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Abstract
We designed Si-based all-dielectric 1 × 2 TE and TM power splitters with various splitting ratios by combining the use of the inverse design of adjoint and numerical 3D finite-difference time-domain methods. The structure of the designed Si-based power splitters contains two Si waveguide branches on a SiO2 substrate that is compatible with CMOS fabrication technology. The proposed devices exhibit ultra-high transmission efficiency above 98 and 99%, and excess losses below 0.1 and 0.035 dB, for TE and TM splitters, respectively. The merits of these devices include a minor footprint of 2.2 × 2.2 µm2 and a flat-broad operating bandwidth of 200 nm with a center wavelength of λ = 1.55 µm. Also, the other advantage of these optical power splitters is the very short optimization time of 2 h for each device. Because of the aforementioned merits, the optimized devices can be crucial candidates for optical integrated circuits.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
All-dielectric metamaterials (MMs) are sub-wavelength dielectric-artificial structures to confine, steer and manipulate properties of incident light in a fraction of wavelength. All-dielectric MMs with high refractive index elements are of great interest for giving additional degrees of freedom in designing shrank optical devices with minimum energy dissipation [1–5]. Among the best candidates for the high refractive index elements is silicon, with its prominent characteristics including a high refractive index (nearly 3.48) in a 1.55 µm communication window, high refractive index contrast between Si and SiO2, and compatibility with CMOS fabrication technology. Another feature of Si is its capability to steer and confine incident light in the λ/n subwavelength, where λ and n denote the wavelength of incident light and refractive index, respectively; these jointly contribute to the application of silicon in all-dielectric MMs. Owing to the benefits of Si-based MMs, numerous optical devices based on MMs have been designed and fabricated to date [6–8].
For designing an optical component, numerous structural simulations are required to find an optimized structure, which enforce high computational costs and time-consuming simulations. To overcome these problems, and design efficient and compact structures, researchers have developed inverse design methods, including artificial intelligence and optimizations. Artificial intelligence contained two subcategories of machine learning (ML) and deep learning (DL). Also, DL is the subcategory of ML that is based on neural networks. ML and DL have a wide variety range of applications in medical imaging [9,10], signal processing [11,12], face recognition [13,14], and so forth. In the past few years, DL, as a strong inverse design algorithm, has been vastly used in designing all-dielectric MMs [15], optical scattering units [16], nanoparticles [17,18], multilayers [19,20], metasurfaces [21–24], and chiral MMs [25].
In 2018, Liu et al. proposed an inverse design of multilayers structure containing SiO2 and Si3N4 materials. In their work layers’ thicknesses and the transmission spectrum were considered as variables for the data set. Their data set contained 750,000 and 5,000 instances for the train and test data of the deep neural network algorithm. Their algorithm showed an average error of 16.0○ for phase delay of the structure [19]. Also, in 2018, Liu et al. used a generative deep neural network model for the inverse design of metasurfaces. The metasurfaces contained a layer of patterned gold on a glass substrate. Their data set contained 6,500 instances. Their results showed an average accuracy of about 90% [22]. In 2019, Tahersima et al. used a deep neural network for the inverse design of integrated power splitters with different splitting ratios. The structure of their power splitters contained etched air holes in a silicon slab on a SiO2 substrate. For the power splitters, they have used 16,000 and 4,000 data sets for training and test instances respectively for the deep neural network algorithm. Their results showed maximum transmission efficiency of 90% [15]. In 2020, the inverse design of plasmonic structure using a convolutional neural network was proposed by Lin et al. The unit cell of their proposed structure contained three gold nanodisks on a substrate. The output and input data respectively are the structural parameters of nanodisks and the absorption spectrum of the metasurface. The training dataset contained 25,000 instances of unit cell simulations. Their inverse design algorithm showed the accuracy of ±8 nm for the structural parameters [21].
Currently, inverse design based on optimization has attracted attentions as an efficient and fast method. Furthermore, a multitude of optical devices has been simulated and fabricated according to these methods, among which are optical power splitters (OPSs) [15,26–31], wavelength demultiplexers [32–34], polarization splitters [35,36], and couplers [32]. There are two major optimization methods for optical design, namely adjoint [32–34,37] and direct binary search [28,29,35]. The major differences are simulation time and efficiency of the optimized devices. The optimization process time and efficiency of the adjoint method are lower and higher, respectively, compared to the direct binary search method. Direct binary search optimizes structures, which can be modeled as zero and one states. In contrast, the adjoint method is capable of being applied to design myriads of structures, including binary [27] and other structures [32–34]. In this study, TE and TM OPSs with various splitting ratios of 50:50, 60:40, 70:30, and 80:20 are designed and simulated using the inverse design of the adjoint method. The adjoint method utilizes Python embedded in the Lumerical FDTD module for inverse design of the OPSs. The inverse adjoint-designed OPSs exhibit great application potential owing to their small footprint, high transmission efficiency, broad bandwidth, and low computational time requirement.
2. Theory and design
Designing an optical integrated circuit (OIC) requires a multitude of basic components such as OPSs, polarization demultiplexers, filters, and wavelength demultiplexers. Although OPSs are basic elements in OICs, those with arbitrary splitting ratios are applicable in signal processing, optical equalization, and feedback circuits [38]. Here, OPSs have been proposed and designed using an efficient and powerful adjoint method, which incorporates a gradient descent algorithm for optimization. In this method, first proposed by Miller to design solar cell structures [39], the structure shape is altered to obtain the desired outputs. It contains three stages, namely, grayscale, binarization, and design for manufacturing [32]. In the grayscale phase, the structure's permittivity is varied between two predefined permittivity values, ɛmax and ɛmin. The binarization stage uses the Heaviside function to change these values to either ɛmax or ɛmin. In the design for the manufacturing process, fabrication constraints such as minimum shape curvature are applied. This method also requires an initial structure region for optimization; to this end, a simple structure has been defined in this study for the OPSs according to Fig. 1. The desired output or figure of merit (FOM) of this method is defined as the integral of the objective function f at each point of $\mathbf{r^{\prime}}$ [27,37,39]:
Based on Eq. (3), for each iteration, two simulations of the forward and adjoint are necessary. As a result, to find the optimal point of the FOM, a gradient-descent algorithm is utilized.
3. Simulation results and discussions
In this section, both 1 × 2 OPSs with TE00 and TM00 modes, respectively, with their electrical fields in y- and z- directions and various splitting ratios are discussed. The prototype structure for both TE and TM OPSs is shown in Fig. 1. The figure shows 1 × 2 OPSs with one input and two output Si waveguide ports with a refractive index of 3.48 and 0.4 µm waveguide width. The footprint of the optimization region is 2.2 × 2.2 µm2 with a thickness of 0.4 µm. The material includes two branches of Si, and the wavelength for the devices is in the 1450–1650 nm range (200 nm bandwidth). The structural parameters of L, W, G, H, and T have been chosen as 2.2, 0.4, 1.2, 3, and 0.4 µm, respectively.
For optimization processes and simulations of the structure, the 3D FDTD module of Lumerical software was employed. Furthermore, the Python embedded in the FDTD module of the software has been applied in the optimization process. The mesh sizes in x-, y-, and z- directions for 3D FDTD simulations have been set to dx = 20 nm, dy = 20 nm, and dz = 50 nm, respectively.
The simulation results of the optimized TE and TM OPSs with splitting ratios of 50:50, 60:40, 70:30, and 80:20 are shown in Figs. 2 and 3, correspondingly. In the following figures, UB and LB stand for upper and lower branches, respectively. The simulation time for optimizing the OPSs is roughly 2 h per device with the computational resource being a PC with a 3 GHz Core-i9 CPU and 128 GB RAM.
Fig. 2. Simulation results for the optimized 1 × 2 TE OPSs. Columns (a)–(d) show the devices with splitting ratios of 50:50, 60:40, 70:30, and 80:20, respectively. The rows (up to down) depict the refractive index profile, transmission spectrum, and the normalized power distribution at λ = 1.55 µm, respectively.
Fig. 3. Simulation results for the optimized 1 × 2 TM OPSs. Columns (a)–(d) show the devices with splitting ratios of 50:50, 60:40, 70:30, and 80:20, respectively. The rows (up to down) depict the refractive index profile, transmission spectrum, and the normalized power distribution at λ = 1.55 µm, respectively.
The total transmission efficiency at both output ports is defined as (T1 + T2), where T1 and T2 denote the transmissions at each output port. As illustrated in Fig. 2, the efficiencies are 98.8, 99.6, 98.23, and 99.11% for TE OPSs with splitting ratios of 50:50, 60:40, 70:30, and 80:20, respectively. Figure 2 indicates that based on the boundary conditions of Maxwell equations, the narrow gap between the output branches in TE mode structures results in discontinuity of the electrical field and subsequently causes electrical field amplification in the gaps. According to Fig. 3, the overall transmission efficiency for TM OPSs with splitting ratios of 50:50, 60:40, 70:30, and 80:20 are 99, 99.59, 99.65, and 99.38%, respectively.
Figure 4 shows the excess loss parameter for the proposed TE and TM OPSs; it is evident that the excess loss is below 0.1 dB for TE and 0.035 dB for TM OPSs. Based on the simulations, the optimized designed structures have ultra-high transmission efficiency, a simple and small structure, fabrication feasibility, and a very short optimization time. In general, three factors aid in the determination of the optimization time, namely, the structure size, proper selection of initial structure for optimization, and selection of an efficient algorithm. The physical phenomenon behind these structures is that the optimized region, which is a Y splitter, acts as a coupler including one input and two output waveguides. The incident light with the specific guided mode is coupled to the output waveguides; the portion of coupled optical waves depends on the geometry of output branches; i.e., based on Snell’s law, the incident light from the input waveguide is transmitted and reflected in the output waveguides, which in turn depends on the topology of output branches, which realizes different splitting ratios. Notably, the guided waves are transmitted through the devices based on total internal reflections. In comparison to the other studies [15,26–32], the proposed OPSs exhibit the highest transmission efficiency (> 98%), the shortest simulation time, and the smallest footprint (2.2 × 2.2 µm2). Further, the proposed OPSs are broadband (200 nm) with a very flat bandwidth spectrum. Moreover, the proposed devices have been designed with different splitting ratios of 50:50, 60:40, 70:30, and 80:20 in either TE00 or TM00 modes.
Fig. 4. Excess loss for (a) TE, and (b) TM OPSs concerning wavelength at splitting ratios of 50:50, 60:40, 70:30, and 80:20.
Figure 5 represents the simulation results of the performance of the 1 × 2 TE and TM OPSs under fabrication imperfections. Owing to the nature of the electric field that is out of the plane in TM OPSs, the fabrication imperfection of ${\pm} 10$ nm does not affect the transmission of the devices. 1 × 2 TE OPSs reveal higher dependence on the fabrication imperfections of ${\pm} 10$ nm.
Fig. 5. Simulation results of fabrication imperfections for the optimized 1 × 2 OPSs. (a)–(d) show the TE OPSs with splitting ratios of 50:50, 60:40, 70:30, and 80:20, respectively. (a)–(d) show the TM OPSs with splitting ratios of 50:50, 60:40, 70:30, and 80:20, respectively.
4. Conclusion
We designed and simulated 1 × 2 TE and TM OPSs with different splitting ratios using the adjoint method. The optimized TE and TM OPSs with splitting ratios of 50:50, 60:40, 70:30, and 80:20 yield high efficiency (> 98% for TE00 and > 99% for TM00 modes), low excess loss (below 0.035 and 0.1 dB for TM00 and TE00 modes, respectively), a minuscule footprint (2.2 × 2.2 µm2), very short optimization time (2 h per each OPS), broad and flat bandwidth (200 nm), operation in either TE00 or TM00 modes, and simple optimized structures. These silicon-on-insulator devices are compatible with CMOS fabrication technology. The adjoint optimization method and 3D FDTD module of Lumerical software have been adopted for designing and simulating the OPSs. Overall, the abovementioned features present the proposed devices as excellent candidates for OICs. Therefore, these optimized devices will be applicable in the mass production of high-density, high-efficiency, and high-speed OICs in the future.
Funding
Samsung (IO201210-08035-01).
Acknowledgments
This work was supported by Samsung Electronics Co., Ltd (IO201210-08035-01) and LG Innotek Co., Ltd.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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