Photonic integrated circuits (PICs) play a critical role in optical computing, optical interconnection, and signal processing [1–5]. Compared with conventional circuits, PICs have larger bandwidths, lower power consumption, and higher data rates [1–5]. The critical basis of PICs is the photonic devices. A large number of basic photonic devices are connected to form complex PICs [3–15]. In recent years, inverse design algorithms have provided a significant impetus for the development of photonic devices. The inverse design algorithms usually enable the resulting photonic devices to have a significantly smaller footprint [7,9,16–18]. Moreover, the inverse design algorithms are highly flexible and are not constrained by conventional design logics [2,19–27]. Over the years, many inversely-designed photonic devices have been proposed, promoting the development of PICs extensively [6,7,28].

However, in the field of inverse-designed photonics, there are three problems to be solved. First, single-function devices cannot adapt to the development requirements of large-scale PICs. With the increasing scale of PICs, the demand for functionality and count of photonic devices also increases rapidly. However, if people separately design a device for each function, the device count in large-scale PICs will be huge, making it difficult to improve their compactness. In addition, the fabricating robustness of devices with different structures is different. Therefore, the performance of many devices in large-scale PICs may deteriorate after fabricating, thus affecting the performance of PICs. Second, the existing inverse design algorithms are incompetent in meeting the requirements of designing ultra-compact complex photonic devices, and their optimization efficiency is insufficient. For example, the brute-force search of the pixelated region in a direct binary search (DBS) algorithm easily leads to local convergence, which makes it hard to design highly integrated devices. For another example, the genetic algorithm (GA) usually produces a large number of individuals with a deteriorated evaluation index in its iteration process, which leads to high computational cost. Third, with the increasing processing ability of PICs, the existing PIC components, such as optical interconnection systems, cannot meet the bandwidth requirements of PICs. Adequate optical communication bandwidth is the basis of high-speed optical computing and optical signal processing. Therefore, there is a strong demand for technologies to increase the bandwidth of PIC components.

For these problems, there are the following three solutions. For the first problem, a practical solution is to design ultra-compact multifunctional integrated photonic devices [15,29,30]. First, these devices can significantly reduce the count of to-be-designed devices in PICs. The multifunctional integrated devices can replace the single-function devices whose functions are covered by their functions. In some situations, a single or a few multifunctional integrated devices can replace a region of a PIC completely, reducing the design cost significantly. Second, these devices can reduce the footprint of PICs by replacing single-function devices or regions. Third, these devices can improve the fabricating robustness of PICs. If multifunctional integrated devices with the same structure are used to replace some single-function devices or regions, thereby reducing the count of device structures in the PIC, the risk of performance deterioration caused by the inconsistent fabricating robustness of multiple devices can be reduced. For the second problem, a new inverse design algorithm with stronger design ability needs to be proposed. An algorithm with high efficiency and robustness is more competent in designing ultra-compact complex devices, including ultra-compact multifunctional integrated devices. For the third problem, in recent years, the proposed mode division multiplexing (MDM) technology provides a new scheme to exponentially increase the bandwidth of optical communication [7–10].

According to the above solutions, we perform the following works. First, we propose a new algorithm named polygon search (PS) algorithm, which has obvious advantages over the existing mainstream inverse design algorithms in terms of effect and efficiency. By introducing the “trial and compare” strategy into the shape-optimizing framework, the PS algorithm can maintain a high search speed in an ample search space. Therefore, it can improve the optimization effect and efficiency simultaneously. In addition, the PS algorithm can take advantage of an approximate gradient or neural network to predict better structures, thus accelerating the optimization process and enhancing its robustness. Neural networks are powerful computational models that have been widely used in photonic device design [31,32]. After appropriate training, the neural networks can give a direct mapping of the input (e.g., structure parameters) and output (e.g., optical response) in a particular range without consuming massive computation to simulate [33]. This advantage can be used not only to optimize the structural parameters of photonic devices directly but also to predict the optimal optical response and the corresponding device structure in a particular range [34–36]. Second, we use the PS algorithm to integrate the functions of two kinds of mode conversions and channel crossings into one device and design an integrated dual-channel mode-conversion-crossing waveguide (DCMC-CW) module. Mode conversion is a highly versatile technique that serves as a crucial foundation for multimode communication, asymmetric optical power transmission, and integrated photonics [37]. Besides, it is an important application scenario for on-chip metasurfaces [38]. In recent years, there have been a variety of mode conversion functions realized by on-chip metasurfaces, proving their wide application prospect. The proposed integrated DCMC-CW module can realize the interconversion between TE0 and TE1, the interconversion between TE0 and TE2, and channel crossing. These three functions are integrated within only 4μm×4μm. The integrated module can not only completely replace a functional region in a PIC in some cases but also be used as a mode converter or a crossing waveguide. Third, we combine the integrated module with several PS-designed mode mixers and PS-designed crossing waveguides to form several multi-channel crossing-mode-division-multiplexing (CMDM) systems. Multi-channel CMDM systems can provide multiple modes on the basis of channel crossing and make the output signal in the same direction be transmitted in parallel within a single output waveguide. These characteristics significantly increase the communication bandwidth and decrease the footprint of PICs.

In this paper, we first explain the theoretical basis and principle of the PS algorithm in detail. Then, we introduce the application of the integrated DCMC-CW module and multi-channel CMDM systems. In addition, we also perform simulations and experimental testing to illustrate their excellent performance. At last, we demonstrate the effect and efficiency advantages of the PS algorithm over the mainstream inverse design algorithms by a comprehensive contrast experiment involving 40 complete design processes.

Before proposing a new algorithm, we analyze the characteristics of the existing algorithms. We find that the “trial and compare” search strategy of the DBS algorithm enables relatively high optimization efficiency by preventing the deterioration of the evaluation index. In addition, the excellent implementation of complex devices of many shape-optimizing algorithms also benefits from the wide search space provided by the continuous device shapes [39–41]. Therefore, we consider introducing the “trial and compare” strategy into the shape-optimizing framework to improve the design efficiency and effect simultaneously. Our main ideas are the following. First, we set the structure of the device to be designed as a polygon prism. Then, we move the first vertex several times to find a position that improves the evaluation index. After these, we move the other vertices in turn. In the search process of each vertex, the evaluation index keeps undeteriorating through the “trial and compare” strategy. In addition, benefiting from the continuity of polygon shape, the optimization effect and efficiency can be improved by “approximate gradient prediction” or “neural network prediction.” After several trials of each vertex, the approximate gradient or neural network can be used to predict a vertex position with a high probability of improving the evaluation index. The specific meaning of the “trial and compare” strategy is described in Appendix A.

Based on the above ideas, we propose the specific principle of the PS algorithm as the following steps.

In the first step, the PS algorithm introduces the initial structure of the device in the design region. The initial structure is a polygon prism with a polygon as the base and the pre-set thickness as the height. The polygon base is connected by several vertices in the x-y plane of the design region. After this, the PS algorithm calculates the figure of merit (FOM) of the initial device as the initial FOM.

In the second step, the PS algorithm moves the first movable vertex to search for a better position near its initial position. When changing the vertex position, the PS algorithm keeps the changed position relatively close to its initial position. Each time the vertex changes position, the shape of the polygon changes, resulting in a new device and a new FOM. This step can be further divided into two sub-steps: first, the PS algorithm moves this vertex near its initial position in the x-y plane several times and calculates the corresponding FOMs; second, according to the information provided by all the positions and their corresponding FOMs, the PS algorithm predicts a position that may improve the FOM, moves this vertex to the predicted position, and calculates its corresponding FOM. Then, the PS algorithm locates this vertex to the position corresponding to the optimal FOM. The prediction can be implemented in many ways. This paper provides two prediction methods: approximate gradient prediction and neural network prediction. Each movement of a vertex is called a “trial.” Each second step operated on a vertex is called a “search.” The schematic of this step is shown in Fig. 1(a).

In the third step, the PS algorithm operates the second step for the following vertices in turn. After all vertices are searched once, the PS algorithm completes a loop.

In the fourth step, the PS algorithm starts from the first vertex again and then operates the second and third steps until each vertex is searched for the pre-set number of loops.

The above optimization steps are shown in Fig. 1(b). In each of the above steps, the specific parameter values or moving directions can be set by the designer. More details are presented in Appendix A.

According to the principle of the PS algorithm, we design an integrated DCMC-CW module, and the specific parameter settings are shown below. The basic simulation model is set as a commercial SOI wafer with a 220-nm-thick top silicon layer and a 3-μm-thick buried oxide layer. The fixed structures are four fixed 1.5-μm-wide waveguides set in the top silicon plane of the SOI. The design region (air filling) is also set in this plane. The center of the operating wavelength is set to 1.55 μm.

For the first step, the initial structure of the integrated DCMC-CW module is a polygon prism with a pre-set polygon as the base and 220 nm as the height. All these elements are shown in the left figure of Fig. 2(a). The polygon base is connected by 52 vertices, as shown in the right figure of Fig. 2(a). The positions of some vertices are fixed to ensure that several contours of the polygon coincide with the fixed waveguides. Throughout the optimization, the height and material of the polygon prism are fixed. The FOM of the initial device is calculated as the initial FOM. For the integrated module, FOM is defined as follows: FOM=TL1((x1,y1),(x2,y2),…,(x52,y52))+TT2((x1,y1),(x2,y2),…,(x52,y52)),(1)where TL1 represents the transmission of TE1 (at right port) when TE0 is input in the left port, and TT2 represents the transmission of TE2 (at bottom port) when TE0 is input in the top port. (x1, y1) represents the coordinate of the first vertex. The mapping between the transmission and coordinates of all vertices is obtained by simulation. When the FOM increases, it means that the mean efficiency of mode conversion in both directions is increased, and the insertion losses of the device decrease.

For the second step, the move method of the first movable vertex is that the PS algorithm moves it 50 nm to the left, right, up, and down of the initial position (these new positions are called “trial position”), respectively. The prediction method is approximate gradient prediction. A new polygon prism and its polygon base after the move are shown in Fig. 2(b).

For the third step, the search count is 44 in a loop. The structure and its corresponding polygon base after the first loop are shown in Fig. 2(c).

For the fourth step, the total number of loops is 20 (4400 computations). The optimized structure and its corresponding polygon base of the integrated DCMC-CW module are shown in the left and right figures of Fig. 2(d).

In the design process, the specific process of the approximate gradient prediction can be described as follows. First, the PS algorithm finds the maximum FOM of the existing five FOMs and its corresponding trial position α, and uses linear or quadratic functions to fit the changes of the FOM along the connecting lines between α and other trial positions, as shown in Figs. 3(a) and 3(b). Then, the PS algorithm calculates the slopes at α of each function. Second, the PS algorithm finds the maximum slope S and the connecting line corresponding to S. The direction in which the FOM increases along this connecting line is set to the direction of the approximate gradient of the FOM at α. Third, starting from α, the PS algorithm moves this vertex with a distance in the direction of the approximate gradient and denotes this trial position as β, as shown in Fig. 3(c). The move distance can be determined by the designers. Fourth, the PS algorithm calculates the FOM corresponding to β and compares it with the FOM corresponding to α. If the FOM corresponding to β is the larger one, the vertex is located at β. Otherwise, it is located at α.

In addition, the specific process of the neural network prediction can be described as follows. First, the PS algorithm finds α. Second, the coordinates of the five trial positions [(x1,y1)−(x5,y5) in Fig. 3(f)] are taken as inputs, and the corresponding FOMs are taken as outputs to train a neural network, as shown in Fig. 3(e). Third, the PS algorithm delimits a prediction region around the five trial positions and samples a certain number of coordinates in the prediction region [(xi,yj),(xm,yn),… in Fig. 3(d)] as input, as shown in Fig. 3(d). Then, the trained neural network is used to calculate the FOM [FOM1−FOMt in Fig. 3(e)] corresponding to each coordinate. Fourth, the PS algorithm finds the max FOM (FOMp) calculated by the neural network and its corresponding position β(xp,yp) and moves the vertex to β, as shown in Fig. 3(f). The remaining process is the same as that in the approximate gradient prediction.

The integrated DCMC-CW module can selectively achieve interconversion between TE0 and TE1 or TE0 and TE2. Therefore, a single or a pair of integrated modules can interconvert these three modes, as shown in Figs. 4(a) and 4(d). Apart from functioning as a multimode converter, the integrated module can act as a crossing waveguide for low-loss channel crossing in multi-channel communication, as shown in Fig. 4(a).

In this section, we verify the performance of the integrated DCMC-CW module in mode conversion and channel crossing by simulation and experimental testing. For a single integrated module, the simulated and experimental tested transmissions of different modes when TE0 is input in different input ports are shown in Fig. 4(f). The corresponding electromagnetic field distribution in the different situations is shown in Figs. 4(b) and 4(c), respectively. In addition, when two integrated modules are used for interconversion between TE1 and TE2, the simulated and experimental tested transmissions of different modes when TE1 is input in the left input port are shown in Fig. 4(g). The corresponding electromagnetic field distribution in these two integrated modules is shown in Fig. 4(e).

According to the simulation results, the insertion losses (ILs) and maximum crosstalks (CTs) of a single integrated module are 0.45 dB and −22.10dB at 1.55 μm when TE0 is input in the left port, and 0.38 dB and −29.21dB at 1.55 μm when TE0 is input in the top port. Another important result is that the ILs and the CTs of a single integrated module are always less than 0.79 dB and lower than −16.77dB in the working band (1.45–1.65 μm) when TE0 is input in any port. When two integrated modules are used for interconversion between TE1 and TE2, the IL is 0.99 dB, and the CT is −17.90dB at 1.55 μm. In addition, the IL is always less than 1.60 dB, and the CT is always lower than −15.51dB in the whole working band. These results represent the performance of a single integrated module in mode conversion and channel crossing. In this paper, the IL is defined as follows: IL=−10×log10(ToutTin),(2)where Tin and Tout are the transmission of the input mode and the target output mode, respectively. Similarly, the CT is defined as follows: CT=10×log10(Tout′Tin),(3)where Tout′ is the transmission of the nontarget output mode.

In summary, the simulation results demonstrate that the integrated DCMC-CW module has a good performance in mode conversion and channel crossing, and the ILs and CTs of its two channels are very low. In addition, in the 200-nm-wide working band, the extinction ratios (ERs) can always reach more than 16 dB for a single integrated module and 13.91 dB for two integrated modules, which proves that the integrated module can operate over a wide band. The ER is defined as follows: ER=−IL−CT.(4)

After the experimental testing, we find that whether for a single module or the combination of two modules, there is a good agreement between the experimental testing and simulation results, proving the correctness of the simulation results. The results are all shown in Figs. 4(f) and 4(g). The simulation and experimental testing results show that the integrated DCMC-CW module performs excellently in mode interconversion and channel crossing, which proves the effectiveness of the PS algorithm.

In addition to their rich set of functions, integrated DCMC-CW modules can achieve various complex functions through combinations. An important application of integrated DCMC-CW modules is constituting multi-channel CMDM systems.

PIC-based optical chips have been widely used in neuromorphic photonics, all-optical neural networks, and photonic reservoir computing [32–49]. However, due to the two-dimensional layout of most PICs, crossing among multiple channels in optical chips is almost inevitable. In addition, with the growth of the processing ability of optical chips, the demand for inter-chip communication bandwidth is increasing [7,9,10,16,50–52]. The integrated DCMC-CW module provides a novel approach to address these two significant challenges about optical chips by constituting multi-channel CMDM systems, as shown in Figs. 5(a) and 5(b). Signals (TE0) generated by different processing modules enter the multi-channel CMDM system from their corresponding input ports. After that, signals are crossed and mode-converted several times in the system and then enter the mode mixer in different modes. Then, these different-mode signals are transmitted in the bus waveguide simultaneously. Compared with the conventional channel crossing regions, the multi-channel CMDM system can provide multiple modes on the basis of channel crossing and make the output signal in the same direction be transmitted in parallel within a single output waveguide, which significantly increases the communication bandwidth and decreases the footprint of PICs. A three- to six-channel CMDM system can be realized by connecting several identical integrated modules and other elements in a pre-set way. As an example, we design a TE0−TE1 mode mixer and a TE0 bending waveguide using the PS algorithm and connect them with several integrated modules to form a kind of configuration of the four-channel CMDM system, as depicted in Fig. 5(b). The concept figures of the other multi-channel CMDM systems are presented in Appendix B.

The performance of this four-channel CMDM system is verified by simulation. The electromagnetic field distribution in this system when all channels have a signal input is shown in Fig. 5(c). The ILs and CTs of every channel in the working bandwidth are shown in Fig. 5(d).

The simulation results show that the four-channel CMDM system has low ILs and CTs in every channel. The average IL and average maximum CT of the system are 0.96 dB and −20dB at 1.55 μm, respectively. In addition, the mode conversion rate of this system also reached a high degree. The average ER of this system is 20.00 dB at 1.55 μm, which shows the practicability of this system. The specific results are presented in Appendix B.

In discussion, we demonstrate the effect and efficiency advantages of the PS algorithm over other main-stream inverse design algorithms: the DBS algorithm, the GA, and the adjoint method (AM) by a comprehensive contrast experiment involving 40 complete design processes of an integrated DCMC-CW module. In this contrast experiment, there are 10 examples of completely designing an integrated DCMC-CW module for each algorithm. In order to make the contrast experiment convincing enough, we set four initial structures, as shown in Appendix C.1. Every initial structure is optimized by each algorithm in the contrast experiment. To exclude the influence of unrelated factors, such as computer performance, we select the calculation count consumed by each algorithm to be the evaluation index for its optimization efficiency. More detailed parameter settings of this contrast experiment are also presented in Appendix C.1. The formulae of FOMs of all algorithms are the same.

After the contrast experiment, the best optimization examples (most efficient or effective) of each algorithm are shown in Table 1. More detailed results of this contrast experiment are presented in Appendix C.2.Table 1.

Best Optimization Examples of Each Algorithm

The specific initial structures of the best examples in Table 1 are shown in Figs. 6(a)–6(f). The final structures optimized by these four algorithms are shown in Figs. 6(g)–6(l). It can be seen intuitively that the PS-optimized module has a relatively simple structure, which is a favorable condition for fabrication. The comparison of the optimization processes of the best examples is shown in Fig. 6(m).

Table 1 and Fig. 6 show that the PS algorithm has a noticeable effect and efficiency advantage compared with the mainstream inverse design algorithms. In each stage of the optimization process, the PS algorithm can maintain a high speed of device performance improvement. Besides, the PS algorithm can improve the device performance to a degree that is difficult for other algorithms to achieve. This performance advantage will increase exponentially when multiple identical devices are cascaded.

The theoretical explanations for the effect and efficiency advantage of the PS algorithm mainly include the following five aspects. (1) If the number of the movable vertices of an initial polygon is N, the dimension of the search space for PS optimization is 2×N. Such a high-dimensional search space ensures a high search freedom of the PS algorithm. (2) The random walk produced by vertex position prediction in each search can enhance the robustness of the PS algorithm to avoid premature local convergence. (3) The shape optimization framework provides certain constraints for the initial optimization stage, so that the PS algorithm searches for the refractive index distribution mainly in the regions near the initial structure at the beginning. This assignment of search weights allows the PS algorithm to focus its search on regions where changes in refractive index distribution are more likely to significantly alter FOM. (4) Compared with pixelated optimization algorithms, the shape optimization framework can make the PS algorithm integrate with the approximate gradient or neural networks better. With the help of these efficient and powerful tools, the PS algorithm can predict better structures more effectively than brute force search. (5) The “trial and compare” strategy can prevent the deterioration of the evaluation index after every search. In addition, although there may be some small regions in the PS-designed results that are difficult to implement in accordance with the original appearance entirely after fabricating, the impact of this issue on the PS-designed devices is small, as detailed in Appendix D.

In this paper, to solve the problems that exist in the field of inversely designed photonics, we perform the following works. First, we propose a new inverse design algorithm named PS algorithm to solve the problems of poor effect and low efficiency faced by the existing algorithms in the design of ultra-compact multifunctional integrated photonic devices. This algorithm possesses multiple theoretical advantages by introducing the “trial and compare” strategy into the shape-optimizing framework. Second, utilizing the PS algorithm, we design an ultra-compact integrated DCMC-CW module. The integrated DCMC-CW module(s) can achieve interconversion between any two of TE0, TE1, and TE2 and enable channel crossing. The footprint of an integrated DCMC-CW module is only 4μm×4μm. Simulations and experimental testing show that the module has low ILs, low CTs, and high ERs. Third, we combine the integrated module with several PS-designed mode mixers and PS-designed crossing waveguides to form several multi-channel CMDM systems. Compared with the conventional channel crossing regions, the multi-channel CMDM system can provide multiple modes on the basis of channel crossing and make the output signal in the same direction be transmitted in parallel within a single output waveguide, which significantly increases the communication bandwidth and decreases the footprint of PICs. At last, we perform a comprehensive contrast experiment to demonstrate the wide practicality and advantages in terms of the optimization effect and efficiency of the PS algorithm. This algorithm, the integrated DCMC-CW module, and the multi-channel CMDM systems are expected to further facilitate the development of inversely-designed integrated photonics.