Integrated Hybrid Mode‐Wavelength Demultiplexers Based on Cascaded Digital Metamaterials

High‐dimensional multiplexing technology is of importance in the on‐chip photonic interconnections and challenging to design within ultracompact footprint. Herein, high‐dimensional demultiplexers are proposed and demonstrated to enable wavelength‐division and mode‐division simultaneously. The functional regions of digital metamaterials are obtained by inverse design individually and are cascaded to work as high‐dimensional demultiplexers. The gradient‐based inverse design is carried out with an efficient method combining finite‐element method, density method, and method of moving asymptotes. The performances are simulated by 3D finite difference time domain with silicon‐on‐insulator configuration. The proposed demultiplexer with four‐channel has ultracompact footprint of 4.1 × 3.65 μm2. Its average transmission efficiency is 38.7% and contrast ratios are higher than 13.0 dB. Besides, the proposed demultiplexer with six‐channel has a footprint of 4.55 × 5.55 μm2. Its average transmission efficiency is 24.3% and contrast ratios are higher than 11.8 dB.

To further enlarge the capacity of on-chip interconnection, high-dimensional multiplexing technologies are of highly importance.Recently, several silicon-based high-dimensional demultiplexers have been proposed and demonstrated.Dai proposed an architecture utilizing cascaded dual-core adiabatic tapers and polarization beam splitters within 6.9 μm Â 20 μm and realized a hybrid mode-polarization demultiplexer. [41]Wang experimentally demonstrated a low-loss hybrid mode-wavelength demultiplexer, by combining microring resonator with asymmetric directional couplers. [42]Zhao designed a reconfigurable optical add-drop demultiplexer with more than 1000 integrated elements.The demultiplexer can achieve a hybrid mode-polarizationwavelength-multiplexing system with low bit error rate. [43]esides, Yang used inverse design to obtain a 6.5 Â 6.5 μm 2 silicon-based mode demultiplexer with 15 THz spectral bandwidth to combine with frequency microcombs, realizing a hybrid modewavelength demultiplexer. [44]These proposed high-dimensional demultiplexer schemes still have great room for improvement in the footprint, due to the introduction of different components or structures.Hybrid mode-wavelength multiplexing through nanophotonic structures only has not been reported.
In this article, we present two digital-metamaterial-based demultiplexers which operate 2/3 mode channels and 2/2 wavelength channels, respectively.The hybrid mode-wavelength multiplexing is realized by cascaded digital metamaterials.The inverse design strategy adopts the gradient-based topology optimization combining finite-element method (FEM), density method (DM), and the method of moving asymptotes (MMA).The footprints of proposed demultiplexers are 4.1 Â 3.65 and 4.55 Â 5.55 μm 2 , respectively.

Scheme and Methods
Figure 1a,b shows the proposed four-channel/six-channel modewavelength demultiplexer, which consists of a two-channel/ three-channel mode demultiplexer and two/three two-channel wavelength demultiplexers.They are entirely based on digital metamaterials, cascaded by 1 um-wide silicon waveguides, and work with TE polarization modes.In both mode demultiplexers, the signals propagate based on mode, regardless of their wavelength.Then, at the waveguides for cascading, the signals are converted to a fundamental mode light, thus entering the wavelength demultiplexers for output by wavelength.As shown in Figure 1c,d, the digital metamaterial region footprints of mode demultiplexers are 1.55 Â 2.2 and 2 Â 4.4 μm 2 respectively.They aim to separate TE polarization modes as many as their channels (i.e., TE0, TE1, TE2) and not to be affected by specific wavelengths 1500 nm/1600 nm.As for the wavelength demultiplexer shown in Figure 1e, it is a primary structure pattern that needs to have a second optimization to overcome the mode distortion that occurs in the cascaded waveguides, so details of six designs are shown in the next part, including the primary optimized design and five second optimized designs corresponding to the five mode demultiplexers in Figure 1a,b, respectively.All mentioned designs have a depth of 0.34 μm and are placed on a silica substrate in 3D model.
The total footprint of the four-channel mode-wavelength demultiplexer is 4.1 Â 3.65 μm 2 , including 8.215 μm 2 metamaterials region, two 1 Â 0.4 μm 2 cascaded waveguides, and air filling the rest.The digital metamaterials region of the mode demultiplexer is divided to 63 Â 89 pixels.Any nonboundary pixel is square with edge length of 25 nm, any pixel on-boundary is rectangular with edge lengths of 25 and 12.5 nm.Similarly, the sixchannel mode-wavelength demultiplexer with a total area of 4.55 Â 5.55 μm 2 consists of 16 μm 2 metamaterials region, three 1 Â 0.4 μm 2 cascaded waveguides, and air filler.This mode demultiplexer includes 81 Â 144 pixels and all wavelength demultiplexers have the same number of pixels as 63 Â 63.All pixels are filled by either silicon (equivalent refractive index = 3 in 2D optimization and refractive index = ffiffiffiffiffi 12 p in 3D simulation) or air (refractive index = 1 in both 2D/3D simulation), forming the designed dielectric refractive index contrast.DM is adopted to describe the digital metamaterial regions by regarding the whole pattern in each region as a single domain. [33,45]We use a simple linear relation to describe and control the dielectric refractive index: Here, θ is a control variable field 0 ≤ θ ≤ 1 (air/silicon corresponding to θ = 0/θ = 1) and evaluated on each FEM mesh node.Therefore, the mesh of FEM is set as square to ensure the distribution of refractive index is nearly pixelated.A projection based on the hyperbolic tangent function θ = tanh(β(θÀθ β )) þ tanh(βθ β )/tanh(β(1Àθ β )) þ tanh(βθ β ) is introduced to reduce intermediate values of θ. [46] The β and θ β are configurable parameters.
[49] The initial values of all primary optimization regions are set as 1 on all nodes corresponding to the situation that the entire region is filled with silicon only.In the second optimization for wavelength demultiplexers, the primary optimized result of θ is set as the new initial values.From the initial values, θ is updated in each iteration based on the sensitivity by MMA.The physics is exactly held at every step without residual.The gradient information is obtained by the adjoint method which needs relatively low additional computation.The optimized variable and its corresponding pattern approach the target design gradually with the optimization process.For all designs involved, a similar objective function based on the spatial integrals of forward power flux (in x-direction) is adopted.All objective functions can be simply summarized as F MC ðλÞ ¼ P i w i ∫ A i P x ðλÞds.λ has two values, 1500 and 1600 nm, w i is the weight of each integral to realize the target modedivision/wavelength-division function and different A i refers to the region of each output port.In the optimization process, the objective function is targeted to reach its minimization, and the number of every primary optimization iteration is below 250 to reach a reasonable solution, while the number of every second optimization iteration is below 30.For the primary optimization, the values of the objective function drop very quickly in the very initial steps (<50 steps), reflecting the high efficiency of this method.It should be noted that the objective functions of both mode demultiplexers also involve 1500 and 1600 nm just like the wavelength demultiplexers.w i of the second optimization has different values with the mode/ wavelength demultiplexers so that satisfactory results can be achieved in less iterations.Then, in order to eliminate any intermediate values between 0 or 1 in the optimization result, a postbinarization method is used, forcing all values to be standard values.All FEM simulations and related optimizations are implemented in COMSOL Multiphysics.In experiments, e-beam lithography and reactive ion etching have the prospects of fabricating the digital metamaterials-based highdimensional demultiplexers. [50,51]

Results and Discussion
In our designs, a mode demultiplexer that can support two center wavelengths is the basis for realization of the hybrid modewavelength multiplexing.The modes of TE0, TE1, and TE2 are the first three modes of TE polarization, and their mode manipulation and multiplexing are basic and reliable.As shown in Figure 2a, the objective function decreases rapidly in the first 50 steps and takes 201 iteration steps for the two-channel mode demultiplexer to achieve a desired result.All black(white) pixels correspond to silicon/air with an equivalent refractive index of 3(1) in the optimization process and in the later 3D simulation the refractive index is adjusted to ffiffiffiffiffi 12 p (still 1).All magnetic fields are perpendicular to the waveguide in this article.Figure 2b shows the z-direction top view of magnetic fields at center wavelength, demonstrating how the modes propagate in the functional region.When the TE0 mode with wavelength 1500 or 1600 nm propagates through the digital metamaterial region, the light outputs in port A, while the TE1 light outputs in port B. The transmission efficiency of TE0 and TE1 mode with the wavelength 1500 nm/1600 nm in 3D simulation is shown in Figure 2c.For the inputs of TE0-1500, TE1-1500, TE0-1600 nm, TE1-1600 nm, the transmission efficiency and the contrast ratio are 56.2%,67.7%, 62.1%, 69.6% and 11.9, 18.6, 16.0, 22.1 dB at the center wavelength, respectively.The transmission efficiency is always higher than 54.4% and has a maximum of 69.6% within 10 nm bandwidth centered on 1500 nm/1600 nm, while the crosstalk is at a low level.
As for the three-channel mode demultiplexer, the objective function shown in Figure 3a has a convergence process similarly with a quick decrease in the initial stage and a total iteration of 165 steps.The magnetic field distribution and power flows of TE0, TE1, TE2 optical signals with wavelength 1500 nm/1600 nm are shown in Figure 3b.Regardless of signals' wavelengths, the TE0 signal outputs in the lower port, the TE1 signal outputs in the middle port, and the TE2 signal outputs in the upper port.Figure 2c shows the transmission efficiency of all situations of 3D simulation.The transmission efficiency is 16.4%, 34.4%, 54.6%, 27.6%, 23.4%, and 56.9% for inputs of TE0-1500, TE1-1500, TE2-1500, TE0-1600, TE1-1600, TE2-1600 nm.The worst contrast ratio, the lowest contrast ratio value of all, is used to describe the performance of demultiplexers with more than two output ports.Though the efficiency is lower than the two-channel mode demultiplexer, the worst contrast ratio is reliable with values of 14.7, 14.2, 17.4, 10.4, 11.7, 17.8 dB.
Through the mode demultiplexers, signals are converted to TE0 light in the cascaded waveguides.Further, these signals of fundamental mode need to be separated according to their wavelengths.Figure 4a shows the primary pattern of wavelength demultiplexer and the convergence curve of normalized optimization objective function.The digital metamaterial region has a footprint of 1.55 Â 1.55 μm 2 and consists of 63 Â 63 pixels which are all filled with silicon or air.Although more iteration steps are involved than the mode demultiplexers, nearly half time is taken because of the λ 2 -level footprint.Shown as Figure 4c, signals with 1500 or 1600 nm propagate to the port A and port B, respectively.As a wavelength demultiplexer, it has already qualified performance, as shown in Figure 3c.At the center wavelength 1500 nm/1600 nm, the transmission efficiency and the contrast ratio are 49.9%/54.5% and 40.7 dB/53.4dB, respectively.Within 10 nm bandwidth centered on 1500 nm/1600 nm, there is no serious decline of the performance, and the transmission efficiency and the contrast rate are around 44-56% and 35-53.5 dB.
However, the optical signals are proved that it is difficult to maintain a pure fundamental mode or any other high-order mode at the cascaded waveguide during the transmission from the mode demultiplexer region to the wavelength demultiplexer regions.The wavelength demultiplexer is optimized in the ideal condition that the input signal is light with pure TE0.Therefore, inevitable performance degradation occurs when the mode demultiplexers and wavelength demultiplexers are cascaded directly as Figure 1a,b.In order to compensate the performance degradation caused by mode distortion, we introduce a second optimization and regard the previous designed wavelength demultiplexers' functional pattern as the domains to be reoptimized.The objective function's weights are rewritten by increasing the positive weights of nontarget output ports with higher crosstalk.As a result, the wavelength demultiplexers of fourchannel/six-channel mode-wavelength demultiplexer are reoptimized together, sharing convergence curves as shown in Figure 5a,b.The number of second optimization steps is both 27, much less and faster than the primary optimization.After the second optimization process, only 59-200 pixels have changed compared to the primary pattern with a rate from 1.49% to 5.04%.The five second optimized wavelength demultiplexers are designed for specific mode distortion in different port of two-channel/three-channel mode demultiplexers, so it is unnecessary to make individual 3D simulations with pure TE0 input for any of them and we don't go in detail with their performance under ideal conditions.
After the optimization of two-channel/three-channel mode demultiplexer and the second optimization of wavelength demultiplexers, we already have all the optical modules to assemble the targeted hybrid mode-wavelength demultiplexers.Figure 6a shows the proposed structure of the four-channel mode-wavelength demultiplexer with detailed size and digital metamaterial patterns.Except for the outermost input/output waveguides, the entire device has a height of 3.65 μm, a width of 4.1 μm, and a depth of 0.34 μm, placed on a silica substate with a refractive index of 1.45.All pixels and regions in black are silicon with a refractive index of ffiffiffiffiffi 12 p and others are air with a refractive index of 1. Through this device, signals of TE0 propagate to the upperwavelength demultiplexer region, while signals of TE1 propagate to the lower-wavelength demultiplexer region, and then all signals propagate for a second time based on their wavelength 1500 nm/1600 nm, as shown in Figure 6b.The transmission efficiency and the worst contrast ratio for the inputs of TE0-1500, TE1-1500, TE0-1600, TE1-1600 nm are 37.6%, 33.4%, 40.3%, 43.8% and 13.0, 13.6, 18.5, 16.0, respectively.Table 1 shows the transmission efficiency and the worst contrast ratio of every port.Figure S1, Supporting Information, supplements the transmission efficiency within 10 nm bandwidth centered on 1500 nm/1600 nm.
The six-channel mode-wavelength demultiplexer has similar structures, as shown in Figure 7a.The volume of the entire device beyond silica substrate is 4.55 Â 5.55 Â 0.34 μm 3 and all    refractive indexes of silicon, air, silica are same as the fourchannel demultiplexer.Compared to the former device, the six-channel mode-wavelength demultiplexer has an additional mode, TE2, to control.All TE0 signals propagate to the lowerwavelength demultiplexer region, TE1 signals propagate to the middle demultiplexer region, while TE2 signals propagate to the upper demultiplexer region, and then each of them is separated into different ports according to its wavelength (see Figure 7b).The six-channel mode-wavelength demultiplexer has an average transmission efficiency of 24.3% and its worst contrast ratios for the inputs of TE0-1500, TE1-1500, TE2-1500, TE0-1600, TE1-1600, and TE2-1600 nm are 16.3, 14.7, 16.0, 11.8, 11.8, and 16.5 dB, respectively.Table 2 shows the transmission efficiency and the worst contrast ratio of every port.Figure S2, Supporting Information, supplements the transmission efficiency from 1495-1505 nm/1595-1605 nm.
To further demonstrate the benefits and importance of the second optimization, we show the performance obtained from 3D FDTD simulation of the two devices without a second optimization in the supplement (see Table S1 and S2, Supporting Information).The worst contrast ratio and the average transmission efficiency are improved, especially for the six-channel modewavelength demultiplexer.
In experiments, e-beam lithography and reactive ion etching are needed to fabricate such inversely designed digital metamaterials whose length of pixels is at the nanometer scale.Major error may occur in the material of these square pixels.To account for this kind of error and verify the robustness of our work, we perform error simulation.In the error simulation, 2% pixels with wrong material are randomly introduced for the four-channel mode-wavelength demultiplexer.As shown in Figure 8, 2% error causes acceptable fluctuations in transmission efficiency.The contrast ratios also maintain high performance in every simulation, indicating the robustness of the proposed design method.Moreover, we perform mirroring simulation of both demultiplexers.In the mirroring simulation, the proposed demultiplexers are used in reverse, as shown in Figure S3 and S4, Supporting Information, to verify that the time-reversal   symmetry in the linear regime is held.Besides, the nanofabrication is expected to be improved further in the near future.Thus, it will be popular to introduce nanophotonic structures with subwavelength feature size, which can effectively help the future development of silicon photonic integrated circuits. [25] Conclusions In conclusion, we have proposed hybrid mode-wavelength demultiplexers based on cascaded digital metamaterial structures.The high-dimensional demultiplexers have ultracompact footprints and support two-mode-two-wavelength and threemode-two-wavelength, respectively.The functions are realized with a two-channel/three-channel mode demultiplexer and two/three two-channel wavelength demultiplexers, all of which are pixelated.Each of the mode/wavelength demultiplexers can be used as an individual demultiplexer with high performance in contrast ratio.All digital metamaterial patterns are determined by an efficient strategy based on density model and MMA.A second optimization is applied to solve the performance degradation caused by the mode distortion of signals in cascading waveguides.The 3D FDTD simulations with SOI configuration demonstrate high optical contrast ratio (over 13.0 and 11.8 dB at least, respectively) and robustness.We expect that the proposed high-dimensional demultiplexers can be helpful to enhance the data capacity of the optical interconnections.

Figure 1 .
Figure 1.a,b) Schematic configuration of the proposed four-channel/six-channel mode-wavelength demultiplexer on SOI platform.c) A two-channel mode demultiplexers that work with TE0 and TE1.d) A three-channel mode demultiplexers that work with TE0, TE1, and TE2.e) A 1500 nm/1600 nm wavelength demultiplexer.

Figure 2 .
Figure 2. a) Convergence curve of normalized objective function and final digital metamaterial of the two-channel mode demultiplexer.b) Magnetic field distribution of different inputs at center wavelengths 1500 nm/1600 nm.c) Transmission efficiency spectra obtained from 3D FDTD simulation with input of TE0-1500, TE1-1500, TE0-1600, TE1-1600 nm.

Figure 4 .
Figure 4. a) Convergence curve of normalized objective function and final digital metamaterial of the wavelength demultiplexer.b) Magnetic field distribution of 1500 nm/1600 nm wavelength demultiplexer at center wavelength.c) Transmission efficiency spectra obtained from 3D FDTD simulation with input wavelength ranges of 1500 nm AE 5 nm and 1600 nm AE 5 nm.

Figure 5 .
Figure 5. a) Normalized convergence curve of second optimization objective function for the four-channel mode-wavelength demultiplexer and final digital metamaterials of the upper-and lower-wavelength demultiplexer regions.b) Normalized convergence curve of second optimization objective function for the six-channel mode-wavelength demultiplexer and final digital metamaterials of the upper-, middle-, and lower-wavelength demultiplexer regions.

Figure 8 .
Figure 8. Transmission efficiencies of four-channel demultiplexer with random errors.

Table 1 .
Performance of the proposed four-channel demultiplexer.

Table 2 .
Performance of the proposed six-channel demultiplexer.