a Artistic representation for an on-chip link between two XPUs utilizing inverse-designed digital metamaterials within future optical compute interconnect systems. Here, the term “XPU” serves as a device abstraction including various computational architectures, including CPUs, GPUs, FPGAs, and other accelerators. These broadband, inverse-designed digital metamaterials facilitate the multiplexing of a substantial number of data channels across both wavelength and mode dimensions. Additionally, the system incorporates an E/O module for converting electrical signals to optical signals and an O/E module for reverse conversion. Signal processing can be supported by co-packaged CMOS electronics. b Strategies for enhancing interconnect capacity across three dimensions: symbol, wavelength, and mode. This study showcases the integration of MDM and DWDM, enabling nearly a hundred wavelength channels per orthogonal mode. Each channel efficiently supports high-order pulse amplitude modulation (PAM) signals, significantly expanding the data capacity per waveguide. c Foundry-compatible inverse-designed digital metamaterials. Our design features a minimum feature size of 120 nm, suitable for robust and large-scale production by commercial foundries. Compared to conventionally designed high-order mode MUX, the inverse-designed MUX significantly reduces the footprint by an order of magnitude.

As high-performance integrated optical frequency combs^13,14,15,16 and multi-wavelength transmitters based on tunable microring modulators^17,18,19 evolve, there is a critical need for innovative chip-based, ultra-compact high-order mode multiplexers (MUX) to achieve broad operation bandwidth, micrometer-scale dimensions, and low-crosstalk mode-multiplexing capabilities. Although phase-matching principles have facilitated broadband and high-order MUXs capable of efficiently converting optical signals into multiple orthogonal modes^{20,21,22,23,24,25}, these designs require cascading several mode converters, substantially increasing their footprint with higher mode counts. Additionally, phase-matching MUXs are prone to horizontal fabrication errors, complicating large-scale production and dense integration²³. To address these challenges, it is necessary to move beyond intuition-first design approaches and analytical theories. Inverse design, as an emerging paradigm in goal-oriented design^{26,27,28,29,30,31,32}, enables the automatic iterative optimization of a large number of design parameters within predefined numerical boundaries through advanced mathematical algorithms. Furthermore, all-dielectric metamaterials^33,34, characterized by their extraordinary ability to manipulate electromagnetic waves within numerous sub-wavelength structures, are particularly amenable to inverse design methods. Inverse-designed all-dielectric metamaterials have been proven effective in a range of integrated photonic applications, such as optical communication^{20,26,28,32,35}, computing²⁷, sensing²⁹, and imaging³¹. Inverse-designed metamaterials can be categorized into digital and analog types based on their permittivity distribution patterns³⁶. Analog metamaterials (AMs) typically display irregular patterns with curvilinear boundaries and isolated islands, while digital metamaterials (DMs) feature regularized etching patterns, such as squares or circles. The inverse design of complex devices, such as high-order mode MUXs, is impractical with the brute-force search algorithms commonly employed for DMs^{9,32,35,37,38,39,40}. In contrast, the gradient-based methods for AMs offer enhanced optimization efficiency and expanded design freedom. However, the challenges posed by the small minimum feature size (MFS) and irregular shapes of AMs present significant fabrication hurdles. Additionally, the reactive ion etching (RIE) lag effect causing variable etching depths for different-sized holes⁴¹, further complicates the accurate replication of the designed local effective index. To date, robust AM designs have only been demonstrated in simple standalone devices with few ports^42,43,44, while DM designs have found broader applications in integrated photonic circuits^9,35,39,45. Due to the outlined challenges, inverse-designed mode MUXs fabricated on the silicon-on-insulator (SOI) platform have been reported to support a maximum of only four modes^{26,35,38,45,46,47}.

Therefore, an advanced inverse design method that can merge the advantages of both analog and DMs is vital for fabricating complicated and robust photonic devices. In this paper, we introduce an edge-guided analog-and-digital optimization (EG-ADO) method, which combines the computational efficiency of topology optimization (TO) with the fabrication reliability of DMs. This is achieved by innovatively integrating edge detection algorithms into the conversion process from analog to DMs. This method enables the design and experimental demonstration of broadband multimode multiplexers (MUXs) on SOI, including a four-mode MUX (TE₀–TE₃), a five-mode MUX (TE₀–TE₄), and a six-mode MUX (TE₀–TE₅) for scalability validation. Building upon the packaged five-mode MDM chip characterized by exceptional uniformity, low loss, and minimal crosstalk, we achieve a MIMO-free high-speed single-λ on-chip interconnect with a line rate exceeding 1.6 Tb s⁻¹ and a record per wavelength per mode channel rate of 324 Gb s⁻¹. By combining DWDM with MDM techniques, we further realize a multi-dimensional ultra-high-capacity on-chip interconnect, supporting 440 channels with a record aggregate net rate exceeding 38 Tb s⁻¹ and a high spectral efficiency of 8.68 bits s⁻¹ Hz⁻¹. Finally, the compatibility of the proposed DMs with mainstream foundry processes highlights the potential for large-scale and cost-effective deployment in future optical compute interconnect applications (Fig. 1), potentially supporting the 32 T and even the 64 T generations⁴⁸.

a The workflow of the EG-ADO method. During each iteration in the first stage, the forward electric fields (????) and adjoint fields (????) for all modes are used to obtain the pixel gradient. During the second stage, a dilation parameter (D) is introduced to regulate the direct conversion rate from analog to digital metamaterials. During the final stage, the TBD pixels are iteratively traversed column by column. The objective functions for Si (???) and SiO₂ (????2) configurations are compared to determine the final pixel material. For any TBD pixels not yet determined, their state remains equivalent to that of the analog pattern. Insets (i) and (ii) depict the evolution of the index pattern for the five-mode MUX during the topology and digital optimization stages, respectively. b Final structure of the DM-based five-mode MUX. The pixel side length is 120 nm. c Simulated mode profiles at the output multimode waveguide for five channels at the wavelength of 1550 nm. The power conversion efficiency is provided along with each 2D profile. d Simulated light propagation process for five channels at the wavelength of 1550 nm. e Computational complexity comparison of state-of-the-art inverse-designed DM-based mode MUXs on the SOI platform. Dashed lines represent the fitted curves. The purple-shaded area denotes the regime that has not been reported in the literature due to the high computational complexity and time consumption of conventional DBS methods. f Average transmission of all modes for two-, four-, and five-mode MUX designed by EG-ADO method under different conversion rates at the wavelength of 1550 nm. For each MUX, changes in the conversion rate are realized by adjusting the dilation parameter D, with values set at D = 0, 2, and 4. A higher D value corresponds to a lower conversion rate due to an increased count of pixels designated as TBD during the index mapping process.

The EG-ADO method consists of three main stages, as illustrated in Fig. 1a. The first stage involves iterative TO of the permittivity distribution within the metamaterial design region using the adjoint method. To accurately calculate the pixel gradients, the pixel size of the design area is set to 20 × 20 nm². During each iteration, we obtain the electric field distributions of the metamaterial design region through one forward and one adjoint electromagnetic simulation for each mode. These fields are used to compute the gradient of the objective function (see Methods) with regard to the permittivity distribution. The permittivity value for each pixel is then updated continuously between ε?? and ε???2 using gradient-based optimizer to enhance the overall device performance. After multiple iterations, an AM-based mode MUX demonstrating optimal performance is obtained. However, challenges in reliable fabrication arise due to the incomplete binary conversion of the index pattern and the presence of notably small gaps and holes, which complicate manufacturing processes. To address these fabrication challenges, the second stage performs an edge-guided conversion from analog to DMs. This is achieved by extracting critical edge information from the material boundaries within the analog pattern using the Canny edge detection algorithm⁴⁹. To accommodate the MFS constraints of commercial foundries, the extracted edge map undergoes dimensional reduction through a max-pooling operation (see Supplementary Note 1), enlarging the pixel size from 20 nm to 120 nm. Following this, the feature map containing edge information is binarized to produce a decision map that guides the subsequent index mapping process. During index mapping, non-edge pixels (black) in the decision map directly adopt the index values from their corresponding positions in the original analog pattern, while the state of edge pixels (white) is temporarily designated as to-be-determined (TBD). Notably, TBD pixels constitute only a small fraction of all pixels, enabling the design to be finalized with a minimal number of iterations in the next stage, thus significantly conserving computational resources. In the final stage, a customized direct binary search (DBS) algorithm identifies the material for each TBD pixel in the converted pattern. For these TBD pixels, the refractive index is alternately set to that of Si or SiO₂, and the objective function is computed for each setting. The material that results in superior performance is then confirmed for each pixel. This stage terminates after a single pass through all TBD pixels, finalizing the design of the DM-based five-mode MUX. The inset (ii) in Fig. 1a distinctly shows that throughout the digital optimization process, irregular gaps and holes within the metamaterial design region are progressively replaced by uniform squares in a column-wise direction. The evolution of device performance during the entire optimization process is provided in Supplementary Note 1.

The final structure of the DM-based five-mode MUX is given in Fig. 2b, featuring a large MFS that is well-suited for ultraviolet lithography fabrication. Insertion loss is quantitatively analyzed by evaluating the conversion efficiency from the incident mode to the target mode based on mode overlap integrals (Fig. 2c). Inter-mode crosstalk is assessed by computing the mode overlap between the obtained mode profiles and other undesired orthogonal modes. The reflection and scattering are also discussed in Supplementary Note 3. The numerical results demonstrate that this device maintains a significantly flat response (see Fig. S5), with loss below 1.96 dB and crosstalk under −15.81 dB across the entire C band. Figure 2d demonstrates the manipulation of the optical field by finely etched holes within the metamaterial design area.

To further explore the generalizability and efficiency of the proposed EG-ADO method and its impact on devices of varying complexities, we have applied this method to design two-, four-, and six-TE-mode MUXs (details in Supplementary Notes 3 and 7). Figure 2e compares the computational complexity of various inverse-designed DM-based mode MUXs^9,35,40,45, measured by the required number of Maxwell solves (RNMS). The fitting results show that the RNMS for the DBS method scales quadratically with the number of modes, leading to excessive resource demand for high-order designs. In contrast, devices designed using the EG-ADO method show a linear scaling of RNMS (see Methods), with more pronounced improvements as the mode number increases. Additionally, for the proposed EG-ADO method, it is noteworthy that increasing the number of TBD pixels by thickening the edges (see Supplementary Notes 1 and 3) does not correlate with an evident performance gain, as shown in Fig. 2f. Here, the conversion rate is defined as the proportion of pixels directly undergoing index mapping. This observation highlights the critical role of edge pixels and indicates a diminished return on optimizing pixels distant from edges, thus reducing unnecessary computational costs.

Optical performance of fabricated devices

We fabricate the four-, five-, and six-mode MUXs on the SOI platform using electron-beam lithography (EBL), with detailed processes provided in Methods. Due to the absence of imaging equipment, we assess the optical performance by cascading two identical devices back-to-back on the chip. The fabricated back-to-back four-mode MDM system (Fig. 3a) includes a pair of four-mode MUX and deMUX linked by a multimode waveguide. The multimode waveguide in the four-mode MUX has a width of 1.8 μm, and the design region occupies an area of 6 × 6 μm². The coupling loss of the grating couplers is ~5.7 dB/facet at the peak wavelength. The transmission spectra of the four-mode MDM circuit are measured and normalized by subtracting the coupling losses, as shown in Fig. 3c. The characterization measurements are conducted using a passive optical component testing platform (detailed in Methods). At 1550 nm, the insertion losses for the four channels of the circuit are approximately 0.94 dB, 2.04 dB, 2.08 dB, and 2.48 dB, respectively. The corresponding measured crosstalks at 1550 nm are −20.09 dB, −19.21 dB, −18.02 dB, and −18.25 dB, respectively.

Fig. 3: Measured optical performance of the fabricated four-mode and five-mode MUXs designed with the EG-ADO method.

a, b Microscope images of the back-to-back (a) four-mode and (b) five-mode MDM circuits. c Measured transmission spectra of the four-mode circuit. The inset depicts the false-color scanning electron microscopy (SEM) image of the four-mode MUX. d–h Measured transmission spectra of the five-mode circuit when the light is incident from I₁ to I₅. The shaded areas represent the C band range (1530–1565 nm).

The micrograph of the fabricated five-mode MDM circuit is shown in Fig. 3b, where the input and output grating coupler arrays are arranged equidistantly at 127 µm intervals, facilitating subsequent coupling and packaging with fiber arrays. Transmission spectra for the five-mode MDM circuit are obtained by launching the fundamental TE mode from ports I₁ to I₅, as shown in Fig. 3d–h. At 1550 nm, the measured insertion losses for the five-mode channels of the circuit are 2.96 dB, 3.62 dB, 2.43 dB, 3.94 dB, and 3.10 dB, with crosstalks of −21.73 dB, −20.72 dB, −21.91 dB, −23.43 dB, and −24.51 dB, respectively. Moreover, the transmission spectra display remarkable flatness across the entire C band, validating its excellent potential for multi-dimensional multiplexing in on-chip optical interconnects that integrate DWDM and MDM technologies. The measurement results of the six-mode MUX for scalability validation are provided in Supplementary Note 7.

High-speed single-λ MDM transmission

To evaluate the capacity of the DM-based MUX for parallel data transmission across different orthogonal modes, we first conduct a high-speed intensity-modulation direct-detection single-λ MDM transmission experiment with the fabricated five-mode MDM circuit. The conceptual diagram of the interconnect setup is shown in Fig. 4a, which consists of the E/O module, the MDM chip, and the O/E module. A single laser is used to generate continuous-wave light in this experiment. For enhanced testing stability, the chip is packaged with a pair of fiber arrays for vertical coupling (see Supplementary Note 6 for detail), with its macrograph presented in Fig. 4b. To assess the impact of inter-mode crosstalk, we compare MDM transmission with single-port transmission. During the MDM transmission experiments, we simultaneously load optical signals into all input channels of the MDM chip, ensuring that all five modes carry decorrelated data concurrently. For each mode channel under test, it serves as the signal branch, while the remaining four form the crosstalk branch. In contrast, the single-port configuration connects only the channel under test to the chip, while the crosstalk branch remains disconnected. As a result, data is carried exclusively by the tested mode, with no interference from other channels. Modulation and propagation losses are compensated by erbium-doped fiber amplifiers (EDFAs) for both branches. At the receiver end, the output optical signal from the mode channel corresponding to the signal branch is converted to the electrical signal by a high-speed photodiode (PD) and captured by an oscilloscope (OSC). The detailed experiment setup is provided in Methods.

Fig. 4: High-speed single-λ MDM on-chip interconnect.

a Schematic of the single-λ MDM transmission based on the packaged five-mode chip. Input continuous-wave light is divided by a splitter into five paths, each modulated by a modulator (MOD) and then coupled to a packaged chip via a fiber array (FA). Each path corresponds to an orthogonal mode in the MDM chip. At the receiver end, the mixed signal is demultiplexed by a mode deMUX and then coupled into photodiodes (PD) through another fiber array, where different mode channels are received and demodulated individually. b Macroscope image of the packaged MDM chip with a pair of fiber arrays. c False-color SEM image of the five-mode MUX. It confirms the successful transfer of the expected digital pattern onto the chip. Etched holes with well-defined square shapes and point-like contacts along diagonals indicate a high-quality fabrication result that accurately replicates the target design. This fabrication robustness stems from our digitally designed pattern and its relatively large MFS. d BER measurements for the MDM and single-port transmission for all five-mode channels under different ROPs. An indicative FEC threshold is given at 0.02, corresponding to a pre-FEC BER for the OFEC threshold. BER, bit-error rate; ROP, received optical power. e BER measurements for MDM transmission for all five channels under different data rates. At the data rate of 324 Gb s⁻¹, all five channels exhibit BER values below the OFEC threshold. f Corresponding eye diagrams for the 108 GBaud PAM-8 signals at different channels.

Figure 4d displays the bit-error-rate (BER) performance for each channel during the transmission of 68 GBaud 8-ary pulse-amplitude modulation (PAM-8) signals, as the received optical power (ROP) before the PD decreases from 4 to −1 dBm. As the received optical power drops, the performance gap between the two cases gradually narrows. This convergence is attributed to the lower signal-to-noise ratio (SNR) at reduced power levels, causing both scenarios to become noise-dominated. Experimental results show that MDM transmission incurs only a modest power penalty at the Open forward error correction (OFEC) threshold⁵⁰, with values of 0.65, 0.35, 1.0, 1.1, and 0.92 dB for MDM CH1 to CH5 respectively, demonstrating effective crosstalk suppression by our device. To further explore the capacity limits of single-λ MDM transmission, we analyze the BER performance of PAM-8 signals under different symbol rates ranging from 48 to 128 GBaud (see Fig. 4e). Under the same FEC threshold, we successfully realize the transmission of 108 GBaud PAM-8 signals for all five channels, with the corresponding eye diagrams displayed in Fig. 4f. Owing to the high channel uniformity, minimal insertion loss, and low inter-mode crosstalk of the inverse-designed five-mode MUX, we achieve the highest single-wavelength, single-mode transmission rate of 324 Gb s⁻¹ in on-chip MDM interconnects, amounting to an aggregate line rate of 1.62 Tb s⁻¹λ⁻¹.

Ultra-high-capacity multi-dimensional interconnect

To validate ultra-high-capacity on-chip optical interconnects, we employ high-order mode-multiplexing-combined with the spectral resources of the C band. We utilize 88 DWDM channels on a dense 50 GHz ITU grid, covering the wavelength range from 1529.94 to 1564.68 nm. It is critical to note that integrating DWDM into MDM does not simply involve multiplying the single-λ data rate by the number of wavelength channels. For a rigorous demonstration of multi-dimensional on-chip interconnect, it is essential to ensure simultaneous transmission of all wavelength and mode channels, which inevitably constrains the optical power per channel and introduces inter-channel crosstalk. A schematic diagram of the multi-dimensional interconnect scheme is shown in Fig. 5a. In this experiment, we simultaneously transmit five modes, each accommodating 88 wavelength channels. Among these, three wavelength channels carry data, while the remaining 85 are filled with amplified spontaneous emission (ASE) noise to emulate a full channel loading scenario^20,51,52. Specifically, the aggregated source module in our experiment generates the multi-λ optical signal at the transmitter side which comprises three parts: target signal, adjacent crosstalk signal, and ASE noise channels. The target channel and its adjacent channels are generated by a tunable laser and modulated by external modulators. High-power ASE noise is generated using cascaded EDFAs and a wavelength selective switch (WSS) to flatten the ASE spectrum and carve out a slot for the laser-generated channels (see Fig. S22). The spectrum of the combined DWDM signal is shown in Fig. 5b, where the blue background area indicates the laser-generated signal channels and the purple area corresponds to the ASE noise used to fill the C band. During the experiment, we measure the transmission performance of all 88 wavelength channels by gradually blue-shifting the slot excavated by the WSS and the corresponding wavelengths of tunable lasers (see Methods for detailed experimental setup). We repeat this procedure for all five-mode channels to comprehensively assess the performance across all 440 data channels.

Fig. 5: Ultra-high-capacity on-chip multi-dimensional interconnect.

a Schematic of the multi-dimensional on-chip interconnect scheme. The spectrally flattened multi-wavelength signal, generated from aggregated sources, is distributed into five branches corresponding to different orthogonal modes on the chip. Each branch utilizes a wavelength demultiplexer (λ deMUX) to segregate different wavelength channels for independent modulation. Subsequently, a wavelength multiplexer (λ MUX) reassembles the WDM signal. Fiber arrays (not depicted for clarity) link the packaged chip with E/O and O/E modules. At the receiving end, the WDM signals are extracted by λ deMUXs and converted into electrical signals by photodiodes. In this experiment, 88 wavelength channels across 5 waveguide modes are employed, facilitating the propagation of a total of 440 parallel signal lanes on-chip. The entropy of each lane is flexibly adjusted based on channel conditions. b Normalized optical spectrum of the combined DWDM signal at the transmitter side. Here, the 44th channel is modulated with a target signal by a Mach-Zehnder modulator (MZM), while its adjacent channels are driven by another MZM. The remaining wavelength channels in the C band are filled with the ASE noise to ensure full spectral utilization. c Measured net data rates (left axis) and entropies (right axis) across 88 wavelength channels for MDM CH1 to CH5. The frequency range of the DWDM signal corresponds to 191.6 to 195.95 THz. d Total AIR and NDR for five-mode channels. AIR achievable information rate, NDR net data rate.

Given the wavelength sensitivity of grating couplers, the loss of the packaged chip fluctuates among different channels. Therefore, the constellation probability shaping (PS) technique^53,54 is implemented to achieve flexible information rate and entropy adjustment for different mode and wavelength channels by optimizing the constellation distribution towards a Maxwell-Boltzmann distribution (see Fig. S23). In order to maximize the single-channel rate, we seamlessly switch among PAM-2, PS-PAM-4, and PS-PAM-8 formats to modulate each channel at a symbol rate of 47 GBaud with a roll-off factor of 0.05, corresponding to utilization of 98.7% of the channel spectrum. Both the achievable information rate (AIR) and net data rate (NDR) are recorded to quantify transmission performance and communication capacity (see Methods for detailed expressions for calculating date rates). Measured NDRs and entropies for all 5 × 88 channels are shown in Fig. 5c, with Fig. 5d presenting the total AIR (7.3 to 9.1 Tb s⁻¹) and NDR (6.8 to 8.6 Tb s⁻¹) across all wavelength channels for each mode. The overall spectral efficiency reaches 8.68 bits s⁻¹Hz⁻¹. Ultimately, our experimental demonstration of the on-chip DWDM-MDM transmission with 440 channels achieves record-setting total AIR and NDR of up to 40.8 Tb s⁻¹ and 38.2 Tb s⁻¹, respectively, validating the capability of our inverse-designed DMs for supporting ultra-high-capacity optical interconnect.