1. Introduction

Silicon photonics offers advantages such as high integration density, low power consumption, and opto-electronic integration, making it promising for applications in optical communications, sensing, and computing [1–3]. Silicon photonics waveguides have a high refractive-index contrast, enabling ultra-high integration density. In addition, silicon photonics based devices can be fabricated using the standard CMOS (Complementary Metal-Oxide-Semiconductor Transistor) technology, allowing for high integration with other integrated circuits (ICs) and significantly reducing manufacturing costs [4,5].

On-chip mode-division multiplexing (MDM) technology based on silicon photonics is a working technique that utilizes the optical mode multiplexing technology to transmit multiple signals simultaneously on the same bus waveguide [6–8]. It has significant importance in improving the performance and reducing the cost of optical communication/interconnect systems due to its characteristics of high-speed, high-capacity, and low-power consumption. To construct on-chip MDM systems, various mode-controlling devices have been proposed to achieve functions such as optical emission, transmission, mode modulation, mode multiplexing/demultiplexing, mode switching, mode exchanger, mode splitting, and so on [9–14]. Among them, the mode multiplexer/demultiplexer (MUX/DeMUX) is a device used to multiplex multiple optical signals over different fundamental modes into different high-order modes within the same bus waveguide at the transmission terminal, and these signals are demultiplexed and restored to their original signals by the mode DeMUX at the receiving terminal [15–17]. The mode MUX and DeMUX have the same structure, but operate in opposite ways, and both are the key components of MDM systems.

In recent years, many structures have been reported for realizing the mode MUX/DeMUXs, including the asymmetric directional couplers (ADCs) [18,19], adiabatic couplers (ACs) [20,21], multimode interference (MMI) couplers [22,23], Y-junctions [24,25], micro-ring resonators (MRRs) [26,27], subwavelength gratings (SWGs) [28,29], inverse-design [30,31], etc. The ADCs based mode MUX/DeMUXs have been theoretically and experimentally reported to achieve low loss and low mode crosstalk at the central wavelength of 1550 nm [18,19]. However, traditional ADC structures require strict mode-to-mode phase matching conditions, leading to tight fabrication tolerances and the need for further bandwidth expansion. Sun et al. experimentally demonstrated an AC based mode MUX/DeMUX, which can achieve the crosstalk of lower than -20 dB and the insertion loss of ∼1 dB over a wavelength range of 75 nm [20]. While structures based on ACs can achieve high performance and larger fabrication tolerances, their mode evolution efficiency is relatively low, resulting in longer device lengths, typically around 200 µm [20,21]. A mode MUX based on an MMI structure proposed by Wang et al. can achieve a working bandwidth exceeding 70 nm with crosstalk levels below −10 dB [22]. However, most conventional MMI-based ones often require excessively long device dimensions to realize mode evolution and conversion [23]. Chen et al. reported a Y-junction based mode MUX/DeMUX, which achieve the best demultiplexing crosstalk of −31.5 dB within a bandwidth from 1537 to 1566 nm [24]. Although Y-junctions can realize low-loss and low-crosstalk mode MUX/DeMUXs, but their device lengths typically exceed 50 µm. The MRR can realize low-crosstalk mode MUX/DeMUXs and is compatible with wavelength-division multiplexing (WDM) systems, but its operating bandwidth is relatively narrow (a typical scale of a few nanometers) [26,27]. The SWG structure based mode MUX/DeMUXs can achieve a relatively large bandwidth of >50 nm and a low crosstalk of < −16 dB, but its design process is complex and has a small feature size of SWG [28,29]. While traditional direct binary search (DBS) algorithm-based mode MUX/DeMUX can reduce the footprint to a few square micrometers, this algorithm's computation time is relatively long [30,31]. Hence, there is a need to propose new structures to further reduce the device size.

The ADC structure has many advantages, such as low loss, simple structure, and easy scalability, and has been widely utilized in mode multiplexing and demultiplexing [32,33]. However, the earliest proposed ADC structure requires strict phase-matching conditions, resulting in a narrow bandwidth and tight fabrication tolerance. To address this issue, a tapered directional coupler was proposed to construct a mode MUX/DeMUX, which includes a fixed-width waveguide and a tapered waveguide [34]. The relatively high fabrication tolerance of phase matching can be achieved by modifying the equivalent refractive index slope. Although the bandwidth and fabrication tolerance have been improved in the single-tapered waveguide based coupler, it still requires the phase matching, leading to limited bandwidth and limited improvement in fabrication tolerance. In 2015, Wang et al. proposed and demonstrated a mode-evolution counter-tapered coupler, which can construct a broadband MDM-Link [35]. This structure consists of two waveguides with counter-tapered cores, and different-order mode conversions can be achieved via the internal mode evolution. The experimentally measured 1-dB bandwidth is greater than 180 nm. In 2017, Li and Dai proposed a dual-core adiabatic tapers structure, which can achieve a low-loss and low-crosstalk mode MUX/DeMUX [36]. The specific mode drop can be achieved by using the supermode evolution inside the two tapered waveguides. The operating bandwidth was measured to be 65 nm. The coupler of the two tapered waveguides is no longer based on the phase matching but based on the mode evolution, which can achieve relatively large operating bandwidth. Nevertheless, the sizes of these device are still large, typically close to or even greater than 100 µm, due to the long evolution distance required for the mode conversion.

More recently, the inverse-designed photonic devices have attracted considerable attention due to their ultra-compact size and simple optimization-design process [37,38]. Several algorithms have been reported for designing photonic devices, including the DBS algorithm [39], particle swarm optimization (PSO) algorithm [40], binary particle swarm optimization (BPSO) algorithm [41], gradient-based optimization algorithm [42,43], etc. Based on the inverse design, the size of photonic devices can be reduced by almost an order of magnitude, providing a feasible solution for constructing ultra-compact mode MUX/DeMUXs. In 2021, Liu et al. proposed an inverse-designed mode MUX/DeMUX based on the DBS algorithm [30]. This device has a footprint of 6.8 µm × 6 µm, insertion loss of <1.4 dB and crosstalk of −15 dB over a 40-nm bandwidth. In 2019, a mode MUX/DeMUX with a footprint of 3.4 µm × 3.9 µm was reported based on the DBS, which can experimentally provide insertion loss of <0.92 dB and crosstalk of −20.43 dB over an 80-nm bandwidth [31]. In 2018, Chang et al. proposed and experimentally demonstrated a two-mode MUX/DeMUX based on an inverse-designed Y-junction [39]. The proposed device is with a footprint of 2.4 µm × 3 µm, and the measured insertion loss and crosstalk are less than 1.0 dB and −24 dB over a 60-nm bandwidth. In 2020, Piggott et al. experimentally demonstrated an inverse-designed two-mode MUX/DeMUX with a compact footprint of 3.55 µm × 2.55 µm, insertion loss of <1.0 dB and crosstalk of −15.6 dB over a 100-nm bandwidth [43]. In 2020, Yang et al. inversely designed and experimentally measured a 4-mode MUX/DeMUX, and the design area is 6.5 µm × 6.5 µm [44]. The measured insertion loss is less than 0.8 dB, and 3-dB bandwidth is wider than 120 nm with the mode crosstalk less than −18 dB. In 2023, Shang et al. demonstrated an inverse-designed mode MUX/DeMUX on the lithium niobate-on-insulator (LNOI) platform [45]. The optimized device is with a footprint of 12 µm × 12 µm, an insertion loss ∼1.5 dB and a crosstalk < −15.8 dB over an 80-nm bandwidth. In 2022, Zhou et al. inversely optimized and experimentally demonstrated a 4-mode MUX/DeMUX, in which the functional region is with a footprint of 4.8 µm × 4.8 µm [46]. The measured results show that the insertion loss and crosstalk are less than ∼1.5 and −13 dB, respectively over a 40-nm bandwidth. In 2021, Chen et al. proposed and investigated a mode MUX/DeMUX based on the PSO algorithm [47]. The optimized length of the proposed device is longer than 27 µm. The measured insertion loss and crosstalk are less than 4.73 and −15.15 dB from 1525 to 1596 nm. From the above-mentioned reports, it is evident that the inverse design can reduce the size of mode MUX/DeMUXs to a few square micrometers. Although the mode MUX/DeMUXs based on reverse design offer a wide operating bandwidth and low mode crosstalk, the inverse design algorithms typically require a significant amount of computation time. Because the computation time required for reverse design algorithms is directly proportional to the size of the functional area of the device. Therefore, proposing new structures to further reduce the size of the device's functional region becomes a challenging aspect of research.

In this paper, we propose and experimentally demonstrate a broadband and compact silicon mode MUX/DeMUX based on an inverse-designed counter-tapered coupler. By embedding functional region composed of subunits between two counter-tapered waveguides, efficient short-range mode evolution can be achieved, overcoming the drawback of the traditional dual-core adiabatic-tapers, which requires long-range supermode evolution. Additionally, by incorporating two counter-tapered waveguides outside the functional region, the required footprint of the functional region can be significantly reduced compared with the earlier works [30,31,39,43–47]. This leads to a substantial reduction in the optimization design time and a significant improvement in the device's operating performance. The functional region was inversely designed using the combination of DBS algorithm and 3D full vectorial finite-difference time-domain (3D-FV-FDTD) method, resulting in a size of only 4 µm × 800 nm. The optimal design results in a compact device size of only 4 × 2.2 µm² for the dual-mode MUX/DeMUX. The experimentally measured 3-dB bandwidths of TE₁ and TE₀ modes are 138 nm and 116 nm, respectively. The proposed structure is scalable and can be used for mode multiplexing and demultiplexing of more mode channels through cascading.

2. Structure, operating principle, and inverse-design

The schematic diagram of the proposed mode MUX/DeMUX based on an inverse-designed counter-tapered coupler is shown in Fig. 1. The proposed structure consists of an input dual-mode waveguide with a port I, two output single-mode waveguides with ports O₁ and O₂, and a functional region composed of subunits embedded between two counter-tapered waveguides. The distance between the two tapered waveguides is fixed, while the upper tapered waveguide gradually decreases in width from supporting dual-mode to supporting single-mode, and the lower tapered waveguide gradually increases in width from nanoscale (50 nm) to supporting single-mode. Traditional dual-core adiabatic-taper based mode MUX/DeMUXs rely on internal supermode evolution for mode multiplexing and demultiplexing, requiring a sufficient length to achieve adiabaticity, which results in a device length typically greater than tens or even hundreds of micrometers. For our proposed inverse-designed counter-tapered coupler, the functional region composed of subunits is introduced and its effective refractive index can be adjusted by inverse optimization of the distribution of square subunits. This functional region can directly guide high-order modes to evolve into the fundamental mode without relying on supermode evolution, greatly reducing the device size to the order of micrometers. Furthermore, the subwavelength subunit-structure based functional region has good working bandwidth characteristics, enabling broadband devices.

 

Fig. 1. Schematic diagram of the proposed mode MUX/DeMUX based on an inverse-designed counter-tapered coupler.

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As shown in Fig. 2, when the TE₀ mode is injected at the port I, it is transmitted to the upper tapered waveguide through the input dual-mode waveguide, where the mode spot size is compressed to the size of the fundamental mode in the single-mode waveguide. Then, it is transmitted to the output single-mode waveguide and finally outputted at the port O₁. When the TE₁ mode is inputted at the port I, the upper tapered waveguide narrows from supporting dual-mode to only supporting single-mode, causing the TE₁ mode to cut off and be “squeezed” into the functional region. After being guided by the functional region, its optical field evolves into the lower tapered waveguide, where the narrow end should be as small as possible to achieve more adiabatic coupling to its interior. Then, the coupled mode gradually evolves into the TE₀ mode in the output waveguide, and finally outputs at the port O₂. In contrast to the traditional dual-core adiabatic-taper, which requires a small waveguide spacing between the two output waveguides (<200 nm) for mode coupling evolution, our proposed inverse-designed counter-tapered coupler has greater redundancy for the output waveguide spacing, which can be set to several hundred nanometers or even micrometers. Therefore, the proposed structure has a lower output-port crosstalk.

 

Fig. 2. (a) Structure and parameters of inverse-designed counter-tapered coupler based mode MUX/DeMUX. (b) The relationship between the transmission (left y-axis) and FOM (right y-axis) with respect to number of iterations.

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The structure and parameters of the inverse-designed counter-tapered coupler based mode MUX/DeMUX are shown in Fig. 2(a). The proposed device is based on the SOI platform, with a core waveguide thickness of h = 220 nm. The input waveguide is a dual-mode waveguide that can support both the TE₁ and TE₀ modes, and its width is chosen to be w₁ = 1 µm. The two output waveguides are both single-mode waveguides, with a width of w₂ = 400 nm, which is a commonly used width for single-mode waveguides. The width of the upper tapered waveguide gradually changes from w₁ to w₂, while the width of the lower tapered waveguide changes from a narrow end width of 50 nm to w₂. To ensure adiabatic propagation of the TE₀ mode in the upper tapered waveguide, the optimized tapered waveguide length is set to L = 4 µm. The size of the square subunit is set to w₃ = 100 nm according to the actual fabrication-process accuracy. The size of the functional region is set to L × g = 4 µm × 800 nm, which includes 40 × 8 square subunits. One major drawback of the DBS algorithm-based inverse design is its lengthy computation time. By introducing an external counter-tapered coupler, we can reduce the entire functional region's area from a maximum of 40.8 µm² to only 3.2 µm² (a reduction of over 12 times) compared with the earlier works [30,31,43]. Consequently, under the same subunit size conditions, the number of subunits required for the DBS algorithm will be significantly reduced, leading to a substantial reduction in the required computation time. The issue of extra loss in the inverse-designed functional region is another drawback. Generally, smaller subunit sizes result in lower extra losses. However, due to the limitation of fabrication processes, subunit sizes within the functional region are typically larger than 100 nm, causing some extra loss. Therefore, reducing the size of the functional region can decrease the magnitude of extra loss it incurs. Because our proposed functional region has the distinct advantage of a small size, its associated extra loss is lower compared to previously reported structures [30,31,43].

The functional region is inversely optimized using the DBS algorithm, which is designed to not affect the transmission of the TE₀ mode and convert the input TE₁ mode to the TE₀ mode. The specific optimization steps are as follows:

The propagating fields of the optimized mode MUX/DeMUX obtained by the 3D-FV-FDTD method are shown in Fig. 3. When the TE₀ mode is input at the port I, the mode field is compressed by the upper taper, and the mode size is reduced from a dual-mode waveguide to a single-mode waveguide without changing the mode order, and finally output from the port O₁. It can be observed from Fig. 3(a) that the influence of the subunit structure in the functional region on the transmission of the input TE₀ mode can be ignored. When the TE₁ mode is input at the port I, the TE₁ mode field is “squeezed” by the upper taper due to the gradually narrowing waveguide, and leaks out of the taper waveguide. Due to the guiding effect of the subunit structure in the functional region, the leaked mode field will propagate along the functional region and then couple with the inverted taper waveguide on the lower side, eventually evolving into the TE₀ mode in the output single-mode waveguide and being output from the port O₂. It can be seen from Fig. 3(b) that although the length of the taper waveguide is only L = 4 µm and the distance between the output waveguides is g = 800 nm, the introduction of the subunit functional region enables efficient conversion between the TE₁ mode and TE₀ mode, overcoming the long-distance defect of traditional supermode evolution.

 

Fig. 3. Propagation H_y fields of the optimized mode MUX/DeMUX based on inverse-designed counter-tapered coupler when (a) TE₀ mode and (b) TE₁ mode are injected at the port I, respectively.

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The relationship between the transmission spectra and the operating wavelength of the optimized mode MUX/DeMUX is calculated using the 3D-FV-FDTD method, as shown in Fig. 4. When the TE₀ mode is input from the port I, the insertion loss and crosstalk are calculated to be 0.87 dB and −14.28 dB, respectively, at the central wavelength of 1550 nm. When the wavelength varies between 1.50 µm and 1.65 µm, the insertion loss and crosstalk of this mode are less than 1.54 dB and −10.89 dB, respectively. When the TE₁ mode is input from the port I, the insertion loss and crosstalk at the central wavelength are 0.56 dB and −21.56 dB, respectively, and the insertion loss and crosstalk within the 150 nm bandwidth range are less than 1.53 dB and −18.06 dB, respectively. These theoretical results indicate that the proposed mode MUX/DeMUX based on the inverse-designed counter-tapered coupler has an ultra-wide operating bandwidth of greater than 150 nm. It can also be noted from Fig. 4 that there is some power unbalance in the transmission spectra of the simulated results. In this study, we did not include the constraint functions in the optimization process. The constraint functions could be included in the algorithm to achieve a better power balance.

 

Fig. 4. The transmission spectra of the optimized mode MUX/DeMUX for input the TE₀ and TE₁ modes.

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3. Fabrication and performance characterizations

The optimized device was fabricated on the SOI platform using two-step etching processes to form the grating couplers and the silicon waveguides, respectively. The TE-type grating couplers with a depth of 70 nm were first fabricated using the electron beam lithography (EBL) lithography and inductively coupled plasma (ICP) etching, followed by a second round of EBL and ICP to produce the core waveguides with a depth of 220 nm. Finally, a 1 µm-thick top cladding layer was deposited using the plasma-enhanced chemical vapor deposition (PECVD). The SOI wafer used in this work has a bottom cladding layer thickness of 3 µm. Microscope images of the fabricated device and a referenced waveguide used for normalization are shown in Fig. 5(a). A back-to-back connection of two identical mode MUX/DeMUXs was used to build an MDM-Link, as shown in Fig. 5(b). SEM images of the fabricated mode MUX/DeMUX and the functional region are shown in Figs. 5(c) and 5(d), respectively. As shown in Figs. 5(a) and 5(b), when the TE₀ mode is launched from the port I_TE0, it propagates along the upper bus waveguide and is directly output from the port O_TE0. When the TE₀ mode is launched from port I_TE1, it propagates along the lower branch waveguide and is converted into the TE₁ mode via the functional region of the mode MUX. And then this TE₁ mode is transmitted to the functional region of the mode DeMUX and converted back to the TE₀ mode, finally being output from the port O_TE1.

 

Fig. 5. (a) Microscope image of the fabricated chip. (b) SEM image of the fabricated MDM-Link. (c) SEM picture of the mode MUX based on inverse-designed counter-tapered coupler. (d) Zoom-in image of the functional region.

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The performance of the fabricated MDM-Link was tested using a self-built vertical coupling test system for silicon-based photonic integrated circuits. The testing system consists of an optical link, an observation system, and an adjustment system. The optical link includes a tunable laser source with an operating wavelength range from 1490 nm to 1640 nm (Agilent 81640B), a polarization controller, and a photodetector. The observation system includes horizontal and vertical cameras and their corresponding displays. The adjustment system includes two six-dimensional manual tuning stages on both sides and a specimen stage in the middle for placing the chip.

The measured and simulated transmission spectra of the MDM-Link as a function of the operating wavelength are shown in Fig. 6. When the optical signal is input from the port I_TE0, the measured insertion loss and crosstalk at the center wavelength of 1550 nm are 2.75 dB and −13.39 dB, respectively, and the 3-dB bandwidth is 116 nm. The crosstalk is less than −10.71 dB within a 138 nm range. As shown in Fig. 6(a), the experimental output power at the port O_TE1 is lower than that in theoretical simulations. The reason for this phenomenon is that the theoretical simulations did not account for the propagation losses within the MDM-Link, whereas in practice, there are propagation losses. This leads to a lower output power at the port O_TE1 compared to the theoretical simulations. When the optical signal is input from the port I_TE1, the measured insertion loss and crosstalk at the center wavelength are 1.68 dB and −18.33 dB, respectively, and the 3-dB bandwidth covers the 138-nm wavelength range from 1500 nm to 1638 nm, with the corresponding crosstalk less than −12.11 dB. Compared with the theoretical results, the measured insertion losses at the center wavelength of the two modes show some degree of degradation, increasing by 1.01 dB and 0.56 dB, respectively. The reasons for this performance degradation include the fact that the theoretical simulation results did not include the propagation losses; the roughness of the sidewalls of the fabricated waveguides and the inherent waveguide propagation losses in the experimental samples caused extra losses. Although the fabrication errors can cause performance degradation, the 3-dB bandwidth of 116 nm and 138 nm demonstrated by the experimental testing confirms that our proposed mode MUX/DeMUX based on the inverse-designed counter-tapered coupler can achieve ultra-wide operating bandwidths.

 

Fig. 6. Measured and simulated transmission spectra of the fabricated MDM-Link based on inverse-designed counter-tapered coupler, when the light is launched at (a) Port I_TE0 and (b) Port I_TE1.

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4. Conclusions

In conclusion, we have proposed, inverse-designed, fabricated, and measured a broadband and high-integration mode MUX/DeMUX based on a functional region assisted counter-tapered coupler. With the joint use of DBS algorithm and 3D-FV-FDTD method, we have obtained a square subunit functional region with a size of only 4 µm × 800 nm through inverse design. Benefitting from the efficient guidance of the functional region for mode evolution, the overall size of the proposed mode MUX/DeMUX has been optimized to be only 4 × 2.2 µm². The theoretical results show that within a bandwidth range of 150 nm, the insertion loss and crosstalk of the TE₀ mode are less than 1.54 dB and −10.89 dB, respectively, while those of the TE₁ mode are less than 1.53 dB and −18.06 dB, respectively. The MDM-Link, containing a pair of back-to-back mode MUX/DeMUX, has been fabricated on the SOI platform. The insertion losses of the two modes at the central wavelength have been measured to be 2.75 dB and 1.68 dB, respectively, and the crosstalks have been tested to be −13.39 dB and −18.33 dB, respectively. The 3-dB bandwidths have been measured to be 116 nm and 138 nm, respectively. Theoretical and experimental results indicate that the proposed mode MUX/DeMUX has both ultra-small size and ultra-wide operating bandwidth, and can expand the number of handling modes through cascading. We believe that the proposed structure has the potential to be applied to on-chip MDM systems, realizing high-performance mode multiplexing and demultiplexing, and providing feasible key components for future hybrid WDM-MDM systems.