In recent years, integrable mode-division multiplexing (MDM) systems have attracted increasing attention due to their great potential in further improving the transmission capacity of on-chip optical interconnection^[1,2]. To assemble a complete MDM system, various key building elements have been proposed, such as mode (de)multiplexers [(De)MUXers]^[3,4], mode filters^[5,6], mode converters^[7,8], and multimode bendings^[9,10]. In addition to these multimode devices, the mode splitter is a crucial component that straightforwardly separates modes of different orders into different channels. Currently, most reported mode splitters are established by implementing the mode (De)MUXers. These (De)MUXers enable mode separation by converting the high-order modes into the fundamental mode and mapping them to the specific channel via a phase-matching mechanism. However, in the case of a mode-sensitive system, mode conversion with mode orders changes would lead to instability and performance degradation of the system^[11,12]. Therefore, a mode splitter without changing the mode orders is highly preferred and can effectively improve the flexibility of the MDM system.

To date, only a few studies have been reported on the mode splitters without changing mode orders, because directly splitting modes with different effective indices within the same waveguide is quite challenging. Liao et al. proposed a mode splitter based on an asymmetric directional coupler (ADC)^[13]. The splitter consists of two slits and a strip waveguide that can separate the TE0 and TE1 modes with cross talk (CT) of <−8dB and insertion loss (IL) of <1dB for a 100 nm bandwidth range. In Ref. [14], a dual-mode mode splitter was designed by introducing small slots in the silicon waveguide to form a symmetric directional coupler (DC), achieving CT of <−15dB and low IL of <0.1dB for the entire C-band. Furthermore, a dual-mode splitter for separating TM0 and TM1 modes was developed using a bridged subwavelength grating-assisted (BSWG) DC and a mode filter, with CT of <−15dB and IL of <1.8dB for a bandwidth of 84 nm^[15]. However, these designs cost a large footprint of hundreds of micrometers squared, and the device structures are also complex. More recently, intelligent design methods for photonic devices have been extensively investigated, offering advantages in reducing device size and improving design efficiency^[16]. However, little research has been conducted on mode splitter design using such methods. In Ref. [17], a mode splitter was designed by using the direct binary search (DBS) algorithm. The optimized device has an ultracompact size but exhibits a high IL of 3.04 dB, which is unsuitable for practical applications. Therefore, designing a high-performance mode splitter in a small footprint remains a challenge.

In this paper, we use a shape optimization method to design a high-performance mode splitter without changing the mode orders. Our design method is able to optimize multiple design objectives simultaneously and requires only four simulations per iteration, effectively improving device performance while increasing device design efficiency. Using this method, we designed and fabricated a mode splitter with a compact footprint of only 14μm×2.5μm, which separates TE0 and TE1 modes with an IL of <0.9dB and a CT of <−16dB over a broad bandwidth range of 1500–1600 nm. To the best of our knowledge, this device achieves the broadest operation bandwidth for this level of performance in such a compact size. Additionally, we demonstrate the generality of the shape optimization method by designing a dual-mode (De)MUXer, with experimental results indicating an IL of below 1 dB and a CT of below −23dB within the 1500–1600 nm bandwidth range.

Figure 1(a) depicts the schematic diagram of the proposed mode splitter, which is designed on the silicon-on-insulator (SOI) platform with a 220-nm core silicon layer, a 2-µm buried oxide layer, and a 1-µm top oxide cladding. The width of the input waveguide W1 is set as 1 µm to support both TE0 and TE1 modes, while the widths of Ports 1 and 2 (W2 and W3) are set as 500 nm and 1 µm to support TE0 and TE1 modes, respectively. The optimization region is labeled with a dotted box with an area of 14μm×2.5μm. As shown in Fig. 1(b), the boundary curves distort when approaching the design target as the shape optimization runs. After optimization, the TE0 mode is output from Port 1, while the TE1 mode from Port 2 without changing the mode orders. Note that a tapered waveguide with a length of 3 µm is introduced between Port 1 and the optimization region. The primary objective is to mitigate backscattering effects and concurrently enhance the optical transmission of TE0 light. The width of the tapered end is set as 500 nm for compatibility with the standard single-mode access, while the width of the initial portion is set as 1 µm to ensure a separation exceeding 200 nm between the tapered waveguide and Port 2.

The flow of designing a mode splitter using the shape optimization method is shown in Fig. 2. The optimization objective is quantified by the figure of merit (FOM). FOM1 and FOM2 are defined as the average transmittance of the target mode for the two output ports in the designed bandwidth, FOM1=1λ2−λ1∫λ1λ2T1(λ)dλ,(1)FOM2=1λ2−λ1∫λ1λ2T2(λ)dλ,(2)where λ1 and λ2 represent the minimum and maximum wavelengths of the target bandwidth, respectively. T1(λ) and T2(λ) are the normalized power of the TE0 and TE1 modes at the respective output ports at wavelength λ. The initial structure of the mode splitter is illustrated in Fig. 2(a). The boundary of the tapered coupler is defined by 140 discrete boundary optimization points S uniformly inserted at the upper and lower boundaries of the coupler. The boundary shape of the coupler is optimized by adjusting the y-axis coordinates of each point. The refractive index perturbation caused by the change of the coupler boundary shape affects the transmission of TE0 and TE1 modes in the device, resulting in mode separation.

The shape optimization method used in this study consists of two stages. The first stage involves simulating the device using the 3D finite-difference time-domain (FDTD) method, as shown in Fig. 2(b). To improve the optimization efficiency, the adjoint method is introduced, which enables the calculation of the gradient of FOM with respect to the geometrical parameters from the solution of Maxwell’s propagation equations and requires only two runs of simulation for each iteration: one for the forward solution and one for the adjoint solution, which greatly improves the design efficiency^[18]. The gradient of FOM1 and FOM2 with respect to the material boundary can be found using the adjoint method^[19], which can be written as $$\begin{gathered} dFOMd[φ(S)]=2Re[(ε2−ε1)E∥(S)·E∥adj(S) \\ +(1ε1−1ε2)D⊥(S)·D⊥adj(S)], \end{gathered}$$(3)where ε1 and ε2 are the relative permittivity of SiO2 and Si, respectively. d[φ(S)] is the size of the deformation at point S along the boundary. E||(S) and E||adj(S) are the tangential components of the electric field at point S obtained from the forward simulation and the adjoint simulation, respectively. D⊥(S) and D⊥adj(S) are the normal components of the electric displacement at point S obtained from the forward simulation and the adjoint simulation, respectively. As a result, the design of the mode splitter using the shape optimization method requires only four simulations per iteration, significantly reducing the number of simulations compared to some brute force optimization algorithms, such as DBS.

The second stage involves optimizing the boundary shape of the coupler using Python, as depicted in Fig. 2(c). The gradients of FOM1 and FOM2 with respect to the optimized point S are computed in Python, and the y coordinate of the point S is adjusted using the gradient descent method to maximize FOM1 and FOM2, respectively, so that each FOM value converges to 1. To ensure that the optimized structure is not too sharp, numerical constraints are added to the y coordinates of the adjacent points to limit their variation within ±0.15μm. The optimized boundary shape is obtained by spline interpolation fitting of the adjusted optimized points. The simulation stage and optimization stage are run iteratively until the device meets the target performance. The final optimized mode splitter is shown in Fig. 2(d). It is worth noting that the two FOMs defined can be optimized simultaneously in each iteration to ensure simultaneous increases in FOM1 and FOM2 during the optimization process.

In addition to the mode splitter, we also designed a dual-mode (De)MUXer that routes either TE0 or TE1 mode to TE0 mode, demonstrating the generality of the design method. Similarly, the two optimization objectives (FOM3 and FOM4) are defined as the average transmittance of the TE0 mode for the two output ports of the mode (De)MUXer over the design bandwidth. The evolution of the boundary shapes and FOMs for both devices during the optimization process is shown in Figs. 3(a) and 3(b), respectively. It can be observed that a balance between FOM1 and FOM2 is maintained throughout the optimization process and is consistently increasing simultaneously. The final geometries for both devices were obtained after around 30 iterations. In order to be able to show the shape of the designed mode splitter and mode (De)MUXer more precisely, we provide the coordinates of the points on the boundary curves of the mode splitter and mode (De)MUXer in a rectangular coordinate system, as shown in Fig. S1 (see Supplementary Material). The detailed coordinates of the 200 points that construct the upper and lower boundaries of the mode splitter/mode demultiplexer are shown in Table S1 (see Supplementary Material). The simulated electric field distributions of the mode splitter and mode (De)MUXer are shown in Figs. 3(c) and 3(d), respectively. As TE0 or TE1 is injected, it propagates to the target port with the desired mode order.

The transmission spectra of the mode splitter and mode (De)MUXer are presented in Figs. 4(a)–4(d), respectively. At 1550 nm, for TE0 mode injection, the IL of the mode splitter is 0.14 dB and the CT is less than −26dB. On the other hand, with TE1 mode injection, the IL is 0.15 dB and the CT is less than −33dB. In the wavelength range from 1500 to 1600 nm, the IL and CT of the mode splitter are less than 0.44 dB and −18dB, respectively, for TE0 mode injection, and less than 0.55 dB and −18dB, respectively, for TE1 mode injection. Similarly, for mode (De)MUXer, at 1550 nm, the IL and CT are 0.06 dB and −39dB, respectively, for TE0 mode injection, and 0.05 dB and −41dB, respectively, for TE1 mode injection. In the wavelength range of 1500 to 1600 nm, for TE0 mode injection, the IL and CT for mode (De)MUXer are less than 0.27 dB and −37dB, respectively, and with TE1 mode injection, IL and CT are less than 0.21 dB and −32dB, respectively.

Fabrication inaccuracies are inevitable during the fabrication process. Therefore, overall tolerance analysis was performed on the deviations of the waveguide widths (ΔW) for both devices, as shown in Figs. 5(a)–5(h). It can be observed that when ΔW deviates by −20nm, the IL of the mode splitter is less than 1.14 dB and the CT is less than −14dB in the wavelength range of 1500–1600 nm. When ΔW deviates by +20nm, in the wavelength range of 1500–1600 nm, the IL of the mode splitter is less than 0.84 dB and the CT is less than −14dB. Similarly, over the 100 nm wavelength range, for mode (De)MUXer, IL is below 0.55 dB and CT is below −26dB when ΔW=−20nm, and IL is below 0.54 dB and CT is below −26dB when ΔW=+20nm. These results demonstrate that the devices designed using the shape optimization method exhibit good fabrication tolerance.

The designed mode splitter and mode (De)MUXer are fabricated on a standard SOI platform with a 220 nm top silicon layer and a 2 µm buried oxide layer. The pattern of the device is defined using 193 nm-deep ultraviolet (UV) photolithography, while the Si layers are etched using inductively coupled plasma (ICP). Finally, a silica upper-cladding was deposited on the structure by the plasma-enhanced chemical vapor deposition (PECVD) process.

Figure 6(a) depicts the microscopic views of the on-chip test structures for the mode splitter. An ADC-based mode (De)MUXer, proposed in Ref. [20], was employed to generate and detect TE0 and TE1 modes. The ADC converts the TE0 light from Port ITE1 into the TE1 in the bus waveguide while maintaining the TE0 light from Port ITE0. At the output, TE0 light is directly output from OTE0, while TE1 light is coupled to TE0 and output from OTE1. Figure 6(c) depicts a microscopic view of the test structure of the mode (De)MUXer. Similarly, TE0 and TE1 light is transmitted using the ADC-based MUXer. Through the fabricated mode (De)MUXer, TE0 light is directly output from O2, while TE1 will be demultiplexed to TE0 output from O3. A reference structure that connects two ADCs back to back was also fabricated for normalizing the test results, as shown in Fig. 6(b). All the ports are connected to TE-type grating couplers (GCs) with a loss of 5 dB/facet for light.

A tunable laser (Santec TSL-550) was used as the light source, and an optical power meter was used to measure the transmission spectra of the fabricated devices. The losses of the ADC-based mode (De)MUXer along with the GCs are deducted from the measured spectrum of the fabricated device; the measurement results are shown in Figs. 7(a)–7(d). As a comparison, the simulation results are also attached in Fig. 7 (dashed curves). The mode splitter shows an IL and CT of below 0.6 dB and −17dB, respectively, for TE0 injection over a broadband wavelength of 1500–1600 nm. On the other hand, when TE1 is injected, the IL and CT are below 0.9 dB and −16dB, respectively. For the mode demultiplexer, the ILs of both TE0 and TE1 are below 1 dB, and the CTs are below −23dB for the same bandwidth. It is worth noting that there are slight variations between the measured results and the simulated results in terms of device performance. This discrepancy can be attributed to random deviations in waveguide sidewall etch roughness and width in fabrication.

A comparison between our work and the previously reported mode splitters is presented in Table 1. It is obvious that the mode splitter designed in this work has better overall performance. Specifically, the fabricated device exhibits the lowest loss and largest bandwidth. Moreover, its footprint is relatively compact, with a length of only 14 µm.

In conclusion, our study presents a mode-splitter design using the shape optimization method, which is experimentally verified to be capable of separating TE0 and TE1 modes without changing their mode orders. The device has a compact footprint of 14μm×2.5μm and demonstrates IL below 0.9 dB and CT below −16dB over a broadband wavelength range of 1500–1600 nm. Furthermore, we demonstrate the versatility of the shape optimization design method by using the same principle to design a dual-mode (De)MUXer, which exhibits IL below 1 dB and CT below −23dB over the same bandwidth. In addition, the two devices designed are able to maintain good performance for a width variation of ±20nm. Overall, the designed mode splitter and mode (De)MUXer could find application in on-chip MDM systems for broadband mode routing. We believe that the shape optimization design method used can be scalable to other photonic devices, providing an effective tool for the development of advanced integrated photonic circuits.