Theory and Experiment of Spatial Light Modulation and Demodulation With Multi-Plane Diffraction and Applications

Spatial light modulation enhances capacity of optical communications by modulating spatial amplitude, phase and polarization degrees of freedom with recent success of orbital angular momentum based architectures. There is a hardware challenge to demodulate large symbol families or high order symbols requiring a general design of spatial light demodulation. Multi-plane diffraction (MPD) recently introduced for improving spatial modulation capabilities in free space optical channels promises utilization at the receiver side as a demodulator. In this article, we theoretically model, numerically simulate and experimentally implement spatial light demodulation based on MPD. Numerical simulations and experimental implementations verify capabilities of MPD for increasing inter-symbol distances at the detector front-end. We obtain approximately two times improvement compared with direct detection for basic design including three diffraction planes as a proof-of-concept and improved performance with increasing number of diffraction planes compared with state-of-the-art single-plane diffraction (SPD) based interferometric receivers. Besides that, we perform, for the first time, experimental implementation of MPD based spatial light modulation. In addition, symmetric-key cryptography application of the proposed system is theoretically presented with low decoder complexity while numerical simulations promise high performance security against intruders. MPD based design is practically applicable and promising for diverse optical architectures including both communications and cryptography as a low-cost, low hardware complexity, passive and high performance design.


I. INTRODUCTION
Spatial light modulation modulates phase, amplitude and polarization of wave function or wave-front [1], [2], [3]. It enhances capacity of optical communications by allowing to use spatial degrees of freedom in various architectures [1], [4], [5], [6], [7]. Communication systems based on spatial light modulation exploit orbital angular moment (OAM) [6], [8], [9], [10], [11] and polarization [12], [13] degrees of freedom in free space and fiber optical communications. In addition, visible light communications (VLC) systems The associate editor coordinating the review of this manuscript and approving it for publication was Guido Lombardi . exploit spatial light modulators to realize spatially variant polarization states [12], [14]. On the other hand, in spatial light modulation based optical communication architectures, detectors include holographic gratings, various interferometer structures, polarizers or additional spatial light modulators with detailed reviews presented in [10] and [15]. Demodulator inverts phase distribution of the transmitted modulation symbol resulting in a hardware based challenge for large number of symbols and high order modulation such as increased complexity for higher mode numbers in OAM receivers [10]. A phase analyzer needs to be utilized to estimate transmitted phase distribution in more general set-ups. Machine learning approaches are also proposed for efficient demodulation capabilities [16]. As a result, high order optical demodulation with low-complexity hardware and diverse applicability for various spatial modulation based systems is an important challenge.
Multi-plane diffraction (MPD) is recently utilized to define quantum spatial modulation in [17] by emphasizing improvement in the symbol family size and spectral efficiency as a method to improve spatial modulation for optical channels [18], [19], [20], [21]. MPD is further extended to radio frequency (RF) channels by showing the promising capabilities of MPD in also RF communications [22]. Furthermore, MPD set-ups with amplitude and phase masks are experimentally realized for defining all-optical machine learning with promising advantages but not for the purpose of optical modulation or demodulation in free space optical communications [23], [24]. There is not any experimental implementation of MPD based spatial light modulation for optical channels for experimental verification of the design. Furthermore, general modeling for modulating both the phase and amplitudes is not considered in [17]. In addition, it is not clear how to demodulate spatial light modulation symbols by exploiting MPD based architectures by maintaining the same hardware simplicity at the receiver side in comparison with challenges in state-of-the-art architectures [10]. Besides that, there are existing architectures to demodulate OAM modulated waves by using single plane diffraction (SPD) apertures by transforming phase information into interferometric intensity differences at the receivers [25], [26], [27], [28]. The major drawback of these architectures is the requirement of N − 1 interferometers to detect N OAM modes resulting in practical complexity for high order modulations [10], [28]. MPD based general spatial light demodulation design, theory and experiment are not available as an alternative to SPD based architectures.
In this article, we theoretically design and experimentally implement a generalized spatial light demodulation by exploiting both phase and amplitude modulating MPD design which is applicable to diverse communication systems while promising the simplicity with low-cost diffraction slit design and passive diffraction architecture. In addition, we perform, for the first time, experimental implementation of free space optical MPD design proposed in [17], i.e., quantum spatial modulation, with simple rectangular slit masks for the purpose of verifying spatial modulation capabilities in optical channels. MPD provides a low-complexity and passive architecture for realizing modulation compared with more complex and active modulation designs including liquid crystal based or digital micro-mirror designs [1], [2], [3].
We numerically simulate and experimentally implement spatial light demodulation to decode symbols with small inter-symbol distances as a proof-of-concept while improving inter-symbol distance by ≈2 times at maximum compared with the case without demodulation for specific set-up parameters including three diffraction planes at the receiver. Furthermore, it is observed that inter-symbol distance increases with increasing number of planes in MPD based demodu-lation architecture both in numerical simulation and experimental studies. Therefore, proposed MPD based demodulation design promises important improvements compared with state-of-the-art SPD based interferometric architectures utilized to demodulate spatially modulated signals. Spatial demodulation creates complex interference patterns with promising applications for large symbol families. In the proposed experimental validation, smooth intensity patterns of symbols with small inter-symbol distances are transformed into higher frequency intensity patterns with larger intersymbol distances by using random spatial masks. Therefore, MPD based interferometric design exploits phase difference of symbols to create more distant interference patterns for different symbols. Therefore, we enhance existing methods based on SPD by proposing MPD to increase the demodulation capability similar to increased spatial modulation capability at the transmitter side presented for quantum spatial modulation [25], [26], [27], [28]. In addition, slit masks are realized with low-cost and low-complexity design, for the first time, based on graphite cylinders allowing to realize complex spatial light modulation and demodulation tasks for free space optical communication and cryptographic applications.
After defining MPD based spatial demodulation for optical communications, we design an all-optical hardware based symmetric-key cryptography system, for the first time, by exploiting only passive MPD design, without any software based decoding complexity. There are various all-optical encryption/decryption mechanisms utilizing symmetric or private-key based mechanism by using spectral shuffling [29] and all-optical logic gates [30], [31]. However, MPD based architecture is not exploited. We numerically simulate protection against an intruder utilizing a random mask at the decoder. The intruder cannot practically find private keys among exponentially large number of possible private masks. Furthermore, our numerical analysis with phase-only modulator masks shows that the intruder observes uniformly distributed phase patterns without any clue to reliably decode if there is no access to private keys.
The contributions presented in this article are summarized as follows: • We design a novel spatial light demodulation method based on MPD introduced previously only for spatial light modulation purposes while achieving a low-cost, low-complexity and passive hardware design. Numerical simulations and experimental results for SPD show improvements of inter-symbol distances by ≈2 times at maximum with three diffraction planes compared with the case without spatial light demodulation. In addition, inter-symbol distances increase with increasing number of diffraction planes by improving performance compared with interferometric state-of-the-art SPD receiver design as a proof-of-concept.
• We improve previous theoretical models for MPD consisting of amplitude modulating slit masks by modeling for both amplitude and phase modulation.
• We, for the first time, experimentally implement both spatial light modulation and demodulation based on MPD. As a proof-of-concept, simple rectangular slits are utilized for experiments.
• We propose a novel diffraction plate design to be utilized for spatial light demodulation purposes by using graphite cylinders with a low-cost and simplified design architecture.
• We present theoretical modeling and numerical simulation of a novel all-optical symmetric-key cryptography system by exploiting MPD based spatial light demodulation design. Experimental challenges include experimental characterization of MPD based complete spatial light modulation and demodulation architectures with solid evidences about the performance including data rates and error rates. Experimental implementations of high order modulation and demodulation set-ups with many consecutive diffraction planes in MPD are left as future works while only proof-of-concept and low-complexity MPD architectures are implemented with the maximum of four diffraction plates.
The remainder of the paper is organized as follows. In Section II, demodulation architecture based on MPD is introduced and theoretically modeled for general phase-amplitude modulation design. In Section III, MPD modeling based on rectangular slits is presented where only the amplitude is modulated with rectangular slits. Performance of demodulation is numerically simulated in Section IV. In Section V, experimental implementations of both spatial light modulation and demodulation based on rectangular slit based diffraction architectures are presented. Application of demodulation for symmetric-key cryptography algorithm including system design, theoretical modeling and numerical simulations is presented in Section VI. Finally, in Section VII, conclusions are presented.

II. SPATIAL LIGHT DEMODULATION WITH MPD BASED PASSIVE DIFFRACTION
The proposed two-dimensional (2D) spatial light modulation and demodulation design is presented in Fig. 1. Assume that spatially modulated optical signal wave-function m (x, y) at the transmitter (Tx) 2D surface for sending information by use of M wave functions with m ∈ [1, M ] is defined as follows: where A m (x, y) and φ m (x, y) are the modulation amplitude and phase, respectively. After through the free space channel propagation of length L c , r,m (x, y) and d,m (x, y) are the received and detected inputs at the surfaces of the demodulator and photo-detector (PD) array, respectively, for a passive demodulation set-up without changing the spatial mask each time with respect to m. MPD multiplicative mask D j (x, y) for j ∈ [1, N ] multiplies incoming wave function on jth plane. In other words, in (x, y) on jth plane is transformed into in (x, y) × D j (x, y). Each plane has K D × K D pixels with each pixel having side length of W D and inter-planar planar distances of L D,j,j+1 between jth and (j + 1)th planes. Modulation is based on quantum spatial modulation defined in [17] exploiting MPD structure where each plane has a mask of M j,m (x, y) for j ∈ [1, V ] depending on m and a total of K M × K M pixels on each plane with pixel side-length of W M where inter-planar distance is L M ,j,j+1 between jth and (j + 1)th planes. L M ,01 denotes the distance of laser plane with the source wave function 0 (x, y) to the first plane in modulation.
Received intensity distribution before demodulation is denoted with I r,m (x, y) = | r,m (x, y)| 2 for varying symbol index m as shown in Fig. 1. Intensity is detected with N P = N x × N y pixels at a longitudinal distance of L d to the N th plane of the demodulator. The resulting intensity distribution is the demodulator output I d,m (x, y) = | d,m (x, y)| 2 which is the input to the detector, e.g., maximum-likelihood detection, for estimating m. Assuming an additive white Gaussian noise (AWGN) model n[k x , k y ] at the PD corresponding to the pixel position (x, y) = (k x , k y ) × T s , the detector input is given as follows: where where s k a ,± ≡ (k a ± 1/2) T s for integer k a gives the boundaries of the detector along the axis a = x or y, P m is the number of photons in a single channel use and r e is PD responsivity. Propagation from the jth plane to (j + 1)th plane with longitudinal distance of L D,j,j+1 λ is calculated as follows: where in (x, y) and out (x, y) denote wave functions on input surfaces of jth and (j + 1)th planes, respectively, and free-space propagation kernel K j (x, y, x 0 , y 0 ) is defined as follows by neglecting distance dependent phase [17]: where k ≡ 2 π / λ is the wave vector and λ is the wavelength of laser source.
Inter-symbol distance between demodulated symbols I d,i [k x , k y ] and I d,j [k x , k y ] for ith and jth symbols, respectively, is defined as follows: In addition, mean values with respect to all symbols are defined as follows: In conventional maximum-likelihood detection for AWGN channel model, decision rule is equivalent to finding the transmitted symbol which is closest to the received symbol. Therefore, in the proposed design, demodulated symbols I d,m [k x , k y ] are analogous to transmitted symbols in simple AWGN models as shown in (2) [33]. Pixel signal-tonoise ratio (SNR) at each PD denoted as SNR p is defined as P av / σ 2 where P av is average of I 2 d,m [k x , k y ], and each PD has a variance of fixed σ 2 . Estimated m minimizes distance as follows: Next, MPD set-up is modeled with simple rectangular slits for more practical experimental implementation. plane is denoted with s j ,k and its width is given by W j ,k . M − 1 diffraction planes include V modulation and N demodulation planes where the distance between j th and (j + 1)th planes is denoted with L j ,j +1 .

III. MPD MODELING WITH RECTANGULAR SLITS
MPD design is simplified by using rectangular slits for practical implementation as shown in Fig. 2. In the proposed experimental set-up, there are M − 1 = V + N diffraction planes and a final monitor plane while in some scenarios an additional thin lens is utilized after the final diffraction plane. V and N correspond to the number of diffraction planes for modulation and demodulation parts, respectively, as shown in Fig. 1. The plane with index j has K j slits for j ∈ [1, M − 1] while the kth slit has the central coordinate of s j,k and width W j,k for k ∈ [1, K j ]. The distance between jth and (j + 1)th planes is denoted with L j,j+1 while the distances of the thin lens to the final diffraction and monitor planes are denoted with L a and L b , respectively.
The wave function on jth plane is denoted by j (x j ) while the source wave function is denoted with 0 (x 0 ). In this case, the subscript j of j (x j ) corresponds to the index of the plane and x j denotes the horizontal axis on the surface of jth plane. The source is assumed to be coherent as the output of a standard laser with the source wave function amplitude of light field in one dimension is described as follows: where σ 0 is beam width of the Gaussian source wave packet. The relation between the wave functions on jth and (j + 1)th planes is defined as follows [17]: where the free space propagation kernel K (x j+1 , x j ) is defined as follows for one-dimensional (1D) modeling by using (5): Assume that the phase and amplitude of the light wave propagating through a narrow slit are assumed not to change significantly through the slit, i.e., assuming a fixed value being equal to j (s j,k ) where s j,k is the center of the kth slit on jth plane. Then, the propagating amplitude of the diffracted wave on the next plane through the diffraction due to kth slit is approximated as follows: where erfc(z) dt is the complementary error function with complex inputs and α ≡ (1 /2 − ı / 2) √ π / 2. Then, total wave function amplitude on the (j + 1)th plane due to diffraction through K j slits on the jth plane is approximated as follows: (20) In addition, we divide each single slit into much larger number of narrower slits in order to increase the accuracy in our calculation. The wave function amplitude on the monitor plane is calculated iteratively through each diffraction plane resulting in the following intensity distribution: If there is lens after the final plane, the effect of the lens on the input wave function in (x) is modeled as follows [32]: where f lens is the focal length of the lens. Therefore, it multiplies the input wave function with the corresponding chirp function depending on the focal length and wavelength. The proposed simplified model with rectangular slits is utilized in experimental implementation in Section V. The general model with both amplitude and phase modulation defined in Section II is utilized for numerical simulation of spatial light demodulation performance in the next section.

IV. NUMERICAL SIMULATION OF SPATIAL LIGHT DEMODULATION
We perform numerical simulation of spatial light demodulation performance in 1D version of the design in Fig. 1 for simplicity for varying number of receiver (Rx) diffraction planes by choosing N between 0 and 2 with parameters as shown in Table 1. MPD based modulation part at Tx is realized with rectangular slits and composed of V = 2 planes including nine different combinations of 3, 4 and 5 rectangular slits where distance between the centers of the slits is D s = 75 µm and slit widths are chosen as W M = 30 µm. Therefore, M j,m (x) representing the modulation mask is set as the following: where K j,m ∈ [3,5] denotes the number of slits on jth plane for the modulation index m and the function rect(x) = 1 for  N = 0 case denotes the case with direct detection where spatial light demodulation is not applied at Rx. The case N = 1 represents state-of-the-art SPD based interferometric demodulation methods while the cases of N ∈ {2, 3} correspond to the MPD based demodulation designs proposed in this article as a proof-of-concept. The extension for N ≥ 4 is left as a future work. Furthermore, we use phase masking of the form D j (x) = e ı φ D,j (x) for j ∈ [1, 3] to preserve total energy in the demodulated signal compared with the case without spatial light demodulation. Phases in each slit width of W D = 50 µm are modulated by either 0 or π with equal probability by mimicking a pseudo-random noise sequence.
L d is chosen as 450 mm for N = 1 while L d + L D,12 + L D,23 = 450 mm for N = 2 and 3 by making the distance between the first diffraction plane at Rx and the monitor plane equal to 450 mm for both cases. L c is chosen as 500 mm for the cases with demodulation for an example of short-range communication channel while L c = 450 mm + 500 mm = 950 mm for the case without demodulation by making the distance between modulator part and the monitor plane the same for both the cases with and without spatial light demodulation.
Each symbol power is normalized by setting r e P m so that total squared intensity at PD array is unity for m ∈ [1,9]  The minimum, average and maximum inter-symbol distances are denoted with d min (m), d mean (m) and d max (m), respectively, for each symbol m for varying N values. The total number of trials is chosen as 10 while the performance is calculated by getting average of the distances in 10 trials with random phase masks D j (x) for j ∈ [1,3] in each trial. Increasing the number of trials is left as a future work due to computational constraints while providing a reliable observation as a proof-of-concept. In Fig. 3, intensity distributions of specific symbols are shown for a single trial to analyze the effect of spatial light demodulation. The minimum of d min occurs between m = 4 and m = 7 corresponding to   Fig. 3(b) with the observations of highly similar intensities and difficulty to decode. In spatial light demodulation, phase difference is exploited with random phase masks to obtain highly interfering diffraction patterns for N = 1, 2 and 3 as shown in Figs. 3(c), (d) and (e), respectively. The difference I d [x] increases with increasing values of N resulting in an increased decoding performance.
It is observed in Fig. 4(a) that spatial light demodulation increases distances significantly for increasing values of N for all symbol indices m ∈ [1,9]. In Fig. 4(b), the distances are calculated by using (10)(11)(12) where the average for all symbol values is calculated, i.e., d val ≡ (1 / 9) 9 m=1 d val (m), for N ∈ [0, 2] where val corresponds to either min, mean or max. All distance values increase with increasing values of N by improving inter-symbol distance at the detector front-end ≈ 2 times at maximum. It is a future work to extend the analysis to larger symbol families and higher values of N .

V. EXPERIMENTAL RESULTS
SPD and the proposed MPD system for spatial light modulation and demodulation are experimentally realized with the optical set-up in Fig. 5(a) where its design is shown in Fig. 5(b). Experiments are based on the practical design shown in Fig. 2 with rectangular slits where the corresponding parameters including set-up identification (ID) numbers, the types of diffraction at Tx and Rx, the total number of diffraction planes M − 1 = V + N , the number of diffraction planes V and N for Tx and Rx, respectively, the number of slits in each plane K j for j ∈ [1, 4], inter-planar distances L j,j+1 for j ∈ [0, 4], whether final lens is utilized, focal length f lens of the lens in centimeters and the distance parameters L a and L b of the lens are presented in Table 2. The plates including rectangular slits are freely moved along x and z axis with micrometer accuracy based on adjustable holding system. The positions of the plates are adjusted by using a rail mechanism allowing to modify inter-planar distances. The monitor plane is composed of black paper reflecting the diffracted wave such that its photo allows to extract the intensity distribution. All the photographs of monitor plane are obtained in total darkness in order to remove the effect of background light. Smart phone camera (Iphone SE 2020) is used to capture the images.
Gaussian laser source with wavelength λ = 635 nm and beam-width σ 0 = 1000 µm is utilized where the power of the compact laser module is 0.91 mW (Thorlabs). Diffraction slits are realized either by using metallic coated glass plates (3B Scientific) as shown in Figs  There are 36 different kinds of experiments performed with the following set-up ID numbers and definitions: • S1 to S8: Spatial light modulation with SPD based Tx • S9 to S12: Spatial light modulation with MPD based Tx • S13 to S20: Spatial light modulation with MPD based Tx and demodulation with direct detection at Rx • S21 to S28: Spatial light modulation with MPD based Tx and demodulation with SPD based Rx • S29 to S36: Spatial light modulation with MPD based Tx and demodulation with MPD based Rx Spatial light demodulation is performed with random slits manufactured by using graphite cylinders as shown in Figs. 5(h) and (i). Concave lens with focal length of f lens = −7.5 cm is utilized in set-up IDs of S5, S6, S7 and S8 in order to obtain patterns with magnified details at the central spatial region for SPD experiments. Next, image processing methods to transform the captured images of light diffraction into intensity distributions are presented.

A. IMAGE PROCESSING METHODS
Images of the intensity pattern on the monitor plane are captured with the camera. After the laser diffracts through the slits, the captured image of the pattern appearing on the monitor is denoted as raw image as shown in the first step in Fig. 6. Photos are captured from the side but not directly in front of the monitor screen. Therefore, there is perspective distortion in the acquired raw image. Warping operation is applied to correct the distortion by obtaining M × N pixels image data. Then, the color image is converted to gray-scale intensity values denoted with matrix A with values A[x, y] along the pixel locations (x, y) by using black and white (BW) filtering. Since the diffraction slits are 1D with lengths of several centimeters along y axis and microscale widths along x axis, the resulting intensity values are summed along y axis vertically in order to obtain 1D intensity distribution denoted with I exp [x] along x axis while each x value corresponds to the VOLUME 11, 2023 discrete pixel location k x T s with a similar notation in Sec. IV. Since the images are taken with camera and the range of the intensity levels which can be captured are limited, intensity saturation is observed depending on the power level of the laser source. Therefore, intensity levels are normalized for different regions around peak values based on theoretically calculated maximum values for each region, i.e., for each R i for i ∈ [1,4]. Resulting normalized intensity distribution is denoted with I exp,n [x]. Raw images for set-up IDs S2, S3, S5 and S8, are shown in Figs. 7(a), (b), (c) and (d), respectively, for SPD while MPD raw images are shown Figs. 7(e) and (f), for S10 and S12, respectively. White regions at the center of the diffraction patterns at the middle of the main peak in Figs. 7(a) and (b) indicate high levels of intensity saturation while smaller level saturation is observed in Figs. 7(c), (d), (e) and (f). Next, image processing steps are explained in detail.

1) WARPING
It is not possible to place the camera exactly in front of the screen since the laser light is blocked as seen in Fig. 5(a). Four corners of the plane is chosen to make projective transformation via MATLAB for realizing perspective correction. Since the images are taken in totally dark environment, glowing phosphors are used to determine four corners of the plane. After this step, an image is obtained as if the camera is positioned in front of the screen as shown in Fig. 6. Warping process decreases the resolution of the warped image to 1600 × 900 pixels expressed as M × N in Fig. 6 although the size of the raw image is 4032 × 2268 pixels.

3) VERTICAL SUMMATION
Diffraction slits utilized in experimental studies are designed for 1D diffraction studies. The slits extend along y axis for several centimeters while having microscale widths along x axis. Therefore, the resulting intensity values are summed along y axis vertically to obtain 1D intensity distribution I exp [x] along x axis as follows:

4) NORMALIZATION
Captured images show saturation while not supporting the exact intensity scale of the spatial distribution of the number of photons. For example, if the number of photons is VOLUME 11, 2023 larger than some specific threshold level, then its intensity is transformed into pure white color in the acquired BW image with maximum intensity level, e.g., 255 for the scale of [0, 255], irrespective of the exact level of the light intensity. Therefore, the levels of peaks for different spatial regions of the distribution need to be rescaled either by using different source intensities or by normalizing with respect to the maximum theoretical values in each region. Spatial regions of the intensity distribution are divided into four different regions denoted with R 1 , R 2 , R 3 and R 4 along x axis as shown in Fig. 6. Then, normalizing coefficients C i in each region R i for i ∈ [1,4] are calculated as follows:  (21) and (22) for free space propagation of wave function and lens based masking, respectively. Its maximum is normalized to unity for better comparison with experimental results. After the normalization process, theoretical and experimental intensity distributions are much more compatible as shown in the last step of Fig. 6 Figs. 9(a), (b), (c) and (d), respectively, while generating different symbols for modulation operation. It is observed that theoretical and experimental results for MPD are in agreement. The distance L 12 and the pairs of [K 1 , K 2 ] are chosen with trial and error to maximize the variation compared with SPD case having the same L 01 and K 1 . Intensity of diffracted laser beam decreases more compared with SPD reducing the effect of saturation. Therefore, peak intensity levels of theoretical and experimental results are very close to each other. It is observed that increasing the number of slits in both plates results in an increase in spatial frequency, e.g., two peaks between the main and side lobes as shown in Fig. 9(d).

D. SPATIAL LIGHT DEMODULATION WITH MULTI-PLANE DIFFRACTION
Spatial light demodulation is experimentally realized for eight different symbols for simplicity and as a proof-ofconcept in a similar manner to the simulation studies in Section IV. Each symbol corresponds to a different combination of the number of slits K 1 and K 2 on the first and second modulation planes, respectively, as shown in Table 2.
The case without spatial light demodulation for N = 0 is compared with the cases exploiting spatial light demodulation for N = 1 and N = 2 for the same pairs of K 1 and K 2 corresponding to different symbols.   Figs. 11(b) and (c) for N = 1 and N = 2, respectively, that as N increases the patterns appear as more randomized and having higher frequency compared with smooth distributions shown in Fig. 11(a). Similarity comparison among symbol intensity patterns is achieved by using inter-symbol distance definition in (6) which is the fundamental mechanism utilized for maximum-likelihood detection as discussed in Section II [33]. The resulting 882 VOLUME 11, 2023    Fig. 12(d), d min , d mean , and d max for N ∈ [0, 2] are shown based on the average values with respect to all symbols. It is observed that mean distance values d min , d mean and d max increase with increasing values of N , e.g., approximately two times improvement with N = 2 similar to the simulation results in Section IV for simple modulation symbols as an experimental proof-of-concept. It is a future VOLUME 11, 2023 FIGURE 11. Intensity distributions of detected eight symbols after vertical summation corresponding to set-up IDs (a) between S13 and S20 for N = 0, (b) between S21 and S28 for N = 1 and (c) between S29 and S36 for N = 2. I d ,m denotes the detected symbol symbol intensity for mth symbol while each of them is normalized to satisfy ||I d ,m || 2 = k x I 2 d ,m [k x ] = 1 while without any normalization based on the theoretical values for fair comparison among different N values with plates composed of randomly distributed slits. work to experiment with much larger number of slits and planes in both modulation and demodulation parts in order to observe the advantages for significantly higher order symbol families. Next, experimental challenges for the proposed design and future works to further improve the experimental accuracy are discussed.

E. EXPERIMENTAL CHALLENGES AND FUTURE WORKS
Three different designs for interference pattern measurement are shown in Fig. 13. The design in Fig. 13(a) is utilized in experimental studies in this article based on measurement of the reflected light from monitor screen in total darkness while the designs in Figs. 13(b) and (c) utilize direct detection of the incoming light as discussed in Section V-E2. In this article, the design shown in Fig. 13(a) is utilized with the following motivation as discussed next while the challenges are discussed in Section V-E1.
Fundamental target in this article is to compare theoretical calculations and experimental measurements for MPD based spatial modulation system design originally defined without any lens in [17] and for large communications distances. We design our experiments for large inter-planar distances on the order of tens of centimeters resulting in large area diffraction patterns requiring additional external lens to focus for detection with camera lenses. The overall system is regarded as a Fourier optics system where the effects of free space and lens on the propagating wave function are theoretically modeled in (16) and (22), respectively. The comparison of theory and experiment for MPD design without external focusing lens is performed to accurately compare the effect of MPD on the wave function for freely propagating wave function. A low-complexity measurement architecture without any external lens is presented for large are diffraction patterns due to long distance communications. As a result, monitor screen based measurement is a low complexity and alternative method fundamentally similar to the camera sensor based measurement in Fig. 13(b) while with various experimental challenges due to the design not utilizing any external lens in total darkness and directly exploiting the reflected light from monitor screen as discussed in Section V-E1. Performed measurements are accurate enough to verify the advantages of spatial light modulation as one of the main targets of the presented article.

1) EXPERIMENTAL CHALLENGES
Firstly, the resolution of the camera is limited to 4032 × 2268 pixels while the active area after warping where the pattern is observed is decreased to 1600 × 900 pixels. Increasing the resolution of camera and dynamic range for reducing the effect of saturation is an open issue to obtain more accurate intensity distributions not only for monitor screen based solution but also for camera sensor based solution in Fig. 13(b).
Secondly, camera is placed at the side of the monitor plane resulting in perspective distortion. Image warping is applied to correct perspective distortion leading to errors at the positions of the peaks and their amplitudes. Glowing phosphors are used to distinguish the four corner points for the image warping process. There are minor errors for warping process itself due to larger size of phosphors and resolution of the camera. Implementing experimental design with smaller phosphors and higher resolution camera improves the accuracy of the experimental measurements as an open issue.
After normalization process, distance between theoretical expectations and experimental results is calculated by using the following distance definition: (27) It is observed that D T ,E is between 0.2483 and 0.6522 for the set-up IDs S1 to S8. It is larger than expected although the peak locations and amplitudes are highly close to each other as shown in Figs. 8 and 9. It is observed that there are small levels of shift at the location of peaks of intensity distributions as shown more clearly in Figs. 9(a) and (b) with peaks not in coincidence leading to larger distance values. The challenges include improving warping operation, increasing resolution of camera and enhancing the quality of the monitor paper. VOLUME 11, 2023 2) EXPERIMENTAL FUTURE WORKS A practical solution to improve experimental accuracy during measurement is to replace black paper based monitor screen with CCD or CMOS based sensors and cameras. Two designs are shown in Figs. 13(b) and (c) for utilizing sensor arrays. In Fig. 13(b), the diffraction pattern is focused on the sensor array of a camera based receiver by using a lens [34]. On the other hand, in Fig. 13(c), either a single photodetector or a sensor array is mechanically positioned at different points while covering the complete area of the diffraction pattern. If the number of planes is significantly large for ultra high order modulation capability and a few photons reach the detector, then the best solution becomes to use a large area single photon detector allowing to measure the distribution of the diffracted photons by photon counting. Resolution is improved by increasing the number of pixels and reducing the pixel width on the sensor array.
On the other hand, sensor based measurements shown in Figs. 13(b) and (c) have also some similar challenges with monitor screen based solution. MPD increases the spatial frequency of the signal significantly as shown in simulation studies in Section IV transforming smooth distributions of the intensity of incoming symbols into intensity patterns which are closer to uniformly distributed patterns. Therefore, focusing high spatial frequency distribution to the pixel area of a camera based receiver as shown in Fig. 13(b) without losing the details may be challenging. On the other hand, the design with mobile sensor array shown in Fig. 13(c) requires mechanical movements of the sensor array to cover large area diffraction patterns.
Next, spatial light demodulation set-up is utilized for alloptical symmetric-key cryptography as a promising application in addition to the presented optical communications applications.

VI. FUTURE APPLICATIONS FOR CRYPTOGRAPHY
A novel cryptographic framework based on spatial light demodulation is shown in Fig. 14. Bob encodes input phaseamplitude distribution with respect to spatial light demodulation set-up of Alice composed of diffraction keys D j (x, y) for j ∈ [1, N ] and integration key p[k x , k y ] ∈ [0, 1] denoting the summation of the intensities at a specific sub-set of PDs with the total count in the sub-set being fixed as N P / 2. We assume that detector intensity is normalized for each lth bit m l ∈ [0, 1], i.e., k x k y I d,m l [k x , k y ] = 1. We define d[l] denoting detector output as follows: Decoding is achieved by using a threshold detector as follows: The output includes signal and noise components as is the integrated noise components of N P / 2 PDs. Since the integrator adds N P / 2 PD outputs, total variance of the noise becomes equal to σ 2 d ≡ N P σ 2 / 2. The details of symmetric-key cryptography is shown in Algorithm VI. In the first step, Alice shares the secret keys D j (x, y) for j ∈ [1, N ] and p[k x , k y ] with Bob. Then, in the second step, Bob initializes M 1,m l (x, y) for each m l ∈ [0, 1] with a random pattern, e.g., for a phase-only modulation mask choosing each phase value at a single pixel with a uniform distribution. Therefore, intruder Charlie observes randomly distributed patterns without any clue to estimate the bit sequence based on the random initialization in the second step. Bob calculates the noise-free signal at the demodulated output of Alice by using private keys. In the third step, he realizes non-linear optimization for adjusting modulation mask M 1,m l (x, y) initialized with a random pattern to maximize d s [l] for m l = 1 and to minimize d s [l] for m l = 0. In Section VI-A, we numerically observe that the optimized patterns are not reliably decoded by Charlie without access to private keys while verifying the efficiency of initialization with random patterns. In the fourth step, Alice decodes the encoded message m l . Assume that, Bob optimizes modulation parameters such that d s [l] equals to fixed d s,m l for m l ∈ [0, 1]. Then, probability of bit error for equally probable symbols is calculated as follows: where p(n d ≥ 0.5 − d s,0 ) and p(n d ≤ 0.5 − d s,1 ) denote probabilities based on the decoding rule in (29) and n d is AWGN with variance of σ 2 d = N P σ 2 / 2. Assume that intensity is uniformly distributed so that average pixel intensity equals to 1 / N P while pixel SNR becomes SNR p = 1 / (N 2 P σ 2 ). Then, P e is easily calculated as follows by using q-function x e −u 2 / 2 du [33]: where it is assumed that the channel is modeled as an AWGN channel with r = √ E b + n while r denotes the received signal, √ E b is the modulation amplitude for a single bit, n is the noise with variance σ 2 n and threshold level is equal to zero leading to P e = Q( E b / σ 2 n ) for the single bit. In our case √ E b is replaced by |0.5 − d s,0 | or |0.5 − d s,1 | and σ 2 n is equal to σ 2 d . Charlie applies a random mask p C [k x , k y ] at the same reception distance with Alice but without any spatial light demodulation. Received intensity with direct detection is denoted as I C d,m l [k x , k y ]. Then, he applies threshold detector by obtaining m C l to estimate m l . We define average probability of demodulator signal output to result in one, i.e., where p C j [k x , k y ] denotes the random mask at jth trial, I t (x) is the detector threshold function with output being equal to 0 for x < 0.5 and 1 for x >= 0.5. It is an open issue to theoretically verify randomness in the pattern for varying symmetric keys.
Assume that each pixel is masked by choosing among G different values depending on the hardware, e.g, 2 for slit based set-ups as in [17]. Then, the worst case complexity of decryption by Charlie requires exponentially growing number of keys N C ≡ N SLD N I where N SLD = G K 2 D N and N I = N P N P / 2 are total number of possible spatial light demodulation set-ups and integration keys, respectively.

A. NUMERICAL SIMULATION PERFORMANCE OF SYMMETRIC-KEY CRYPTOGRAPHY
Symmetric-key cryptography design in Fig. 14 is simulated for N = 1 for simplicity in order to verify capabilities of SPD with further improvement based on MPD with promising results obtained in Sections IV and V. Design parameters for modulation, demodulation and detection showing the widths and total numbers of slits and pixels are shown in demodulator phase only mask slit width is chosen as W D = 10 µm with much larger number of K D = 800 slits. Fixed spatial mask extends between [−2, 2] mm by clipping the received signal before spatial light demodulation. N x = 400 PD pixels along the clipped region have T s = 10 µm. p[k x , k y ] is chosen as a random sequence composed of zeros and ones. We choose M r = 10 5 for simplicity to calculate Prob { m C l = 1}. The channel and demodulator distances are chosen as L c = 500 mm and L d = 150 mm, respectively.
We simulate transmission of 64 bit message sequence for simplicity with equal probability of 0 and 1 source bits indexed by l ∈ The message input sequence m l is sorted and the corresponding decoded output d s [l] with optimized masks M 1,m l [k x ] are shown in Fig. 15(a). The optimization of the masks is achieved with non-linear optimization toolbox of MATLAB. It is observed that the optimized modulator mask achieves mean values of d s,0 and d s,1 being equal to ≈ 0.18 and ≈ 0.67, respectively, based on the obtained values for 64 messages by creating a large noise margin compared with the normalized total intensity of unity. The distance to the threshold is observed to be not equal, i.e., |0.5 − d s,0 | = |d s,1 − 0.5|, requiring further optimization in the design in order to realize more balanced key generation as an open issue.
On the other hand, Prob { m C l = 1} for the random mask of an intruder is shown in Fig. 15(a) with the value of ≈ 0.5 producing a random sequence at the output without access to private keys D 1 (k x ) and p[k x ]. P e is calculated for varying SNR p as shown in Fig. 15(b) with high performance due to the receiver diversity with many pixels.

VII. CONCLUSION
In this article, theoretical modeling, numerical simulation and experimental implementation of MPD based passive spatial light modulation and demodulation designs are presented. Numerical simulations and experimental implementation of demodulation provide proof-of-concept performance improvement compared with direct detection of spatial phase modulated signals, e.g., improving inter-symbol distances by ≈2 times at maximum for specific set-up parameters. In addition, MPD based demodulation provides higher performance with increasing number of diffraction layers by improving state-of-the-art interferometric demodulation architectures exploiting only SPD design. We experimentally show the capabilities of randomly distributed slits to demodulate signals with similar intensity but different phase distributions. Besides that, quantum spatial modulation method based on MPD previously defined for improving spatial modulation capability in optical communications is experimentally implemented as a proof-of-concept with simple rectangular slit based diffraction architectures. Furthermore, we apply spatial light demodulation for designing an all-optical free space symmetric-key cryptography system as a novel application of MPD based design architecture. Numerical simulations of symmetric-key cryptography verify security against decoding without access to private keys, respectively. The proposed modulation and demodulation design based on MPD promises important applications in optical communications and cryptography with low-cost, low-complexity and high performance system design. Open issues include designing methods to improve experimental accuracy and verification of complete communication and cryptographic communication systems based on proposed modulation and demodulation mechanisms.