2.1.

Theory and Strategy

For an inverse design, there are two key points to improve the design efficiency. First, the search space dimension should be minimized as much as possible. Second, the search space should be well-defined so that the final optimal design is included. As a result, in this paper, we propose an inverse design method with unique features that are useful for efficiently developing advanced passive silicon photonic devices, as described below.

2.1.1.

Device geometry definition

The device structure is basically defined geometrically with some waveguide branches separated by subwavelength grating (SWG) metamaterial structures, as shown by the example given in Fig. 1(a). Definitely, the devices with more ports have more branches. Here each branch [see the deep-blue part in Fig. 1(a) which consists of several segments longitudinally, and each segment is defined by its corner locations (i.e., the black pillars in Fig. 1(a)]. As shown in Fig. 1(a), an SWG structure is usually defined by the period $P$ and the fill factor $\mathrm{\Lambda }$, which can be preset or adjusted freely as desired. The SWG structures enable the dispersion engineering to realize broadband or wavelength-sensitive photonic devices when desired.^23,28 These to-be-designed geometry parameters are included in the individual for the optimization.²³

Fig. 1

Proposed inverse design strategy for passive photonic devices. (a) The design framework is demonstrated by an example of a photonic device with one input port and two output ports. Here, the SWG has a period of $P$ and a fill factor of $\mathrm{\Lambda }$. (b) Design flow chart with manual interventions. (c) Typical manual intervention operations.

This device geometry definition is derived from our previous work,²³ where a universal strategy of on-chip multimode excitation/interference was demonstrated. In this work, the optimization of the design region enables flexible mode manipulation and further realizes on-chip light manipulations. The present design strategy is potentially useful as a universal approach for developing various optical functional devices even without any special initialization. This is different from those previously reported works based on multisectional multimode interferences (MMIs) optimized with appropriate initialization.^24,29 In particular, the multibranch structure proposed here as the initialization can build up a relatively smooth connection between the input/output ports and the design region. As a result, it is possible to achieve excess losses (ELs) lower than those of previous inverse design devices with random nanostructures.

Apparently, the search space dimension is decided by the number of the lengthwise sections in the design region. Any design region with $n$ sections can be redefined with $2n$ sections when needed. Such flexibility in the design makes it convenient to gradually adjust the search space dimension. For example, a low search space dimension with, e.g., $n=10$ only can be adopted at the first stage of the optimization, so that one can achieve a suboptimal solution with a low computational cost. The suboptimal solution can be further used to initialize the second stage of optimization for which the search space dimension is improved to be with $n=\sim {10}^2$. In this way, the present approach has much higher capability for the optimization of complex photonic devices even with limited computational resources than those of previous inverse design approaches.

2.2.

Realization of a Six-channel Mode (de)multiplexer

A mode (de)multiplexer is the fundamental device in mode-division-multiplexing systems. Typically, high-performance mode (de)multiplexers are often realized by utilizing multi-stage asymmetric directional couplers.^36,37 Recently, ultracompact mode (de)multiplexers have been demonstrated by using the inverse design methods.^12–14 However, these inverse-designed modes (de)multiplexers have no more than four channels. The reason is that the inverse design becomes much more difficult due to the increased complexity when it is desired to be with more channels (more ports).

Here, we propose and demonstrate a six-channel mode (de)multiplexer for the first time by utilizing the inverse design method. As shown in Fig. 2(a), the initial geometry framework is based on an ordinary $1\times 6$ branch configuration without special considerations. In the optimization region, there are six polygon waveguides defined in the following way. Along the lengthwise direction, there are $n$ sections defined with the lengths $\mathit{L}=[l_1,l_2,\dots ,l_n]$. At the $yz$ cross-sectional interface between the ($n-1$)’th and the $n$’th sections, the parameters ${\mathit{D}}_{\mathit{n}}=[d_{n1},d_{n2},\dots ,d_{n12}]$ define the lateral corner location in the $y$ direction [see Fig. 2(a)]. The metamaterial SWG structure has a period $P$ and a fill factor $\mathrm{\Lambda }$. Specifically, here we choose $P=200\mathrm{nm}$ and $\mathrm{\Lambda }=50\%$. For the input/output (I/O) waveguides, $w_{\text{in}}$ and $w_{\text{out}}$ respectively denote the waveguide widths, $d_{\text{in}0}$ and $d_{\text{out}0}$ respectively denote the positions of Port #1 and Port #7 in the $y$ axis, and $d_{\text{out}}$ is the separation between the adjacent output waveguides. The parameter $l_{\text{in}}$ ($l_{\text{out}}$) denotes the length of the waveguide taper connecting the optimization region and the input (output) waveguide. The parameter set ${\mathit{S}}_{\mathbf{\text{total}}}$ to be optimized can combine flexibly any related parameters. For example, one might have ${\mathit{S}}_{\mathbf{\text{total}}}=[{\mathit{D}}_{\mathbf{1}},{\mathit{D}}_{\mathbf{2}}\dots {\mathit{D}}_{\mathit{n}},{\mathit{D}}_{\mathit{n}+\mathbf{1}}]$ or ${\mathit{S}}_{\mathbf{\text{total}}}=[{\mathit{D}}_{\mathbf{1}},{\mathit{D}}_{\mathbf{2}}\dots {\mathit{D}}_{\mathit{n}},{\mathit{D}}_{\mathit{n}+\mathbf{1}},\mathit{L},l_{\text{in}},l_{\text{out}}]$.

Fig. 2

Inverse design of a six-channel mode (de)multiplexer on silicon. (a) Top view ($xy$ plane), 3D view, and cross-sectional view ($yz$ plane) at the interface between section $n-1$ and section $n$ (not to scale). (b) FOM and the simulation time cost as functions of the iteration generation for air-slot and ${\mathrm{SiO}}_2$-slot devices. Inset: the milestone structures in the optimization flow. (c) and (d) The 3D FDTD simulation performance of the designed ${\mathrm{SiO}}_2$-filled mode (de)multiplexer including the transmissions (c) and the light propagation fields for different input modes at 1550 nm (d). Modes #1 to #6 are, respectively, the ${\mathrm{TE}}_0$ to ${\mathrm{TE}}_5$ modes supported by the bus waveguides (Port #1), and modes #7 to #12 are, respectively, the ${\mathrm{TE}}_0$ mode of Ports #2 to #7. Here the silicon photonic waveguides with a 220-nm-thick silicon core and a ${\mathrm{SiO}}_2$ upper-cladding are used.

For the scattering matrix of a photonic device, the element $S_{ij}$ denotes the coupling coefficient from mode $\#j$ to mode $\#i$. For the six-channel mode (de)multiplexer considered here, the bus waveguide at Port #1 supports six modes (i.e., the ${\mathrm{TE}}_0-{\mathrm{TE}}_5$ modes denoted as modes #1–#6). The output waveguides at Ports #2–#7 support the corresponding ${\mathrm{TE}}_0$ modes (denoted as modes #7–#12). Ideally, a six-channel mode (de)multiplexer should have the transmissions of ${|S_{ij}|}^2=1$, where $|i-j|=6$, $1\le i\le 6$, and $7\le j\le 12$. The FOM for optimization is defined by $\mathrm{FOM}=-\frac{1}{6}\sum _{i=1}^{6}20\lg |{\mathrm{S}}_{(i+6)i}|$. Even though the optimization was run with a single operation wavelength of $\lambda =1550\mathrm{nm}$, fortunately, the optimized photonic device is very possible to work very well with a broad bandwidth, which is attributed to the dispersion-insensitive multimode excitation/interference processes in the SWG waveguides, as demonstrated previously.²³

Figure 2(b) shows the FOM and the total computation time cost as the iteration generation updates. The computation time was evaluated on an ordinary personal computer (PC; details in Appendix, Sec. 4.2). The milestone structures generated during the optimization are also depicted in Fig. 2(b). One can see that an ordinary branch waveguide structure is used as the initial individual. In terms of the fabrication process, the ${\mathrm{SiO}}_2$ filling in the SWG slots usually depends on the ${\mathrm{SiO}}_2$ deposition technology. In the design, we consider the cases with air slots or ${\mathrm{SiO}}_2$ slots. The air-slot device was designed first with a regular linear initial structure [Fig. 2(b)]. The EME method was used at the early stages, and the FOM reaches 0.42 dB after 384 generations of iteration (80.6 h). The FDTD simulation with two types of mesh precision (see the details in Table S1 in the Supplementary Material) was then used in the following stages of optimization. Finally, the FOM is about 0.69 dB, obtained with 104 generations (41.5 h). This corresponding design was then used as the initialization for further optimizing the air-slot and ${\mathrm{SiO}}_2$-filled devices with high-precision FDTD simulation. As shown in Fig. 2(b), the optimization of the air-slot device finishes at the 507th generation with an FOM of 0.55 dB, costing 247.1 h in total. And the optimization of the ${\mathrm{SiO}}_2$-filled device finishes at the 590th generation with an FOM of 0.62 dB, costing 419.2 h in total (including the 122.1 h for the optimization of the air-slot device initially). For such multistage designs, manual intervention processes were carried out. For example, the section number $n$ changes from 6 to 24. More details are given in Sec. S1 in the Supplementary Material. The two types of devices have similar theoretical performances, providing excellent options for the labs/fabs with different fabrication technologies. In the following section, we focus on ${\mathrm{SiO}}_2$-filled devices regarding the fabrication technologies available.

According to the high-precision 3D FDTD simulations with very fine meshes (26 mesh points per effective-wavelength scale), the designed ${\mathrm{SiO}}_2$-filled device (with detail geometric parameters in Table S2 in the Supplementary Material) has a footprint as compact as $7.5\times 18\mu {\mathrm{m}}^2$, low ELs from 0.53 to 1.03 dB, and low cross talks (CTs) $<-15.1\mathrm{dB}$ in the wavelength range from 1500 to 1600 nm for all mode channels, as shown in Fig. 2(c). In Fig. 2(d), the simulated light propagation at 1550 nm is given, showing impressive mode (de)multiplexing behaviors. The devices were fabricated by the regular processes from Applied Nanotools Inc.³⁸ (details in Appendix, Sec. 4.3). Figure 3 shows the experimental results. Here the measured PIC consists of a pair of mode (de)multiplexers and fiber-to-chip grating couplers [see Fig. 3(a)]. Figure 3(b) shows the scanning electron microscope (SEM) image of the fabricated device, and the normalized transmissions of the PIC are shown in Fig. 3(c). It can be seen that all channels for the fabricated mode (de)multiplexers have a low EL of $<\sim 1\mathrm{dB}$ and low CT of $<-10\mathrm{dB}$ in the wavelength range from 1520 to 1610 nm experimentally. The slight ripples in the transmission spectrums may be attributed to the nonzero reflection of the mode (de)multiplexers.

Fig. 3

Fabricated devices and measured results. (a) Microscope picture for the fabricated silicon PIC consisting of a pair of mode (de)multiplexers with six input ports ($I_{\mathrm{TE}0}-I_{\mathrm{TE}5}$) and six output ports ($O_{\mathrm{TE}0}-O_{\mathrm{TE}5}$). (b) SEM image of a mode (de)multiplexer. (c) Normalized transmissions of different port pairs.

2.3.

Realization of a 90 deg Hybrid

A 90 deg hybrid is the key device in optical coherent communication systems. Basically speaking, a 90 deg hybrid has two input ports [i.e., Port #1 for the signal and Port #2 for the local oscillator (LO)] and four output ports (#3, #4, #5, and #6), as shown in Fig. 4(a). Generally speaking, the optimal design of a 90 deg hybrid is pretty challenging because not only the amplitude responses but also the phase responses at the four ports should be considered.

Fig. 4

Inverse design of a 90 deg hybrid on silicon. (a) The 3D schematic diagram. LO, local oscillator. (b)–(f) The simulated performances of the final design (${\mathrm{SiO}}_2$ slot device): the simulated light propagation fields with varied phase difference $\mathrm{\Delta }{\theta }_{21}$ between Ports 2 and 1 at (b) 1550 nm, (c) the transmissions, (d) CMMRs, (e) the Els, and (f) phase errors (f). Here the transmission is given by the $S$ parameter (i.e., $S_{ij}$ channel is given by $20{\log }_{10}|S_{ij}|$). CMMR, common mode rejection ratio.

For a 90 deg hybrid, the scattering matrix of the electric field amplitude should satisfy eight requirements of ${|S_{ij}|}^2=1/4$ ($j=\mathrm{1,}2$; $i=\mathrm{3,}4,\mathrm{5,}6$), while the phase differences between the output ports should satisfy the condition of $\{{\theta }_3-{\theta }_6,{\theta }_4-{\theta }_6,{\theta }_5-{\theta }_6\}=\{-90\mathrm{deg},90\mathrm{deg},180\mathrm{deg}\}$, where ${\theta }_i=\arg (S_{i2})-\arg (S_{i1})(i=\mathrm{3,}4,\mathrm{5,}6$. In this problem, there are 11 target values, including eight amplitude requirements and three phase requirements. One should notice that the correlation of these target values is weak in the optimal design of a 90 deg hybrid. Specifically, the amplitude-target-value-set $\{{|S_{ij}|}^2,j=1\}$ for Port #1, the amplitude-target-value-set $\{|S_{ij}{|}^2,j=2\}$ for Port #2 and the phase-target-value-set $\{{\theta }_3-{\theta }_6,{\theta }_4-{\theta }_6,{\theta }_5-{\theta }_6\}$ have almost no correlation among them. In this case, the optimization is often likely to be trapped locally, and it is hard to reach the globally optimal design. Nevertheless, the present high-efficiency inverse design works well and gives a high-performance 90 deg hybrid successfully. As shown by the details in Sec. S2 in the Supplementary Material, the FOM definition can be adjusted manually in the multistage optimization process to ensure global optimization for all the target values.

Finally, the 90 deg hybrid is designed with a footprint of $\sim 13\times 5\mu {\mathrm{m}}^2$ (with detail geometric parameters in Table S3 in the Supplementary Material), and one has $\{{\theta }_3,{\theta }_4,{\theta }_5,{\theta }_6\}\approx \{0\mathrm{deg},-180\mathrm{deg},90\mathrm{deg},-90\mathrm{deg}\}+133\mathrm{deg}$. As a result, Ports #3 and #4 are the in-phase ($I$) channels, while Ports #5 and #6 are the quadrature ($Q$) channels. Figure 4(b) shows the simulated light propagation when the ${\mathrm{TE}}_0$ modes with different phase differences $\mathrm{\Delta }{\theta }_{21}$ are, respectively, injected into the two input ports. One can clearly see that the interference cancellation happens at one of the four output ports according to the phase difference. It proves that the present design method is effective even when both the amplitude and phase responses are required to be considered. Figures 4(c) and 4(d) show the simulated amplitude and phase responses in the wavelength range from 1530 to 1570 nm. It can be seen that the transmission losses are 6.6 to 7.4 dB and the ELs²⁹ of the signal- and LO-ports are $<0.96\mathrm{dB}$, while all the phase errors are within $-4.6\mathrm{deg}\sim 4.6\mathrm{deg}$, and the common mode rejection ratio (CMMR²⁹) are $>26.3\mathrm{dB}$ for all $I/Q$ channels.

Figure 5 shows the measurement results of the fabricated 90 deg hybrid. In our experiments, the transmissions were measured with the testing PIC [Fig. 5(a)], while the CMMRs and the ELs are extracted from the measured transmissions. As shown in Fig. 5(b), the measured transmissions are $-4.8\sim -9.0\mathrm{dB}$ in the wavelength range from 1530 to 1570 nm. The CMMRs of both $I/Q$ channels are below $-10.2\mathrm{dB}$, and the measured ELs are below 1 dB, as shown in Fig. 5(c). The phase responses were measured by the testing PIC consisting of a $1\times 2$ power splitter, a pair of asymmetric interference arms, a 90 deg hybrid, as well as grating couplers (see Fig. S1 in Sec. S2 in the Supplementary Material), which has been widely used before.²⁹ The measured transmissions at the four outports are presented in Fig. 5(d), and the extracted phase errors are given in Fig. 5(c), which shows that the phase error in the wavelength range from 1530 to 1570 nm is within the range from $-14.90\mathrm{deg}$ to 10.63 deg.

Fig. 5

Experimental results of the fabricated 90 deg hybrid. (a) Measured transmissions. (b) Measured CMMRs. (c) Measured ELs. (d) The port-to-port optical transmission are spectra measured by the phase-test PIC. (e) Phase error extracted from the measured results of the phase-test PIC.

2.4.

Realization of a Two-channel Flat-top Wavelength-Division (De)multiplexer

A wavelength (de)multiplexer³⁹ is a key element to separate/combine different wavelength channels in wavelength division multiplexing (WDM) systems. Here, a two-channel wavelength (de)multiplexer is designed optimally with the present inverse design method. Currently, there have been several inverse designs for the wavelength (de)multiplexers with two channels,⁹ three channels,^33,40,41 and six channels,⁴² and most of them are narrowband. As it is well known, a flat-top response is often desired in practice because it becomes tolerant of some random wavelength variation. For the design of a wavelength (de)multiplexer with a flat-top spectral responses, the inverse design is usually challenging because it is required to include multiple objective values. In this example, the two wavelength-channels from 1290 to 1330 nm and 1470 to 1570 nm are considered.

Basically, the two-channel wavelength (de)multiplexer has an input Port (#1) and two output Ports (#2 and #3), as shown in Fig. 6(a). The 1550 nm channel and the 1310 nm channel are launched from Port #1 and are then separated to Port #2 and Port #3, respectively. Here we choose several key wavelengths with a spacing of 20 nm for each channel, and the condition for the optimization is given by ${|S_{21}({\lambda }_a)|}^{2 }=1$, ${\lambda }_a\in \{\mathrm{1470,1490,1510,1530,1550,1570}\}\mathrm{nm}$ and ${|S_{31}({\lambda }_b)|}^2=1$, ${\lambda }_b\in \{\mathrm{1290,1310,1330}\}\mathrm{nm}$. The optimization costs 302.5 h (see more details in Sec. S3 in the Supplementary Material). Finally, the designed device has a compact footprint of $3.1\times 12.5\mu {\mathrm{m}}^2$ (with detail geometric parameters in Table S4 in the Supplementary Material). Figure 6(b) shows the simulated light propagation for the two central wavelengths by using the 3D FDTD method. As shown in Fig. 6(c), the designed wavelength (de)multiplexer has ELs of $<1\mathrm{dB}$ and ERs of $>10\mathrm{dB}$ in the wavelength ranges from 1250 to 1330 nm and 1470 to 1610 nm. The ELs at 1310 and 1550 nm are, respectively, 0.67 and 0.36 dB, while the corresponding ERs are, respectively, 15.02 and 30.33 dB. The fabricated wavelength (de)multiplexer [see the inset in Fig. 6(d)] was then measured with the help of the PICs with different grating couplers working at the corresponding wavelength bands of 1310 and 1550 nm. From the measured transmissions shown in Fig. 6(d), it can be seen that the EL is $<1\mathrm{dB}$ and the ER is $>10\mathrm{dB}$ in the wavelength ranges from 1256 to 1344 nm and 1482 to 1591 nm.

Fig. 6

The inverse-designed two-channel wavelength multiplexer on silicon. (a) 3D schematic diagram. (b) Simulated light propagation in the designed devices. (c) Calculated transmissions. (d) Measured transmissions. Inset: SEM image of the fabricated wavelength demultiplexer.

2.5.

Comparisons and Perspective

Table 1 gives a summary of various related silicon photonic devices reported. The state-of-the-art devices developed by conventional (forward) design methods and inverse design methods are listed. Here, the devices are evaluated by their performance within the whole bandwidth except those non-flat-top wavelength (de)multiplexers and those labeled with “@ ${\lambda }_0$.” Here, “@ ${\lambda }_0$” is for the performance at the central wavelength ${\lambda }_0$. The conventional (forward) design methods usually have advantages in device performance and the design scalability. For these devices, however, the difficulty of footprint shrinking is the key issue. In contrast, inverse designs have natural advantages in footprint compactness, while the bottleneck is the design efficiency for complex functional devices with a large number of to-be-optimized objective values, such as multichannel devices or flat-top wavelength-selective devices. As shown in Table 1, the present mode (de)multiplexer exhibits high theoretical performances of low ELs, low CT, and large bandwidths comparable to or even better than those of conventional mode (de)multiplexers.^37,43 The measured CT for the fabricated mode (de)multiplexers is slightly inferior to conventional mode (de)multiplexers, and the footprints depend on the channel numbers. More recently, a four-channel mode (de)multiplexers using tilt waveguide junctions was reported with footprint of $2\times 50\mu {\mathrm{m}}^2$,⁴³ which represent the state-of-the-art level of the conventional methods. Definitely, the device footprint will increase greatly if the channel number increases from 4 to 6. Furthermore, the performance degradation may occur as usual. To the best of our knowledge, we present the first six-channel mode (de)multiplexers developed with inverse design, owing to the high efficiency of our inverse design strategy. Moreover, our devices have advantages in the performances of the EL and the bandwidth when compared to the other inverse design counterparts.

Table 1

Summary of the state-of-the-art mode (de)multiplexers, 90 deg hybrids, and wavelength (de)multiplexers on silicon.

For conventional 90 deg hybrids, the state of the art is with an EL below 0.5 dB, a CMMR over 30 dB, and phase errors below 3 deg in the C-band (1530 to 1565 nm) while the footprint is $21.6\times 27.9\mu {\mathrm{m}}^2$.²⁹ In contrast, the present 90 deg hybrid has a footprint shrunk by ∼tenfold and similar performances, including low ELs, high CMMRs, low phase errors, and large bandwidth. Our present 90 deg hybrid has shown much better performance on the EL, the imbalance, the phase error, and the bandwidth than those of inverse-designed 90 deg-hybrid with 2D code structures.¹² The broadband characteristic of our 90 deg hybrid also reflects the high-efficiency capability of our approach for the optimization with a large number of objective values (e.g., with multiple wavelengths involved).

Nowadays, there have been many wavelength (de)multiplexers designed conventionally,³⁹ in which high performance and footprint compactness usually conflict. As shown in Table 1, the two-channel flat-top wavelength (de)multiplexers designed with conventional methods are usually based on Mach–Zehnder interferometers,⁴⁵ waveguide Bragg gratings,⁴⁶ or wavelength-selective waveguide couplers,⁴⁷ and they have lengths in scale of $\sim {10}^2\mu \mathrm{m}$. In contrast, the inverse-designed wavelength (de)multiplexer in this work has an ultracompact footprint of $3.07\times 12.46\mu {\mathrm{m}}^2$. In recent years, there have been several inverse-designed wavelength (de)multiplexers.^9,33,40–42 However, they most are nonflat-top.^33,41,42 The inverse-designed wavelength (de)multiplexer in Ref. 9 is indeed flat-top while the EL is higher than $1.8/2.4\mathrm{dB}$ for the central wavelengths of $1.31/1.55\mu \mathrm{m}$. In contrast, our device has flat-top responses as well as a low EL of $<1\mathrm{dB}$. More recently, a three-channel flat-top wavelength (de)multiplexer was reported with a bandwidth of $\sim 30\mathrm{nm}$ for an EL of $\sim 1\mathrm{dB}$ and an ER of $>15\mathrm{dB}$.⁴⁰ In contrast, our wavelength (de)multiplexer enables flat-top responses with an ultrabroad bandwidth from 80 to 110 nm. Furthermore, the present devices can be scaled easily for more channels by structure cascading when needed.

In summary, for practical applications (e.g., optical interconnects^1,2), the devices are often required to have decent performance with low losses, high extinction ratios, multiple channels, broad bandwidths, etc. As shown in Table 1, the previously reported inverse design devices usually cannot satisfy the requirement with ELs of $<1\mathrm{dB}$ in a broad bandwidth, which fortunately can be achieved in this work. Our designs need computation time of $\sim 200\mathrm{h}$ (see Supplementary Material) when using a single PC. This computation resource requirement is affordable even for the six-channel mode (de)multiplexer and 90 deg hybrid, which are beyond the capability of previous inverse designs with regular computational resources. As shown in Table 1, the experimental results for the CT/ER/CMMR/IM are usually inferior to the simulation ones, mainly due to fabrication errors. In our case, all the geometry parameters including ${\mathit{D}}_{\mathit{n}}=[d_{n1},d_{n2},\dots ,d_{n12}]$ and the SWG feature size are chosen to be sufficiently large (e.g., $>80\mathrm{nm}$) according to the fabrication technology. Unfortunately, there exist some acute-angle structures at some corners, which leads to some performance degradation for the fabricated devices. In the future work, the device performance (e.g., the CT) could be possibly improved by introducing smoothened subwavelength structures. The optimization efficiency should be improved further for realizing the devices with very high complexity, such as the mode/wavelength (de)multiplexers with many channels.

4.1.

Covariance Matrix Adaptation Evolution Strategy (CMA-ES)

CMA-ES is based on a multi-individual search in which the search individuals of each generation are sampled from multivariate normal distributions. The individuals with good FOMs are used for guiding the evolution of the normal distribution parameters. The introduction of normal distribution follows the maximum entropy principle, making this algorithm powerful. The algorithm flow is described as follows. The individuals of each generation are obtained by sampling of the normal distribution $N(m,({\sigma }^{(g)}{)}^{2(c)})m^{(g)}+{\sigma }^{(g)}N(0,C^{(g)})$, where $m^{(g)}$ is the mean value of the search distribution at generation $g$, ${\sigma }^{(g)}$ is the step-size, and $C^{(g)}\in {\mathbf{R}}_{n\times n}$, is the covariance matrix of the normal distribution $N(0,C^{(g)})$ at generation $g$. The FOM of each individual is evaluated by EM solving and then ranked. After that, part of the individuals with good FOMs (usually the better 50%) are used for updating the parameters $[m^{(g+1)},{\sigma }^{(g+1)},C^{(g+1)}]$ by covariance matrix adaptation algorithm for next-generation computation (details in Ref. 30). As the generation increases, the population (including the mean value and the sampled individuals) will approach to the promising area covering good solutions. CMA-ES has been known as one of the most efficient evolutionary approaches, which is effective for the nonseparable, illconditioned black-box problems with high dimensionality. In this work, the start sample is assigned to $m^{(1)}$, the initial step size ${\sigma }^{(1)}$ is set from 2 to 7 nm, depending on the specific problems, and $C^{(1)}$ is an identity matrix.

4.2.

Electromagnetic Field Numerical Simulation Details

The 3D FDTD simulations and EME simulations were performed by using Lumerical FDTD Solutions and Mode Solutions. In this work, the simulation time was tested on a personal computer (CPU: Inter Core i7-11700 @2.5 GHz, RAM: 64 G). The EME solver was carried with configuration of one process and 16 threads. The SWG metamaterial is equivalent to homogeneous anisotropic metamaterials.²³ The FDTD solver was carried with configuration of four processes and one thread. The final performance confirmations of the devices were performed with dense meshes (i.e., mesh accuracy index of 6 or 8, corresponding to 26 or 34 mesh points per effective-wavelength scale).

4.3.

Device Fabrication

All the devices were fabricated on a silicon-on-insulator (SOI) wafer with a 220 nm-thick top-silicon layer and a $2\mu \mathrm{m}$-thick buried dioxide layer. The silicon photonic waveguides were patterned by the processes of electron-beam lithography and inductively coupled plasma dry-etching. Then the $2.2\text{-}\mu \mathrm{m}$-thick silicon dioxide cladding was deposited by using chemical vapor deposition.