Open Access
Add to BibSonomy
Abstract
Nanophotonic devices, which consist of multiple cell structures of the same size, are easy to manufacture. To avoid the optical proximity effect in the ultraviolet lithography process, the cell structures must be maintained at a distance from one another. In the inverse design process, the distance is maintained by limiting the optimized range of the location. However, this implementation can weaken the performance of the devices designed during transmission. To solve this problem, a self-adjusting inverse design method based on the adjoint variable method is developed. By introducing artificial potential field method, the location of one cell structure is modified only when the distances between this cell structure and other cell structures are smaller than a threshold. In this case, the range of the location can be expanded, and thus the performance of the designed devices can be improved. A wavelength demultiplexer with a channel spacing of 1.6 nm is designed to verify the performance of the proposed method. The experiment reveals that the transmission of the designed devices can be improved by 20%, and the self-adjusting inverse design process is 100 times faster than the inverse-design process based on the genetic algorithm.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In the field of nanophotonics, the inverse design method has achieved remarkable results in designing nanophotonics devices [1–3]. Compared with other devices designed with this method, devices consisting of multiple cell structures, such as round perforations on a dielectric slab, are easier to manufacture, with ultraviolet (UV) lithography as the common manufacturing technique. The designation of these devices can be realized with genetic algorithm (GA) [4], direct binary search (DBS) method [5,6], and adjoint-based inverse design method [2,7] which is also referred to as the inverse design method based on the adjoint variable method [8–10].
However, an inverse design process based on the GA or DBS method is significantly slower than the adjoint-based inverse design process [4,11–14]. In addition, the design time greatly increases for larger the design areas. Therefore, the design of large-scale devices based on GA or DBS method in a limited period is difficult.
With the adjoint-based inverse design method, the inverse design process can be accelerated and larger devices can be designed. To ensure that the designed device can be manufactured with UV lithography, each cell structure must be maintained at a certain distance from the other to avoid the optical proximity effect [15–17]. To this end, a limitation is applied to the parameter range of location [18]. However, this implementation degrades the performance of the designed devices.
To improve the performance of the designed nanophotonic devices, a self-adjusting inverse design method based on the artificial potential field method is developed. The artificial potential field method is generally used to prevent overlapping in motion planning [19]. With the artificial potential field method, the locations of the cell structures are adjusted to maintain a safe distance only when the distance between different cell structures is smaller than a threshold. Consequently, the range of the location of the cell structures can be expanded, and the performance of the designed nanophotonic designed devices can be improved.
In the self-adjusting inverse design method, the design area is also considered a parameter to accelerate the self-adjusting inverse design process. The traditional inverse-design process is repeated multiple times to obtain high-performance devices and minimize the footprint simultaneously. This reduces the efficiency of the design process. By adjusting the design area in the inverse design process, the footprint of the devices can be minimized in one inverse design process.
In the proposed method, the depth or height of each cell structure is optimized along with the location and binarized to ensure that only one etch depth is required in the manufacturing process. Compared with the level-set method [20], the filtering-threshold scheme [21] is computationally inexpensive and easier to be implemented, maintaining the binarization of optimized parameters in the inverse design process [2]. Therefore, the depth or height is binarized with the filtering-threshold scheme in this study.
The inverse design process is composed of an optimization process and a simulation process [1,2]. The finite-difference time-domain (FDTD) method and the finite-difference frequency-domain method (FDFD) can be used to execute the simulation process [22–24]. The FDFD method is faster, more efficient than the FDTD method, with less memory consumption [22]. Therefore, the electromagnetic simulation of the experiment in this study is based on the FDFD method.
The wavelength demultiplexer, also known as the wavelength router [3], widely used in optical communication and optical computing, is difficult to design through the inverse design process when the channel spacing is narrower than 40 nm [25,26]. To evaluate the performance of the developed self-adjusting inverse design method, a wavelength demultiplexer with a channel spacing of 1.6 nm is designed. The initial structure affected the performance of the designed wavelength demultiplexer considerably. To improve the performance of the designed wavelength demultiplexer, the structures of multi-one-dimensional (1D) conjugate topological photonic crystals [1,27,28] with different Zak phases are introduced as the initial structure. The conjugate topological photonic crystal was based on a dielectric slab that had multiple perforations of the same size.
The remainder of this paper is organized as follows. Section 2 describes the basic theory and principle of the self-adjusting inverse design method. Section 3.1 presents the initial structure of the designation. Sections 3.2 and 3.3 demonstrate the performance of the designed nanophotonics demultiplexers. In Section 3.4, the comparison of convergence between different optimization processes is described, and the limitation of the self-adjusting inverse design method is analyzed.
2. Theory and working principle
2.1 Adjoint-based inverse design method
The adjoint-based inverse design process is based on a simulation system and an adjoint simulation system. The simulation system based on the FDFD method can be written as
where Ce and Ch represent the matrices for the curl operator on the electric and magnetic fields, respectively; µ and ε(p) are permeability and permittivity, respectively; p is the optimized parameter in the inverse design process; E represents the distribution of the electric field; ω represents the frequency; and J corresponds to the electric current source densities. This study considers only nonmagnetic materials. Hence, µ = 1.To simplify the illustration, Eq. (1) can be written as follows:
whereTo compute the gradient in the inverse design method, an adjoint simulation system is constructed, which is in the form of
where $\bar{E}$ and ba are the adjoint field and adjoint source, respectively. The construction of the adjoint source ba is based on the figure-of-merit (FOM) function F(E) and E. where F(E) represents the expected performance of the designed device.Equations (2) and (4) can be used to execute the adjoint-based inverse design process. In the experiment, the adjoint-based inverse design process is based on a limited memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) method.
The schematic of the process is illustrated in Fig. 1. A detailed description of the adjoint-based inverse design process is as follows.
- 1) The simulation system, the adjoint simulation system and the optimized parameter p are initialized.
- 2) The simulation system E, expressed in Eq. (2), is solved and |1-F(E)| is calculated.
- 3) If |1-F(E)| is smaller than the convergence threshold (CT), we can obtain the optimized parameter p and use ε(p) to represent the designed device that satisfies our requirements. Otherwise, the adjoint field $\bar{E}$ is calculated based on the adjoint simulation system expressed in Eq. (4).
- 4) With the adjoint field $\bar{E}$ and electric field E, the gradient in the optimization process can be calculated.$$\begin{array}{l} \frac{{\partial F(E )}}{{\partial p}} = \frac{{\partial F(E )}}{{\partial E}}\frac{{\partial E}}{{\partial p}} = \frac{{\partial F(E )}}{{\partial E}}\left[ { - A{{(p )}^{ - 1}}\frac{{\partial A(p )}}{{\partial p}}E} \right]\\ ={-} \left[ {\frac{{\partial F(E )}}{{\partial E}}A{{(p )}^{ - 1}}} \right]\frac{{\partial A(p )}}{{\partial p}}E\\ ={-} {[{{{({A{{(p )}^T}} )}^{ - 1}}{b_a}} ]^T}\frac{{\partial A(p )}}{{\partial p}}E\\ ={-} {{\bar{E}}^T}\frac{{\partial A(p )}}{{\partial p}}E\\ = {{\bar{E}}^T}{\omega ^2}\frac{{\partial \varepsilon (p )}}{{\partial p}}E .\end{array}$$
Steps 2) to 5) are repeated until the designed device satisfies the experimental requirements.
2.2 Self-adjusting inverse design method
In the self-adjusting inverse design method, the optimized parameter p contains the locations of each perforation xqj,k and yqj,k, the depth of each perforation hqj,k, and the design areas wq1, wq2, wq3, and wq4, where j and k are corresponding to the row and column of the perforation in the design space; q represents number of iterations in the inverse design process. To simplify the illustration, a nanophotonic device with two channels was designed using the proposed method (Fig. 2). A mode source was set on the left, and the two ports on the right were set as the output ports. We assume that the design area contains m×n holes. In the inverse design process, the computations of F(E), adjoint source ba, and gradient ${{\partial \varepsilon } / {\partial p}}$ are different for different devices with different optimized parameter p. Therefore, the computations of F(E), ba and ${{\partial \varepsilon } / {\partial p}}$ in the self-adjusting inverse design process are described, respectively.
The FOM is related to the transmission in the two channels, and it can be written as follows:
Based on the F(E) expressed in Eq. (8), the adjoint source ba for the adjoint simulation system can be obtained as
whereThe adjoint field $\bar{E}$ can be obtained by solving Eq. (4).
Generally, ${{\partial \varepsilon } / {\partial p}}$ is obtained from the numerical gradient, and it can be written as follows:
In the self-adjusting inverse design method, ${{\partial \varepsilon } / {\partial p}}$ must be calculated for different optimized parameters. In this section, the optimized parameters p1, p2, and p3 are used to represent the location and depth of each perforation and the design area, while ${{\partial \varepsilon } / {\partial p}}1$, ${{\partial \varepsilon } / {\partial p}}2$, and ${{\partial \varepsilon } / {\partial p}}3$ are computed separately.
To avoid the optical proximity effect in the ultraviolet lithography process, an artificial potential field function is introduced, which can be written as follows:
The spatial distribution of the artificial potential field is depicted in Fig. 3(b). We assume the location of one perforation as the original target (xq-1j,k+dxj,k, yq-1j,k+dyj,k) after updating based on Eq. (7), xq-1j,k and yq-1j,k are the locations of each perforation since the previous update, and dxj,k and dyj,k are the offset of each perforation’s locations in the x- and y- directions. When the original target is within the range of the artificial potential field, the perforation will move from the original target to the actual one by adding axj,k and ayj,k to xq-1j,k+dxj,k and yq-1j,k+dyj,k. Consequently, each perforation can maintain a distance with the rest, and the optimized parameter range can be expanded, as illustrated in Fig. 3(c). As etching the boundaries of the nanophotonic devices was difficult, an artificial potential field was also set on the boundary.
Fig. 3. (a) Threshold distance between different perforations. (b) Artificial potential field method to maintain distances between different perforations. (c) Scheme of maintaining the distance.
As axj,k and ayj,k constitute the artificial potential field of one perforation, the total artificial potential fields required to maintain distances between multiple perforations were calculated using
The function for the location with the artificial potential field method can be written as follows:
To ensure the ease of fabrication of the devices, the location of each cell structure was discretized on a 5-nm mesh (Fig. 4). The discretized process is based on a filtering-threshold scheme, which is as follows:
Fig. 4. (a) M(x) under different β. (b) ∂M(x)/∂x under different β. (c) Discretized mesh to which the location is applied.
The derivatives ∂ε/∂p1 can be represented by
The depth of each perforation p2 must be binarized to ensure only one etch depth is required in the manufacturing process. In this study, the binarization was realized with a filtering-threshold scheme, which can be written as follows:
To calculate the gradient for depth with the filtering-threshold function, ∂ε/∂p2 can be rewritten as follows:
The gradient for depth can be calculated by substituting ∂ε/∂p2 into Eq. (6).
In the self-adjusting inverse design process, four parameters wq1, wq2, wq3, and wq4 were used to describe the design area (Fig. 6). The calculation of the gradient for the four parameters is based on Eqs. (12) and (6). A new perforation will be generated as the size of the designed device increases.
The total gradient in the self-adjusting inverse process can be written as follows:
By substituting Eq. (22) into Eq. (6), the gradient in the self-adjusting inverse design process can be obtained. By calculating the gradient and the FOM function with the optimized parameter p, the self-adjusting inverse design process can be realized.
3. Numerical experiment
In this section, all the computations were performed on a computer with four graphics processing units (GPUs; NVIDIA TITAN V 12 G) and two central processing units (CPUs; Intel Xeon Platinum 8280). Software was based on the Python platform with CUDA 10.0 package.
The research objects were wavelength demultiplexers. To accelerate the simulation process, a biconjugate gradient-stabilized (BICGSTAB) solver was used as the system solver.
The initial structure in the inverse design process significantly impedes the performance of the designed devices. To augment the performance, the structure of a 1D topological photonic crystal was adopted as the initial structure.
3.1 Initial structure of the designation
One-dimensional topological photonic crystal is used as a high-Q filter and is highly sensitive to different wavelengths. The performance of the designed devices in the channel space can be improved with this crystal as the initial structure of the designation. For this crystal, the lattice constant, diameter of each perforation, and period number are the undetermined variables.
In this study, we considered the UV lithography process with 45-nm fabrication accuracy [4–6]. To maintain accuracy, each perforation in this section was drilled to a diameter of 90 nm. The lattice constant was calculated from the transmission peak. As the period number was related to the channel space of the devices, the channel space was considered while setting its value.
The 1D crystal is the combination of two types of photonic crystals with different Zak phases [26–28]. The general equation for calculating a Zak phase is as follows:
A 1D conjugate topological photonic crystal is a special case in which two types of photonic crystals have the same structure, but different starting points in the integration [29,30]. We focus on this type of crystal owing to its higher transmission performance at high period numbers.
To calculate the lattice constant, we designed a 1D topological photonic crystal with a transmission peak of approximately 1,550 nm. In this experiment, when the lattice constant was 301 nm, the transmission peak for the topological photonic crystal was approximately 1,550 nm. To analyze the Zak phase and spatial mode field of the crystal, the FDFD and plane-wave expansion (PWE) methods were used. As illustrated in Fig. 7, the energy band with the FDFD method matches that with the PWE method. The subtle difference between the two energy bands is from the difference between the dielectric constants produced by the Fourier transform in the PWE method.
Fig. 7. Schematic of the one-dimensional (1D) conjugate topological photonic crystals. (a) Energy band for PC1. (b) Energy band for PC2.
Traditional conjugate topological photonic crystals must be centrosymmetric to ensure that the Zak phase is either 0 or π. However, this would require the fabrication of semicircular structures, which is difficult. Previous studies have shown that after the displacement of the starting point in the integration, the topological boundary states can also be excited [27]. To simplify the fabrication process, the integration range was shifted from PC1 and PC2 to PC3 and PC4 (Fig. 8).
The conjugate topological photonic crystals have a property that the half-peak width reduces as the period number N increases. This characteristic can be used to determine the period number from the channel spacing of the wavelength demultiplexer. To better exploit this feature, the half-peak widths of photonic crystals under different period numbers were calculated (Fig. 9). We emphasize that the devices in this simulation have only one perforation in the y-direction, whereas the topological photonic crystals should have infinite perforations in this direction. The reason is that the devices can have a limited number of perforations in the real space, and one perforation in the y-direction is the lower limit. As the photonic crystal in this simulation was not a standard conjugate topological photonic crystal, the transmission peak moved as the period number increased.
Fig. 9. (a) Comparison of the photonic crystal devices. (b) Transmission of the photonic crystal devices with one perforation in the y-direction under different wavelengths.
3.2 Three-channel wavelength demultiplexer
In this section, a three-channel wavelength demultiplexer with a channel spacing of 1.6 nm is designed. The application of the wavelength demultiplexer is focused on the C-band, and its transmission peaks for the three channels were set to 1,546.8, 1,548.4, and 1,550 nm, respectively. The designation of the three-channel wavelength demultiplexer was based on a three-dimensional FDFD simulation, with the refractive index from the Palik database [31].
Because the channel spacing of the devices was 1.6 nm, the initial device contained a photonic crystal with a full width at half maximum of approximately 0.8 nm and a period number of 40. In the experiment, the devices performed better when the number of perforations in the x-direction was similar to that in the y-direction. In this case, 90 perforations were set in the y-direction of the initial devices. As the ports on boundaries generally exhibit lower transmission performance, two perforations were added to the boundary in the y-direction. Overall, the initial device of the wavelength demultiplexer contained 80 × 94 perforations, while the footprint was 24 µm × 18.8 µm. The incorporation of the self-adjusting inverse design process reduced the device footprint to 23 µm × 18 µm. The structure and field distribution of the wavelength demultiplexer are depicted in Fig. 10.
Fig. 10. Structure and field distribution of the designed wavelength demultiplexer. (a) Structure of wavelength demultiplexer. (b) Electric field intensities of the wavelength demultiplexer under transmission peaks of 1,546.8 nm. (c) 1,548.4 nm. (d) 1,550 nm.
To verify the simulation result of the wavelength demultiplexer, it was compared with the results of the commercial simulation software Lumerical FDTD module. As shown in Fig. 11, the simulation result of the wavelength demultiplexer with the FDTD simulation matches that obtained with the FDFD simulation. The differences in the transmission and transmission peaks were caused by two main reasons. Firstly, in the FDTD simulation, the Fourier transform process is introduced to obtain the electromagnetic field in the frequency domain. However, the Fourier transform process usually leads to uncertainty in peak amplitude because of scalloping loss or picket fence effect [32,33]. Secondly, since the implementation of the mesh refinement setting is unknown in Lumerical FDTD module, the distribution of the permittivity in x, y, and z directions with Lumerical FDTD module might be different from the FDFD simulation. Differences in the permittivity of the simulation system lead to differences in the simulation result.
Fig. 11. Transmission comparison of wavelength demultiplexer based on FDFD simulation and Lumerical FDTD module.
3.3 Four-channel wavelength demultiplexer
To verify the performance of the self-adjusting inverse design method for large-scale problems, two-dimensional simulation with an effective refractive index [34] was used in the design process. A four-channel wavelength demultiplexer with a channel spacing of 1.6 nm and transmission peaks of 1,546.8, 1,548.4, 1,550, and 1,551.5 nm was designed. The wavelength demultiplexer contained 80 × 124 perforations and had a footprint of 24 µm × 24.8 µm. The structure and field distribution of the wavelength demultiplexer are depicted in Fig. 12.
Fig. 12. Structure and field distribution of the designed wavelength demultiplexer. (a) Structure of wavelength demultiplexer. (b) Electric field intensities of the wavelength demultiplexer under transmission peaks of 1,546.8 nm. (c) 1,548.4 nm. (d) 1,550 nm. (e) 1,551.6 nm.
To verify the simulation result of the four-port wavelength demultiplexer, the simulation result with the FDFD simulation was compared with that of the Lumerical FDTD module. The simulation results of the FDFD simulation matched with that of the Lumerical FDTD module. The designation of the 4-channel demultiplexer is based on the two-dimensional simulation with an effective refractive index, which is calculated by using a waveguide mode eigenvalue solver, because designing a 4-channel demultiplexer with three-dimensional simulation requires more system resources than we can afford. The two-dimensional simulation results are not as accurate as three-dimensional simulation results but can be used as an approximation of the three-dimensional simulation result [34]. We can observe that there is a certain difference in transmission between FDFD simulation and Lumerical FDTD module in Fig. 13.
Fig. 13. Transmission comparison of wavelength demultiplexer based on FDFD simulation and Lumerical FDTD module.
3.4 Discussion
To verify the performance of the self-adjusting optimization process in terms of speed, a comparison of the time required in the design process between the GA and self-adjusting inverse design method was demonstrated. The simulation process was based on the effective refractive index method [3,34]. As illustrated in Fig. 14, the proposed method is 100 times faster than the method using GA. As a slab with more perforations would require a longer design time, designing devices with numerous perforations within a limited time would be difficult. A high-performance nanophotonic device, such as the wavelength demultiplexer, with a narrow channel spacing with limited perforations is difficult to develop. Therefore, the self-adjusting inverse design method can expand the range of applications. The devices designed with the self-adjusting inverse design method also performed significantly better than the one with the GA in terms of transmission, as shown in Table 1.
Fig. 14. Comparison of time required in the design process between genetic algorithm and self-adjusting inverse design method.
Table 1. Transmission of a two-port wavelength demultiplexer designed with different optimization processes
To verify the performance of the self-adjusting optimization process in terms of speed, a comparison of convergence speed between different optimization processes was demonstrated. The optimization process for depth and location was compared with the self-adjusting optimization process. To verify the effect of the artificial potential field, both the optimization processes for location with artificial potential field method and mesh limitation are depicted in Fig. 2. In the comparison, the experimental target was a two-port wavelength demultiplexer.
As shown in Fig. 15, the convergence speed of the proposed process is faster than the rest. Furthermore, in the location optimization with the artificial potential field, the convergence accelerates periodically. This phenomenon occurred because the search range was updated every ten steps in the optimization process.
To verify the design result, the transmission performance of a two-port wavelength demultiplexer designed with different optimization processes is summarized in Table 2. As observed in the table, the devices designed with the self-adjusting optimization process performed markedly better than the rest.
Table 2. Transmission of a two-port wavelength demultiplexer designed with different optimization processes
Based on the adjoint variable method and artificial potential field method, self-adjusting inverse design method is highly effective in designing wavelength demultiplexers with a defined distance between different perforations. However, the performance of the designed devices reduced as the distance between perforations increased (Table 3). The blockmin are related to the manufacturing technology. If the manufacturing technology is inferior, the blockmin is large and designing high performance devices with any design method is difficult.
Table 3. Transmission of the four ports wavelength demultiplexer designed with self-adjusting inverse design method using different artificial potential fields
4. Conclusions
In this study, a self-adjusting inverse design method was developed, in which the location, height of each cell structure, and design area were set as optimized parameters. With the artificial potential field method, each cell structure maintained a certain distance to avoid the optical proximity effect in the UV lithography process, if the distances between them were shorter than a threshold. The experimental results showed that using the self-adjusting inverse design method can improve the performance of the nanophotonic devices during transmission.
A wavelength demultiplexer was designed to verify the capability of the self-adjusting inverse design method. The self-adjusting inverse design method can also be used to design other nanophotonic devices, such as beam splitters, T-junctions, and optical hubs.
The self-adjusting inverse design method is developed to ensure that the designed devices can be manufactured with UV lithography and maintain high performance.
In addition to the performance enhancement, the self-adjusting inverse design method can be used to develop a mathematical model between transmission peak and location of the cell structure. This will be a viable direction of future research.
Funding
National Natural Science Foundation of China (61935003).
Acknowledgments
This research was supported by the National Natural Science Foundation of China.
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.
References
1. S. Molesky, Z. Lin, A. Y. Piggott, W. Jin, J. Vucković, and A. W. Rodriguez, “Inverse design in nanophotonics,” Nat. Photonics 12(11), 659–670 (2018). [CrossRef]
2. J. Wang, Y. Shi, T. Hughes, Z. Zhao, and S. Fan, “Adjoint-based optimization of active nanophotonic devices,” Opt. Express 26(3), 3236–3248 (2018). [CrossRef]
3. ZL Dong, JF Qiu, YC Chen, LH Wang, HX Guo, and J Wu, “Ultracompact and ultralow-loss S-bends with easy fabrication by numerical optimization,” Opt. Lett. 47(10), 2434–2437 (2022). [CrossRef]
4. Z. Liu, X. Liu, Z. Xiao, C. Lu, H.Q. Wang, Y. Wu, X. Hu, Y. C. Liu, H. Zhang, and X. Zhang, “Integrated nanophotonic wavelength router based on an intelligent algorithm,” Optica 6(10), 1367 (2019). [CrossRef]
5. L. Lu, D. Liu, and F. Zhou, “Inverse-designed single-step-etched colorless 3 dB couplers based on RIE-lag-insensitive PhC-like subwavelength structures,” Opt. Lett. 41(21), 5051 (2016). [CrossRef]
6. L. Luluzi, Minming Lu, F. Zhou, W. Chang, J. Tang, D. Li, X. Ren, Z. Pan, M. Cheng, and D. Liu, “Inverse-designed ultra-compact star-crossings based on PhC-like subwavelength structures for optical intercross connect,” Opt. Express 25(15), 18355 (2017). [CrossRef]
7. E Khoram, X Qian, and M Yuan, “Controlling the minimal feature sizes in adjoint optimization of nanophotonic devices using b-spline surfaces,” Opt. Express 28(5), 7060–7069 (2020). [CrossRef]
8. T.W. Hughes, M. Minkov, I.A.D. Williamson, and S. Fan, “Adjoint method and inverse design for nonlinear nanophotonic devices,” ACS Photonics 5(12), 4781–4787 (2018). [CrossRef]
9. ZL Dong, JF Qiu, YC Chen, C Liu, HX Guo, WJ Zhang, ZY He, and J Wu, “Ultra-compact X-shaped waveguide crossings with flexible angles based on inverse design,” Opt. Express 29(13), 19715–19726 (2021). [CrossRef]
10. KY Wang, XS Ren, WJ Chang, LH Lu, DM Liu, and MM Zhang, “Inverse design of digital nanophotonic devices using the adjoint method,” Photonics Res. 8(4), 528–533 (2020). [CrossRef]
11. A.Y. Piggott, J. Lu, K.G. Lagoudakis, J. Petykiewicz, T.M. Babinec, and J. Vuckovic, “Inverse design and demonstration of a compact and broadband on-chip wavelength demultiplexer,” Nat. Photonics 9(6), 374–377 (2015). [CrossRef]
12. C. M. Lalau-Keraly, S. Bhargava, O. D. Miller, and E. Yablonovitch, “Adjoint shape optimization applied to electromagnetic design,” Opt. Express 21(18), 21693 (2013). [CrossRef]
13. N.V. Sapra, D. Vercruysse, L. Su, K.Y. Yang, J. Skarda, A.Y. Piggott, and J. Vuckovic, “Inverse design and demonstration of broadband grating couplers,” IEEE J. Sel. Top. Quantum Electron. 25(3), 1–7 (2019). [CrossRef]
14. H. Chen, H. Jia, T. Wang, and J. Yang, “A Gradient-oriented binary search method for photonic device design,” J. Lightwave Technol. 39(8), 2407–2412 (2021). [CrossRef]
15. Y. Ooka, T. Tetsumoto, and A. Fushimi A, “CMOS compatible high-Q photonic crystal nanocavity fabricated with photolithography on silicon photonic platform,” Sci. Rep. 5(1), 11312 (2015). [CrossRef]
16. M. Peckerar, D. Sander, A. Srivastava, A. Foli, and U. Vishkin, “Electron beam and optical proximity effect reduction for nanolithography: New results,” J. Vac. Sci. Technol., B: Microelectron. Nanometer Struct.--Process., Meas., Phenom. 25(6), 2288–2294 (2007). [CrossRef]
17. K. Hitomi, S. Halle, M.L. Miller, I. Graur, N. Saulnier, D. Dunn, N. Okai, S. Hotta, A. Yamaguchi, H. Komuro, T. Ishimoto, S. Koshihara, and Y. Hojo, “Improvement of optical proximity-effect correction model accuracy by hybrid optical proximity-effect correction modeling and shrink correction technique for 10-nm node process,” J. Micro/Nanolithogr., MEMS, MOEMS 15(3), 034002 (2016). [CrossRef]
18. M. Minkov, I.A.D. Williamson, L. C. Andreani, D. Gerace, B. Lou, A.Y. Song, T.W. Hughes, and S. Fan, “Inverse design of photonic crystals through automatic differentiation,” ACS Photonics 7(7), 1729–1741 (2020). [CrossRef]
19. Y. Tian, X. Zhu, D. Meng, X. Wang, and B. Liang, “An overall configuration planning method of continuum hyper-redundant manipulators based on improved artificial potential field method,” IEEE Robot. Autom. Lett. 6(3), 4867–4874 (2021). [CrossRef]
20. N. Lebbe, C. Dapogny, E. Oudet, K. Hassan, and A. Gliere, “Robust shape and topology optimization of nanophotonic devices using the level set method,” J. Comput. Phys. 395, 710–746 (2019). [CrossRef]
21. M. Zhou, B. S. Lazarov, F. Wang, and O. Sigmund, “Minimum length scale in topology optimization by geometric constraints,” Comput. Methods Appl. Mech. Eng. 293, 266–282 (2015). [CrossRef]
22. H. Luan, Y. Geng, Y. Yu, and S. Guan, “Three-Dimensional Transient Electromagnetic Numerical Simulation Using FDFD Based on Octree Grids,” IEEE Access 7, 161052–161063 (2019). [CrossRef]
23. W. Shin and S. Fan, “Choice of the perfectly matched layer boundary condition for frequency-domain Maxwell’s equations solvers,” J. Comput. Phys. 231(8), 3406–3431 (2012). [CrossRef]
24. W. Shin and S. Fan, “Accelerated solution of the frequency-domain Maxwell’s equations by engineering the eigenvalue distribution of the operator,” Opt. Express 21(19), 22578–22595 (2013). [CrossRef]
25. L. Su, A. Y. Piggott, N. V. Sapra, J. Petykiewicz, and J. Vuckovic, “Inverse design and demonstration of a compact on-chip narrowband three-channel wavelength demultiplexer,” ACS Photonics 5(2), 301–305 (2018). [CrossRef]
26. W. Gao, X. Hu, C. Li, J. Yang, Z. Chai, J. Xie, and Q. Gong, “Fano-resonance in one-dimensional topological photonic crystal heterostructure,” Opt. Express 26(7), 8634–8644 (2018). [CrossRef]
27. L. Gu, B. Wang, Q. Yuan, L. Fang, Q. Zhao, X. Gan, and J. Zhao, “Fano resonance from a one-dimensional topological photonic crystal,” APL Photonics 6(8), 086105 (2021). [CrossRef]
28. Y. C. Lin, S. H. Chou, and W. J. Hsueh, “Robust high-Q filter with complete transmission by conjugated topological photonic crystals,” Sci. Rep. 10(1), 1 (2020). [CrossRef]
29. M. Xiao, Z. Q. Zhang, and C. T. Chan, “Surface Impedance and Bulk Band Geometric Phases in One-Dimensional Systems,” Phys. Rev. X 4(2), 130–136 (2014). [CrossRef]
30. X. Shi, C. Xue, H. Jiang, and H. Chen, “Topological description for gaps of one-dimensional symmetric all-dielectric photonic crystals,” Opt. Express 24(16), 18580–18591 (2016). [CrossRef]
31. E. D. Palik, Handbook of Optical Constants of Solids II (Academic Press, 1985).
32. R. Lyons, “Reducing FFT Scalloping Loss Errors without Multiplication,” IEEE Signal Proc. Mag. 28(2), 112–116 (2011). [CrossRef]
33. YF. Li and KF. Chen, “Eliminating the picket fence effect of the fast Fourier transform,” Comput. Phys. Commun. 178(7), 486–491 (2008). [CrossRef]
34. C. Li, X. Hu, W. Gao, Y. Ao, S. Chu, H. Yang, and Q. Gong, “Thermo-optical Tunable Ultracompact Chip-Integrated 1D Photonic Topological Insulator,” Adv. Opt. Mater. 6(4), 1701071 (2018). [CrossRef]
