Arbitrarily routed mode-division multiplexed photonic circuits for dense integration

On-chip integrated mode-division multiplexing (MDM) is an emerging technique for large-capacity data communications. In the past few years, while several configurations have been developed to realize on-chip MDM circuits, their practical applications are significantly hindered by the large footprint and inter-mode cross talk. Most importantly, the high-speed MDM signal transmission in an arbitrarily routed circuit is still absent. Herein, we demonstrate the MDM circuits based on digitized meta-structures which have extremely compact footprints. 112 Gbit/s signals encoded on each mode are arbitrarily routed through the circuits consisting of many sharp bends and compact crossings with a bit error rate under forward error correction limit. This will significantly improve the integration density and benefit various on-chip multimode optical systems.

T he orthogonal guiding modes in integrated optical waveguides are considered as important degree of freedom to provide the flexibility and scalability for a wide scope of on-chip applications, such as optical interconnect [1][2][3][4] , quantum information science 5,6 , nonlinear optics 7,8 , and so on. Recently, mode-division multiplexing (MDM) is emerging as a powerful technique to boost the bandwidth of the photonic chip. Compared with wavelength division multiplexing, MDM only needs monochromatic light source, and precise wavelength control is not necessary 9 . A typical optical system requires a variety of multimode devices such as (de)multiplexer [10][11][12] , grating coupler 13 , switch [14][15][16] , mode filter 17 , splitter 18,19 , and many other building blocks [20][21][22] . The interconnections among these elements have to rely on multimode waveguides with a few orthogonal and co-propagating modes. This is quite challenging, since the signal transmission in such multimode photonic circuits is vulnerable to sharp bending and cross connection due to radiation leakage and inter-mode coupling. To achieve adiabatic wave propagation in a bent waveguide, the required footprint is too large 23 . The modeindependent crossing is also complicated for higher-order modes since the MDM signals needs to be demultiplexed to single modes prior to waveguide crossing 24 . For both cases, the MDM signal is routed at the cost of large chip area, which eventually limits the integration density.
To arbitrarily route the multimode waveguide, the bending and crossing are two key elements. Recently, various waveguide structures have been proposed to achieve low-loss and low-intermode-crosstalk (CT) waveguiding for the bend and cross connection scenario [25][26][27][28][29][30][31] . However, most of the demonstrations only achieve dual-mode bending and crossing. Recently, Gabrielli et al. have shown a three-mode bending with low inter-mode coupling, but the bending radius is large and gray-scale lithography is needed 29 . A three-mode crossing has also been reported based on Maxwell fish eye lens structure with a footprint of 11.6 × 11.6 μm 2 where a precisely designed focusing taper is required 31 .
In this article, we demonstrate ultra-compact and arbitrarily routed circuits for three modes via the unprecedentedly small bending (3.9-μm radius) and crossing (8 × 8 μm 2 ). The sharp bending and direct cross connection for TE 0 , TE 1 , and TE 2 modes are achieved by the sophisticated waveguiding, which is realized by a discretized meta-structure with a number of nanoholes. Optimization algorithm is utilized to design the complex structure of nanohole distribution. The tailored index profile allows for mode matching between the multimode waveguide and the bending/crossing region. Furthermore, our proposed devices can be compatible with standard silicon photonics foundry fabrication process (see Supplementary Note 5). We have achieved a three-mode bending with 0.71-0.95 dB loss and a crossing with 0.28-0.82 dB loss over 80 nm bandwidth. Finally, 112 Gbit/s signals encoded on each mode are successfully routed along the arbitrarily designed circuits with a bit error rate (BER) under forward error correction (FEC) limit.

Results
Three-mode bending. The micro-bend with a digital metastructure (DMS) is designed and illustrated in Fig. 1a. The bending region is divided into discrete pixels with minimum feature size of 130 nm. The device platform is silicon-on-insulator (SOI) with 220 nm top silicon and 2 μm buried oxide. We define a binary material state for each pixel, i.e., it can be either silicon or air (by etching away the silicon). This allows for a local index contrast of Δn = 2.48 to manipulate the wave propagation. The waveguide width is chosen to be 2.3 μm, which allows for enough number of pixels. The nanohole (air pixel) distributions can be determined by various optimization algorithms [32][33][34][35][36][37] . Here we use a direct binary search (DBS) algorithm to optimize the structure (see "Methods"). First, we have designed an extremely compact bend with an inner curvature radius of only 2.75 μm (effective radius: 3.9 μm) for a three-mode division multiplexed system. The bend can support transmission of TE 0 , TE 1 , and TE 2 mode with low-loss and low-inter-mode CT. The optical field distributions for different modes are simulated by three-dimensional finite difference time domain (FDTD) method and are shown in Fig. 1b-d. It can be seen that the optical waves for all the modes are squeezed and focused to the corner region of the bend by a few nanoholes near the inner sidewall of the bent waveguide. The outer bound nearly reflects the beam to the output waveguide with preserved mode profile. Though this is not an absolute adiabatic process, the inter-mode interference can be well suppressed to a negligible level. We also quantitatively study how these nanoholes affect the device performance by filling some of the holes with silicon and re-simulate the transmission spectra. We found that the three isolated holes near the outer curvature had nearly negligible impact on the transmission of all the modes. The efficiency drops <2% without these air holes. However, the other nanoholes are critical to the device performance. The simulated spectra of the device from 1500 to 1580 nm are shown in Fig. 1e-g. The simulated insertion losses (ILs) are 0.4, 0.7, and 0.84 dB for TE 0 , TE 1 , and TE 2 modes, respectively. The CTs are <−20 dB for all the modes over the 80-nm wavelength range, which is sufficient for negligible signal-to-noise ratio (SNR) penalties 2,4 .
The device fabrication process can be found in "Methods". To demonstrate the sharp bending of a few-mode waveguide, we fabricated an MDM circuit consisting of a mode multiplexer (MUX), two cascaded bends, and a mode demultiplexer (DEMUX), as shown in Fig. 2a. The MUX is also a compact DMS with a footprint of only 3.4 × 3.9 μm 2 , which is described in details in Supplementary Note 1. As illustrated in Fig. 2a, the circuit has three input ports labeled as I1 (TE 0 mode), I2 (TE 1 mode), and l3 (TE 2 mode) and three output ports labeled as O1-O3 (TE 0 -TE 2 modes), respectively. In the meantime, as shown in Fig. 2b, an extra reference circuit with a back-to-back mode MUX is fabricated to normalize the transmission of the device and to extract the bending loss. The transverse electric (TE)-polarized grating coupler is used to interface the fiber and the waveguide. The coupling loss is~6 dB per facet. The DMS-based bent waveguide is well fabricated as we designed, which can be seen from the zoom-in scanning electron microscope (SEM) image shown in Fig. 2c. The measurement set-up for the transmission spectra of the device are described in "Methods". The normalized transmission spectra measured at O1-O3 ports are plotted in Fig. 2d-f when the continuous wave is launched from input ports I1-I3, respectively. The bending loss is measured to be 0.71, 0.74, and 0.95 dB for TE 0 , TE 1 , and TE 2 modes, respectively. The measured CTs are <−20 dB for all the guiding modes from 1500 to 1580 nm, which is quite consistent with the simulation results.
Three-mode crossing. The on-chip routing inevitably needs to address the waveguide cross-connect problem. Here we design a three-mode crossing with a footprint of only 8 × 8 μm 2 . The structural schematic diagram of the device is described in Fig. 3a. The device has a quadrature symmetry with the minimum feature size of 130 nm. The input/output waveguide widths of the crossing are set to be 1.4 μm to support TE 0 , TE 1 , and TE 2 modes. Each quadrature area is axial symmetric and occupies a chip area of only 4 × 4 μm 2 . This design region is divided into 25 × 25 discrete pixels. The optimization process is similar to the design of bending. Figure 3b-d show the simulated in-plane optical field profiles of the nanostructured crossing when the input waves are TE 0 , TE 1 , and TE 2 modes, respectively. There are several nanoholes located at the corner region of the crossing, which provides a forbidden gap with low index to avoid mode expansion. Hence, the mode profile of each mode can be well preserved when propagating through the crossing region. There is a small amount of mode leakage into the orthogonal waveguide. This can be further reduced by setting an additional optimization objective to reduce such leaked power. To quantitatively investigate the performance of the crossing, we calculate the transmission efficiency and the mode CT taking mode overlap integral into account. We experimentally characterize the crossing performance by fabricating the three-mode cross-connect structure. The microscope image of the circuit is shown in Fig. 4a. The devices were fabricated using the same process as the bending. Owing to the ultra-low loss of an individual crossing, we cascade different numbers of crossing to accurately extract the IL of the device.  -TE0  TE1-TE0  TE2-TE0  TE2-TE1  TE2-TE2   TE1-TE1  TE1-TE2   TE0-TE1  TE0- Figure 4b shows the microscope image of a four-stage cascaded crossing circuit, which is more measureable than a single crossing. In Fig. 4c, the zoom-in SEM image of the nanostructured crossing confirms that the desired pattern has been successfully transferred to the device layer. The transmission spectra of the cascaded crossing are measured and normalized. The measured ILs and CTs for TE 0 -TE 2 modes of a single crossing are shown in Fig. 4d-f, respectively. The ILs are 0.28, 0.68, and 0.82 dB for TE 0 , TE 1 , and TE 2 modes, respectively. The CTs for all modes are characterized by measuring the optical power at the other output ports. The spectra indicate that the CTs are <−20 dB for all the three guiding modes.  TE1-TE0   TE2-TE0  TE2-TE1  TE2-TE2  TE1-TE1  TE1-TE2 TE0-TE1 TE0-TE2 allows both sharp bending and cross connection, which can be scaled to cascaded structure or waveguide array scenarios. We define input ports I1-I3 and output ports O1-O3 for TE 0 -TE 2 modes, respectively. Fig. 5b, c show the measured IL and CT spectra for all the modes after transmission through the circuit. The losses come from the MUX, DEMUX, two times cross connection, and four bending. The average IL is~8 dB and the CTs are <−20 dB within the wavelength range from 1500-1580 nm.
MDM circuit 2 consists of a MUX, ten bends, and a DEMUX. The waveguide is configured in a spiral route and can be further scaled to a dense spiral waveguide. This circuit is used to confirm the ability to continuously change the routing direction. The microscope image of MDM circuit 2 is shown in Fig. 6a. The measured IL and CT spectra of circuit 2 are shown in Fig. 6b, c. The average IL for all the mode channels is 12 dB from 1500 to 1580 nm. The CT of MDM circuit 2 is measured to be <−20 dB within the same wavelength range.
In addition to simply measuring the circuit loss and CT, we verified the capability of high-speed signal routing via the proposed MDM circuits. Each stream of 112 Gbit/s discrete multi-tone (DMT) signals with identical bit allocation are transmitted through each channel of the MDM circuits one by one. Though we measure each mode channel of signal transmission separately, we would expect the potential of simultaneous transmission of the mode-multiplexed signals. This is because the additional signal penalties mainly come from the inter-mode CT of the MDM circuits. According to the theoretical predictions 2,4 , the additional penalties induced by CT are insignificant for the demonstrated routing circuits. The detailed experimental set-up and the digital signal processing technique can be found in Supplementary Note 4. The measured SNR responses at the back-to-back (B2B) case and MDM transmission are plotted in Fig. 7a. The SNR responses are roughly unchanged after the MDM circuit transmission with very slight difference compared with B2B case, and the differences imply the diverse transmission performance among the three channels due to the IL, CT, and even the circumstance instability. The SNR response measured at the B2B case is used to generate a DMT signal by bitallocation algorithm, shown in Fig. 7b. The maximum bit allocation is 5 which corresponds to 32-QAM. The same DMT signals are then routed in different mode channels one by one. The spectrally efficient DMT technique allows us to achieve 112 Gbit/s single lane rate under a bandwidth-constrained condition. The constellations with a bit index of 5 (32-QAM) and 4 (16-QAM) under different subcarrier bands are plotted as the inset of Fig. 7b, which indicates a good signal quality for high-order modulation formats. BER measurement is performed, and the results are shown in Fig. 7b. The measured BER curves of 112-Gbit/s DMT are plotted in Fig. 7c, which are mostly below the 20% FEC limit 38 . Thus the ultra-compact signal routing of highspeed signals is successfully achieved. Besides, the characterization has also been performed at lower speed of 80-Gbit/s 4-level pulse-amplitude modulation (PAM-4) (See Supplementary Note 4).

Discussion
We demonstrate the MDM circuit that can be arbitrarily routed using extremely compact bending and crossing. The performances of the devices are summarized in Supplementary Note 2.  I1-O2  I2-O1  I3-O1  I1-O3  I2-O3  I3-O2   1500  1520  1540  1560  Manipulating the optical wave in such a tiny structure relies on the index engineering in a deep subwavelength scale. An optimized index profile is the key to mode matching, which helps us to avoid mode leakage and inter-mode coupling. The index contrast of the nanoholes with air cladding is n = 2.48. However, a thick oxide cladding is normally deposited on the silicon layer, which reduces the index contrast by 20%. Here we have confirmed in the simulations that the oxide cladding does not affect the design of such compact MDM device. The digitized meta-structure for bending and crossing with oxide cladding are designed following IMEC's design rule (see Supplementary Note 5).
It is important to verify the contributions from those discretized nanoholes. We have numerically investigated the performance of the bending and crossing structure if all the etched pixels are filled by silicon again. For such a sharp bend and compact crossing waveguide, the ILs and CTs increases significantly as we expect (See Supplementary Note 6). This in turn proves the evidence of the effectiveness of index engineering via inverse design.

Methods
Device design. First, discretization is applied to the design area of the device. The geometry of each pixel can be circle, square, or other complex shapes. The minimum feature size can be determined according to the fabrication capability. Here each pixel can be either high refractive index dielectric or low index cladding. The DBS method is used to find a proper distribution of the binary pixels that meets the design targets. For the initial pattern of the device, all the pixel states are chosen to be silicon. Then the material state of each pixel is altered one by one, and the figure-of-merits (FOMs) are inspected. Here the FOMs are ILs and CTs for TE 0 -TE 2 modes, which are calculated by 3D FDTD with 30 nm × 30 nm × 30 nm grid size. If the FOMs improve, the pixel state alteration will be effective and be saved. The improvement can be verified by the following condition: and where η j,i is the transmission efficiency of the jth mode from 1500 to 1580 nm in the ith iteration (j = 1 for TE 0 , j = 2 for TE 1 , and j = 3 for TE 2 ), and β k,i is the kth CT from 1500 to 1580 nm in ith iteration (k = 1 for TE 0 -TE 1 , k = 2 for TE 0 -TE 2 , k = 3 for TE 1 -TE 0 , k = 4 for TE 1 -TE 2 , k = 5 for TE 2 -TE 0 , and k = 6 for TE 2 -TE 1 ). σ j,i is the radius of convergence for efficiency of the jth mode in the ith iteration, and ρ j,i is the radius of convergence for the kth CT in the ith iteration, which is a monotone decreasing function.
One iteration ends after all the pixel states are inspected. Then this process goes over again to further improve the FOMs to design objectives. It takes several iterations to meet the condition of convergence where σ and ρ converge to zero: 39 and where ε and δ are the convergence factor, η j,objective is the transmission objective of the jth mode, and β k,objective is the transmission objective of the kth CT. We perform the device design and optimization using an eight-core desktop. For the bending device, it takes~50 h on average to get the convergent results after 4 iterations. For the crossing device, it takes~40 h after 3 iterations.
Device fabrication process. The devices are fabricated on a SOI wafer with 220 nm top silicon device layer on 2-μm buried oxide. The positive photo resist ZEP 520A is used as the soft mask on SOI. The devices are patterned by the electron beam lithography (Raith eLINE), which operates at 30 kV, then the top silicon layer was fully etched to a depth of 220 nm by using a single-step inductively coupled plasma dry etching.
Measurement set-up and methods. For device characterization, the set-up consists of a sweep laser (Yenista T100S-HP/SCL), a fiber-chip coupling stage, and a benchtop power meter. The continuous wave output of the laser is TE polarized via a polarization controller. Grating couplers are used to interface the single-mode fibers and the silicon waveguides. A tilt angle of 10 degrees with respect to the surface normal direction is designed to suppress the back-reflection. The coupling loss of each grating coupler is measured to be~6 dB. The transmission spectra are obtained by measuring the chip output via a benchtop power meter. For high-speed experiments, the set-up is described in Supplementary Note 4.

Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.