1. Introduction

In recent years, silicon photonics (SiP) has experienced tremendous advancements in both integration level and device performance due to its high compatibility with complementary-metal-oxide-semiconductor (CMOS) fabrication techniques [1–3]. These improvements have led to the expansion of application areas of SiP into various fields such as optical communication [4–6], optical computing [7–10], LiDAR [11,12], and quantum technologies [13,14]. In order to further decrease the device footprint while enhancing device capabilities, the inverse design method has been proposed and widely applied to the design of both passive [15–25] and active components [26]. Conventionally, the design of SiP components is performed through a forward design method, which starts with a complex physical model derived from Maxwell's equations, followed by the estimation of geometric parameters and manual parameter adjustment through experiments or electromagnetic simulations. In contrast, the inverse design method avoids the complicated process of physical modeling and manual fine-tuning [27,28] by directly generating desired structures with the assistance of artificial intelligence (AI) or optimization algorithms (OA). Furthermore, the inverse design method increases the number of degrees of freedom (DoF) in the parameter space by about two orders of magnitude or more, enabling a more thorough search of the full parameter space [28].

The digital meta-structure (DMS) [18,24,25,29–31] is a major category of inverse-designed structures that mainly consists of a densely arranged binary pixel array (BPA). Compared with the analog meta-structure (AMS) [32–35], DMS offers the benefits of simpler implementation, minimal mathematical prerequisites, and enhanced compatibility with foundry fabrication processes [36]. The state and arrangement of the BPA can be flexibly modified to achieve various functionalities. The principle behind manipulating the optical field with the DMS, which is typically shaped like QR codes, is the capability to provide precise control over the effective refractive index, waveguide dispersion, and optical modal confinement in sub-wavelength scale, thereby exerting a significant impact on the optical field at a 2D level. With advancements in OA and AI, many inverse-designed DMS structures have been reported, such as bend waveguides [37,38], waveguide crossings [25,38,39], wavelength (de)multiplexers [18,29,39], mode (de)multiplexers [38,40], power splitters [31], and grating couplers [17].

However, previous reports on OA-based inverse design have overlooked two main aspects. Firstly, they have primarily focused on the algorithms and optimization process, ignoring the impact of various “hyperparameters”. For instance, when designing DMSs, a common practice is to employ a square plate with a length-width ratio of 1 as the design area [31,25]. However, other choices of length-width ratio may enhance the optimization performance. Other factors such as the initial etching density of BPA can also significantly influence optimization. We refer to these parameters as “hyperparameters” due to their similarities to hyperparameters in machine learning, which are set during initialization and cannot be altered until the current optimization or training process terminates. Secondly, most of the inverse-designed DMSs are unifunctional and merely improved versions of conventional devices, which limits the expansion of the SiP device library. Recent work has explored the potential of inverse design to create multifunctional devices that are difficult to achieve through forward design, such as grating coupler wavelength demultiplexers [23], and dual-channel multimode wavelength demultiplexers [24]. However, photonic integrated devices that can perform both wavelength demultiplexing and power splitting have not yet been reported. This type of devices can increase the networking flexibility in optical communication systems. For example, the coexistence of O-band and C-band passive optical networks (PONs) has become particularly interesting, given recent developments in C-band coherent PONs [41,42]. Thus, technologies for integrated dual-band wavelength demultiplexing and power splitting will be required during the development stage of next-generation PONs [43].

In this work, we present a series of ultra-compact dual-band wavelength demultiplexing power splitters (WDPSs) that simultaneously perform wavelength demultiplexing for the O-band and C-band and 1:1 optical power splitting. The WDPSs are designed on a standard silicon-on-insulator (SOI) platform using a novel binary optimization method named two-step direct binary search (TS-DBS) method, which successfully addresses premature convergence by utilizing two steps of optimizations with different figure of merits (FoMs). Our focus during the design process was on examining how hyperparameters affect optimization performance, whose evaluation criteria include fabrication error tolerance (FET), insertion loss (IL), and channel crosstalk (XT). We divided the hyperparameters into two categories: structure-related hyperparameters (Type I) and algorithm-related hyperparameters (Type II). We investigated four Type I hyperparameters and one Type II hyperparameter as a proof of concept. It was found that the optimization performance is greatly influenced by the hyperparameters as they act as the boundaries within which the inverse design searches for the optimal structure. For each hyperparameter, we conducted a thorough analysis of its impact and underlying mechanisms. Additionally, we used a forward-designed WDPS as the baseline device and compared it with our inverse-designed WDPS. Simulation results show that the inverse-designed WDPS is 12.77 times smaller in footprint, while still displaying a slight improvement in device performance. Throughout the design process, we adhered to mainstream foundry fabrication constraints, with a minimum feature size of 130 nm. When we reduced the minimum feature size to 65 nm and applied the optimal combination of hyperparameters, the inverse-designed WDPS achieves ultra-low ILs of 0.36 dB and 0.37 dB and XTs of -19.91 dB and -17.02 dB at 1310 nm and 1550 nm, respectively. Finally, the generalizability of our findings regarding the hyperparameters was also validated through numerical or theoretical analysis. This study offers insights into the considerations of multifunctionality and manipulation of hyperparameters in DMSs, thereby providing inspiration for the advancement of more effective and efficient inverse design methodologies for digital or even analog topology optimizations [22,32,44].

2. Principles and methods

2.1 Dual-band wavelength demultiplexing power splitters (WDPSs)

The schematic of the WDPS is shown in Fig. 1. The WDPS is based on a 1 × 4 multimode interference (MMI) coupler on a standard SOI platform with a 220 nm thick top silicon layer and a 2 µm buried oxide layer (silica). The cladding material can be silica or air, depending on the etching material. The WDPS consists of three parts: the coupling area, the input port, and the output ports. In the coupling area, as seen in Fig. 1(a), the multimode waveguide is divided into discrete pixels of a unit size of 260 × 260 nm², each with a binary state of “0” or “1”, as shown in Fig. 1(b). A state of “1” indicates that the pixel is etched by a cylindrical rod with a diameter of 130 nm and a depth of 220 nm (with refractive index n_etch), while a state of “0” indicates that the pixel remains unetched (with refractive index n_Si). The local index contrast (Δn = n_Si - n_etch) enables the DMS to manipulate the optical field. All binary pixels form a binary pixel array (BPA), which is randomly generated and will be modified during the optimization process. The width (W_ca) and length (L_ca) of the coupling area may change based on the configuration of the BPA (e.g., 20 × 30 configuration, as seen in Fig. 1). As for the input port and four output ports (OP₁-OP₄), five identical linear tapered waveguides are employed to connect the coupling area with input/output single-mode waveguides (not shown in Fig. 1). The taper structure reduces the excess loss caused by mode mismatch between the single-mode waveguides and the coupling area. The length of the tapered waveguides (L_t) is set to 4 µm to ensure smooth conversion of optical modes. The narrower end of the tapered waveguides is fixed to 500 nm wide, as seen in Fig. 1(c), to match the single-mode waveguides, while the width of the wider end (W_t) increases with W_ca to reduce backscattering and insertion loss. The channel spacing between adjacent tapered waveguides (S_t and S_c) must be greater than 200 nm to avoid the formation of sharp angles. The specific parameters of the coupling area, taper waveguides, and channel spacing will be presented in Section 3.

 

Fig. 1. Schematic of the WDPS device. (a) Top view of the WDPS. (b) Schematic of two pixels with state “1” and “0”, respectively. (c) Cross-section of the single-mode waveguides connected to the input tapered waveguide and output tapered waveguides.

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The working principle of the WDPS can be explained as follows. When a beam of light composed of two different wavelength components of 1310 nm (referred to as 1310 nm light) and 1550 nm (referred to as 1550 nm light) enters the coupling area via the input port, both wavelengths are split into two parts and emerge from the four output ports. The 1310 nm light is equally divided and guided to OP₁ and OP₄, while the 1550 nm light is equally divided and emitted from OP₂ and OP₃, respectively. Thus, the WDPS is capable of performing wavelength demultiplexing of O-band and C-band while also achieving 1:1 power splitting, offering multifunctionality with a single device.

2.2 Workflow and principles

2.2.1 Workflow of the inverse design process

The process of the inverse design of WDPSs is shown in Fig. 2. Unlike traditional inverse design methods for DMSs, our approach places a greater emphasis on the systematic consideration and manipulation of hyperparameters. In Fig. 2(a), we provide an example of four types of hyperparameters that are carefully adjusted and recorded before each optimization. These hyperparameters include the duty cycle of the etching hole, the initial etching density, the length-width ratio of the BPA configuration, and the type of etching material. After these structural hyperparameters are defined, an initial BPA pattern is generated. Next, we employ the TS-DBS method to optimize the BPA pattern. TS-DBS can effectively address the premature convergence which may occur during the optimization. The principle of TS-DBS is presented in next part while the optimization performance is offered in Section 3.

 

Fig. 2. Inverse design process of WDPSs based on TS-DBS algorithm. (a) Schematic diagram of several hyperparameters: etching duty cycle, initial etching density, length-width ratio, and etching material. (b) Scan scheme of TS-DBS optimization. The symmetrical axis of the coupling area is marked with a white dash-dotted line. (c) Simulated electric field intensity at 1310 nm and 1550 nm and (d) final device structure after a particular optimization process.

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This new inverse design scheme utilizes two cascaded FoMs during the optimization. The detailed procedure is illustrated in the flowchart of Fig. 2 and can be explained as follows: (1) FoM₁ is first used to evaluate the performance of the WDPS during the optimization process. During each iteration, the first pixel of the BPA is switched from either “0” to “1” or “1” to “0” and the change in FoM₁ is calculated using the finite-difference time-domain (FDTD) method. If the FoM₁ value increases, the switch is preserved; otherwise, the pixel is switched back. This process is repeated for all pixels in the BPA and is performed by scanning the pixels row-by-row, as depicted in Fig. 2(b). The final BPA pattern of each iteration is implemented as the initial pattern for the next iteration. (2) After several iterations, FoM₁ reaches convergence and FoM₂ is then adopted as the FoM for optimization. The subsequent convergence of FoM₂ marks the end of the optimization, and the final BPA pattern is recorded for further analysis. The equations of the cascaded FoMs (FoM₁ and FoM₂) aimed at optimizing performance at two single frequency points of 1310 nm and 1550 nm are given by:

According to the S-parameters given above, the expression for insertion loss (IL) and channel crosstalk (XT) can be derived here. The ILs for the 1310 nm and 1550 nm channels can be written as:

2.2.2 Principles of TS-DBS method

Conventional methods of designing a device with the DBS method typically use a single, fixed FoM corresponding to the optical transmission of each output port as the optimization objective. However, using a single FoM for inverse design can severely limit the final device performance for several reasons: (1) The optimization process may easily become trapped in a local optimum, and (2) some components included in the FoM may reach an acceptable level early on in the optimization process, but continue to contend for optimization space in subsequent iterations, thus reducing the overall optimization performance.

To address these issues, we propose the TS-DBS method for the inverse design of WDPSs. The optimization objective is to maximize FoM₁ and FoM₂. In an ideal scenario where FoM₁ and FoM₂ are both equal to 1, all of the 1310 nm light would output from OP₁ and OP₄, and all of the 1550 nm light would output from OP₂ and OP₃, resulting in zero crosstalk and loss. As seen in Eqs. (1) and (2), FoM₁ includes both the output transmission terms (S21_λ and S31_λ) and backscattering term (S11_λ), so the first step of the TS-DBS method aims to reduce both XT and IL. Meanwhile, FoM₂ eliminates the S11_λ term and thus the second step focuses only on improving the transmission spectra of the output ports. The significance of removing the backscattering term lies in the fact that, in most cases, the backscattering term can be rapidly reduced to an ultra-low level (see Fig. S3 in Supplement 1), but still competes with the output transmission terms (S21_λ and S31_λ) for optimization space, hindering their optimization potential and leading to early convergence of the optimization. By utilizing two cascaded FoMs and eliminating fast-converging elements in the second FoM, we can successfully address the issue of premature convergence. The optimization performance of TS-DBS and a comparison with the conventional single-step DBS method will be presented in Section 3.

2.3 Fabrication errors

To evaluate the performance of WDPSs under different hyperparameters, we introduce two cases of fabrication errors. The first case, as shown in Fig. 3(a), involves a misalignment of pixels in the BPA. This error occurs when the actual etching holes are shifted from their intended position during fabrication. To simulate this error in our FET test, we randomly shift the position of each etching hole in either the x or y direction, and then monitor the fluctuations and degradation of device performance as we vary the offset value. The second case involves variations in the radius of the etching holes, as depicted in Fig. 3(b). In the FET test, the radius of the etching holes is varied from the shrunk case to the expanded case to gather the spectra data necessary for the FET evaluation. Detailed information regarding the parameters of the position offset and radius variation can be found in the subsequent section.

 

Fig. 3. Two cases of fabrication error. (a) Pixel misalignment of BPA. (b) Radius variation of the etching holes. The dashed circles are the ideal etching holes, while the solid circles represent the actual etching holes affected by fabrication errors.

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3. Results and analysis

3.1 Optimization performance of TS-DBS method

To demonstrate the effectiveness of TS-DBS method, we present a comparison of the device performance between WDPSs optimized using the conventional single-step DBS (using FoM₁ only) and TS-DBS (using FoM₁ and FoM₂ in sequence) methods in Fig. 4. The device performance can be directly characterized by the FoM, where a higher FoM indicates better device performance. To ensure fairness, we first calculate the FoM using Eq. (1) for both WDPSs. From Fig. 4(a), it can be observed that the FoM value significantly improves by approximately 22.9% when TS-DBS is employed. In order to better illustrate the impact of the TS-DBS method on the individual components of the FoM, we extract the S-parameters from the simulation results to characterize the forward transmission strength and the backscattering suppression. The forward transmission strength is characterized by S21_1310nm and S31_1550nm, where a higher value of S21_1310nm indicates a greater amount of 1310 nm light being guided to OP₁ and OP₄, and a higher value of S31_1550nm indicates a greater amount of 1550 nm light being guided to OP₂ and OP₃. This signifies better achievement of wavelength division multiplexing and power splitting functionalities. Additionally, we define the backscattering suppression of the WDPS as $- 10 \cdot {\log _{10}}[{{{({S{{11}_{1310nm}} + S{{11}_{1550nm}}} )} \left/ 2\right.}} ]$, where a higher value indicates a better suppression effect on multi-bounce and backscattering within the device. The backscattering strength is then defined as ${{({S{{11}_{1310nm}} + S{{11}_{1550nm}}} )} \left/ 2\right.}$.

 

Fig. 4. Comparison of the device performance characterized by (a) FoM calculated using Eq. (1), (b) S21_1310nm, (c) S31_1550nm, and (d) backscattering suppression between WDPSs optimized using the single-step DBS and TS-DBS methods.

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From Fig. 4(b) and (c), it is evident that the TS-DBS method leads to significant improvements in the transmission performance of both the 1310 nm and 1550 nm wavelength channels. Specifically, the 1310 nm channel exhibits a substantial enhancement in forward transmission strength (approximately 0.3), while the 1550 nm channel shows a more modest improvement (approximately 0.04). This discrepancy in improvement suggests that the backscattering term primarily occupies the optimization space of the transmission strength of the 1310 nm channel. This observation can be attributed to the fact that the output ports associated with the 1310 nm channel, namely OP₁ and OP₄, are positioned at the top and bottom of the device, requiring a larger deflection angle within the design area to guide the 1310 nm light from the input port to the output ports. A larger deflection angle inevitably increases the likelihood of multi-bounce, thereby intensifying the backscattering effect. Thus, there exists a stronger competition for the optimization space between the backscattering term and the transmission term at the 1310 nm channel. Regarding backscattering, it is expected that the removal of the backscattering term in FoM₂ would result in a reduction in the backscattering suppression capability of the WDPS. Indeed, the backscattering suppression is decreased from 21.4 dB to 19.6 dB, corresponding to a minimal increase of 3.7 × 10^−3­ in backscattering strength. In general, the backscattering suppression of the WDPS still remains at a relatively high level, indicating a strong ability to mitigate backscattering effects.

The above discussions demonstrate that the implementation of the TS-DBS method yields noticeable enhancements in the device performance of WDPSs compared to the single-step DBS method.

3.2 Discussion on hyperparameters of inverse design

In deep neural networks (DNNs), hyperparameters usually refer to the parameters associated with the network structure which are not acquired by the network but set manually. In our design of WDPSs, however, we find that the hyperparameters of inverse design can be grouped into two categories: (I) hyperparameters of physical structures and (II) hyperparameters related to the algorithm itself. Figure 2(a) presents a brief overview to four type I hyperparameters. These hyperparameters relate to the actual physical structure of WDPSs, such as shape or material. To be specific, the length-width ratio determines the ratio of pixel number in each row to that in each column of the BPA. The etching material can be any material permitted by the fabrication technique and its selection may result in different device performances due to the variation of the refractory index. The etching duty cycle is defined as the ratio of the hole diameter to the pixel length. Moreover, the optimization begins with an initial BPA pattern, where the initial etching density is set to control the ratio of pixels in the state “1” to all pixels. In terms of the type II hyperparameters, we discuss and demonstrate the effect of TS-DBS optimization bandwidth on device performance as a proof of concept. In this Section, we will study the type I parameters under the condition that the TS-DBS is only concerned about the transmission of two single frequency points, namely 1310 nm and 1550 nm. Then, we will progressively expand the range of frequency points that the TS-DBS algorithm focuses on.

3.2.1 Length-width ratio

For a BPA with M rows and N columns (named as the M × N configuration), the length-width ratio (LWR) is defined as:

To investigate the effect of the LWR on the optimization performance, we carry out two groups of simulations. The parameters used for FDTD simulations are shown in Table 1. In the first group, we fix the row number to 20 and alter the column number to 10, 20 and 30 respectively. To minimize contingency, we run six independent optimizations under each configuration with various initial etching densities, and calculate the mean FoMs of the six optimizations to evaluate the optimization performance of each configuration. Figure 5(a) shows the evolution of the mean FoMs of 20 × 10, 20 × 20 and 20 × 30 configurations with FDTD simulation runs. It is worth noting that due to the symmetric optimization method we can only focus on the upper half of the pixels, thus one iteration of the M × N configuration requires (M × N)/2 runs. Among this group of configurations, the 20 × 30 configuration achieves the highest mean FoM (0.744), and the evolution trend clearly demonstrates that higher LWRs result in better device performance. To further understand the mechanism behind this trend, we record the number of iterations required for convergence for each configuration, as shown in the inset of Fig. 5(a). The results reveal that the 20 × 30 configuration requires the maximum number of iterations to converge, providing more opportunities for optimization and resulting in a growth of the FoM.

 

Fig. 5. (a) Evolution of the mean FoMs in the first group of simulations. The error bars of the curves represent the range between the maximum and the minimum FoMs. The inset shows the number of iterations required for final convergence under different configurations. The bar presents the mean value and the error bar shows the range between maximum and minimum values. (b) Evolution of FoMs in the second group of simulations. The insets are the 3D view diagrams of the initial structures of 30 × 20 and 40 × 15 configurations, respectively. The 20 × 30 configuration is shown in Fig. 1.

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Table 1. Device Parameters in the Investigation of LWR

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However, the setting of configurations in the first group changes not only the LWR, but also the total number of pixels, resulting in an alteration in the optimization DoF. For a fairer comparison, we fix the number of pixels to 600 in the second group of simulations. We evaluate the optimization performances of three BPA configurations with different LWRs: 20 × 30 (LWR of 1.50), 30 × 20 (LWR of 0.667), and 40 × 15 (LWR of 0.375). For each configuration, we perform one optimization with an initial etching density of 0%. The optimization results show that the final FoM of the 20 × 30 configuration (0.761) significantly surpasses the 30 × 20 and 40 × 15 configurations by 45.51% and 61.57% respectively (Fig. 5(b)). This confirms that higher LWRs result in better optimization performance, even under the same DoF condition. Furthermore, the FoM of the 20 × 20 configuration (0.628) is higher than the 30 × 20 and 40 × 15 configurations (0.523 and 0.471 respectively), indicating that higher LWRs can even enhance the inverse design of WDPSs with less DoF. The insets of Fig. 5(b) show that low LWRs can lead to larger steering angles, particularly for 1310 nm light that is supposed to output from the top and bottom ports. This large steering angle can cause high scattering, resulting in degraded overall optimization performance.

3.2.2 Initial etching density

For the inverse design of the WDPS, an initial BPA pattern with specified initial etching density (IED) must be generated before optimization. First, we need to generate a state matrix (SM) to determine and store the state of each pixel in the BPA. The size of the SM is M/2×N for the M × N configuration due to the symmetry. The initial elements of SM are random numbers that are uniformly distributed between 0 and 1. Next, all elements with values less than IED will be reassigned to 1 and the rest will be reassigned to 0. The SM therefore becomes a binary matrix and the initial BPA pattern will be produced according to the SM, where the pixel state of the i-th row and j-th column is determined by the value of SM_ij. To better quantify and record the evolution of etching density (ED) during the optimization process, we define the ED as:

In the first group of simulations of LWR, we run six independent optimizations for each configuration, with IEDs of 0%, 20%, 40%, 60%, 80% and 100%, respectively. However, there is no clear correlation observed between the FoM and IED (see Supplement 1). Therefore, we primarily focus on analyzing the FET of WDPSs under various IEDs. We analyze the simulation results of the 20 × 30 configuration and find that both the final device structure and performance are highly sensitive to the IED. To be specific, the final etching density of the BPA pattern increases with the IED (Fig. 6(a)), leading to variations in the FET. Here, two cases of fabrication error mentioned in Section 2.4 are introduced to test the FET under different IED condition. We first record the transmission fluctuation of output ports when the pixel misalignment is applied to the WDPSs. For each IED, we carry out 26 FDTD simulations of the final optimized WDPS with pixel misalignment, of which the position offset is swept from 0 to 10 nm. We present a box plot for the normalized S21_1310nm in Fig. 6(b) and conclude that the fluctuation range increases from 3.5% to nearly 20% as the IED increases from 0% to 100% (see Supplement 1 for more detailed information of the transmission fluctuation). It is noteworthy that the fluctuations in S21_1310nm are much more significant than in S31_1550nm (Fig. 6(b)) because OP₁ is further away from the central line of the coupling area than OP₂. As a result, the 1310 nm light requires a greater angle of deflection to be guided to OP₁, and is therefore more susceptible to scattering caused by the index mismatch between the etching hole and the silicon waveguide. The total fluctuation also reveals a positive correlation with IED, proving that higher etching density leads to worse FET.

 

Fig. 6. Simulation results for six different IEDs ranging from 0 to 100%. The BPA configuration is 20 × 30. (a) Evolution of ED with the optimization iterations. Only the results of first 9 iterations and the final BPA pattern are given because the ED becomes stable in the later stages of optimization. (b) Box plot of the normalized S21_1310nm. Each box contains the middle 50% of data points among each group of 26 simulations. The black line in each box shows the median value of each group. The error bar extends from the box to mark the range between the minimum and maximum values. The minimum values of the different groups are connected by a red dashed line. (c) Normalized S21_1310nm under different radius variation. Only the data points from -10 nm to 10 nm are plotted for simplicity. (d) FET bandwidth (marked in red) and transmission fluctuation (marked in blue) under different IED.

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We then test the FET bandwidth by applying the radius variation of etching hole to each WDPS with different IED. We run 101 simulations to sweep the radius variation from -20 nm to 20 nm for each IED. As in the first case of fabrication error, the S21_1310nm shows a much more dramatic change than S31_1550nm under different radius variation. The FET bandwidth is then defined as the radius variation range when the normalized S21_1310nm is reduced by 3 dB (Fig. 6(c)). The red line in Fig. 6(d) shows a downward trend in the FET bandwidth as the IED rises from 0% to 100%, indicating that higher IED leads to a deterioration of the FET, consistent with the findings from the first case of fabrication error.

3.2.3 Duty cycle

Another type I hyperparameter is the duty cycle (DC) of WDPSs, which is defined as:

 

Fig. 7. Final BPA patterns under different DC: (a) 25%; (b) 33.3%; (c) 50%; (d) 66.7%; (e) 75%; (f) 100%. The etching density is labeled in each pattern.

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The device performances including FoM and FET of WDPSs under different DC are shown in Fig. 8, while several key simulation results obtained via FDTD method are provided in Table 2. Figure 8(a) demonstrates that as the DC increases from 25% to 100%, the final FoM starts from 0.424, reaches its highest value of 0.761 at a DC of 50%, and eventually decreases to 0.633 at a DC of 100%. The decline in FoM when the DC strays from the optimal value of 50% is due to the limited ability to manipulate the optical field, which is either caused by too few etching holes (as shown in Fig. 7(f)) or by an excessive distance between the etching holes (as seen in Fig. 7(a)). As for the FET, Fig. 8(b) displays the transmission fluctuations in S21_1310nm and S31_1550nm (blue bars) when the radius variation is swept from -10 nm to 10 nm and the normalized EA (red line) under different DC. The findings indicate that the fluctuation range could be determined by the etching area, for the normalized EA shows a clear positive correlation with the transmission fluctuation. This concurs with the earlier discussions on IED, where it is noted that the FET degrades as the etching density increases. The fabrication error acts on the etching holes rather than the silicon waveguide core, therefore, as the etching area expands, the likelihood of being affected by the fabrication error increases as well. Considering both the FoM and the FET, it appears that a DC of 50% is still the optimal choice to achieve the optimal balance between etching density and etching hole size.

 

Fig. 8. (a) Evolution of FoM under different DC. (b) Transmission fluctuations in S21_1310nm and S31_1550nm and normalized etching area under different DC.

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Table 2. Key Results of the Device Performance of Inverse-designed WDPSs under Different DC

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3.2.4 Etching material

The final type I hyperparameter that we evaluate is the etching material. We explore the effect of three different materials on optimization performance: silica (used in all the designs above), air and silicon nitride. The material indices obtained from [45] are listed in Table 3. The cladding material is silica for the silica-etch and the silicon-nitride-etch cases, while the air cladding is applied for the air-etch case due to the fabrication constraints. The other hyperparameters are set to the optimal choices identified before: LWR = 1.5 (20 × 30 configuration), IED = 0% and DC = 50%. Before starting the inverse design, we directly run 50 FDTD simulations of randomly initialized WDPSs without optimization for each material to explore the intrinsic differences brought by the index contrast. For each simulation, the BPA pattern is reinitialized randomly with an etching density of 50%. Figure 9(a) shows the mean value of the total transmission of OP₁ and OP₂ in each group of 50 simulations with an input source of the 1550 nm light. The 1310 nm light will produce a similar result for the unoptimized WDPSs and thus will not be discussed here. The transmission of the unetched WDPS is also presented in Fig. 9(a) as a reference. Results reveal that the air-etch case has the smallest mean transmission which corresponds to the highest loss while the silicon-nitride-etch case achieves the best transmission. The reason is that a larger index mismatch between the etching hole and the silicon multimode waveguide will lead to severer scattering and, therefore, a larger insertion loss.

 

Fig. 9. (a) Summed transmission of 1550 nm at OP₁ and OP₂ of randomly generated WDPSs without optimization. The bar height represents the mean transmission of 50 simulations while the error bar indicates the range between the minimum value and maximum value. (b) S21 and S31 at 1310 nm and 1550 nm of the final optimized WDPSs with different etching materials. (c) The broadband S21/S31 in a wavelength range from 1200 nm to 1700nm. The S21/S31 parameter represents the channel crosstalk and optical bandwidth qualitatively. (d) Normalized S21_1310nm under different radius variation. (e) Transmission fluctuation and FET bandwidth of the final optimized WDPSs with different etching materials.

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Table 3. Refractory Indices of Silicon, Silica, Air and Silicon Nitride

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Afterward, we start the optimization process for each etching material based on TS-DBS method. Optimization results show that the FoMs of three cases are very close (Fig. 9(b)), with the S parameters staying at almost the same level, which means that the inverse design based on TS-DBS method can overcome the inborn poor performance brought by large index mismatch. However, since the TS-DBS does not focus on the FET, the FET is instead determined by the structural and material properties. The optimal etching material is selected by conducting FET tests on each optimized WDPS with different etching materials as their FoMs are close. The FET tests include 61 runs with the radius variation swept from -30 nm to 30 nm to calculate the FET bandwidth and another 26 runs with the pixel position offset increased from 0 to 10 nm. The results, shown in Fig. 9(d), are consistent with previous random tests and indicate that the air-etch case has the smallest FET bandwidth and highest transmission fluctuation, while the silicon-nitride-etch case achieves the best FET performance. Silicon nitride is therefore chosen as the optimal etching material.

3.2.5 Optimization bandwidth

The inverse designs above are all carried out under narrowband optimization condition, which means the FoM only contains the transmission data of two single wavelength points: 1310 nm and 1550 nm. However, the optical bandwidth of WDPSs may be constrained by this narrowband optimization method. In this part, we focus on the broadband optimization of TS-DBS, expanding the optimization bandwidth (OB) from 0 nm to 100 nm to explore the trade-off between peak transmission and optical bandwidth. To enable broadband optimization, the expressions for FoM₁ and FoM₂ must be altered to include the effect of OB, which can be rewritten as:

We conduct four independent optimizations with OBs of 0 nm, 7 nm, 50 nm and 100 nm, respectively. The type I hyperparameters of the WDPSs are set the same as the silicon-etch case in part D. After optimization, we extract the broadband response of each WDPS via FDTD simulations and evaluate the optical bandwidth corresponding to each OB, as shown in Fig. 10. The 3 dB bandwidth of S31 around 1550 nm (S31_{1550 nm}) exceeds 100 nm when using narrowband optimization, and the further expansion of optimization bandwidth only flattens S31 rather than increasing its bandwidth. However, the S21 parameter around 1310 nm (S21_{1310 nm}) experiences a dramatic amplification in both its 3 dB bandwidth and flatness when the OB is increased to 50 nm and 100 nm. The evolution of FoMs under different OB is presented in Fig. 11(a). However, the final FoMs under different optimization bandwidth should not be directly compared since the FoM of broadband optimization is determined by the worst transmission point in the relevant wavelength range. As a result, we use the narrowband expressions (1) and (2) to recalculate the FoMs (named as effective FoMs) which represent the peak transmission of 1310 nm and 1550 nm as shown in Fig. 11(b). The 3 dB bandwidth of S21_{1310 nm} is also shown in Fig. 11(b) to quantify the change in the optical bandwidth. We find that the effective FoM and optical bandwidth both decline when the OB is increased from 0 nm to 7 nm, indicating that an OB smaller than the original optical bandwidth of the narrowband-optimized WDPSs will impair the device performance. The bandwidth enhancement is noticeable when the OB is large enough, while the effective FoM gradually drops due to the trade-off between the optical bandwidth and the peak transmission.

 

Fig. 10. Broadband response of S21 and S31 under different OB: (a) Narrowband optimization (OB = 0 nm); (b) OB = 7 nm; (c) OB = 50 nm; (d) OB = 100 nm.

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Fig. 11. (a) Evolution of FoMs under different optimization bandwidth. (b) Effective FoM (red line) and 3 dB bandwidth of S21_1310nm (blue bars). The bandwidth for WDPS with an OB of 100 nm exceeds the wavelength range of FDTD simulation and covers ∼140 nm in the simulation range.

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3.3 Performance comparison with a forward-designed WDPS

Since the concept of WDPS is introduced for the first time to the best of our knowledge, we design an additional WDPS using the traditional forward design method as the baseline device for performance comparison with the inverse-designed WDPS (iWDPS). The forward-designed WDPS (fWDPS) comprises of one 2 × 2 MMI and two 1 × 2 MMIs, all based on the 220 nm SOI platform with silica cladding, as depicted in Fig. 12. The forward design of each MMI is based on the self-imaging theory [46], which states that the optical modes in the waveguide interfere with each other, leading to the periodic formation of single-images and multiple-images of the input optical field along the propagation direction. We first set the width of MMIs to 2.6 µm and the length of each MMI can be then calculated based on the corresponding imaging length (see Supplement 1). The 2 × 2 MMI works as a wavelength demultiplexer, directing the 1550 nm light to the upper waveguide and the 1310 nm light to the bottom output waveguide, respectively. The demultiplexer MMI has a length of 49 µm, which is the single-image length of 1550 nm. A relatively small footprint can be obtained using this configuration, while the IL at O band will be slightly degraded (Fig. 12(b)). The output ports of the demultiplexer MMI are connected to two 1 × 2 MMIs which function as the 3 dB power splitters at O band and C band respectively. The lengths of power-splitter MMIs are set to 29.6 µm and 24.5 µm, corresponding to the double-image length of 1310 nm and 1550 nm respectively (see Supplement 1).

 

Fig. 12. (a) Forward-designed WDPS and inverse-designed WDPS. The output ports of the forward-designed WDPS are marked as Port I∼IV. (b) Broadband response and (c) field profiles at 1310 nm and 1550 nm of the forward-designed WDPS.

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To evaluate the performance of fWDPS, we run a simulation via FDTD method and extract the transmission data. This device features a relatively symmetric transmission spectrum and large optical bandwidth (Fig. 12(b)), while the device footprint far exceeds the iWDPS (Fig. 12 (a)). To better compare the device performance and footprint of the iWDPS and fWDPS, we present the simulation results including average insertion loss (Avg. IL), average channel crosstalk (Avg. XT) and final FoM as well as the structure parameters in Table 4. The iWDPS presented here is the silicon-etched 20 × 30 WDPS using the narrowband optimization method, with an IED of 40% and a DC of 50%. It is worth mentioning that both the iWDPS and fWDPS can be fabricated by the mainstream commercial foundries. As seen from Table 4, the performance of both devices is comparable, with the iWDPS even achieving 1% higher FoM than the fWDPS. The device length of the fWDPS is 10.7 times larger than the iWDPS. The width of the fWDPS is obtained using straight single-mode waveguides as the connection waveguides between the demultiplexer MMI and the power-splitter MMIs. However, this close positioning of the power-splitter MMIs can lead to unwanted mode coupling between the two multimode waveguides. A more practical choice is to implement a pair of bent waveguides as the connection waveguides to broaden the distance between the two power splitters. Therefore, the total footprint of the fWDPS is at least 12.77 times larger than that of the iWDPS.

Table 4. Comparison of the Inverse-designed and Forward-designed WDPSs

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3.4 Releasing fabrication constraints

After all the discussion on the hyperparameters above, we choose to apply the optimal hyperparameters to the inverse design of a WDPS with a minimum feature size reduced to 65 nm. The footprint of this released WDPS (rWDPS) maintains the same as the previous 20 × 30 configuration. With the reduction of the minimum feature size by half, the pixel length can be reduced to 130 nm, allowing for an expansion of the rWDPS configuration to 40 × 60, producing a four-times larger DoF compared to the 20 × 30 configuration. The specific hyperparameters of the rWDPS are outlined in Table 5. The rWDPS can only be fabricated using electron beam lithography (EBL) instead of ultra-violet lithography (UVL) due to the ultra-small feature size.

Table 5. Hyperparameters of the Constraint-released WDPS

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Figure 13(a) displays the evolution of insertion loss and wavelength isolation for the 1310 nm light and 1550 nm light, respectively. The wavelength isolation (WI) is defined as the opposite of XT, which can benefit the illustration of performance evolution in Fig. 13(a). It takes 12 iterations for the FoM to converge. During the first 1000 simulations, IL and WI undergo substantial changes, while during the subsequent optimization process, IL shows a stable decline trend while WI exhibits a stable upward trend. The simulation results indicate that the ILs of the rWDPS at 1310 nm and 1550 nm are 0.36 dB and 0.37 dB, respectively, and the XTs are -19.91 dB and -17.02 dB, respectively (Fig. 13(b)). We use the narrowband TS-DBS method to achieve the highest FoM (0.90), while the broadband TS-DBS can also be implemented depending on the desired optimization objective. We also investigate the change of transmission spectrum when the optimization bandwidth increases (see Supplement 1). It is worth noting that the performance of the two wavelength channels is not only improved but also more balanced compared to the iWDPS and fWDPS presented in Table 4. These results demonstrate that with the appropriate selection of hyperparameters and a high DoF, the performance of the WDPS can be boosted using the TS-DBS method, enabling multifunctionality and ultra-compact footprint simultaneously.

 

Fig. 13. (a) Evolution of insertion loss and wavelength isolation of the released WDPS during the optimization process. (b) Simulated broadband S-parameter of the optimized rWDPS. (c) Field profiles at 1310 nm and 1550 nm of the optimized rWDPS.

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4. Discussion and outlook

4.1 Discussion

4.1.1 Underlying influence of hyperparameters

In the inverse design of DMSs with hundreds of pixels, the dimension of the parameter space is nearly infinite, making it difficult to find the optimal solution without proper guidance and limitations. The hyperparameters, however, serve as the boundaries of inverse design by dividing the parameter space into separate, non-overlapping regions. Each combination of hyperparameters corresponds to a unique region in the parameter space. An appropriate selection of hyperparameters actually confines the search space to a region that is more likely to contain the optimal solution. Therefore, the setting of hyperparameters is crucial for the inverse design of DMSs based on optimization algorithm.

4.1.2 Generalizability of hyperparameter conclusions

Although our goal in this work is not to find a universally optimal set of hyperparameters for different DMS devices, we have made efforts to validate the generalizability of our findings, including numerical simulation and theoretical analysis.

Due to the constraints of time and computational resources, we only carried out numerical validations on the generalizability of two hyperparameters: length-width ratio and etching duty cycle. Specifically, an additional inverse design was performed to create a 1310/1550 nm wavelength demultiplexer based on DMS. The results of this supplementary analysis are in line with the conclusions of the WDPSs (see Supplement 1 for detailed information on the design process and results of the demultiplexer).

The validations for other hyperparameters are carried out through theoretical analysis, which are illustrated as follows. (1) As for the etching materials, the generalizability can be explained through theoretical analysis. While a higher refractive index mismatch can lead to increased scattering, the inverse design method actually utilizes the scattering resulting from local refractive index variations to achieve effective control over light guidance. This specific type of scattering is referred to as “expected scattering” and is carefully managed and optimized through the use of the TS-DBS algorithm. However, the presence of fabrication errors can cause deviations in the local refractive index profiles, deviating from the optimal results achieved through TS-DBS optimization, resulting in “unexpected scattering”. When materials with a larger refractive index mismatch in comparison to silicon waveguides are employed, the levels of unexpected scattering tend to be higher, consequently reducing the FET and affecting overall performance. Therefore, based on the above analysis, we can conclude that a larger refractive index mismatch between the etching material and the silicon waveguide results in poorer FET performance of the device. Since this conclusion is driven by the inherent characteristics of the materials rather than specific device functionalities, we believe that this finding can be applied to inverse design of other devices, showcasing its strong generalizability. (2) In terms of the initial etching density, we find that a higher initial etching density results in a worse FET. The explanation is similar to the previous discussion on etching materials, which pertains to the degree of unexpected scattering. More specifically, a larger initial etching density leads to a higher final etching density due to the nature of the TS-DBS algorithm, which seeks an optimal solution closest to the initial conditions. A higher etching density implies the presence of more interfaces between silicon and the etching material, which are prone to scattering. Consequently, when fabrication errors occur, WDPSs with higher etching densities are more susceptible to unexpected scattering, leading to a more significant degradation in performance. The generalizability of this conclusion extends beyond the specific functionality of the device. Hence, it is highly potential to be applied to the inverse design of other DMSs, exhibiting a robust generalizability. (3) Another notable hyperparameter is the optimization bandwidth, which is the only type II hyperparameter discussed in this study. The intrinsic generalizability of this hyperparameter is evident, as its implementation serves as an intentional constraint applied during the optimization process, rather than a conclusion derived from parameter sweeping.

Based on the above discussions, a general conclusion can be offered as follows. The findings regarding the impact of etching materials and initial etching density can be extrapolated to the inverse design of various common DMS devices. The insights on length-width ratio and etching duty cycle are specifically applicable to forward-propagating DMS devices, including our proposed WDPS, power splitters, wavelength demultiplexers, and mode demultiplexers. Regarding the discussion on optimization bandwidth, we consider it as a novel optimization approach that involves controlling the device bandwidth by imposing constraints within the FoM.

4.1.3 Application of WDPS in next-generation PON

Our proposed WDPS is the first reported nanophotonic device that can perform wavelength demultiplexing and power splitting at the same time. As for the application of this multifunctionality, WDPS hold potentials for supporting the development of next-generation PON.

Specifically, traditional PON systems based on intensity-modulation and direct detection (IMDD-PON) operating in the O-band (centered at 1310 nm) exhibit limitations in sensitivity characteristics and communication capacity. The adoption of coherent PON (CPON) technology operating in the C-band (centered at 1550 nm) presents a potential solution to overcome these limitations, enabling the realization of 200 Gbps-PON in next-generation systems [41]. The transition from IMDD-PON to CPON is expected to occur gradually, necessitating devices capable of functioning in both the O-band and C-band during this transition period [43].

In general, our proposed ultra-compact dual-band WDPS, which can work as a dual-band integrated power splitter, offers three main advantages for supporting the development of next-generation PON. Firstly, the 3 dB power splitting functionality of WDPS allows for efficient power allocation among different optical network units (ONUs) by cascading multiple WDPSs. Secondly, its dual-band characteristics enable a smooth and efficient transition from IMDD-PON to CPON, requiring only the replacement or upgrade of equipment at the optical line terminator (OLT) and ONU ends, while the infrastructure in the transmission part does not require reinstallation or replacement, resulting in substantial cost and complexity savings during network upgrades. Finally, its compact size enhances scalability and cost-effectiveness of PON deployments, supporting the development of on-chip integrated systems for next-generation PON.

4.2 Outlook

In this study, we analyze the impact of hyperparameters based on the spectra data. However, to gain a deeper understanding, the simulated spectra and their corresponding hyperparameters can be used as input for an AI-based learning machine. The AI-based learning machine should be capable of uncovering patterns in the hyperparameters that would be difficult to discern through traditional methods. More specifically, leveraging transfer learning technique as the AI learner, optimal hyperparameter values obtained from one specific case can be directly employed or utilized as preconditioners for other cases. This technique can significantly accelerate the inverse design process and amplify its efficacy and efficiency. Therefore, our future work includes a more comprehensive investigation of the optimal hyperparameters for various DMS devices, such as wavelength demultiplexers, mode demultiplexers, power splitters, grating couplers, and others.

Furthermore, it is worth mentioning that recent studies focusing on the in situ programmable inverse design of nanophotonic devices have predominantly utilized DMS structures [47,48]. Therefore, our findings can be readily extended and applied to this promising area of research.

Another noteworthy aspect is how to reduce the backscattering strength of the inverse-designed DMSs. DMSs typically contain numerous regions characterized by abrupt changes in refractive index, which are prone to generating multi-bounce effect and increasing backscattering strength. Excessive backscattering can lead to significant crosstalk between different paths and increased noise, thereby resulting in performance degradation. In our work, the proposed WDPSs exhibit relatively low backscattering strengths ranging from approximately -20 to -30 dB. To further mitigate the backscattering effects and achieve truly reflectionless signal routing, our future work will involve exploring the applicability of a recently proposed method named as reflectionless scattering modes (RSM) based signal routing in the field of nanophotonic inverse design [49].

5. Conclusion

In this study, we designed a series of WDPSs using the inverse design method based on the TS-DBS algorithm, which can effectively address the premature convergence of optimizations. Our findings on the impact of different hyperparameters related to the physical structure (Type I hyperparameters) and the optimization algorithm (Type II hyperparameters) were discussed in detail. By varying the hyperparameters, we discovered that a higher LWR and a DC closer to 50% lead to improved XT and IL, while a lower IED and lower index mismatch between the etching material and silicon improve FET. The influence of the optimization bandwidth of the TS-DBS on the device performance was also investigated and a trade-off between peak transmission and optical bandwidth was found. In addition, we compared the device performance and footprint of the inverse-designed WDPS with those of a WDPS designed using the conventional forward design method. The inverse-designed WDPS resulted in a 12.77 times reduction in footprint and 1% improvement in FoM compared to the forward-designed WDPS, both of which can be manufactured by commercial foundries. We further applied the best combination of hyperparameters to the inverse design of a WDPS released from fabrication constraints, resulting in ultra-low ILs of 0.36 dB and 0.37 dB and XTs of -19.91 dB and -17.02 dB at 1310 nm and 1550 nm, respectively. These results demonstrate that our proposed WDPSs extend the SiP device library and offer a novel solution for optical communication systems. The generalizability of our findings was also validated through numerical or theoretical analysis. In future work, we aim to use AI-based learning machine to learn the hidden patterns of hyperparameters for a more comprehensive analysis, and validate the generalizability of our findings on the inverse design of other functional SiP devices.