The progress of integrated optical communication requires mode division multiplexing (MDM) technologies to bolster communication capacity and establish comprehensive networks [1–6]. With wavelength and mode demonstrating physical independence, MDM, when coupled with wavelength division multiplexing [7–9], holds promise in reshaping multi-dimensional multiplexing, optimizing the utilization of limited spectrum resources [10]. Conventional multi-dimensional multiplexers often employ separate devices for mode (e.g., liquid crystals [11,12]) and wavelength (e.g., gratings [13–15]) channels, resulting in bulky system footprints. To meet the space-saving needs of integrated communication systems, an optical wavelength-mode (de)multiplexing chip emerged for multi-dimensional information processing. On-chip wavelength-mode demultiplexers typically manipulate light using cascade total internal reflection structures, such as tapered adiabatic micro-ring resonators [16–19] and directional coupler connected array waveguide gratings [20,21]. However, their inherent complexity, stemming from multi-level mode conversion based on phase matching conditions within the cascaded structure [22,23], restricts their compactness. Moreover, interference among nodes arises from coherent intervention in the shared time slots, hindering duplex communication [24–26] when time is divided into slots for data transmission among communicating parties. While the utilization of identical communication paths for both transmission and reception, employing the same frequency and mode, is attainable, achieving compatibility with full-duplex capabilities and high integration poses considerable challenges.

We overcome these challenges by employing a photonic-like crystal-connected bidirectional micro-ring resonator array (PBMRA), supporting duplex mode and wavelength (de)multiplexing. The micro-ring resonator array [27–29] utilizes an energy feedback principle to facilitate resonant wavelength multiplexing. Additionally, the subwavelength-scale photonic crystal structure significantly enhances the interaction between the mode field and material based on Bragg’s law of diffraction, reducing the energy conversion length [30,31], owing to its effective control over the mode field phase. In our duplex multi-dimensional multiplexing system based on a PBMRA, the directional independence of total internal reflection [32,33] of the micro-rings allows simultaneous coupling and separation of wavelength channels. Utilizing the inverse design algorithm, the photonic crystal structure only retains crucial units [34–36], enabling multiple mode conversion processes within the operational bandwidth through scattering accumulation effect. Consequently, this approach enables simultaneous co-frequency, co-mode full-duplex communication for a single-pair node and bidirectional wavelength-mode signal transmission among multiple pairs without interference. Moreover, the pixelated area of the PBMRA streamlines the cascaded mode conversion structure through precise phase compensation. This single-level mode conversion strategy reduces the device footprint and complexity, culminating in a highly integrated multi-dimensional full-duplex communication system.

This work leverages the principle of total internal reflection directional independence of the micro-rings and the robust mode field phase modulation capacity of the photonic crystals to engineer a bidirectional nine-channel multi-dimensional (de)multiplexer. Operating within an 80μm×80μm footprint, it effectively multiplexes and demultiplexes QPSK-OFDM signals at a bidirectional transmission rate of 216 Gbit/s. With a bit error rate below 10−4 and crosstalk lower than −15.5dB, this confirms the orthogonal separation of duplex multi-dimensional signals, meeting the requirements for practical full-duplex communication multiplexing. The photonic crystal’s mode conversion length, approximately 5 μm, facilitates simultaneous conversion of three modes, reducing the beat length and enabling single-level mode conversions, thereby enhancing device integration. This integration successfully combines high integration and full-duplex characteristics in on-chip multi-dimensional multiplexing communication. Moreover, our proposed micro-ring resonator acts as a versatile component, functioning not only just as a transceiver port and filter but also as a transmitter or receiver for dual-channel operation due to its co-ring transmission attributes. When combined with the orthogonal features of the wavelength and mode channels, it facilitates interference-free simplex communication via dual physical transmission paths, effectively doubling the transmission rate and capacity density among multi-pair nodes in optical networks.

The schematic of the on-chip duplex multi-dimensional mode (de)multiplexer using a PBMRA is depicted in Fig. 1. It comprises two integral components: an inverse-designed photonic crystal (IPC) and bidirectional micro-ring resonator array (BMRA). These components facilitate bidirectional mode conversion and wavelength selection, enabling effective full-duplex communication. During unidirectional signal transmission or reception, the system maintains orthogonal wavelengths and modes. However, during simultaneous transmission and reception, bidirectional signals remain distinct without interference.

Utilizing bidirectional micro-ring resonators as user transceiver ports and wavelength filters, our system enables simultaneous transmission and separation of wavelengths. The operation principle of the BMRA is illustrated in Fig. 1(a), supporting a broader range of wavelength channels. The unit bidirectional micro-ring resonator integrates two straight coupling waveguides, two add ports, two drop ports, and a micro-ring [37–39]. The resonant wavelength adheres to the following resonance condition: neff(λ)×2πλ×(2πR+2L)=2mπ,(1)where m, L, neff, and λ represent the positive integer, the coupling length, the effective refractive index, and wavelength, respectively. R is the radius of the micro-ring. The straight waveguide with optimized length L connecting the micro-ring expands the arcuate coupling area, thus boosting energy conversion efficiency. The coupling coefficient is 60.33mm−1 and the calculation process is shown in Appendix A. Precise adjustment of the gap (G) between the straight waveguide and micro-ring ensures critical coupling, balancing coupling loss with the micro-ring’s inherent loss, thereby maximizing optical energy containment and amplification. When light waves propagate within a waveguide and encounter a boundary, total internal reflection will occur due to the difference in refractive index, and it will be effectively confined within the waveguide. Within the micro-ring, resonance is achieved when the frequency of the incident light is aligned with the characteristic frequency of the micro-ring. The resonance is maintained by constructive interference of light on the loop, and this process is independent of the chosen input port. This independence ensures the generation of resonant states from any coupling point along the ring against interference. Consequently, the frequency selection is attained by fine-tuning the micro-ring’s size to meet the resonance condition.

To incorporate the mode dimension for full-duplex communication and ensure optimal optical energy conversion within a confined space, we utilize photonic crystal structures optimized through inverse design strategy, illustrated in Fig. 1(b). Here, we select the TE mode as the multiplexing mode. The predominantly planar electric field distribution of the TE mode contributes to improved field confinement and elevated Q factor within the micro-ring resonator. Moreover, the TE mode exhibits reduced mode dispersion in photonic crystals, facilitating enhanced stability and uniformity in the control of multi-wavelength signals [40]. Appendix B delineates the inverse design methodology of the photonic crystal structure. The well-structured arrangement of air pillars collectively modulates all input light waves exhibiting the TE0 fundamental mode through scattering accumulation effect, denoted as ∑i=1,j=1x,yPi,j. Based on the Bragg’s law of diffraction, adapting to subwavelength sizes in photonic crystals matches the dimensions closely with the C-band wavelength, thereby increasing the efficiency of light–matter interaction within the crystal. This enhancement allows for more precise control over optical wave modes, facilitating more effective phase compensation required for mode conversion. Subsequently, the intermodal coupling effects within the same single-level mode conversion region are suppressed, ensuring different mode conversions with multi-modal orthogonal phase distribution. Therefore, the orthogonal modes (TE0, TE1, and TE2) can be demultiplexed and separated. Operating across the broad bandwidth, the mode multiplexer IPC seamlessly integrates with the BMRA’s trunk waveguide via bending and crossing waveguides, ensuring compatibility with numerous closely spaced resonant wavelengths.

In our design of the multi-dimensional mode multiplexer encompassing three modes (TE0−TE2) and three wavelengths (1545.5 nm, 1550.0 nm, and 1554.5 nm) with bidirectional physical transmission paths, we employed an optimization approach using finite-difference time-domain (FDTD) methods. Details outlining the relationship between effective refractive indices and waveguide widths for TE0−TE2 in the 1540.0 nm to 1570.0 nm range are provided in Appendix C. The optimized widths, ensuring effective performance, were determined for both single-mode (ws=0.50μm) and tri-mode (wm=1.75μm) waveguides. The waveguide parameters for the air medium pillars include a diameter (d) of 120 nm, with a 180 nm spacing between adjacent air columns, encompassing a pixelated structural region measuring 5.4μm×5.4μm. The bidirectional micro-ring resonator array comprises three units. Leveraging FDTD simulations, we fine-tuned the cavity perimeter to regulate wavelength, ensuring uniform spacing. The coupling lengths for the three micro-ring resonator arrays corresponding to the resonant wavelengths are 6.17 μm, 6.13 μm, and 6.08 μm, respectively. In the simulation, the gap (G) between the single-mode waveguides and the radius (R) of the MMR are defined as 0.18 μm and 2.8 μm, respectively. The simulation utilized a value of m as 48. This setup facilitates concurrent multiplexing and demultiplexing across three wavelengths and three TE mode channels.

We fabricated the on-chip bidirectional multi-dimensional (de)multiplexer utilizing SOI with a 220 nm Si layer. For details about the fabrication process, please refer to Appendix D. Figure 2 illustrates the top-view structure of the PBMRAs, showcasing its ability for bidirectional wavelength and mode (de)multiplexing. The scanning electron microscopy images in Figs. 2(a) and 2(b) depict the BMRR and IPC, respectively, highlighting integration areas of approximately ∼50μm×11μm and ∼5.4μm×5.4μm, respectively. The entire demultiplexer footprint is approximately 80μm×80μm. Compared with the ordinary ADC multiplexing structure (around 100 μm) [32,33,41], the footprint of the proposed IPC multiplexing structure is reduced by 1 to 2 orders of magnitude. The connection between the BMRA trunk waveguide and the IPC is based on a crossing waveguide (with a footprint of ∼10μm×10μm), as shown in Appendix E, and bent waveguides (with radii of 1.4 μm and 2.9 μm), as shown in Appendix F. By integrating bidirectional micro-ring resonators as user transceiver ports and wavelength filters, our system achieves simultaneous transmission and separation of wavelengths. It facilitates the transmission and reception of multi-dimensional multiplexing signals through the left and right ports simultaneously.

In Fig. 3(a), the simulated intensity distribution illustrates the BMRA’s response at wavelengths 1545.5 nm, 1550.0 nm, and 1554.5 nm, respectively. Irrespective of the incident light’s direction (from the left or right end), resonance occurs within the micro-ring, resulting in distinct exits through various drop ports. This capability allows for the simultaneous transmission of dual-wavelength signals through co-ring resonance. Meanwhile, Fig. 3(b) shows the simulated intensity distribution concerning TE0, TE1, and TE2, respectively. The maps demonstrate that the various order modes are almost entirely converted back to the TE0 mode, owing to the phase matching condition in the coupling region. However, there is a slight mode leakage beyond the pixelated region, which can be further mitigated by introducing optimization objectives aimed at minimizing this leaked power. The simulated transmittance curves of the BMRR and IPC are shown in Appendix G. Furthermore, bent waveguides with radii of 1.4 μm and 2.9 μm were used in this integrated device, and their insertion losses are 0.12 dB and 0.03 dB, respectively, indicating high efficiency and minimal loss in optical signal transmission.

In Fig. 4(a), the experimentally measured transmission spectrum (with a sampling interval of 0.5 nm) of the six wavelength channels within the BMRA exhibits nearly identical spectral shapes for different channels within the same bidirectional micro-ring resonator. The insertion losses for the resonant wavelengths are measured as 1.3 dB, 1.0 dB, 1.8 dB, 1.9 dB, 1.5 dB, and 1.2 dB, respectively. The resonance peak jitters compared to the simulation results, and this nonuniformity in excess loss may partially result from fabrication errors. Crosstalk among wavelength channels operating in the same transmission direction remains below −20dB. Crosstalk level denotes the power difference between the desired channel (detected at port P0) and all unwanted channels (N0), calculated as Crosstalk=−10lg(P0/N0). Widening the spacing between resonant wavelengths could further decrease overall crosstalk. Figure 4(b) illustrates the experimentally measured transmission spectrum of the three mode channels (TE0, TE1, and TE2) in the IPC. Multiplexing and demultiplexing through the pixelated structure result in losses of 6.9 dB, 7.4 dB, and 7.9 dB, respectively, with corresponding crosstalk levels at −15.9dB, −15.3dB, and −13.9dB at the 1550.0 nm wavelength. The insertion losses are about 5 dB higher than the simulation results, while the crosstalk is improved by 2 dB compared with the simulation results. The broad working bandwidth accommodates BMRA wavelengths. By adjusting channel spacing and bandwidth through micro-manufacturing technology, signal transmission through TE0, TE1, and TE2 mode channels on the chip becomes viable within the wide 1540 nm to 1560 nm waveband. Furthermore, the increased losses and crosstalk observed in the TE2 mode can be attributed to two primary factors: the complexity of modal field distribution and increased sensitivity to phase mismatches during mode conversion. The complexity arises as higher-order TE modes exhibit more intricate spatial field configurations, characterized by numerous nodes and antinodes. This expanded modal volume heightens the likelihood of scattering and other loss-inducing interactions with material imperfections and waveguide boundaries. Moreover, efficient mode conversion for higher-order modes necessitates stringent phase matching conditions. More precise control over the waveguide’s geometric and material attributes is necessary to ensure effective phase compensation. Manufacturing errors such as air column distortion and varying column radius particularly affect higher modes, as shown in Appendix H. In silicon-based waveguide device research using photonic crystal structures, deliberate introduction of device errors, multiple simulations, and fabrication tests are common practices to adjust device parameters and mitigate loss of functionality resulting from fabrication errors.

To evaluate the performance of PBMRA devices and prepare for subsequent assessments of signal transmission quality, we conducted measurements of transmission efficiency and crosstalk in the PBMRA. Figure 5 illustrates the transmission characteristics of the nine duplex transmission channels, comprising three TE modes (TE0, TE1, and TE2) combined with three wavelengths (1545.5 nm, 1550.0 nm, and 1554.5 nm) within the BMRA. The optical energy loss measures range from 7.4 dB to 8.9 dB (the system loss after removing the fiber-to-chip loss), while the channel crosstalk ranges from −15.4dB to −13.3dB.

Furthermore, we conducted an evaluation of its bidirectional communication performance using QPSK-OFDM multi-dimensional multiplexing signals. The experiment setup is shown in Appendix G. Employing QPSK-OFDM signals with a single channel transmission rate of 24 Gbit/s, we present the communication results in Fig. 6. Figure 6(a) demonstrates the BER performance of signals transmitted from the left end to the right end, while Fig. 6(b) illustrates the BER performance of signals transmitted from the right end to the left end. Our designed device, showcasing a communication capacity of 216 Gbit/s, sustains a BER lower than 10−4 at a received power of −16.5dBm. Accounting for variances in design and fabrication across diverse mode-crossing devices, the maximum sensitivity difference among the nine mode channels approximates 2 dB, meeting the requisite criteria for practical full-duplex communication multiplexing. For clearer insights, Fig. 6(c) displays the constellations of bidirectional multi-dimensional multiplexing signals transmitted across the nine multi-dimensional channels at a received power of −16.5dBm. A constellation diagram graphically represents the mapping and quality of digital modulated signals, using I/Q components to depict the amplitude and phase of each symbol. These diagrams calculate the error vector magnitude (EVM), crucial in gauging signal quality where smaller values signify better quality by showcasing the difference between the actual signal and the ideal signal. Notably, all the channel constellation diagrams exhibit strong convergence, with EVMs below 20%. These results underscore the compatibility of wavelength and mode dimensions, affirming their utility in multi-dimensional full-duplex communication.

In the realm of on-chip multi-dimensional optical networks, balancing system compactness with achieving full-duplex functionality emerges as a pivotal strategy, enhancing both signal transmission efficiency and communication capacity density within multi-dimensional multiplexing systems. Our study introduces a novel approach, the PBMRA, meticulously crafted to enable full-duplex transmission of wavelength-mode multiplexing signals. By optimizing a photonic crystal region using a binary search algorithm, this system adeptly manages multi-mode conversion within the operational bandwidth. Harnessing the bidirectional signal circulation inherent in the bidirectional micro-ring resonators, our PBMRA showcases remarkable bidirectional filtering prowess for spatial mode light, allowing seamless full-duplex transmission among node pairs, free from interference. Noteworthy is its significant reduction in space required for mode conversion compared to conventional cascaded tapered adiabatic micro-ring resonators and directional coupler connected array waveguide gratings, courtesy of the subwavelength-size photonic crystal with robust mode field modulation capacity. Furthermore, its ability to achieve single-level mode conversions through precise phase compensation significantly reduces system redundancy, propelling the integration of multi-dimensional full-duplex multiplexing systems.

Our achievement of a communication capacity of 216 Gbit/s in single-directional multi-dimensional multiplexing signal transmission makes a significance in optical communication. Leveraging the principle of directional independence within total reflection, the bidirectional micro-ring resonator transforms from an optical signal transceiver into a dual-space channel sender or receiver, allowing simultaneous dual-wavelength transmission via co-ring resonance [42]. When coupled with the orthogonal attributes of the wavelength-mode joint channel, it enables simplex communication at identical frequencies and modes across dual physical transmission paths, regardless of their entry direction (left or right port), as shown in Fig. 7(a). Figure 7(b) illustrates the measured BERs across 18 channels in multi-dimensional multiplexing signal transmission as a function of received optical power. The achieved communication transmission rate of 432 Gbit/s maintains a BER below the FEC threshold (3.8×10−3). These findings underscore the potential for doubling the communication capacity density within on-chip optical networks using the co-ring transmission characteristic. Furthermore, the system’s maximum sensitivity difference, nearing 2 dB, enables the capture of weaker signals, ensuring stable connections and enhancing transmission distance.

This study serves as a proof of concept for high-density multi-dimensional full-duplex communication. The refinement level within the PBMRA pixel region and the obtained refractive index distribution, manipulated via the inverse design algorithm and variations in calculated depth, dictate the number of supported mode conversions. Furthermore, the size of the dielectric column plays a critical role in determining the precision of phase compensation. However, achieving a delicate balance between these factors becomes imperative during the design process, especially considering the influence of fabrication tolerance limits.

This study introduces a compact solution for bidirectional multi-dimensional mode (de)multiplexing using a PBMRA on SOI. It allows bidirectional transmission and reception of wavelength multiplexing signals through the total internal reflection directional independence of the micro-rings. Moreover, the robust mode phase modulation of subwavelength-scale photonic-like crystals introduces the mode dimension, streamlining the mode conversion region. Our development includes a successful bidirectional nine-channel multi-dimensional (de)multiplexer, accommodating three wavelengths and three TE modes, within a compact footprint of approximately 80μm×80μm and a mode conversion length of 5.4 μm. With (de)multiplexed crosstalk below −15.5dB, achieving a transmission rate of 216 Gbit/s and bit error rates lower than 10−4, this underscores the system’s compatibility with full-duplex capability and high integration in muti-dimensional communication systems. Leveraging the co-ring transmission attributes and capitalizing on the orthogonal features of the wavelength-mode channels, this system holds the potential to double the communication capacity density in simplex communication within on-chip optical networks.