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Abstract
An ultracompact broadband dual-mode 3 dB power splitter using inverse design method for highly integrated on-chip mode (de) multiplexing system is proposed and experimentally demonstrated. A dual-mode convertor based on subwavelength axisymmetric three-branch waveguide is utilized to convert TE0 and TE1 to three intermediate fundamental modes. The axisymmetric topology constraint of the nanostructures enables the optimized device to achieve a strict 50:50 splitting ratio over a broad wavelength range from 1.52 to 1.60 µm. The fabricated device occupied a compact footprint of only 2.88 µm × 2.88 µm. The measured average excess losses and crosstalks for both modes were respectively less than 1.5 dB and −20 dB from 1.52 to 1.58 µm for both TE0 and TE1, which are consistent with the numerical simulations.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Mode-division multiplexing (MDM) on silicon-on-insulator (SOI) platform, as a more promising and attractive technology, has attracted substantial attention to further increase the transmission capacity of on-chip optical interconnect [1]. To realize on-chip MDM systems, various key building blocks have been reported, such as mode (de) multiplexers ((DE) MUX) [2–6], mode switch [7, 8], multimode waveguide crossing [9, 10] and multimode bent waveguides [11–13]. However, a dual-mode power splitter with compact footprint and broad bandwidth, as an essential component for realizing the densely integrated MDM optical interconnect network, has been rarely addressed due to the complex mode coupling for high order modes. Recently, an integrated dual-mode 3 dB power coupler based on tapered directional coupler was present [14], which could work at the wavelength of 1.55 μm for both TM0 and TM1. However, it is bandwidth-limited even though the tapered directional coupler is adopted. To achieve the broadband property, multimode power splitters based on symmetric Y-junctions were proposed [15, 16]. In [15], an input Nth-order mode has to be converted to a 2Nth-order mode using cascaded mode convertors before power division by a symmetric Y-junction. In [16], an adiabatic coupler was used in the input stage to convert the input mode to an intermediate high order supermode and realize 3 dB power splitter by an S-bend based Y-junction. However, adiabatic components employed in both types of devices lead to a large device footprint, which may prevent further potential application in dense and large-scale on-chip photonic integration.
Recently, free-form metamaterials based on inverse design potentially offer an effective approach to design ultracompact and highly functional devices simultaneously [17–21]. In contrast to the traditional device design, the inverse design approach can flexibly engineer the refractive index distribution and manipulate light field at a deep subwavelength scale to realize a variety of ultracompact integrated devices. However, due to the lag effect of plasma etching process, etching patterns with small and random feature sizes of the inverse designed nanostructures may probably introduce large fabrication errors. To implement nanostructured devices robust to fabrication tolerances, one may use the photonic-crystal-like (PhC-like) subwavelength structure [22] or fabrication-constrained topology optimization method [23] for nanophotonic inverse design.
In this work, we propose a broadband dual-mode 3 dB power splitter based on axisymmetric three-branch waveguides, and realize its ultracompact version based on the PhC-like subwavelength structure using an inverse design approach. The basic idea of the proposed device is that we first convert the two input normal modes to six intermediate fundamental modes (IFMs), and finally synthesize the two output normal modes in each output waveguide from the phase-shifted IFMs, respectively. To further decrease the footprint, we employ PhC-like subwavelength structure to design the device by inverse design method. We find that, in the inverse design of dual-mode 3 dB power splitter using axisymmetric topology constraint, the optimization iteration based on random initial patterns usually converges to optimized subwavelength structures which are equivalent to the proposed waveguide device model. In addition, the axisymmetric optical field distribution contributes to a strict 50: 50 splitting ratio for both modes. The footprint of the fabricated device is 2.88 × 2.88 µm2, which may be the most compact dual-mode 3 dB power splitter that has been reported to the best of our knowledge. The measured excess losses (ELs) and crosstalks (XTs) for both modes are less than 1.5 dB and ‒20 dB from 1540 to 1580 nm, respectively.
2. Design, optimization, and simulations
2.1 Dual-mode 3 dB power splitter based on conventional waveguide
Figure 1(a) schematically shows the dual-mode 3 dB power splitter, based on conventional waveguides, which is composed of three axisymmetric three-branch waveguides, three symmetric Y-junctions and two π phase shifters.
Fig. 1 (a) Dual-mode 3 dB power splitter assisted with DMCs based on conventional waveguides. (b) The proposed dual-mode power splitter assisted with DMCs, based on subwavelength structure.
The structure of the proposed device can be divided into three stages. In the stage I, the axisymmetric three-branch waveguide works as a dual-mode convertor (DMC), which converts the fundamental transverse electric mode (TE0) to an intermediate fundamental mode in the middle branch (IFM1m), and splits the first-order transverse electric mode (TE1) into two anti-phase IFMs in the upper and lower branches (IFM1u and IFM1l), respectively. In the stage II, the IFM1u is further split into two anti-phase IFMs via one symmetric Y-junction and one π phase shifter, then both anti-phase IFMs are respectively injected into the upper and lower branches of the upper mirrored DMC in the stage III to excite the TE1 mode in the upper output waveguide. Meanwhile, the IFM1m is split into two in-phase IFMs using a Y-junction in the stage II and the upper IFM is injected into the middle branch of the upper mirrored DMC in the stage III to excite the TE0 mode in the upper output waveguide. The two normal modes in the lower output waveguide are excited in the similar way. Notably, benefiting from the axisymmetric structure, a strict 50: 50 power splitting ratio for both modes can be achieved.
The DMC is a key component of the dual-mode 3 dB power splitter. As shown in Fig. 2(a), the layout of DMC is axisymmetric. Each DMC consists of a dual-mode stem and three single-mode branches with different widths. The waveguide widths of the two lateral branches and the middle one are W0 and W1, respectively. The gap between adjacent branches increases from 0 to Wgap gradually, forming an adiabatic coupled waveguide system with a coupling length of L. Due to close modal effective refraction index match, TE0 and TE1 are first evolved into the corresponding even supermode TSM0 and odd supermode TSM1 in the adiabatic coupled waveguide system region, respectively. Then the TSM0 and TSM1 are converted to the IFM in the middle output waveguide and two anti-phase IFMs in the two lateral output waveguides. To avoid the modal coupling, the Wgap should be larger than 1 μm. The effective indices of TSM0 and TSM1 as a function of W0, W1 and Wgap are presented in Figs. 2(b)-2(d), respectively. The effective indices of TE0 and TE1 in the stem waveguide are 2.6 and 2.28, respectively. Thus, to guarantee only supporting single mode in the multi-branches and satisfy close modal match, W0 and W1 are chosen to 550 nm and 450 nm, respectively. The high-efficiency mode conversion in a short adiabatic coupling length could be achieved in this way. We also simulate the adiabatic evolution in the DMC with different adiabatic coupling lengths. The calculated mode conversion efficiencies for the TE0 and TE1 launched into the stem waveguide as a function of the coupler length are shown in Fig. 2(f). Therefore, L should be more than 100 μm to achieve mode conversion with a high efficiency up to 98%. The other components of the dual-mode 3 dB splitter including broadband single-mode symmetric Y-junctions, broadband 2 × 2 waveguide crossings and broadband π phase shifters could be designed and optimized based on the conventional waveguide [24–26].
Fig. 2 (a) The schematic of DMC based on conventional waveguide. (b) For Wgap = 200 nm, W1 = 450 nm, mode effective index with different W0 (c) For Wgap = 200 nm, W0 = 550 nm, mode effective index as a function of W1 (d) For W0 = 550 nm, W1 = 450 nm, mode effective index with different Wgap (f) Transmission spectra as a function of L.
Theoretically, a broadband dual-mode 3 dB power splitter could be achieved with these waveguide components, as illustrated in Fig. 1(a). However, a long adiabatic evolution length (> 100 μm) is usually required for each DMC and the total length of the device based on the conventional waveguide model may be larger than 200 μm, which prevents its potential application in dense and large-scale on-chip photonic integration.
2.2 Dual-mode 3 dB power splitter based on subwavelength structure
In this work, the PhC-like subwavelength structure is used for inverse design of an ultracompact dual-mode 3 dB power splitter. For an ideal dual-mode 3 dB power splitter with an axisymmetric layout, the distributions of the electric field and the dielectric permittivity inside the device should also be axisymmetric. Therefore, the axisymmetric topology constraint is enforced during the inverse design process, which maintains an axisymmetric pattern in each round of iteration.
As shown in Fig. 1(b), the proposed 3 dB power splitter based on the PhC-like subwavelength structure is designed on a silicon-on-insulator platform with 220 nm-thick air-cladded top silicon layer. The widths of the input waveguide and output waveguides are 900 nm. The gap between the two output waveguides should be larger than 1 μm to avoid modal crosstalk. The inverse design region composed of 24 × 24 discrete pixels occupies a compact footprint of only 2.88 × 2.88 µm2. Each pixel is a square of 120 × 120 nm2 with a circular hole. The hole has a radius of 45 nm and a depth of 220 nm. Each hole can be occupied by silicon or air. The nonlinear direct-binary-search (DBS) optimization algorithm is utilized to decide the material of each hole to be silicon or air one by one and the figure-of-merit (FOM) is introduced to evaluate the optimized performance [18]. We randomly chose one pixel to change its occupied material (silicon or air), and calculate the FOM. If the FOM increases, then the changed material will be reserved. Otherwise, the pixel goes back to the original material. A 3D finite-difference time-domain (FDTD) method via a commercial software (Lumerical FDTD Solutions) is used to calculate the FOM [27]. The FOM in our simulation is defined as:
where t1 and t2 are the transmittances of TE0 and TE1, respectively. Likewise, x1 and x2 are the conversion efficiencies of input TE0-output TE1 and input TE1-output TE0, respectively. M denotes the number of wavelength and three wavelengths over an operating bandwidth of 80 nm are taken into consideration in simulations. Actually, the second and third terms in the right of Eq. (1) are used to optimize average excess losses and crosstalks for both modes, respectively. α, a weighted coefficient over a range from 0 to 1, is utilized to achieve a tradeoff between excess losses and crosstalks. For an ideal dual-mode 3 dB power splitter, the FOM is 1.Considering the axisymmetric topology constraint, only the upper half of the pattern need to be optimized in each round of optimization based on the DBS algorithm. When the FOM exhibits no great improvement (< 1% for our case), the optimization process terminates. We first set α = 0 to optimize the EL performance separately. Using the optimized pattern under α = 0 as the new initial pattern, we set α = 0.25 and then re-optimize the device based on the same algorithm to get the final optimized pattern which could reach a compromise between ELs and XTs. The inverse design takes about 28 hours on a computer with an 8-core central processing unit (Intel Xeon E5-2637).
As illustrated in Figs. 3(a) and 3(b), the working mechanism of the inverse-designed subwavelength device is equivalent to that of the conventional waveguide-based 3 dB power splitter in Fig. 1(a), while the footprint of the subwavelength scheme is reduced by about two orders of magnitude. The optical field evolution of both TE0 and TE1 in the optimized nanostructures can be also divided into three stages. In the stage I, the nanostructured DMC converts TE0 to an IFM and splits TE1 into two anti-phase IFMs, respectively. In the stage II, the IFM is further split into two in-phase IFMs by a nanostructured Y-junction while the anti-phase IFMs are further split into two pair of anti-phase IFMs via two axisymmetric nanostructured Y-junctions and two π phase shifters. In the stage III, these six IFMs finally synthesize the two output normal modes in each output waveguide via the mirrored nanostructured DMC. Moreover, the axisymmetric topology constraint of the nanostructures could ensure the optical field distribution to be absolutely axisymmetric. A strict 50: 50 power splitter for both TE0 and TE1 can be obtained. Thus, an ultracompact broadband 3 dB splitter for both modes is achieved in this way. The simulated performance is shown in Figs. 3(c) and 3(d). The ELs of both TE0 and TE1 are less than 0.9 dB. The XTs for TE0 and TE1 at output port are lower than −20 dB and −24 dB over the wavelength range from 1520 to 1580 nm, respectively.
Fig. 3 (a) and (b) Simulated optical field evolutions of Hz for TE0 and TE1, respectively. (c) and (d) Simulated transmission spectra for the MUX and the MDM system, respectively.
Generally, optimization algorithms for inverse design are sensitive to the initial pattern, and the optimized patterns usually seem like irregular and random [17, 18]. Here, under the axisymmetric topology constraint, we simulate and characterize several inverse-designed optimization patterns with random-generated and different initial patterns. The random initial patterns are generated by MATLAB. First, we use the rand function to obtain a matrix with uniformly randomly distributed values from 0 to 1. Then we convert it to a binary matrix: if the matrix element value is less than 0.5, it will be set to 0 (The corresponding occupied material is air). Otherwise, it will be set to 1 (The corresponding occupied material is silicon). The initial patterns and corresponding optimized patterns are shown in Figs. 4(a)-4(b), Figs. 4(e)-4(f) and Figs. 4(i)-4(j), respectively. The out-of-plane magnetic fields (Hz) at 1560 nm calculated for both TE0 and TE1 transmitted through the device are illustrated in Figs. 4(c)-4(d), Figs. 4(g)-4(h) and Figs. 4(k)-4(l), respectively. The FOMs for all the patterns after every iteration are given in Fig. 4(m). The corresponding transmission spectra for TE0 and TE1 of the three optimized devices are shown in Figs. 4(n) and 4(o), respectively. We select the pattern with best performance as the final pattern of the subwavelength dual-mode 3 dB power splitter. For all inverse-designed devices based on different random initial patterns, we can find that the optimization patterns vary widely and their performance are also different. But it is very interesting that the optical field evolution of the two modes in all optimized patterns is almost identical and equivalent to the mode evolution of the conventional waveguide model in Fig. 1(a).
Fig. 4 (a)−(d), (e)−(h) and (i)−(l) The initial and optimized pattern pictures and the corresponding simulated optical field evolutions of Hz for TE0 and TE1 for different random initial patterns, respectively. (m) The calculated FOMs after every iteration for different random initial patterns. (n) and (o) The corresponding simulated ELs for TE0 and TE1 for different random initial patterns, respectively.
We also simulated the mode evolution in inverse-designed subwavelength dual-mode 3 dB power splitters with different footprints of 2.88 × 2.16 μm2, 2.88 × 2.88 μm2 and 2.88 × 3.6 μm2. The initial patterns are also random. The optimized patterns are shown in Figs. 5(a)-5(b), Figs. 5(e)-5(f) and Figs. 5(i)-5(j), respectively. Besides, the corresponding optical field evolution of Hz for TE0 and TE1 are illustrated in Figs. 5(c)-5(d), Figs. 5(g)-5(h) and Figs. 5(k)-5(l), respectively. As expected, the working mechanisms of all the ultracompact power splitter with different footprints also converge to the proposed equivalent model based on subwavelength structure DMC. The calculated FOMs for all the patterns after every iteration are also illustrated in Fig. 5(m). The corresponding transmission spectra for TE0 and TE1 of the three optimized devices with different footprints are shown in Figs. 5(n) and 5(o), respectively. The simulation results indicate that the optimized devices exhibits better performance as the length of device increases. However, there is a tradeoff between the calculation time and device footprint. As a result, the footprint of the device is finally chosen as 2.88 × 2.88 μm2, which may be the most compact footprint for a dual-mode 3 dB power splitter to the best of our knowledge.
Fig. 5 (a)−(d), (e)−(h) and (i)−(l) The initial and optimized pattern pictures and the corresponding simulated optical field evolutions of Hz for TE0 and TE1 for the different footprints of 2.88 × 2.16 μm2, 2.88 × 2.88 μm2 and 2.88 × 3.6 μm2, respectively. (m) The calculated FOMs after every iteration for the different footprints. (n) and (o) The corresponding simulated ELs for TE0 and TE1 for the different footprints, respectively.
3. Experiment results
We fabricated and experimentally demonstrated the ultracompact dual-mode 3 dB power splitter based on subwavelength structure DMC. Firstly, the optimized nanopattern was formed on SOI platform with a 220 nm-thick top silicon layer using electron-beam lithography (EBL) system (Vistec EBPG 5000 Plus). Then an inductively coupled plasma (ICP) etcher (Plasma lab System100) was utilized to transfer the mask to the silicon device layer. To generate and characterize the TE0 or TE1 signals separately, subwavelength (DE) MUXs were also designed and fabricated [28]. Besides, a reference MDM system was also fabricated on the same chip to evaluate the performance of the dual-mode 3 dB power splitter.
Figure 6(a) shows the top-view scanning electron microscope (SEM) picture of the fabricated test system composed of a dual-mode 3 dB power splitter, a MUX and two DEMUXs. The reference MDM system is shown in Fig. 6(d). Figures. 6(c) and 6(d) illustrate the detailed SEM images of the fabricated dual-mode 3 dB power splitter and DEMUX, respectively. Here, a broad amplified spontaneous emission (ASE) light source and an optical spectrum analyzer (Yokogawa AQ6370C-20) were utilized to measure the transmission spectrum of the fabricated device. From the spectral transmission scans for each combination of input and output ports, the performance of the reference MDM system was characterized and presented in Fig. 6(e), in which, for example, “1-2” denotes the spectral transmission from input port 1 (I1) to output port 2 (O2). TE0 and TE1 are excited when the light is launched in input port I3 and I4, respectively. The measured insertion losses and XTs were less than 1.5 dB and lower than −24 dB for both TE0 and TE1 from 1.52 μm to 1.6 μm. Similarly, the transmission spectra of the fabricated dual-mode 3 dB power splitter were characterized and obtained, as shown in Figs. 6(f)- 6(h). The measured average ELs and XTs (normalized to the referenced MDM system) for both modes were less than 1.5 dB and −20 dB over a wavelength range from 1520 to 1580 nm, respectively. The ELs for the fabricated devices were measured by three times, as shown in Figs. 6(f) and 6(g). Shaded areas indicate minimum and maximum measured values across the fabricated devices, and solid lines indicate the average values. The measured average standard deviations for the transmission spectrum of ‘1-2’, ‘1-3′, ‘2-1’ and ‘2-4’ are 0.32 dB, 0.16 dB, 0.21dB and 0.15 dB, respectively. Moreover, the measured average EL imbalances for TE0 and TE1 were 0.08 dB and 0.07 dB, respectively. The measured results indicate that the fabricated nanostructured device has an approximate 50:50 power splitter for both modes from 1520 to 1580 nm. The consistent experimental performance with the simulated results exhibited that the PhC-like nanostructured devices were robust to fabrication errors.
Fig. 6 (a) SEM image for the entire fabricated device composed of a dual-mode 3 dB power splitter and three (DE) MUXs. (b) and (c) The detailed SEM images for dual-mode 3 dB power splitter and DEMUX. (d) SEM picture for the reference MDM system. (e) The normalized measured transmission spectra for the fabricated reference MDM system. (f) and (g) The measured ELs for both modes for the fabricated 3 dB power splitter, respectively. Shaded areas indicate minimum and maximum measured values across the fabricated devices, and solid lines indicate the average values. (h) The measured XTs for both modes for the fabricated 3 dB power splitter.
4. Conclusion
We propose and experimentally demonstrate an ultracompact broadband dual-mode 3 dB power splitter based on subwavelength structure DMC using the inverse design method. Numerical simulations indicate that the inverse-designed nanostructures could enable one to flexibly manipulate modal optical field evolution and implement ultracompact and high performance dual-mode 3 dB power splitter. The axisymmetric topology constraint enables the nanostructures with random-generated initial patterns or different footprints usually converge to our proposed equivalent conventional waveguide model and realize a strict 50: 50 splitting ratio for both modes. The fabricated device exhibits high performance with average ELs less than 1.5 dB and XTs lower than ‒20 dB from 1.52 to 1.58 μm for both modes. Meanwhile, an approximate 50:50 power splitter for both modes from 1.52 to 1.58 μm was achieved, which was consistent with the numerical simulations. The footprint of the fabricated device was only 2.88 × 2.88 µm2, which is two orders of magnitude smaller than that of conventional one. The ultracompact broadband dual-mode 3 dB power splitter will show greatly potential application in densely integrated photonic MDM systems for on-chip optical interconnect.
Funding
National Natural Science Foundation of China (61775069); Fundamental Research Funds for the Central Universities, Huazhong University of Science and Technology (HUST) (2017KFXKJC002).
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