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Simulated annealing algorithm with neural network for designing topological photonic crystals

Yaodong Liao, Tianen Yu, Yueke Wang, Boxuan Dong, and Guofeng Yang

Open Access Open Access

Abstract

In this work, we utilize simulated annealing algorithm with neural network, to achieve rapid design of topological photonic crystals. We firstly train a high-accuracy neural network that predicts the band structure of hexagonal lattice photonic crystals. Subsequently, we embed the neural network into the simulated annealing algorithm, and choose the on-demand evaluation functions for optimizing topological band gaps. As examples, designing from the Dirac crystal of hexagonal lattice, two types of valley photonic crystals with the relative bandwidth of bandgap 26.8% and 47.6%, and one type of pseudospin photonic crystal with the relative bandwidth of bandgap 28.8% are obtained. In a further way, domain walls composed of valley photonic crystals (pseudospin photonic crystals) are also proposed, and full-wave simulations are conducted to verify the valley-locked (pseudospin-locked) edge states unidirectionally propagates under the excitation of circularly polarized source. Our proposed method demonstrates the efficiency and flexibility of neural network with simulated annealing algorithm in designing topological photonic crystals.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

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Supplementary Material (1)

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Supplement 1       Supplemental Document

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.

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Figures (7)
Fig. 1.
Fig. 1. Design space. (a) The unit cell constructed with Si (blue) and Air (grey) is controlled by ${\boldsymbol{\mathcal{D}}}$   = (x1, y1, r1, x2, y2, r2). (b) Band structure including the first six bands, and the Brillouin zone is shown in the inset when ${\boldsymbol{\mathcal{D}}}$  = (0.0278a, 0.2405a, 0.1658a, 0.3611a, 0.2084a, 0.0812a) for the schematic unit cell. 30 wave vectors are uniformly selected along the high-symmetry boundaries of the Brillouin zone.

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Fig. 2.
Fig. 2. Structure and result of the forward model. (a) The network structure consists of 9 fully connected layers, with node numbers of 6-12-24-48-144-288-576-360-180. ReLU activation function is applied after each layer and Adam optimizer is used with a learning rate of 3*10−4, and the training batch size is set as 32. (b) MSELoss of each epoch, blue dashed line and red line represent the MSEloss of the training set and test set, respectively. The inset shows the network performance of the last 20 epochs. (c) The predicted bands (red line) of the example structure shown in Fig. 1(a), which almost completely overlaps with the actual bands (blue line).

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Fig. 3.
Fig. 3. The flow chart of simulated annealing algorithm with forward model.

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Fig. 4.
Fig. 4. Starting point of designing TPC. (a) The Dirac PC consists of six silicon cylinders (b), and the first six bands of (a).

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Fig. 5.
Fig. 5. Designed TPC with QVHE by SAFM. (a) The evaluation function in the iteration. The inset shows unit cell structures with different parameters ${\boldsymbol{\mathcal{D}}}$ . (b) Unit cell of VPC with ${\boldsymbol{\mathcal{D}}}$ optimal and the corresponding band structure. The phase profiles of second and third bands at K points, indicating the topological properties of the band gap. (c) The projected band structure (left panel) of the zigzag interface constructed by VPCs with opposite valley Chern numbers, and the eigen electrical field of the edge state at 0.512 c/a (right panel). (d) Electrical fields of valley-locked topological edge states excited by sources carrying positive (or negative) orbital angular momentum at 0.512 c/a.

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Fig. 6.
Fig. 6. Designed TPC with QVHE by SAFM. (a) The evaluation function varies in the iteration. The inset shows unit cell structures with different parameters ${\boldsymbol{\mathcal{D}}}$ . (b) Unit cell of VPC with ${\boldsymbol{\mathcal{D}}}$ optimal and the corresponding band structure. The phase profiles of fourth and fifth bands at K points. (c) The projected band structure (left panel) of the zigzag interface constructed by VPCs with opposite valley Chern numbers, and the eigen electric field of the edge state at 0.716 c/a (right panel). (d) Electrical fields of valley-locked topological edge states excited by chiral sources at 0.716 c/a.

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Fig. 7.
Fig. 7. Designed TPC with QSHE. The convergence process of the evaluation function for optimizing (a) topological nontrivial PC and (b) trivial PC. The inset shows unit cell structures during search process. (c) The unit cell and (d) band structure of topological nontrivial PC with ${\boldsymbol{\mathcal{D}}}$ optimal . (e) The eigen electrical field of the second, third, fourth and fifth bands at Г of topological nontrivial PC. (f) The unit cell and (g) band structure of topological trivial PC with ${\boldsymbol{\mathcal{D}}}$ optimal . (h) The eigen electrical field of the second, third, fourth and fifth bands at Г of topological trivial PC. (i) The projected band structure of the zigzag interface constructed by the topological nontrivial and trivial PC. The red and blue lines correspond to the pseudospin-up and pseudospin-down edge state modes, and band inversion of edge states happens at the kx = 0. (j) Electric fields of the pseudospin-locked edge states under the chiral excitation source with frequency f = 0.536c/a.

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Equations (3)

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M S E L o s s = 1 N K ∑ n , k N , K ( ω n , k , p r e d i c t e d − ω n , k , a c t u a l ) 2
Q ( n ) = Δ ω n ω n ¯
Q  =  f max ′ − f min ′ + f max − f min − | f min ′ − f min | − | f m a x ′ − f m a x | ) 2 ( f m i n + f max )
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