On-chip metamaterial-enabled high-order mode-division multiplexing

Abstract. Mode-division multiplexing (MDM) technology enables high-bandwidth data transmission using orthogonal waveguide modes to construct parallel data streams. However, few demonstrations have been realized for generating and supporting high-order modes, mainly due to the intrinsic large material group-velocity dispersion (GVD), which make it challenging to selectively couple different-order spatial modes. We show the feasibility of on-chip GVD engineering by introducing a gradient-index metamaterial structure, which enables a robust and fully scalable MDM process. We demonstrate a record-high-order MDM device that supports TE0–TE15 modes simultaneously. 40-GBaud 16-ary quadrature amplitude modulation signals encoded on 16 mode channels contribute to a 2.162  Tbit  /  s net data rate, which is the highest data rate ever reported for an on-chip single-wavelength transmission. Our method can effectively expand the number of channels provided by MDM technology and promote the emerging research fields with great demand for parallelism, such as high-capacity optical interconnects, high-dimensional quantum communications, and large-scale neural networks.


Introduction
To manage the explosive growth of data streams, optical interconnect has emerged as a promising solution with the merits of large bandwidth, low transmission losses, and inherent parallelism enabled by various physical dimensions of the optical carrier. 1 Mode, as an important degree of freedom, can be leveraged to construct parallel data streams to accommodate a projected 30-fold data traffic growth 2 for the optical interconnect systems 3 and provide scalability for a wide range of on-chip applications, such as nonlinear optics 4 and optical sensing. 5 also contributes to emerging fields of research where parallelism is in great demand, such as quantum communications 6,7 and AI computing. 8For instance, high-dimensional Hilbert spaces can be encoded in terms of waveguide modes, leading to high-capacity and noise-robust quantum communications. 9or large-scale neural networks, it is desired to process operations in parallel by mapping the weight and vector elements onto different modes for hardware accelerating. 8These applications indicate the great potential of multimode operation in future high-capacity optical networks and novel integrated photonic circuits.
3][24] High data rate MDM transmissions were also demonstrated by employing spatial modes to carry high-speed optical signals. 21,25,26MDM is among the most important techniques for multimode photonics, capable of generating and multiplexing orthogonal spatial modes.Most reported onchip mode multiplexers (MUXs) are based on phase-matching approaches, 3,27 by designing structural parameters to match the propagation constants of waveguide modes for efficient mode coupling.However, the phase-matching condition is determined by the group-velocity dispersion (GVD), 4 which varies significantly between waveguide modes, thus imposing challenges to achieve an efficient mode coupling and limiting the scaling of MDM technology toward high-order modes and more parallel spatial channels.In photonic waveguides, the GVD can be manipulated either by engineering the waveguide confinement or using new materials. 4Most of the previous MDM works were based on waveguide dimension engineering approaches. 28More specifically, adiabatic tapers are employed to relax the stringent phase-matching conditions for MDM, at the cost of large footprints, which impose challenges for high-density integration. 14he introduction of new materials may be of great interest, which provides a new degree of freedom for engineering GVD between waveguide modes by transmitting signals in different materials, thus enabling robust mode couplings and the highorder MDM.However, limited by the available material choices 29 and the feasibility of the fabrication process, 30 most of the reported MDM works are based on single material platforms, where GVD largely limits the achievable MDM order.
Enabled by the increasingly high resolution of advanced lithography, metamaterial has emerged as a new candidate for designing silicon-on-insulator (SOI)-based devices. 31Researchers have demonstrated the use of all-dielectric metamaterials for optical cloaking, 32 neuromorphic computing, 33 and transformation optics applications. 34,35In these works, silicon metamaterials construct various artificial effective medium devices with tailorable refractive indices and dispersion properties, which make them possible to guide light in a desired manner without introducing materials other than silicon.Thus, it is feasible to manipulate the GVD for waveguide modes using metamaterial engineering, offering unprecedented flexibilities for multimode device design.In previous works, metamaterial structures were employed to introduce dielectric perturbations and tune the mode overlap within a multimode waveguide, which enable arbitrary mode conversions [36][37][38] and multimode routings. 39However, most reported metamaterial-based MDM devices operate exclusively in the conventional phase-matching regime, where the GVD between waveguide modes still exists, thus limiting the achievable MDM orders and channel count.To the best of our knowledge, the maximum number of reported MDM channels is eight prior to our recent work, where cascaded asymmetrical directional couplers are utilized to (de)multiplex signals on different modes. 38In our previous work, subwavelength grating (SWG), as one of the simplest metamaterial architectures, [40][41][42] has been introduced to engineer the GVD between modes and relax the stringent phase-matching conditions, thus enabling an 11-channel MDM. 43However, the performance of the device is largely determined by the fabrication accuracies, and the channel counts are challenging to be further scaled, limited by the available duty cycle choices defined by the lithography resolutions.
In this paper, we demonstrate a robust and fully scalable mode coupling method by leveraging the gradient index metamaterial (GIM) structure.Different from previous phasematching approaches, the coupling strength in the proposed scheme can be effectively tailored by engineering the parameters of the GIM waveguides.This scheme has been proposed to achieve an inter-layer coupling between the active InP layer and the passive optical layer. 44Adiabatic coupling enabled by the GIM coupler can help to relax the stringent phase matching condition encountered by conventional DCs and provides a new degree of freedom for designing an efficient mode coupling process and achieving a record 16-channel (TE 0 -TE 15 ) on-chip MDM device. 45Here, we conduct a detailed theoretical analysis, provide a full characterization of the device, and carry out a high-speed data transmission experiment.The measured transmission capacity reaches 2.16 Tbit∕s, which is the highest data rate ever reported for an on-chip single-wavelength transmission.The record MDM channel count and the total data rate demonstrated in this paper highlight the large potential of MDM technology for the arising fields of research, such as high-capacity optical interconnects, high-dimensional quantum communications, large-scale neural networks, and full characterization sensing applications.

Gradient-Index Metamaterial Enabled Adiabatic Coupling
Figure 1(a) shows the conceptional illustration of the GIM waveguide.Silicon nanorods are arranged on a predefined pattern with the duty cycle DðzÞ varying along the propagation direction (z axis).With the adoption of the subwavelengthscale nanostructure, it is feasible to tailor the refractive index distribution nðzÞ as well as the anisotropy properties by engineering the structural parameters.This provides large design flexibilities for controlling the fundamental attributes of electromagnetic waves, such as amplitude, phase, polarization, and wavefront.Here, we focus on the coupling between a GIM waveguide and a nanowire waveguide, as shown in Fig. 1

(b).
The GIM waveguide is designed with a linearly decreasing duty cycle DðzÞ along the propagation direction, which corresponds to a linearly decreasing refractive index distribution nðzÞ.We calculate the equivalent material refractive index of the subwavelength-scale structure using effective-medium theory 42 (detailed information can be found in Sec. 1 in the Supplementary Material).Figure 1(c) shows the simulated electric field distributions of GIM waveguides with different duty cycles using a finite-difference eigenmode (FDE) solver.
It can be noted that the GIM waveguide with a larger duty cycle exhibits a much stronger electric field confinement at the waveguide center, which can be attributed to the larger refractive index contrast between the effective-medium core and the SiO 2 cladding.This property makes it feasible to manipulate the electric field and the refractive index distributions for planar waveguide devices by varying the duty cycle of the GIM waveguide, as shown in Fig. 1(d), which enables a new degree of freedom for on-chip coupling strength engineering.We characterize the coupling strength of the GIM coupler by calculating the coupling coefficient κðzÞ using coupled mode theory. 46The coupled mode equations can be written as where A and B are the amplitudes of the supported waveguide modes a and b, respectively.β a and β b are the propagation constants, and κ ab and κ ba are the coupling coefficients.Since the refractive index nðzÞ and the electric field distributions of the GIM-based coupler are all z-dependent, we calculate κ ab ðzÞ and κ ba ðzÞ at selected evenly distributed z positions: where E a ðx; yÞ and E b ðx; yÞ are the electric field distributions of the waveguide modes a and b, respectively.S is the cross section of the GIM-based coupler, and Δnðx; y; zÞ is the refractive index contrast between the cladding (SiO 2 ) and the equivalent material of the GIM waveguide.To illustrate the benefits of introducing the GIM structure, we calculate the coupling coefficients as functions of the propagation distance for the single mode GIM coupler, conventional DC, and tapered DC, respectively.The dashed lines in Fig. 1(e) mark the coupling lengths we choose to enable a full power transfer from a 400-nm-wide nanowire waveguide to (i) a waveguide tapered from 250 to 550 nm; (ii) a GIM waveguide with its duty cycle varying from 30% to 70%; and (iii) a 400-nm strip waveguide, respectively.The gaps for the three architectures are chosen as 200 nm.
Figure 1(e) shows the calculated coupling coefficients.The conventional DC exhibits the most compact footprint but is sensitive to fabrication deviations.The GIM coupler outperforms the other two types of DCs with much larger coupling coefficients, mainly due to the low optical field confinement of the GIM structure, which results in a compact footprint.In addition, the coupling coefficient tuning range of the GIM coupler is significantly larger than that of the taper structure, which enables large flexibilities to manipulate κðzÞ by engineering the DðzÞ distribution.

GIM Coupler-Based High-Order Mode-Division Multiplexing
By taking advantage of the adiabatic coupling and fabrication tolerant properties of the GIM coupler, we show its applications in high-order mode generation and multi-channel MDM. Figure 2(a) shows the schematic configuration of the GIM coupler-based mode (de)multiplexing design for selective mode coupling.The duty cycle of the GIM waveguide DðzÞ linearly increases from 30% to 70% while maintaining a fixed period value Λ ¼ 320 nm.The feature size of the device is calculated as 320 × 0.3 ¼ 96 nm, which is feasible to be fabricated using E-beam lithography.We choose the width of the GIM waveguide as w G ¼ 0.5 μm and the gap between two waveguides as g ¼ 200 nm.The bus waveguides that support high-order modes are designed to operate under weak-confinement conditions, where the effective refractive indices are chosen as ∼1.76 to ensure a sufficient mode overlap at the coupling region without introducing excess losses.The widths of the silicon bus waveguides for different spatial modes are calculated using an FDE solver.Here, we take the TE 5 mode as an example to illustrate the high-order mode excitation process.Since the two waveguides are placed closely, evanescent fields can interact with each other according to the coupled mode theory.We calculate the coupling coefficients between the Bloch mode in the GIM waveguide and all the supported eigen-modes in the bus waveguide for a GIM coupler with a bus waveguide width of 2.09 μm.It can be noted that the coupling coefficient κ ba linearly increases along the propagation direction with the incorporation of the GIM structure, as shown in Fig. 2(b), which facilitates the power transfer from the TE 5 mode in the bus waveguide to the Bloch mode in the GIM waveguide.The coupling between other waveguide modes to the Bloch modes is suppressed due to the negligible coupling coefficients, which can be attributed to the well-confined mode profiles.We optimize the structural parameters of the GIM-based coupler for selective mode coupling separately using the 3D-finite difference time domain method (ANSYS Lumerical).The coupling length L is 25 μm to meet the adiabatic criteria, and we use SWGbased waveguide bends to gradually separate the coupling region with a bending radius of 15 μm.Here, we consider the mode demultiplexing process by injecting a high-order mode from the bus waveguide and monitoring the output from the GIM waveguide.Figure 2(c) shows the simulated electric field distribution for the TE 5 mode demultiplexing process.The injected TE 5 mode is adiabatically coupled to the fundamental Bloch mode in the GIM waveguide.Further information of the simulations and optimizations of the field distributions and the optimized structural parameters for other waveguide modes can be found in Sec. 3 in the Supplementary Material.To verify the feasibility of the GIM coupler to achieve a robust high-order mode coupling, we simulate the TE 15 mode demultiplexing process using the GIM structure, the conventional DC, and the tapered DC, respectively, as shown in Fig. 2(d).It can be noted that the conventional DC is very sensitive to fabrication deviations; even a 1% dimension variation leads to significant performance degradations.The GIM coupler exhibits comparable tolerance performance to the tapered DC when the waveguide dimensions vary by AE1%.However, when the waveguide deviation is > AE 3%, the GIM coupler shows some performance advances, which can be attributed to its larger refractive index tuning range.These results show the feasibility for achieving a very-high-order MDM by employing the GIM structure.These results show the feasibility for achieving a very-high-order MDM by employing the GIM structure.

Device Fabrication and Characterization
To verify the feasibility of the MDM technique in scaling the transmission capacity of optical waveguides, we cascade the GIM couplers designed for different spatial modes and construct a 16-channel MDM device, as shown in Figs.3(a)-3(c).GIM waveguides are placed along a strip bus waveguide with a varying width to (de)multiplex selected waveguide modes, as shown in Fig. 3(a).We use adiabatic tapers to connect bus waveguides To characterize the optical performance of the device, we use a continuous-wave tunable laser source (Keysight 81960A) to generate laser output ranging from 1507 to 1600 nm.A customized 34-channel fiber array is used to couple the light between fibers and the 16 input ports (I 0 -I 15 ) and 16 output ports (O 0 -O 15 ) of the device.We fix the input port and record the power output from all 16 output ports using an optical power meter (Keysight N7744A).A micro-electro-mechanical system (MEMS) 1 × 16 optical switch is used to switch between channels.The wavelength sweep program runs repeatedly so the data are recorded for all 16 output ports when the MEMS switches between channels.Measured transmission results of the fabricated 16-channel mode deMUX are provided in Fig. 4. Light is injected from one input port to excite a corresponding high-order mode in the bus waveguide.Optical signals output from all 16 output ports are recorded as the transmission and crosstalk for the selected mode channels.The transmission spectra of the mode deMUX are normalized to that of the identical grating couplers fabricated on the same chip with a 6.5 dB∕facet coupling loss.Measured insertion losses range from 0.96 to 6.95 dB, and the crosstalk values vary between −10.30 and −16.23 dB for all 16 channels at 1550 nm.Note that the TE 3 mode experiences the largest insertion loss of 6.95 dB while exhibiting an acceptable crosstalk of −12.5 dB.This performance degradation can be attributed to the relatively low fiberto-chip coupling efficiency for this channel, caused by the defects observed at the corresponding grating coupler, which can be optimized through multiple iterations.
To verify the feasibility of the proposed mode MUX for supporting parallel mode channels, we perform a high-speed data transmission experiment.The experimental setup is shown in Fig. 5(a).At the transmitter side, an external cavity laser (ECL) operated at 1550.12 nm with a 15 kHz linewidth is utilized as the light source.A 100-GSa/s dual-channel digital-toanalog converter (DAC) (MICRAM DAC10002) with a 3-dB bandwidth of 44 GHz is employed to generate a 40-GBaud Nyquist 16-ary quadrature amplitude modulation (16-QAM)  signal with a roll-off factor of 0.01.After being amplified by two electrical amplifiers (SHF S807 C), the electrical 16-QAM signal drives a 22-GHz in-phase and quadrature modulator (Fujitsu FTM7961EX), which is biased at its transmission null.The generated optical 16-QAM signal is amplified and then time gated via an acousto-optic modulator with a duty cycle of 11.8% and a frequency of 12.5 kHz.To avoid transient power boost induced by signal gating, an amplified spontaneous emission (ASE) source is spectrally shaped by a wave shaper and    47 when crosstalk performance is limited, researchers are also working on the crosstalk mitigation approach to enable an MIMO-free multi-channel transmission.We may further optimize the mode deMUX design to suppress undesired mode coupling to achieve a WDM-compatible MDM device.

Conclusion
We have proposed a high-order mode coupling process by introducing metamaterial structures.It is feasible to mitigate the GVD between waveguide modes by employing GIM waveguides for coupling strength engineering, which contributes to a robust and fully scalable MDM.We have demonstrated a record-high-order MDM supporting TE 0 -TE 15 mode channels and performed a high-speed data transmission experiment.A net data rate of 2.162 Tbit∕s has been achieved at a single wavelength.This high-order MDM approach is fully compatible with conventional WDM and PDM technologies.The effective MDM channel scaling may boost the arising fields of research where parallelism is in great demand, such as high-capacity optical interconnects, high-dimensional quantum communications, large-scale neural networks for hardware accelerating, and full characterization sensing.

Code, Data, and Materials Availability
Data underlying the results presented in this paper can be obtained from the authors upon reasonable request.

Fig. 1
Fig. 1 Planar-metamaterial-enabled refractive-index-distribution manipulation and GIM-enabled adiabatic coupling.(a) Conceptional illustration of a 1D metamaterial structure with a linearly decreasing duty cycle DðzÞ, which can be regarded as a strip waveguide with a varying refractive index n eq ðzÞ.(b) Schematic configuration of a GIM-based mode coupling process.The GIM waveguide is tailored with a linear duty cycle distribution DðzÞ along the propagation direction.(c) Simulated electric field distributions of GIM waveguides with different duty cycles.(d) Crosssection view of the GIM-based direction coupler.The Gaussian curves illustrate the power distributions of the modes supported by the GIM waveguide.(e) Calculated coupling coefficients for the tapered DC, GIM coupler, and conventional DC as functions of the propagation distance.

Fig. 2
Fig. 2 GIM-based coupler for 16-channel MDM.(a) Schematic configuration of the GIM-based coupler designed for selective mode coupling.The duty cycle of the GIM waveguide DðzÞ linearly increases from 30% to 70% along the propagation direction for constructing a linear refractive index distribution.(b) Calculated coupling coefficients κ ab and κ ba and the normalized transmission responses for the TE 5 mode in the strip waveguide (mode b) and the Bloch mode in the GIM waveguide (mode a), as a function of propagation distance.(c) Simulated electric field distribution for the TE 5 mode demultiplexing.(d) Simulated TE 15 mode demultiplexing process using GIM structure, conventional DC, and tapered DC, respectively, when the waveguide dimension varies.

Fig. 3
Fig. 3 GIM-based couplers for 16-channel MDM.(a) Schematic configuration of the 16-channel mode MUX.(b) Microscope image and (c) SEM image of the 16-channel GIM-based mode MUX and deMUX.(d)-(f) Magnified SEM images of the coupling regions.

Figure 5 (
c) shows the calculated bit error ratios (BERs) of 40-GBaud 16-QAM signals for 16 modes after the MIMO-FFE and maximum likelihood sequence detection (MLSD) algorithm.The BERs for all 16 channels are below the 15% forward error correction (FEC) threshold.Considering the frame redundancy and 15% FEC overhead, we achieve the highest on-chip net data rate of 2.162 Tbit∕s [40 GBaud × 4 bits per symbol × 16 modes × 97.1% / (1 + 15%)] at a single wavelength by employing 16 parallel mode channels.
Figure 5(d) shows the constellations of all 16 modes, which indicate that all 16 channels have good signal quality.A potential net spectral efficiency of 43.24 ðbit∕sÞ∕Hz [2.162 ðTbit∕sÞ∕50 GHz] can be obtained by further combining the WDM technology with 50 GHz channel spacing.While it is possible to recover signals using MIMO DSP as long as the transmission matrix is unitary

Fig. 4
Fig. 4 (a)-(p) Measured transmission responses of the 16-channel mode deMUX.Each subplot shows the transmission and crosstalk recorded at 16 output ports for a selected input channel.