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Abstract
In this paper, we investigate the theoretical models and potential applications of spatial diffractive neural network (SDNN) structures, with a particular focus on mode manipulation. Our research introduces a novel diffractive transmission simulation method that employs matrix multiplication, alongside a parameter optimization algorithm based on neural network gradient descent. This approach facilitates a comprehensive understanding of the light field manipulation capabilities inherent to SDNNs. We extend our investigation to parameter optimization for SDNNs of various scales. We achieve the demultiplexing of 5, 11 and 100 orthogonal orbital angular momentum (OAM) modes using neural networks with 4, 10 and 50 layers, respectively. Notably, the optimized 100 OAM mode demultiplexer shows an average loss of 0.52 dB, a maximum loss of 0.62 dB, and a maximum crosstalk of -28.24 dB. Further exploring the potential of SDNNs, we optimize a 10-layer structure for mode conversion applications. This optimization enables conversions from Hermite-Gaussian (HG) to Laguerre-Gaussian (LG) modes, as well as from HG to OAM modes, showing the versatility of SDNNs in mode manipulation. We propose an innovative assembly of SDNNs on a glass substrate integrated with photonic devices. A 10-layer diffractive neural network, with a size of 49 mm × 7 mm × 7 mm, effectively demultiplexes 11 orthogonal OAM modes with minimal loss and crosstalk. Similarly, a 20-layer diffractive neural network, with a size of 67 mm × 7 mm × 7 mm, serves as a highly efficient 25-channel OAM to HG mode converter, showing the potential of SDNNs in advanced optical communications.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the advancement of optical communication technology, various multiplexing techniques such as wavelength-division multiplexing (WDM) [1,2], time-division multiplexing (TDM) [3], polarization-division multiplexing (PDM) [4], and advanced modulation methods [5–9] have been utilized to make full use of different dimensions of light waves, including wavelength, time, polarization, and complex amplitude. However, the capacity of optical communication systems is limited by noise and the Shannon limits [10,11], hindering significant further increment. To overcome these limitations, space-division multiplexing (SDM) [12–14] has emerged as a promising approach to enhance the capacity and spectral efficiency of optical communication systems by utilizing spatial dimension resources. Mode-division multiplexing (MDM) technology [15,16], as an important branch of SDM, has attracted significant attention in recent years. MDM technology utilizes optical modes that exhibit spatially overlapping and mutually orthogonal characteristics, including linearly polarized (LP) [15–20] modes, orbital angular momentum (OAM) modes [21–32], and vector modes [33–36].
Similar to the core components in WDM systems such as multiplexers, demultiplexers [37,38], and photonic switches [39,40], it is widely recognized that mode manipulation components, including mode multiplexers, demultiplexers, and mode converters, play a crucial role in the commercialization of MDM systems. These mode manipulation components are designed to possess key features such as large channel numbers, low crosstalk, low loss, small size, and low cost. By achieving these key features, they enable powerful mode manipulation capabilities, facilitating the establishment of high-speed and large-capacity optical communication systems with sustainable capacity expansion.
The lack of photonic integration in many previously reported MDM core component schemes often resulted in constraints regarding size, stability, and scalability [41–50]. On the other hand, a range of MDM core component schemes based on photonic integration [32,51–55] consistently demonstrated limitations concerning loss, crosstalk, or expandability. Despite the presence of large channel numbers and low loss, there remains a dearth of mature photonic integration schemes proficient at effectively controlling crosstalk.
To simultaneously fulfill these performance requirements, the SDNN offers an innovative solution, leveraging its extensive adjustable dimensions. The single-phase plane scheme [56] and the multi-plane scheme [57–59] have been proposed and show similarities with our SDNN scheme, which could be broadly termed an SDNN structure. However, in a stricter sense of SDNN, the structure should be entirely parallel and consist of at least two planes. The differences result in the broader interpretation of the SDNN structure still exhibiting scalability limitations due to shallow planes. This issue is an inherent limitation resulting from the reflective cascaded structure and the inefficient algorithm. The reflective cascade structure induces spot amplification and accumulative inclination angles. The inefficient algorithm, on the other hand, leads to high time consumption and rapid exhaustion of computing resources.
While certain design and optimization methods have been utilized to aid the application of shallow SDNN structures, a thorough discussion concerning deep structures is notably lacking. Regarding optimization algorithms, the implementation of a deep network introduces an exceptionally large number of parameters via the SDNN structure, posing significant challenges. The conventional global gradient descent algorithm [60,61] in tandem with a standard diffraction calculation method [61] is insufficient to optimally support the deep SDNN structure. Therefore, there's an urgent need to develop a new optimization algorithm based on the gradient flow of spatial diffractive neural networks. From a component-building perspective, the material system most conducive to achieving superior performance remains an open question. Spatial light modulators (SLM) [62] based on silicon-based liquid crystals are widely used in optical mode manipulation. However, they exhibit a relatively large size, limited resolution, high cost, and most notably, difficulties in cascading and forming deep networks due to their reflective structure. Therefore, exploring alternative material schemes is a worthwhile endeavor.
In this paper, we propose a simulation and optimization method for SDNN structures that can accommodate vast parameter numbers and extensive layers of deep networks. We examine the factors influencing the mode manipulation capabilities of SDNN, asserting that a deeper network provides enhanced manipulation capacity. Furthermore, we optimize the SDNN structures to facilitate efficient mode multiplexing and conversion applications. To address the essential requirements of mode manipulation components, we introduce a spatial diffractive neural network scheme featuring a photonic integrated device assembly on a glass substrate. This configuration distinctly offers benefits such as compact size, low loss, easy alignment, scalability, and cost-effectiveness. We evaluate the efficiency and process of multi-task mode manipulation, demonstrating favorable performance.
2. Concept of the spatial diffractive neural network (SDNN)
Figures 1(a) and 1(b) depict the schematic structures of a shallow and a deep multi-layer spatial diffractive neural network, respectively. The proposed deep and high-resolution multi-layer spatial diffractive contains a sufficiently long diffraction distance and a lot of parallel equidistant phase planes. Phase plans bring a large number of phase units into a structure which significantly increases adjustable parameters for optimization. Sufficient diffraction distance enhances the intensity manipulation caused by phase adjustment. As a result, the proposed deep SDNN structure is expected to enhance mode manipulation capabilities due to a greater number of possible optimized states and a more pronounced optical coherence effect than other structures.
Fig. 1. (a) Structure schematic diagram of shallow spatial diffractive neural network. (b) Structure schematic diagram of deep multi-layer spatial diffractive neural network.
On each phase plane of the proposed deep SDNN structure has a large number of neurons, and the number of diffracted full connections physically existing between the two layers will be on the order of the fourth power of the number of discrete samples. Specifically, if the number of discretization expands by 8 times, the number of neurons increases by 64 times, and the number of connections will expand by 4096 times. This expansion will significantly reduce the computational speed of the system that simulate performance of the deep SDNN and the required memory will increase substantially. A deep optical neural network optimization method modified by a new deep learning technology is needed to greatly improve the optimization speed and alleviate the limit of maximum design layers number caused by computing power constraints. Our proposed deep learning method starts from the physical optics theory of space diffraction transmission, and combines with the traditional mathematical methods of discretization, matrix and fast Fourier transform. Adopting the theoretical improvement aims at realizing deep special diffraction neural networks, optimizing structure performance, improving the parallelism, and promoting the development of the optical mode manipulation field.
3. Simulation and optimization method of SDNN
The propagation and diffraction of light in SDNN structures could be regarded as many multiple repeated processes from a previous plane to the next plane as shown in Fig. 2. Firstly, the principle diagram of the forward transmission process reveals that each period of the periodic structures comprises a diffraction distance and a single phase plane modulation. In this setup, every light wave unit that emerges from one phase plane is directly connected through diffraction to the subsequent light wave unit before it encounters the next phase plane. This creates a continuous and complete connection between the units of two adjacent light wave planes in the diffraction transmission process. Significantly, the coefficient defining this diffraction connection is determined solely by the distance between the two light wave units and the intrinsic properties of the light wave.
Fig. 2. Principle diagram of the forward transmission process of the electric field going through several parallel phase planes with zoom in principle diagram of matrix multiplicative connections and matrix in simplified forward transmission calculation of 2 periods of SDNN structure.
One period of the processes can be expressed by
where ${E_i}$ and ${E_{i - 1}}$ are field distributions at the previous and i-th plane. ${U_{i - 1}}$ is used to represent one time of diffraction transmission function from the previous plane to the i-th plan. The symbol of⊙ represents dot multiplication. ${P_i}$ represents the phase units matrix of the i-th plane, exp() means an exponential function based on the natural constant e. In one of the periodic processes, the simulation could be separated into two steps. In the first step, the propagation and diffraction between two planes is simulated. In the second step, phase modulation of the traversing plane is simulated. In the situation that the next plane is parallel to the previous plane with no reflection in every plane, the light field simulation problem could be further simplified through a diffraction formula written byThe precise electromagnetic field in every plane can be calculated step by step. As the electric field is the primary component, we will focus solely on the formula for the electric field in further studies, considering that the formula for the magnetic field follows a similar pattern. By separating the different spatial-frequency components in the preceding plane-wave through the Fast Fourier Transform (FFT) method, propagation and diffracted light fields before the next plane can be swiftly computed using a dot product, maintaining low computational complexity. Subsequently, the Inverse Fast Fourier Transform (IFFT) method is employed to return the calculation scene to the spatial domain, as illustrated by:
where $\mathrm{{\cal F}}$ is two-dimension Fourier transform, ${\mathrm{{\cal F}}^{ - 1}}$ is a two-dimension inverse Fourier transform. Herein the phase modulation values could be quickly calculated using the dot product operation, maintaining low computational complexity. It's important to note that convolution operations are generally more complex than spatial-frequency transformation and dot product operations, especially in scenarios requiring high plane precision and numerous layers. By employing the FFT and IFFT methods interchangeably, we can eliminate the need for convolution operations, which are typically associated with high computational burden and time consumption. Furthermore, akin to the fundamental principle of the angular spectrum method, the FFT and IFFT processes can be further simplified in terms of matrix multiplication. This simplification is based on the following formula:Finally, the simulation method is well simplified to efficiently calculate forward transmission in SDNN structures. Benefiting from the simplification of the forward transmission simulation method, only matrix multiplication and dot product operation are involved without the convolution operation of a large matrix. So, a neural network gradient descent algorithm could be used to optimize deep SDNN structures as shown in Fig. 3.
Fig. 3. Principle diagram of gradient backward transmission method in SDNN and the flow direction of both electric field (forward) and gradient (backward).
The gradient reverse transmission in diffraction optics is more complicated than the normal gradient reverse transmission in computer science. The gradient of the electromagnetic field is composed of intensity and phase part. Therefore, both the real and imaginary parts of light field should be taken into account. There are two key approaches for deriving the gradient backpropagation formula under operations with complex matrices. The first is to easily decompose simple matrix operations between complex matrices into slightly more complicated operations between two real matrices using the rules of complex arithmetic. The second is that after a series of optimizations, the diffraction transmission process has been transformed into a combination of pure matrix operations. And then, though differentiating the formula (12), the backpropagation process of the gradient matrix is expressed by
It can be further simplified by
To avoid local optimal solutions and speed up convergence, Nesterov and Adam methods are used to improve the optimization method. Through these modified methods, the global gradient of final loss in every phase unit could be accurately calculated, which has taken the multi-group of aim optical field into account. Finally, combining global gradient and learning rate, the change value of phase units in each round of optimization could be determined and is beneficial to most of the stargets.
In addition, we evaluate the optimization method using an NVIDIA 1080ti graphics card. The significant advancements in our approach can largely be credited to two key developments. The first was the incorporation of gradient descent algorithms from the realm of deep learning, combined with the acceleration capabilities of GPUs. The second enhancement involved leveraging alternating two-dimensional Fourier and inverse Fourier transforms, as well as the matrix-based implementation of fast Fourier transforms, to avoid the convolution computations characterized by high computational complexity, or in other terms, dense diffractive convolutional connections. Due to the limitation of the initial approach, we compared performance before and after the implementation of the second enhancement strategy using different resolution. We selected a lower resolution (64 × 64) before the improvement and a higher resolution (512 × 512) after. Due to the inability of initial approach to produce results in high-layer scenarios, a two-layer diffractive neural network is used for comparison. With a single batch and a grid precision of 512 × 512, our method is 2083.7 times faster than before the improvement in a two-stage diffraction scenario. When using 20 batches, it is 16230.7 times faster.
4. Analysis of the influencing factors of SDNN's mode manipulation ability
The proposed deep SDNN structures are designed to incrementally enhance mode manipulation capabilities by adding more layers. Our first exploration focuses on the mode manipulation ability of a 10-layered SDNN orbital angular momentum (OAM) mode demultiplexer, capable of supporting 11 OAM modes, and then we compare it with structures containing fewer layers. For the calculations, the distance between phase planes is fixed at 15 mm, while the grid size and number for each phase plane are set to 8 μm and 256 × 256, respectively. The wavelength of 1550 nm was used in our simulations to align with the experimental conditions of our research group, providing support for future experimental work. Additionally, the input orthogonal OAM modes are formed with a relatively constant radius, corresponding to the OAM demultiplexing scenario. To ensure a fair comparison, we maintain the same number of channels, altering only the layer number, and focus on the differences in maximum channel crosstalk, maximum loss, and average loss. Those performance metrics are employed to showcase the SDNN's performance, because practical communication systems require high-performance mode demultiplexers, for which commonly used performance parameters include device loss and crosstalk [63,64].
The maximum channel crosstalk is referring to the maximum value observed in the set of crosstalk measurement for any of the activated channels on the output plane during the demultiplexing application in optical modes. The maximum channel crosstalk $Crosstal{k_{max}}$ is defined as:
where the crosstalk value for each other channel, in other words $i \ne j$, denoted as $crosstalk({i,j} )$, is calculated by summing all the discrete optical intensity values within the area of that output channel, as expressed in the following formula:Including the situation of $j = i$, $crosstalk$ matrix could be generalized to channel coupling coefficients matrix $coupling\_m$. The term loss refers to the optical intensity loss at the location of the Gaussian light spot on the output plane when any single channel is activated, with the mode light being the unit of the total input light intensity. The maximum loss $Los{s_{max}}$ and the average loss $Los{s_{ave}}$ are defined as the maximum and average values of the losses across all channels respectively, as illustrated in the two formulas below:
Notably, through optimized SDNN structures, output Gaussian light spots are evenly dispersed along the diagonal when input OAM modes are set to be coaxial. Hence, the maximum channel crosstalk, maximum loss, and average loss can directly represent the mode manipulation capability of the specific multiplexing systems. In this paper, the thresholds for loss and crosstalk are set to 1 dB and -15 dB, respectively. For simplicity, we do not detail the intensity and phase distribution of OAM modes in this section, which resemble the input light field distribution in the next section.
Figure 4(a) presents the recorded maximum channel crosstalk, maximum loss, and average loss as functions of the layer number for optimized SDNN structures. When the layer number exceeds 4, loss and crosstalk decrease slightly and linearly with the increment in layer number. Conversely, when the layer number is less than 4, loss and crosstalk exhibit unstable and marked changes with increasing layer numbers. The minimum loss and crosstalk at 10 layers are nearly 0 and -21 dB, respectively. It is evident that increasing the layer number of SDNN structures can yield better performance. Figure 4(b) and (c) illustrate the histograms of channel coupling coefficients when the layer number is set to 5 and 10, respectively. It is observed that the 5-layered SDNN incurs almost 3 dB more loss than the 10-layered SDNN. Despite the minor degradation of crosstalk, a 5-layered SDNN can still be employed in mode manipulation applications where minor loss degradation is acceptable.
Fig. 4. Simulated results of the well-optimized SDNN structure for 11 OAM modes demultiplexer. (a) Maximum channel crosstalk, maximum loss and average versus the number of the layers. (b)(c) Histograms of channel coupling coefficients for (b) 5-layer structure and (c) 10-layer structure.
Given the unstable performance when the layer number is less than 4, a deeper analysis considering the mode number is necessary. We subsequently optimize the proposed SDNN structures with layer numbers of 2, 3, 4, and 5 and mode numbers of 4, 5, 6, 7, 8, 9, and 10. The optimized structure geometries (phase planes distance: 15 mm, grid size: 8 µm, phase units number: 256 × 256) and output Gaussian light spot distribution are maintained constant. Figure 5(a) depicts the recorded maximum channel crosstalk, maximum loss, and average loss as a function of the OAM modes number for optimized 2-layer SDNN structures. Figure 5(b), (c), and (d) present the results of 3, 4, and 5-layer SDNN structures, respectively. It is evident that, with 2 layers, loss and crosstalk results for all mode numbers are significantly above the performance threshold. The best result appears at the mode number 4, still showing a high average loss of 5 dB and unacceptable crosstalk. When the layer number is increased to 3, the loss and crosstalk results for all mode numbers exhibit an improving trend, particularly at mode numbers 4 and 5. The crosstalk result improves to nearly -10 dB at a mode number of 5. At 4 layers, both maximum and average loss results decrease to less than 1 dB, while crosstalk results decrease to below -20 dB at mode numbers 4 and 5. As the mode number increases, crosstalk remains below -15 dB at a mode number of 6 and continues to stay below -14 dB at mode numbers of 7, 8, 9, and 10. With 5 layers, the crosstalk further decreases and ranges between -19 dB and -22 dB. After comprehensive consideration of the results from the four types of SDNN structures, we conclude the following: 1) crosstalk and loss decrease as the number of layers increases; 2) 4-layer SDNN structures are adequate to support demultiplexing of 5 OAM modes; 3) demultiplexing more than 6 OAM modes necessitates at least 5 layers.
Fig. 5. (a)-(c) Calculated loss performance of shallow spatial diffractive neural network versus OAM modes number when adopting (a) 2-layer structure, (b) 3-layer structure, (c) 4-layer structure, and (d) 5-layer structure.
5. Simulation of mode demultiplexing applications based on SDNN
To incorporate scalable mode-division multiplexing in practical designs, the substantial challenge of designing SDNN structures with high mode manipulation ability should be considered. As we presented in the previous section, there exists a tradeoff between mode manipulation ability and the layer numbers of the SDNN structure, resulting in limited performance when the loss and crosstalk are rationally controlled to be lower than 1 dB and -15 dB, respectively. In this section, we optimize three demultiplexers, which can efficiently separate 25, 64, and 100 modes with high efficiency, low loss, and low crosstalk.
Figure 6 illustrates the basic structure and mode manipulation principle of the proposed SDNN multiplexer and demultiplexer. Figure 6(a) shows building blocks of the overall structure, which comprise the input layer, fully connected layers, hidden layers for multiplexing, a middle transmission layer for multiplexed modes, hidden layers for demultiplexing, and the output layer. The input and output layers are used to define the start and end positions of the SDNN multiplexer and demultiplexer. The fully connected layers describe the process of diffraction transmission. We distinguish two kinds of fully connected layers based on diffraction distance. For the connection between hidden layers, we adopt a small distance of about 5∼20 µm to realize Rayleigh-Sommerfeld diffraction, which accentuates the distribution difference of diffraction transmission coefficient across the horizontal plane. For the connection between hidden layers and the input or output layer, we use a larger distance to achieve Fresnel diffraction. The purpose of distinguishing these two types of full connections is to make the spatial location of the input layer, middle layer, and output layer more flexible and to compact the structure of the hidden layer. Each hidden layer comprises square grid phase units that can be set arbitrarily between 0 and 2π. The dense phase units and deep network increase adjustable dimensions and optimization complexity dramatically, but also enhance mode manipulation ability.
Fig. 6. Basic structure and mode manipulation principle of the proposed SDNN multiplexer and demultiplexer. (a) Building blocks of the overall structure. (b) Full connection of spatial transmission. (c) Full connection of short-distance diffraction.
We again select the OAM modes basis as an example to study the mode manipulation ability limitation of a deep SDNN. The layer number is set at 50 for all 25, 64, and 100 modes situations. Meanwhile, these input orthogonal OAM modes are still formed with a relatively constant radius. To fully utilize the spatial resources and balance the loss and crosstalk of every output Gaussian light spot, we adopt a uniform square lattice distribution. As shown in Fig. 7(a), (b), (c), in the 25 OAM modes demultiplexing situation, the mixed modes are successfully demultiplexed to 25 Gaussian spots of uniform square lattice distribution in the output plane. Figure 7(d) displays the histograms of channel coupling coefficients, showing that the loss and crosstalk are almost negligible with an average conversion loss of 0.017 dB, maximum loss of 0.027 dB, and maximum crosstalk of -23.02 dB.
Fig. 7. Simulated results of the well-optimized 50-layers SDNN structure demultiplexer for 25 OAM modes with phase units gird of 256 × 256. (a) Intensity distribution and (b) phase distribution among the horizontal plane of input modes. (c) Intensity distribution among horizontal plane of Gaussian light spots. (d) Histogram of channel coupling coefficients.
Compared to the 10-layer SDNN OAM mode demultiplexer, the 50-layer SDNN shows significant improvements in mode manipulation ability. These improvements stem from a deeper network and better light spot distribution. To further explore the potential of 50-layer SDNN structures with a phase units grid of 256 × 256, we increased the mode number to 64, which directly results in a smaller separation distance of Gaussian light spots and more parallel tasks. The intensity and phase distributions across the horizontal plane of input modes are shown in Fig. 8(a), (b). Using the same optimization method and settings, the optimized 50-layer SDNN demultiplexer successfully separates 64 orthogonal OAM modes into 64 Gaussian spots of uniform square lattice distribution in the output plane, as shown in Fig. 8(c). Figure 8(d) presents the histograms of channel coupling coefficients, showing an average conversion loss of 0.15 dB, a maximum loss of 0.21 dB, and maximum crosstalk of -14.91 dB. Although these results meet the standards, they exhibit higher loss and crosstalk than in the 25 OAM modes situation. This doesn't necessarily suggest that the structure has reached its limit. Notably, only crosstalk seems to have reached a limit, predictably restricted by the separation distance of Gaussian light spots.
Fig. 8. Simulated results of the well-optimized 50-layer SDNN structure demultiplexer for 64 OAM modes with phase units gird of 256 × 256. (a) Intensity distribution and (b) phase distribution among the horizontal plane of input modes. (c) Intensity distribution among horizontal plane of Gaussian light spots. (d) Histogram of channel coupling coefficients.
Given the finite area of each phase plane, the separation distance of Gaussian light spots also significantly influences the SDNN's mode manipulation ability in demultiplexing applications. We increased the unit grid to 512 × 512 for a fair comparison, maintaining the separation distance within a small range of variation. To further explore the potential of 50-layer SDNN structures, we increased the modes number to 100. Using the same optimization methods and settings, the optimized 50-layer SDNN demultiplexer successfully separates 100 orthogonal OAM modes into 100 Gaussian spots of uniform square lattice distribution in the output plane, as shown in Fig. 9(a). Figure 9(b) displays the histograms of channel coupling coefficients, indicating an average conversion loss of 0.52 dB, a maximum loss of 0.62 dB, and maximum crosstalk of -28.24 dB. The high level of crosstalk validates that crosstalk is mainly influenced by the spatial distribution of Gaussian light spots. The loss in the 50-layer 512 × 512 SDNN OAM modes demultiplexer also suggests the potential for supporting more multiplexing modes. For realizing an ultra-large-scale SDNN modes demultiplexer, the unit grid is a key factor, limited by computing power. With increasing computational power, especially with advancements in GPU acceleration technology, cloud computing, quantum computing, and other technologies, it is foreseeable that larger-scale SDNN modes demultiplexers will become possible.
Fig. 9. Simulated results of the well-optimized 50-layer SDNN structure demultiplexer for 100 OAM modes with phase units gird of 512 × 512. (a) Intensity distribution among horizontal plane of Gaussian light spots. (b) Histogram of channel coupling coefficients.
6. Simulation of mode conversion applications based on SDNN
Significant mode mismatching between different transmission mediums often leads to substantial coupling loss and restricted performance. Addressing the challenge of achieving compatibility across various MDM mediums — including multi-mode fiber, few-mode fiber, ring-core fiber, and silicon waveguides — is paramount, especially when accommodating diverse MDM techniques based on Hermite-Gaussian, Laguerre-Gaussian (LG), and Orbital Angular Momentum (OAM) mode bases. In this context, developing compatible mode converters based on SDNN technology becomes crucial. We have successfully optimized two SDNN converters for efficient mode conversion.
Figure 10 illustrates the structure and principle of MDM converters based on SDNN structures. On the left side, the input mode basis adopts Hermite-Gaussian mode profiles as an example. On the right side, the output mode basis adopts OAM mode profiles as an example. The ellipsis in Fig. 10 indicates that the optimized converter can convert multiple inputs into multiple outputs simultaneously.
To evaluate the performance of these proposed MDM converters, Fig. 11 illustrates simulated results for a 10-layer SDNN-based converter, transitioning from HG to LG modes. Figures 11(a) and (b) display the intensity and phase distribution of 11 input HG modes. Unlike optimization in multiplexing applications where phase considerations are less critical, mode converter optimization necessitates careful attention to the phase of light. The intensity and phase distribution of the output 11 LG modes are depicted in Figs. 11(c) and (d), confirming the successful realization of the desired mode manipulation. For the HG to LG mode conversion, our findings indicate a maximum loss of 0.68 dB and an average loss of 0.11 dB. These outcomes affirm the capability of a 10-layer SDNN structure to adeptly convert 11 orthogonal HG modes into 11 orthogonal LG modes.
Fig. 11. Simulated results of the well-optimized 10-layer SDNN structure HG mode to LG mode converter for 11 channels with phase units gird of 256 × 256. (a) Intensity distribution and (b) phase distribution among the horizontal plane of input HG modes. (c) Intensity distribution and (d) phase distribution among the horizontal plane of output LG modes.
Figure 12 presents simulated results of a second 10-layer HG mode to OAM mode converter based on SDNN structures, optimized using the same method. As shown in Fig. 12(a) (b), the same intensity and phase distribution of 11 HG modes are given as input modes. In Fig. 12(c) (d), the intensity and phase distribution of output 11 OAM modes are recorded, indicating the achievement of the intended mode manipulation effect. Figure 12(c) illustrates that the radial distribution of a single circle has been achieved. Furthermore, Fig. 12(d) shows the phase distribution exhibiting a periodic spiral change along the angle. It's noteworthy that $\textrm{OA}{\textrm{M}_{ - 5}}$, $\textrm{OA}{\textrm{M}_{ - 4}}$, $\textrm{OA}{\textrm{M}_{ - 3}}$, $\textrm{OA}{\textrm{M}_{ - 2}}$, $\textrm{OA}{\textrm{M}_{ - 1}}$ display counterclockwise spiral angle variations of 5, 4, 3, 2, and 1 period, respectively. The phase of the $\textrm{OA}{\textrm{M}_0}$ mode is entirely zero in all regions of the light spot. $\textrm{OA}{\textrm{M}_1}$, $\textrm{OA}{\textrm{M}_2}$, $\textrm{OA}{\textrm{M}_3}$, $\textrm{OA}{\textrm{M}_4}$, $\textrm{OA}{\textrm{M}_5}$ shows clockwise spiral angle variation of 1, 2, 3, 4, and 5 periods, respectively. These results suggest that a 10-layer SDNN structure can successfully convert 11 orthogonal HG modes into 11 orthogonal OAM modes.
Fig. 12. Simulated results of the well-optimized 10-layer SDNN structure HG mode to OAM mode converter for 11 channels with phase units gird of 256 × 256. (a) Intensity distribution and (b) phase distribution among the horizontal plane of input HG modes. (c) Intensity distribution and (d) phase distribution among the horizontal plane of output OAM modes.
7. Photonic integrated assembly SDNN mode manipulation components
Despite advantages in terms of low loss, scalability, and powerful mode manipulation ability of the proposed deep SDNN structure, there remain some uncertainties in the design toward practical applications. Two principal issues are how to align multiple phase surfaces and which material platform to choose for device fabrication.
Glass substrate offers unique advantages such as low loss, low cost, good transmissivity in the communication band, and ease of processing. Leveraging the simulation results and considering potential real MDM applications, we design and study a photonic integrated 10-layer SDNN mode manipulation component to realize the mode demultiplexing application of 11 orthogonal OAM modes. Figure 13(a) displays the complete 3D structure of the photonic integrated SDNN demultiplexer. The component comprises a 3D printed base, five glass substrates, and ten phase surfaces of microstructure processed on each side of the glass substrates. The glass substrates fit into the grooves of the 3D printed base's ridges, ensuring stable alignment among the ten-phase surfaces. The phase surfaces are fabricated by calculating the optical path difference for each phase unit and etching blocks of varying depth. The optimized phase parameters ${P_i}$ are transformed to etching depth d at the wavelength of λ through the following formula:
where the ${n_{Si{O_2}}}$ and ${n_{Air}}$ are the refractive index of the glass substrate and air, respectively. At the communication wavelength of 1550 nm, the maximum vertical etching dimension can be determined to be on the µm scale (3.74 µm) as calculated using Eq. (22). This scale is less susceptible to the collapse of the waveguide block and the slanting of the waveguide walls when the etching aspect ratio is not large. Figure 13(b) illustrates the structure of the fifth microstructure phase plane, which has three marked areas to provide a closer view of the structure. Figure 13(c) provides a detailed look at these marked areas, showcasing the etching depth differences of partial blocks.Fig. 13. Schematic diagram of the proposed photonic integrated assembly 10-layer SDNN demultiplexer for 11 modes with phase units gird of 256 × 256. (a) Schematic diagram of the 3D structure of the glass substrate assembly SDNN demultiplexer. (b) Zoom in on the 5-th microstructure phase plane. (c) Further zoom in the structure of the 5-th microstructure phase plane including 3 detail phase block arrays of 10 × 10 phase units.
We further examine the mode manipulation performance of the proposed mode demultiplexing component. Figure 14 shows the simulated results of the demultiplexer. Parameters considered include a phase unit size of 4 µm, a unit number of each phase plane of 512 × 512, a groove distance of 6 mm, a groove width of 5 mm, a groove depth of 2.5 mm, and a glass substrate thickness of 3 mm. As depicted in parts a and b of Fig. 13, 11 orthogonal OAM modes are used as the input light modes to optimize the component. Figure 14(c) illustrates the successful demultiplexing of the 11 orthogonal OAM modes into 11 Gaussian spots arranged diagonally on the output plane. Figure 14(d) shows the histograms of channel coupling coefficients, revealing an average conversion loss of only 0.12 dB with a maximum crosstalk of -26.68 dB.
Fig. 14. Simulated results of the well-optimized 10-layer SDNN structure demultiplexer for 11 OAM modes with phase units gird of 256 × 256. (a) Intensity distribution and (b) phase distribution among the horizontal plane of input modes. (c) Intensity distribution among horizontal plane of Gaussian light spots. (d) Histogram of channel coupling coefficients.
We further design and study a photonic integrated 20-layer SDNN mode manipulation component to realize the mode conversion from 25 orthogonal OAM modes to 25 orthogonal HG modes. Figure 15(a) presents the complete 3D structure of the photonic integrated SDNN mode converter. Figure 15(b) illustrates the structure of the eighth microstructure phase plane, while Fig. 15(c) provides detailed pictures of select areas The thickness of the glass substrate, phase unit size and other structural parameters of the optimized 20-layer converter are the same as the optimized 10-layers demultiplexer, while the distance between the flutes has changed from 6 mm to 3 mm.
Fig. 15. Schematic diagram of the proposed photonic integrated assembly 20-layer SDNN OAM mode to HG mode converter for 25 channels with phase units gird of 256 × 256. (a) Schematic diagram of the 3D structure of the glass substrate assembly SDNN demultiplexer. (b) Zoom in on the 8th microstructure phase plane. (c) Further zoom in on the structure of the 8-th microstructure phase plane including 3 detail phase block arrays of 10 × 10 phase units.
Figure 16 shows simulated results of the optimized photonic integrated assembly 20-layer SDNN mode converter. The input modes of the optimized converter are OAM modes with OAM order from -12 to +12, which are shown in Fig. 16(a),(b). The calculated output modes after transmission contain 25 HG modes, which are shown in Fig. 16(c), and (d). The optimized loss of the mode conversion process is only 0.33 dB.
Fig. 16. Simulated results of the well-optimized 20-layer SDNN structure HG mode to OAM mode converter for 25 channels with phase units gird of 256 × 256. (a) Intensity distribution and (b) phase distribution among the horizontal plane of input OAM modes. (c) Intensity distribution and (d) phase distribution among the horizontal plane of output HG modes.
8. Discussion
The results obtained in Figs. 4–16 indicate the successful implementation of a new type of mode manipulation components based on the diffraction principle, specifically the deep SDNN structure. A notable characteristic of this deep SDNN structure is its exceptionally large number of adjustable units and diffractive connections between layers. Consequently, by optimizing the values of the phase units, the deep SDNN structure has the potential to enable large-scale multichannel mode manipulation, which can significantly contribute to optical communication multiplexing systems. In addition to mode demultiplexers, this deep SDNN structure can also be utilized to realize other crucial multichannel components such as mode converters and mode switches. However, there is a trade-off between the number of channels and performance parameters, including loss and crosstalk. As the channel number is increased substantially, the risk of experiencing high levels of loss and crosstalk becomes more pronounced, ultimately degrading communication performance.
To achieve both an expansion in the channel number and negligible loss and crosstalk, it is necessary to optimize the spatial distribution or expand the scale of the SDNN structure. Currently, the expansion of the channel number is limited by the available computing power. Therefore, striking a balance between channel count and performance remains a crucial consideration in the design and optimization of deep SDNN structures. The bulk part of the calculation mainly comes from the convolution operation in each diffractive connection. Fortunately, the matrix of the diffractive coefficient is always the same when the diffractive distance has been set. Beyond that, convolution operation in the space domain could be transformed into a dot product in the space-frequency domain. Meanwhile, two-dimensional Fourier transformation could also be replaced by several matrix multiplications. Combining the above replacements, the complicated diffractive calculation is simplified to a series of simple matrix calculations.
Similar to the optimization methods used in deep learning in computer science, the direction of gradient flow in deep SDNN structures is opposite to the calculation connection in forward transmission. We can develop a simplified gradient reverse transmission method by simplifying the forward transmission calculation. It is worth noting that while computer science typically deals with real numbers, diffractive optics involves complex numbers. We have derived a complex gradient reverse transmission method to address this difference. As a result, the proposed optimization method for deep SDNN structures continues to benefit from advancements in deep learning technology. The field of deep learning constantly evolves, and the experiences gained from it can be applied to further improve the optimization methods for deep SDNN structures. The ongoing development of technologies such as GPU acceleration, cloud computing, quantum computing, and other acceleration technologies will contribute to refining and enhancing the optimization methods used in deep SDNN structures.
It is intriguing to consider that, within the domain of computer science, deeper neural networks with an increased number of parameters typically indicate enhanced intelligence. For instance, the large language model ChatGPT: the prevailing trend in its development is to expand the model's parameter count, as emergent phenomena tend to manifest only within large-scale models—a subject that merits significant research. In this work, we present an open method hoping that it will initiate the exploration of deep, high-precision SDNN structures and aid in the discovery of the fundamental physical principles associated with such phenomena.
The proposed deep SDNN structure consists of multiple etched glass plates that are assembled on a 3D-printed substrate. The fabrication of the phase planes utilizes well-established techniques on glass plates and could be potentially low cost with mass production. Moreover, the optimization of the deep SDNN structure draws inspiration from the optimization methods employed in deep learning within the field of computer science. Additionally, our simplification of the forward transmission algorithm further enhances the feasibility and practicality of the proposed deep SDNN structure.
9. Conclusion
In summary, our study focuses on a novel deep SDNN structure consisting of multiple parallel phase planes. This configuration offers a compact design and employs large-scale phase units, significantly enhancing the optical coherence effect within a short diffractive region. To enhance the optimization speed of the SDNN structure, we propose two methods: a forward transmission simulation method and a complex gradient reverse transmission method. We thoroughly investigate the factors that influence the mode manipulation ability of a well-optimized SDNN and our results suggest that as the number of layers increases, crosstalk and loss decrease. Additionally, we identify a minimum number of layers necessary to achieve tolerable crosstalk and loss when a certain multiplexing mode number is targeted. By employing a more reasonable spatial distribution, larger unit grids, and increasing the number of layers to 50, we achieve powerful deep SDNN mode converters. These converters support 100 modes, and exhibit an average conversion loss of 0.52 dB, a maximum loss of 0.62 dB, and a maximum crosstalk of -28.24 dB. Furthermore, we develop compatible mode converters based on SDNN, specifically an HG-to-LG mode converter and an HG-to-OAM mode converter. We also propose and optimize two photonic integrated assembly deep SDNN structures for multiplexing applications and mode conversion applications. These structures show favorable multitask performance with high conversion efficiency. The results obtained from the optimized SDNN, featuring a large channel number, low loss, and crosstalk, indicate potential applications in MDM communication systems.
Funding
National Natural Science Foundation of China (62125503, 62261160388); Natural Science Foundation of Hubei Province (2023AFA028); Fundamental Research Funds for the Provincial Universities of Zhejiang (QRK23018); Key University Construction Project of Zhejiang (0020XJ060703); Key R&D Program of Zhejiang Province (2023C01039); Major Project of Natural Science Foundation of Zhejiang (D24F020006).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available but could be obtained from the authors per reasonable request.
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