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Abstract
Digital micromirror device (DMD)-based 4f-systems, a type of coherent optical information processing system, have become a powerful tool for optical convolutional neural networks taking advantage of their fast modulation speed and high-resolution capability. However, proper high bit-depth image information processing remains challenging due to the optical diffractions that arise from the binary nature of DMD operation. In this paper, we first characterize the diffraction phenomena that cause irradiance defects, namely the nonlinear grayscale and unintended dark lines. Then to resolve the issues, we propose a DMD operation method and a modified structure of the 4f-system based on blazed diffraction grating theory and numerical calculation of the Rayleigh–Sommerfeld propagation model. As a demonstration, we implement high bit-depth image information processing with an optimized optical 4f-system using DMDs and a collimated coherent light source.
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1. Introduction
Artificial intelligence (AI) and machine learning, with the capability to imitate the recognitive ability of humans, have been spotlighted in a large number of applications such as autonomous driving and the internet of things (IoT) as wireless communication continues to advance [1–3]. Most AI systems to date, designed to compute various tasks mimicking how the human brain works, are composed of artificial neural networks (ANNs) based on traditional electronic computer systems. As the amount of computation that ANNs perform grows exponentially, the computation speed of regular electronic computers barely meets the demands of industry, with limitations expected in the near future. This is because the von Neumann computer structure was developed to execute predetermined sequential programs and is thus not optimized for ANNs [4]. Plus, the energy efficiency of regular computers for ANN computation is another issue, as ANNs require a relatively large amount of energy compared to typical computer systems [5,6].
Recent research has demonstrated ANNs based on optical computing systems to resolve the issues that typically occur with regular electronic computers [7–11]. Even though most of them are hybrid systems composed of optical and electronic computing systems, such works have demonstrated that optical computing is a promising method to outperform regular electronic computers in ANN applications with highly efficient energy consumption and a high level of parallelism in their optical nature [12–16].
Among the ANNs, convolutional neural networks (CNNs) are one of the most widely applied for image information processing and object recognition, where the neural layers of linear calculation in the CNN occupy a large amount of computation power [17–20]. Miscuglio et al. demonstrated a hybrid CNN system combining an optical 4f-system for the convolutional neural layer with an electronic computer for the rest of the neural layers including activation, normalization, and fully connected layers, to realize partially optical computation with an ANN [21]. In the system, the input images are binary-like images, such as letters or simple objects. Such systems mostly employ digital micromirror devices (DMDs) for spatial light modulation to take advantage of their fast switching speed compared to liquid crystal spatial light modulators [22–24]. However, DMDs are basically designed for binary modulation by tilting micromirrors to ‘on’ and ‘off’ positions and have a periodic structure, two factors that drive diffraction of the beams which in turn causes not only low energy efficiency in the optical system but also irradiance defects [25–27]. The coaction of this binary nature and diffraction of DMDs in 4f-systems working with a coherent collimated light source generates defects in terms of the linearity of the grayscale and the irradiance between areas with certain radiance levels when high bit-depth images are processed. These defects cause errors in the convolved data, which may be amplified in the subsequent layers of the CNN. Therefore, correcting the detects is expected to help minimize the errors in the convolution layer, thereby leading to more accurate result. Furthermore, as ANNs advance, high bit-depth image information processing as well as high resolution are required, so these issues are an unavoidable challenge that should be resolved [28].
In this paper, we characterize the diffraction phenomena that cause irradiance defects when high bit-depth image information processing is performed with a 4f-system using DMDs. One of the irradiance defects is an improper linearity of the grayscale. In the DMD system, the grayscale is expressed by the on-off states of the DMD micromirrors, but slight deviations from the exact on-off positions of the mirrors affect the grayscale linearity in 4f-systems where one of the diffracted beams is chosen to be processed [29]. The other irradiance defect is the appearance of dark lines at certain radiance levels due to diffraction when the DMDs operate with a collimated light source. Since the effect of diffraction by the DMDs becomes more pronounced when the incident light source is coherent and collimated, these two issues are crucial for coherent optical information processing, where the degree of coherence is related with the accuracy and amount of information processing [30]. Plus, beams divided into several directions by diffraction will deteriorate the energy efficiency of the 4f-system as the number of diffracted beams increases; therefore, energy efficiency is another issue that should be taken into account when dealing with diffraction. In Section 2 of this work, we introduce the structures and characteristics of typical optical 4f-systems with blazed diffraction grating theory. The irradiance defects in terms of grayscale linearity and dark lines are then analyzed in Sections 3 and 4, respectively, where we propose a DMD operation method and a modified structure of the optical 4f-system based on blazed diffraction grating theory and numerical calculation of the Rayleigh–Sommerfeld propagation model to resolve the above issues. Finally, we demonstrate the implementation of high bit-depth image information processing with an optimized optical 4f-system using DMDs and a collimated coherent light source. Section 5 concludes the paper.
2. Optical 4f-system using DMDs and issues for grayscale image information processing
Most optical 4f-systems use DMDs as spatial light modulators for their fast switching speed and high resolution. Figure 1 shows a typical 4f-system using two DMDs and two convex lenses. In the system, a He-Ne laser generates collimated and coherent light at a wavelength of 633 nm, and an objective lens with a pinhole filter converts the beam intensity profile into a Gaussian profile with divergence. The beam is then collimated by a double concave lens (DCV) and a double convex lens (DCX). DMD 1 generates the input image by reflecting the incident beam by tilting micromirrors to on-off positions. When the light is reflected on the DMDs, diffraction of the beam occurs due to the periodic structure of the micromirrors. The beam is divided into several directions by this diffraction, similar to blazed diffraction gratings, with the angle of the diffracted beam following the equation below,
where a, ${\theta _m}$, ${\theta _i}$,$m$, and $\mathrm{\lambda }$ are the period of blazed grating, angle of the diffracted beam, angle of the incident beam, order of diffraction, and wavelength of the incident light, respectively. Since the tilt axis of the micromirrors has a diagonal direction of the rectangular active area of the DMD, the DMDs are installed at an angle of 45° so that the beam path is parallel with the optical table and the hinge axis is perpendicular to the table surface. The incident beam is aligned to have an incidence angle of 24° with respect to the DMD surface, and one of the diffracted beams propagates along an almost orthogonal direction with respect to the DMD surface when the micromirror is at the ‘on’ position. As one diffracted beam is chosen to be imaged in the system, the rest of the diffracted beams are dumped, consequently leading to a low energy efficiency of the optical system.An optical Fourier transform of the input image generated by DMD 1 is made by the DCX and multiplied with the kernel function on DMD 2. Then a Fourier transform is made by the next DCX again on the camera sensor surface, after which a convolution is performed by the optical system. A wedge prism is used to compensate the path difference between the rays propagating inside and outside of the beam path for proper imaging.
In such systems using DMDs, there are three issues to be solved for high bit-depth image computation. The first is the low energy efficiency of the system by diffraction from the DMDs, where the energy efficiency of the system decreases with an increasing number of diffraction beams. The second issue concerns the grayscale of the image when a high bit-depth image is computed, where grayscale nonlinearity is caused by the diffraction and the position inaccuracy of the DMD micromirrors. These first two issues can be minimized by optimizing the angle of the incident beam on DMD 1 to control the number of diffracted beams and the angle of diffraction. The last issue is the appearance of dark lines at certain radiance levels in high bit-depth grayscale images, a phenomenon that occurs when a coherent and collimated beam is applied in a DMD system based on bit-plane operation. These issues are covered in the following two sections.
3. Grayscale linearity of a high bit-depth image
For proper operation of an optical 4f-system using DMDs for high bit-depth image information processing, the linearity of the grayscale of the image generated by DMD 1 is one of the most critical factors in terms of computation error. Diffraction from the DMDs and the inaccuracy of the on-off positions of the DMD micromirrors distort the grayscale. Figure 2(a) shows a measured image using the optical 4f-system (Fig. 1) from an input image [inset in Fig. 2(a)] with a gradient from 0 to 255 (8-bit) in the lateral direction generated by DMD 1. For the measurement, DMD 2 is set to be totally white, meaning all of the micromirrors are at the ‘on’ position. The measured image shows a quite different result from the input image even at a glance: the irradiance is not linear, and vertical dark lines appear periodically on the measurement image. The irradiance with respect to the lateral position along the dashed line in Fig. 2(a) is plotted in Fig. 2(b), where it can be seen that the measured irradiance differs significantly from the linear function (red dashed line). It should be noted that this issue is likewise present for imaging systems in which one camera, one DMD, and one lens are each 2f apart. As a note, the camara linearity is calibrated by the manufacturer in advance (Ximea, model CB019MG-LX-X8G3).
Fig. 2. (a) Measured image from an input image with a gradient from 0 to 255 when DMD 2 is set to white (inset: input image on DMD 1). (b) Plot of irradiance with respect to the lateral position of the measured image.
In order to look deeper into the diffraction phenomenon, the intensities of the beams diffracted from DMD 1 were measured in the time domain using a photodetector. For this, the input image on DMD 1 was a constant gray image with a radiance level of 232 that is arbitrarily chosen (the radiance level is between 0–255 for 8-bit operation); Fig. 3(a) shows an image of the diffracted beams projected on a white screen. As the envelope of diffraction power includes high-order diffracted beams, several images are projected on the screen. The brightest four images are the closest to the center of the optical power envelope, and their optical power in the time domain was measured. The result in Fig. 2 is the imaging of the beam at the + x position among the diffracted beams shown in Fig. 3(a). The hinge axis of the micromirrors is parallel with the y-axis when the propagation direction is defined as the z-axis. Figure 3(b) shows the optical power in the time domain of the beams at + y and –y positions. Both signals are measured to behave similarly, and there is no significant deviation between the two signals. However, the signals at + x and –x positions behave quite differently, as shown in Fig. 3(c). Both optical power signals vary significantly with time at each bit-plane for 8-bit depth image information processing, and the direction of their variation is opposite. Based on the results, the linearity problem in Fig. 2(a) is inferred to come from inaccuracy of the micromirror positions deviating from the exact on-off positions at each bit-plane, since the micromirror angles determine the angle of maximum optical power of the diffraction. We believe that it is quite complex to get rid of the deviation of the micromirrors at each bit-plane, and thus the 4f-system should be re-designed to tolerate the deviations. To reduce the variation in the optical power signals with time, the number of diffracted beams should be minimized to not divide the beams into several paths. While optimizing the diffraction angle is a well-known method for energy efficiency, we expect that it will also help to improve the linearity of the irradiance for the diffracted beam [31–33]. Equation (2) shows the optical power envelope ($I$) of diffracted beams with respect to the diffracted angle (${\theta _m}$) and incidence angle (${\theta _i}$) [32,33],
Fig. 3. (a) Image of diffracted beams projected on a white screen when ${\theta _i} = 24^\circ $. (b) Optical power signals in the time domain from diffracted beams at + y and –y, and (c) those at + x and –x.
Fig. 4. (a) Power ratio between two diffraction orders having the highest and second-highest power. (b) Grating condition and optical power envelope function when the incidence angle is 24° and (c) 44°.
Figure 5(a) shows the diffracted beams projected on a white screen with a DMD 1 incidence angle of 44°. The input image is a constant gray image with a radiance level of 240 in 8-bit depth. The number of diffracted beams is quite reduced, and particularly, the number of diffracted beams on the x-axis reduced to one. Figure 5(b) shows the optical power signal with time of the diffracted beam at the center of the screen. The optical power signal is constant for the picture time of the DMD compared to the results in Fig. 3(c), even though the deviations of the micromirror positions still exist.
Fig. 5. (a) Image of diffracted beams projected on a white screen when ${\theta _i} = 44^\circ $. (b) Optical power signal in the time domain from the diffracted beam at the center of the screen.
Based on the results, we modified the 4f-system with the optimized diffraction angle on DMD 1 for the appropriate implementation of linear grayscale imaging as shown in Fig. 6. The incidence angle was set to 44° and the diffracted beam (m=3) from DMD 1 having an angle of 20° was chosen to be processed. A wedge prism was used between DMD 1 and the DCX to steer the beam for perpendicular incidence on DMD 2. The rest of the components are identical with the setup in Fig. 1. Figure 7(a) shows the measured gradient image from 0 to 255 in the lateral direction using the modified optical 4f-system. As shown in Fig. 7(b), the grayscale image irradiance was measured to be linear except for a few notches at certain radiance levels. The notches correspond to the vertical dark lines in Fig. 7(a). We discuss the factors and figure out a solution to this phenomenon in the next section.
Fig. 6. Modified optical 4f-system with an optimized diffraction angle for appropriate implementation of linear grayscale imaging.
Fig. 7. (a) Measured gradient grayscale image from 0 to 255 in the lateral direction using the modified optical 4f-system. (b) Irradiance profile along the white dashed line in (a).
4. Dark lines between two different radiance levels
As shown in Fig. 7(a), dark lines are captured when a DMD generates an 8-bit grayscale image even when the diffraction angle is optimized for a linear grayscale. To find out the factors behind this phenomenon, we consider the operation mechanism of the DMDs for high bit-depth images, as shown in Fig. 8(a) for an 8-bit depth grayscale image. As a well-known method, eight bit-planes switch ‘on’ and ‘off’ to express 0 to 255 radiance levels with a different time duration of each bit-plane, namely multiples of two when the input light source is continuous wave (CW). Figure 8(b) shows numerically calculated images at each bit-plane with the 2f imaging model when a DMD generates an 8-bit gradient image using the method shown in Fig. 8(a). The 2f imaging model uses the Rayleigh–Sommerfeld equation (Eqs.3)–35,37 in Ref. [34]) for its propagation and calculates the field amplitude for its final image [34–37]. The final numerically calculated image with time integration for the bit-planes is shown in Fig. 9(a). The result seems to have a linear grayscale, but there are a few vertical dark lines at certain radiance levels. The major factor of this phenomenon is the diffraction of the image at each bit-plane. Figure 9(b) shows the irradiance profile at the border between ‘on’ and ‘off’ positions with the light source having a wavelength of 633 nm and various focal lengths of the lens. Even though the original radiance function at each bit-plane is a step function, the captured image shows continuous variation and an irradiance that is lower than half of the original value at the edge of the ‘on’ area. This means that the combination of the numerically calculated images of each bit-plane will have a lower irradiance at the borders of ‘on’ and ‘off’, where the resulting dark lines are more severe in the bit-planes with longer time durations. The three dark lines that are the most easily noticeable in Fig. 9(a) correspond to the borders of ‘on’ and ‘off’ in the bit-planes of bit-6 and bit-7 having the longest time duration among the bit-planes.
Fig. 8. (a) Typical 8-bit grayscale operation method of DMDs and (b) numerically calculated images of each bit-plane for a horizontal gradient image from the Rayleigh–Sommerfeld propagation model.
Fig. 9. (a) Numerically calculated result of a horizontal gradient image with the Rayleigh–Sommerfeld propagation model. (b) Irradiance profile at the on-off border of the micromirrors with a 633 nm wavelength light source and various focal lengths of the lens.
To reduce the effect of diffraction in the image of each bit-plane, the time duration of the on-off border staying at a certain position should be minimized. Figure 10(a) shows the proposed DMD operation method to minimize the diffraction and get rid of the dark lines from the final image. In our method, we can control the radiance level of each pixel (except for the radiance level of 0) by changing the time when the micromirrors turn to the ‘off’ position from an initial state of all micromirrors at the ‘on’ position. Accordingly, the time duration of the border staying at a certain position can be minimized. However, due to mechanical movement limitations, the time duration to generate an image will increase, and as a consequence, the optical information processing speed will be deteriorated. We suggest a new moderate solution for DMD operation as follows. We sliced an 8-bit image (256-level, 0–255 radiance level) into a total of 17 4-bit (16-level, 0–15 radiance level) images with respect to the radiance level of each pixel. Each 4-bit image contains the radiance level information from 0 to 15, 16 to 31, 32 to 47, and so on. The radiance level information out of range for each image will stay at the ‘on’ or ‘off’ position, i.e., the 4-bit image having radiance level information from 32 to 47 will have black pixels (‘off’ position) where the radiance level is lower than 32 and white pixels (‘on’ position) where the radiance level is higher than 47. Figure 10(b) shows a horizontal gradient image divided into 17 4-bit slides with respect to radiance level. In this case, the DMDs will project the success of the 17 4-bit image slides for generation of a single 8-bit image. It should be noted that the number of divided images can be reduced with higher bit-depth sliced images when the diffraction at the edge of the on-off border is low enough. Figure 10(c) shows the captured image with the modified optical 4f-system shown in Fig. 6 adopting the suggested DMD operation method. The resulting image shows a horizontal gradient without dark lines, and as shown in Fig. 10(d), the irradiance profile along the white dashed line is linear without notches.
To compare the captured images with and without the diffraction optimization suggested in this paper, we projected an 8-bit grayscale peppers image and captured it with the two different methods. Figure 11(a) shows the original input image, and Fig. 11(b), (c) show the captured images with and without diffraction optimization, respectively. As shown in Fig. 11(b), the captured peppers image without diffraction optimization has dark lines at certain radiance levels, and the grayscale seems to be unnatural compared to the original image. On the other hand, the captured peppers image with diffraction optimization shows no dark lines and no grayscale defect, as shown in Fig. 11(c). The red dashed box in the captured pepper images is magnified and shown in the insets of Fig. 11(b), (c) for a clear comparison.
Fig. 10. (a) Proposed DMD operation method to minimize the edge diffraction at the on-off border of the micromirrors. (b) Horizontal gradient image (left) and divided images (right) with respect to radiance level. (c) Captured horizontal gradient image with the modified 4f-system following the suggested method. (d) Irradiance profile along the white dashed line in (c).
Fig. 11. (a) Original 8-bit peppers image, (b) captured peppers image with the 4f-system without optimization, and (c) that with the suggested methods in this paper.
It should be noted that the experiments in this work were conducted to correct defects for improved linearity of the irradiance signal, even though typical optical 4f-systems operate on field amplitude. We believe that the linearity of the field amplitude can be guaranteed by defect correction for the linearity of the irradiance signal based on their mathematical relation, where the irradiance signal is proportional to the square of the field amplitude value.
5. Conclusion
The diffraction phenomena and factors that can cause irradiance defects and unintended dark lines in DMD-based optical 4f-systems have been studied. We revealed that the distortion of the grayscale is caused by deviation from the exact on-off positions of the micromirrors, and also that one of the major factors of the dark lines at certain radiance levels is diffraction at the edge of the micromirrors in each bit-plane. Based on the numerical calculation of blazed diffraction grating equations and the Rayleigh–Sommerfeld propagation model, the incidence and diffraction angles were optimized to generate a proper linearity of the grayscale image, and a new DMD operation method was proposed for implementation of optical information processing without dark lines. The operation method is based on dividing the high bit-depth image into sliced images in terms of radiance levels to reduce the effect of micromirror edge diffraction. By optimizing the incidence angle for proper optical information processing, the energy efficiency of the optical system can also be increased. Applying our findings, we demonstrated a modified optical 4f-system using DMDs without any defects in the image information. We believe that the presented optimization approach to the optical characteristics is highly applicable for convolutional neural networks using DMD-based optical 4f-systems.
Funding
Institute of Information & Communications Technology Planning & Evaluation (IITP) (No.2021-0-00019).
Acknowledgments
This work was supported by an Institute of Information & Communications Technology Planning & Evaluation (IITP) grant funded by the Korean government (MSIT) (No.2021-0-00019, Research on Optical Learning Technology for AI).
Disclosures
The authors declare no conflicts of interest
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request
References
1. J. J. Hopfield, “Artificial neural networks,” IEEE Circuits Devices Mag. 4(5), 3–10 (1988). [CrossRef]
2. A. K. Jain, J. Mao, and K. M. Mohiuddin, “Artificial neural networks: a tutorial,” Computer 29(3), 31–44 (1996). [CrossRef]
3. A. Krogh, “What are artificial neural networks?” Nat. Biotechnol. 26(2), 195–197 (2008). [CrossRef]
4. K. Sun, J. Chen, and X. Yan, “The Future of Memristors: Materials Engineering and Neural Networks,” Adv. Funct. Mater. 31(8), 2006773 (2021). [CrossRef]
5. T. Zanotti, F. M. Puglisi, and P. Pavan, “Energy-Efficient Non-Von Neumann Computing Architecture Supporting Multiple Computing Paradigms for Logic and Binarized Neural Networks,” J. Low Power Electron. Appl. 11(3), 29 (2021). [CrossRef]
6. A. Ganguly, R. Muralidhar, and V. Singh, “Towards Energy Efficient non-von Neumann Architectures for Deep Learning,” in Proceedings of IEEE International Symposium on Quality Electronic Design (IEEE, 2019), pp. 335–342.
7. M. Miscuglio, A. Mehrabian, Z. Hu, S. I. Azzam, J. George, A. V. Kildishev, M. Pelton, and V. J. Sorger, “All-optical nonlinear activation function for photonic neural networks [Invited],” Opt. Mater. Express 8(12), 3851 (2018). [CrossRef]
8. Y. Zuo, B. Li, Y. Zhao, Y. Jiang, Y.-C. Chen, P. Chen, G.-B. Jo, J. Liu, and S. Du, “All-optical neural network with nonlinear activation functions,” Optica 6(9), 1132 (2019). [CrossRef]
9. J. Feldmann, N. Youngblood, C. D. Wright, H. Bhaskaran, and W. H. P. Pernice, “All-optical spiking neurosynaptic networks with self-learning capabilities,” Nature 569(7755), 208–214 (2019). [CrossRef]
10. R. Hamerly, L. Bernstein, A. Sludds, M. Soljačić, and D. Englund, “Large-Scale Optical Neural Networks Based on Photoelectric Multiplication,” Phys. Rev. X 9(2), 021032 (2019). [CrossRef]
11. J. Bueno, S. Maktoobi, L. Froehly, I. Fischer, M. Jacquot, L. Larger, and D. Brunner, “Reinforcement learning in a large-scale photonic recurrent neural network,” Optica 5(6), 756 (2018). [CrossRef]
12. S. Xiang, Y. Han, Z. Song, X. Guo, Y. Zhang, Z. Ren, S. Wang, Y. Ma, W. Zou, B. Ma, S. Xu, J. Dong, H. Zhou, Q. Ren, T. Deng, Y. Liu, G. Han, and Y. Hao, “A review: Photonics devices, architectures, and algorithms for optical neural computing,” J. Semicond. 42(2), 023105 (2021). [CrossRef]
13. X. Sui, Q. Wu, J. Liu, Q. Chen, and G. Gu, “A Review of Optical Neural Networks,” IEEE Access 8, 70773–70783 (2020). [CrossRef]
14. F. Duport, B. Schneider, A. Smerieri, M. Haelterman, and S. Massar, “All-optical reservoir computing,” Opt. Express 20(20), 22783 (2012). [CrossRef]
15. T. Wang, S.-Y. Ma, L. G. Wright, T. Onodera, B. C. Richard, and P. L. McMahon, “An optical neural network using less than 1 photon per multiplication,” Nat. Commun. 13(1), 123 (2022). [CrossRef]
16. S. Xiang, Z. Ren, Z. Song, Y. Zhang, X. Guo, G. Han, and Y. Hao, “Computing Primitive of Fully VCSEL-Based All-Optical Spiking Neural Network for Supervised Learning and Pattern Classification,” IEEE Trans. Neural Netw. Learn. Syst. 32(6), 2494–2505 (2021). [CrossRef]
17. R. Tang, W. Wang, Z. Tu, and J. Lin, “An Experimental Analysis of the Power Consumption of Convolutional Neural Networks for Keyword Spotting,” in Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 2018), pp. 5479–5483.
18. T.-J. Yang, Y.-H. Chen, J. Emer, and V. Sze, “A Method to Estimate the Energy Consumption of Deep Neural Networks,” in Proceedings of Asilomar Conference on Signals, Systems, and Computers (IEEE, 2017), pp. 1916–1920.
19. T. Ujiie, M. Hiromoto, and T. Sato, “Approximated Prediction Strategy for Reducing Power Consumption of Convolutional Neural Network Processor,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition Workshops (IEEE, 2016), pp. 870–876.
20. S. Xu, J. Wang, R. Wang, J. Chen, and W. Zou, “High-accuracy optical convolution unit architecture for convolutional neural networks by cascaded acousto-optical modulator arrays,” Opt. Express 27(14), 19778 (2019). [CrossRef]
21. M. Miscuglio, Z. Hu, S. Li, J. K. George, R. Capanna, H. Dalir, P. M. Bardet, P. Gupta, and V. J. Sorger, “Massively parallel amplitude-only Fourier neural network,” Optica 7(12), 1812 (2020). [CrossRef]
22. J. Liu, Q. Wu, X. Sui, Q. Chen, G. Gu, L. Wang, and S. Li, “Research progress in optical neural networks: theory, applications and developments,” PhotoniX 2(1), 5 (2021). [CrossRef]
23. J. Chang, V. Sitzmann, X. Dun, W. Heidrich, and G. Wetzstein, “Hybrid optical-electronic convolutional neural networks with optimized diffractive optics for image classification,” Sci. Rep. 8(1), 12324 (2018). [CrossRef]
24. Z. Gu, Y. Gao, and X. Liu, “Optronic convolutional neural networks of multi-layers with different functions executed in optics for image classification,” Opt. Express 29(4), 5877 (2021). [CrossRef]
25. Y. Meuret and P. D. Visschere, “Contrast-improving methods for Digital Micromirror Device projectors,” Opt. Eng. 42(3), 840–845 (2003). [CrossRef]
26. M.-C. Park, B.-R. Lee, J.-Y. Son, and O. Chernyshov, “Properties of DMDs for holographic displays,” J. Mod. Opt. 62(19), 1600–1607 (2015). [CrossRef]
27. S. Scholes, R. Kara, J. Pinnell, V. Rodríguez-Fajardo, and A. Forbes, “Structured light with digital micromirror devices: a guide to best practice,” Opt. Eng. 59(4), 041202 (2019). [CrossRef]
28. D. Woods and T. J. Naughton, “Photonic neural networks,,” Nat. Phys. 8, 257–259 (2012). [CrossRef]
29. J. Liang, S.-Y. Wu, R. N. Kohn, M. F. Becker, and D. J. Heinzen, “Grayscale laser image formation using a programmable binary mask,” Opt. Eng. 51(10), 108201 (2012). [CrossRef]
30. D. Benton, “Aberration and coherence effects with a micromirror array,” Proc. SPIE 11867, 118670D (2021). [CrossRef]
31. A. Sandmeyer, M. Lachetta, H. Sandmeyer, W. Hübner, T. Huser, and M. Müller, “DMD-based super-resolution structured illumination microscopy visualizes live cell dynamics at high speed and low cost,” bioRxiv 797670 (2019).
32. L. Li, J. Chandezon, G. Granet, and J.-P. Plumey, “Rigorous and efficient grating-analysis method made easy for optical engineers,” Appl. Opt. 38(2), 304 (1999). [CrossRef]
33. R. Casini and P. G. Nelson, “On the intensity distribution function of blazed reflective diffraction gratings,” J. Opt. Soc. Am. A 31(10), 2179 (2014). [CrossRef]
34. Sommerfeld, Lectures on Theoretical Physics (Academic Press, 1954).
35. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996).
36. M. Born and E. Wolf, Principles of Optics (Pergamon Press, 1999).
37. S. Mehrabkhani and T. Schneider, “Is the Rayleigh-Sommerfeld diffraction always an exact reference for high speed diffraction algorithms?” Opt. Express 25(24), 30229 (2017). [CrossRef]
