrary polarization states. The designed polarization-multiplexing holograms are implemented with the help of LC-SLM and an azimuthal polarization converter was used to convert an incident linear polarization into azimuthally polarized vector beams (APVB).

5. Conclusion

LC-SLMs are essential optical components for the experimental implementation of modern optical imaging techniques, viz., computational imaging [4, 60], ghost imaging [61, 62], structured light-based imaging [63, 64], orbital angular momentum manipulation [65, 66], etc. The complicated structure and subtle working principle require the SLM calibration prior to its utilization for various imaging applications. SLM calibration methods can be broadly divided into two categories, i.e., interferometric techniques, and polarimetric techniques. In interferometric calibration techniques, phase modulation characteristics of LC-SLM can be calibrated by quantifying the relative phase-shift of the recorded interference patterns, which can be formed using traditional interferometry. In such methods, reference beams can be either considered off-axis (out-referenced) or generated from the LC-SLM itself (self-referenced). Self-referenced interferometric methods are easy to implement as they offer compactness and stability to the experimental set-up meanwhile off-axis referenced interferometric methods require tedious experimental set-ups as compared to the earlier case but offer wide applicability in the form of spatially resolved modulation characteristics, pixel-crosstalk, etc. On the other hand, SLMs are polarization-sensitive devices and hence polarization of light plays a vital role in SLM operation. In this context, polarization modulation characteristics of SLM can be accounted for using the polarimetric calibration methods. The polarimetric methods revolve around two polarimetric techniques, i.e. Jones matrix imaging and Mueller matrix imaging. In general, Jones matrix imaging is suitable for carryout polarimetric measurements of LC-SLM due to their homogeneous structure. However, SLMs are not perfectly homogenous and hence can exhibit depolarization as well [12, 49]. In such a scenario, Mueller matrix imaging can be employed to investigate the polarimetric characteristics of LC-SLM.

In summary, a comprehensive literature study of LC-SLMs (including their construction, working principle, calibration methods, and applications) has been revisited in this chapter. We have reviewed the detailed polarimetric characteristics of LC-SLM (Holoeye, LC-R 720) concerning its grayscales using polarization-sensitive digital holographic technique followed by recent investigations on the spatial anomalies in the LC-SLM using Mueller-Stokes polarimetry. Further, a few crucial applications of LC-SLMs have been quickly discussed within the framework of polarization holography. The present work is an attempt to consolidate the major SLM calibration methods and is expected to serve as a manual guide for LC-SLMs along with their applications in advanced imaging domains.

Acknowledgments

Vipin Tiwari acknowledges support from DST-INSPIRE (IF-170861).