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Abstract
The design of photonic crystals using novel materials is of great significance for the construction of high-performance, next-generation photonic crystal devices. We propose a universal Band structure-Transmission optimization-Band structure method based on moving asymptotic (MMA) method, which can be widely applied to photonic crystal structures. In this paper, we use the method to optimize the band structure of high temperature superconducting photonic crystal, and obtain a wider photonic bandgap and better band flatness in a specific frequency band. This method avoids the disadvantages of traditional scanning methods such as low efficiency and high resource consumption, allows multi-parameter optimization, and improves the accuracy and effectiveness of band modulation based on the iterative process of numerical calculation. The study provides some insights for the design of novel wide-bandgap optical devices.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Photonic crystals (PCs) are artificial structures made of materials with different dielectric constants arranged periodically [1–6]. By adjusting their band structures and periodicity, functions such as optical filtering [7–10], optical sensing [11–15], detection [15], and power distribution [16] can be realized. It provides a new idea and performance optimization approach for the design and implementation of optical devices such as optical switches [17,18], optical waveguides [19,20], sensors [11–15] and high-Q resonators [21–23] as well as logic devices [13].
Similar to the band gap in traditional solid electronic materials, photonic band gap (PBG) refers to the specific frequency range in which light cannot propagate in a PC. It is related to the geometrical structure of the crystal, duty cycle, and refractive index difference of dielectric materials. By designing the structure and parameters of PCs, the optical properties (reflection, transmission, etc.) can be precisely regulated based on topology optimization. Compared with some traditional solid materials, PCs provide greater flexibility in the design and performance optimization of related devices [24,25]. Additionally, since the dielectric constants and optical properties of superconducting materials [26–29] can be effectively adjusted by temperature and other parameters, the tunability of PBG in superconducting photonic crystals (SPCs) can be realized [30–32], giving them great prospects in tunable PCs [33–36].
The structure of photonic crystals typically involves a large number of units and a huge parameter space, which will make traditional parameter scanning methods inefficient [37,38]. It becomes challenging to obtain the optimal design using such approaches. A common approach is to formulate the problem in terms of appropriate objective functions and constraints and then utilize topological optimization techniques [39] to solve it. Topological optimization is an iterative process based on numerical calculation [40], which gives rise to the proposed multi-parameter optimization method. However, due to the complexity and degeneracy of the band structure, the optimization process based on band calculation may encounter issues with local optimal solutions or convergence difficulties in the parameter space, resulting in unstable design outcomes and complex computations. Relevant studies [41,42] have shown that there is a correspondence between transmission spectrum and band structure, where the peak intensity and position are determined by the allowed propagating states in the band structure, and the attenuation reflects the PBG. This insight offers a novel approach to optimize the band structure. Based on this theory, we propose a band optimization method that relies on transmission spectrum calculation. This approach allows for effective control and optimization of the transmission rate, ultimately achieving the optimization goal of the band structure.
In this paper, for two-dimensional SPCs with different structures, we use the proposed Band structure-Transmission optimization-Band structure method to achieve a wider photonic band gap and better flatness of the energy band structure in a specific frequency range, and expand the design and application space of new materials PC devices.
2. Models and theoretical methods
In Fig. 1 and 2, we draw the structure diagram of two different 2D lattice structures (square and triangular) SPCs and the first irreducible Brillouin zone respectively. Where a is the lattice constant, the material of the dielectric hole is air, and R is its radius. The background material is the cuprate superconductor YBa2Cu3O7 (YBCO). Its dielectric constant can be tunable in both superconducting and non-superconducting states by adjusting the temperature and applying external electric or magnetic field, providing greater flexibility for the application of SPCs [43–45]. In addition, the YBCO material is assumed to be isotropic and homogeneous in this paper, with a refractive index of ns.
Fig. 1. Model diagram of the square structure superconducting photonic crystal(SPC). Wherein, the background material is YBCO, lattice constant a = 800 nm, and air hole radius R = 200 nm. Electromagnetic waves are incident from the left and emitted from the right, setting Scattering boundary conditions on the upper and lower boundaries. The lower right corner is the first Brillouin zone of the square structure.
Fig. 2. Model diagram of the triangular structure SPC. The background material is YBCO, the lattice constant a = 1200 nm, and the air hole radius R = 400 nm. Electromagnetic waves are incident from the left and emitted from the right, and Scattering boundary conditions are set at the upper and lower boundaries. The lower right corner is the first Brillouin zone of the triangular structure.
2.1 Plane-wave expansion method
The plane-wave expansion method (PWE) [46] uses Bloch’s theorem to expand Maxwell’s set of equations in reciprocal space in the form of plane waves, turn it into the eigen equation, and solve for the eigenvalue of this equation. The relationship between the obtained eigenfrequencies and the dispersion of light propagating in the PC is the energy band distribution. If the PC is in passive space and consists of an isotropic dissipation-free nonmagnetic medium, and in the case where the electromagnetic field in space is time-harmonic, the eigen equation of the PC can be derived from the Maxwell equation as:
Both the relative permittivity $\mathrm{\varepsilon }({\vec{\mathrm r}} )$ and the magnetic field intensity vector ${\vec{\mathrm H}}({\vec{\mathrm r}} )$ are periodic functions of the spatial displacement vector ${\vec{\mathrm r}}$, which can be expanded into Fourier series in the reciprocal space as:
Therefore, for a two-dimensional PC, the eigenequation in TE mode can be written as:
The method is applicable to calculate the photonic bands of a wide class of physical systems.
2.2 Transmittance calculation method
The S-parameter is generally used for networks operating in the microwave and RF frequency bands, and it describes the electrical behavior of a linear electrical network that is in various steady states when excited by an electrical signal. In this paper, we use the S-parameter for the calculation of the reflection coefficient at the input port (Port1) and the transmission coefficient at the output port (Port2) with the following equations:
2.3 Method of moving asymptotes (MMA)
Topology optimization is a complex structural optimization problem characterized by many design variables and large computational scale. The effective optimization algorithm is the key to solving topological optimization problem, and MMA is the main optimization algorithm often used in variable density method. The method approximates the objective function and constraint function as some convex sequence subproblems by mathematical programming. Solving the optimization problem based on Lagrange duality method. Then, the gradient method is used to obtain the optimal design variables of the subproblems. For the general optimization problem, it can be expressed as [47]:
2.4 Band structure-transmission optimization-band structure method
The high-performance requirements of PCs necessitate band modulation, and topological optimization of the band structure is a common approach for achieving this modulation. However, traditional computational methods such as the PWE method have certain limitations. Previous studies have shown that the energy band structure of PC can correspond to band positions of the transmission spectrum. In the frequency range where there is a bandgap in the band structure, exhibits regions of significantly lower transmittance or even complete lack of transmittance. Therefore, we propose the Band structure-Transmission optimization-Band structure Method to achieve topological optimization of the band structure, as shown in Fig. 3.
Fig. 3. The flowchart of the Band structure-Transmission optimization-Band structure method based on transmission spectrum optimization.
In general, this method is divided into three steps. Firstly, a basic unit is used to calculate the energy band of the whole structure. Band structure calculation is performed to determine the approximate location of the bandgap, which is used to define the frequency range of the transmission spectra; Secondly, the transmission spectrum is computed within the corresponding frequency range. Then, the objective function is defined and shape optimization features such as free-shape domain are used to broaden the low transmittance band based on the MMA method; Finally, band structure calculation is carried out on the optimized structure to analyze the improvement of the energy band, thus validating the effectiveness of the topology optimization method. This approach completes a closed-loop process from band structure to transmission spectrum and back to the band structure, which to a certain extent achieves accuracy and effectiveness in research.
We employ the Band structure-Transmission optimization-Band structure Method to investigate the transmission characteristic of electromagnetic wave in PC. To simulate the propagation properties of light wave exiting the PC, it is necessary to define the objective function at the output port [48]. The objective function is formulated as the integral function of electric field, approximating the variations in transmittance. Specifically, it takes the following form:
When the frequency of light is within the band gap range of PC, the light can difficultly be transmitted, and the transmission spectrum exhibits a low transmission intensity. Generally, a wider band gap implies more design space and better performance for applications. In order to broaden the band gap and optimize the energy band structure, the transmittance within a specific frequency range should be as low as possible. Therefore, the objective function should be minimized.
2.5 Theory of superconductors
Due to the presence of a persistent supercurrent at the surface, superconductors are completely resistant to magnetism. This surface current flows in a very thin thickness layer called the London magnetic field penetration depth [29], which is the depth at which the magnetic field diminishes to 1/e of the strength at the surface of the superconductor. In conventional superconductors, the dependence of ${\lambda _L}$ on temperature can be characterized by the following equation:
where ${T_c}$ is the critical temperature of the superconducting material and the value of p is taken with respect to it (2 for ${T_c}$ > 77 K and 4 for ${T_c}$ < 77 K). In addition, ${\lambda _L}(0 )$ is the London penetration depth at T = 0 K. The refractive index of superconducting materials is generally described by the GorterCasmir two-fluid model [49]:In this equation, ω is the frequency and c is the velocity of the electromagnetic wave in vacuum. For YBCO, the superconducting critical temperature is 92 K and the London penetration depth ${\lambda _L}(0 )$ = 200 nm [50].
3. Results and discussion
To demonstrate the topological optimization method for energy bands based on transmission spectra, we introduce the YBCO high-temperature SPCs structure. Initially, we calculate the energy band structures by the basic units in different two-dimensional lattice configurations (square and triangular). Using PWE method with periodic boundary conditions, we calculate the transverse electric (TE) polarized photonic band structure. The environmental temperature is set at 88 K, closely approaching the superconducting critical temperature. As cuprate superconductors typically operate in the terahertz frequency range, we focus our investigation on this domain. For the square lattice structure, with the lattice constant a of 800 nm and air holes of radius R at 200 nm, Fig. 4(a) reveals the existence of two PBGs between 300 THz and 1000 THz. The bandgap below 475 THz represents the cutoff frequency, which prompts us to focus on broadening and optimizing the narrower bandgap situated around 550 THz. On the other hand, the triangular lattice structure features the lattice constant a of 1200 nm and air holes of radius R at 400 nm. In comparison to the square lattice, the triangular structure exhibits more band gaps. Figure 4(b) demonstrates five photonic band gaps, ranging from 300 THz to 850 THz. Similar to the square lattice, the bandgap below 435 THz also corresponds to the cutoff frequency. The other four band gaps are situated near the frequencies of 500 THz, 525 THz, 590 THz, and 600 THz, respectively. Our primary focus here is to broaden and optimize the narrow bandgap in the frequency band around 500 THz.
Fig. 4. Energy band structures of two different 2D photonic crystals composed of background material YBCO, air hole (where the gray area is the band gap) (a) square structure; (b) triangular structure.
According to the proposed optimization method, we need to calculate and optimize the transmission spectra in the corresponding frequency range. Since the period number of the SPC structure has little influence on the transmission spectrum, for the subsequent electric field distribution and the observability of the results, the 10 × 8 structure is selected for the transmission spectrum calculation of the square lattice structure, as shown in Fig. 1. Figure 5 shows the transmission spectra of two structures before and after optimization, where blue and red lines correspond to the transmission curves before and after topology optimization, respectively. For the square lattice structure, as shown in Fig. 5(a), we can observe a near-zero transmittance within the range from 510 THz to 570 THz. In addition, there is a small transmission peak between 530 THz and 540 THz, corresponding to the close proximity of two bands in the band diagram. For the triangular lattice structure, as shown in Fig. 5(b), similarly, blue line reflects the transmission before optimization, and there is a considerable transmission capacity after the 520 THz band. We can observe that the transmission spectra of two structures exhibit a significant correspondence with their respective calculated band structures, which further confirms the theory correlating band structures with transmission spectra.
Fig. 5. The transmission spectrum of two different photonic crystals before and after topology optimization in the corresponding frequency band range, where the blue and red lines indicate the transmission before and after optimization (a) the transmission corresponding to the square structure; (b) the transmission corresponding to the triangular structure.
Furthermore, we need to perform topological optimization of the structures. Firstly, we define the integral of electric field at the output port as the objective function, minimizing it to achieve the goal of low transmittance. For square and triangular lattice structures, the functions are further minimized within frequency ranges of 500 THz to 580 THz and 480 THz to 540 THz, respectively. Secondly, we achieve the optimization by setting the free-shape domain, free-shape boundary, and moving the position of each air hole independently. In this case, the topology of the geometry is fixed to facilitate gradient-based optimization and the shape of the air holes is also fixed. To avoid error messages due to collision situations, the air holes are restricted to move at most 0.05 µm in the x and y directions. Thirdly, we select the MMA optimization algorithm, and the maximum number of iterations is set to 50, and the optimization tolerance is set to 0.001. In each iteration, based on the “moving asymptotic” strategy, we move on the approximation surface of the objective function to find a new optimization direction, and update the design variables to obtain better optimization results until the maximum number of iterations is reached. After optimization, for the square lattice structure, the transmittance is almost zero in the frequency band range of 500 THz to 580 THz, indicating that photons cannot propagate within this frequency range, achieving the effect of bandgap widening. As for the triangular lattice structure, the transmittance approaches zero in the range of 480 THz to 540 THz, especially after 485 THz, where the transmission efficiency is extremely low, demonstrating the successful optimization of the bandgap. The optimized PCs can have higher transmission efficiency, lower loss, and better propagation control, which is of significant importance for designing PCs with specific band gaps.
Then, we further verify the electromagnetic wave propagation before and after the optimization by examining the electric field intensity of the PC structure at specific frequencies. Figure 6 shows the electric field distribution of the square lattice structure PC before and after the topology optimization at a frequency of 535 THz. As depicted in Fig. 6(a), at 535 THz frequency, a clear transmission with significant propagation characteristic can be observed at the right output port. On the other hand, Fig. 6(b) shows that after optimization, the light wave near this frequency is effectively suppressed, and its transmission at the output port is difficult to detect. Additionally, Fig. 6(a) and Fig. 6(b) correspond to the transmission at specific frequencies represented by the blue and red curves are shown in Fig. 5(a), respectively, which also reflects the optimization effect on the energy band. Furthermore, Fig. 6(c) and Fig. 6(d) display the electric field distribution of the triangular lattice structure PC before and after the topology optimization at the frequency of 523 THz. Similar to the square lattice structure, the distribution of the electric field intensity also indicates the effective suppression of electromagnetic wave propagation at this frequency.
Fig. 6. The electric field distribution of the photonic crystal at a specific frequency, where (a)(b) corresponds to the square structured photonic crystal before and after topology optimization at frequency = 535 THz, respectively, and (c)(d) corresponds to the triangular structured photonic crystal before and after topology optimization at frequency = 523 THz, respectively.
Next, we present the movement of air holes before and after optimization for both structures. For the square lattice structure, as shown in Fig. 7(a), the approximate relative movement of each air hole can be observed, where colored circles represent the positions of the air holes before optimization, and black circles represent the positions after optimization. To further investigate the specific movement of each air hole, we filter out the upper left part of the periodic structure using a filter, as shown in Fig. 7(b), and number all the air holes within it for further calculation of their relative displacement coordinates. Finally, in Table 1, we provide the relative displacement distances in the x and y directions corresponding to each of the 20 air holes. Similarly, for the triangular lattice structure, we follow a similar procedure. The filtered-out 16 air holes are numbered, as shown in Fig. 8(b), and their relative displacements on both directions are given in Table 2.
Fig. 7. Topology optimization of the two-dimensional square structure SPC, where the black circles indicate the positions after optimization. (a) Schematic diagram of the topology optimization displacement of the whole SPC. (b) Schematic diagram of the topology optimization displacement of the part of the SPC marked by the purple dashed line.
Fig. 8. Topology optimization of the two-dimensional triangular structure SPC, where the black circles indicate the positions after optimization. (a) Schematic diagram of the topology optimization displacement of the whole SPC. (b) Schematic diagram of the topology optimization displacement of the partial SPC marked by the purple dashed line.
Table 1. Displacement of the air hole moving in the x and y directions after topology optimization of the square structure SPC
Table 2. Displacement of the air hole moving in the x and y directions after topology optimization of the triangular structure SPC
Due to the limitations of experimental equipment, it is difficult to simulate the energy band of PC structures containing many units, so we choose a relatively simple 3 × 3 structure for further study and verification. The construction of the structure is shown in Fig. 9(a). Firstly, periodic boundary conditions are set for the structure to complete the energy band simulation. It is worth noting that, compared with single-cell structure, the 3 × 3 type structure itself has periodicity, which will cause unreasonable overlapping and interleaving of bands at different wave vectors. Therefore, bands should be selected according to the eigenfrequencies and electric field distribution to facilitate further research. The results are shown in blue line in Fig. 10(b). Next, the transmission near the band gap is simulated, as shown by blue line in Fig. 10(a). In the range of 500 to 580 THz, due to the action of the surrounding bands, there is a certain transmission capacity, and a small transmission peak appears near 530 THz, which is also corresponding to transmission before the optimization in Fig. 5(a).
Fig. 9. The structure diagram of the 3 × 3 square structure SPC and the position diagram after topology optimization. (a) SPC structure diagram of square structure. (b) Displacement diagram after topology optimization, where the black circle indicates the optimized position.
Fig. 10. Transmission spectrum and band diagram of 3 × 3 type square structure SPC in the corresponding frequency band range before and after topology optimization, where the blue and red lines respectively represent the situation before and after optimization. (a) Transmission spectrum before and after optimization. (b) Band structure before and after optimization.
Then, according to the method mentioned above, the relative position of each air hole is independently changed on a global basis to achieve topological optimization of the transmittance. The results, as shown in red line in Fig. 10(a), show that the transmittance is significantly reduced and the smoothness is improved to some extent. Meanwhile, the movement of holes is shown in Fig. 9(b). According to the optimized arrangement, the band structure calculation is carried out, as shown in red line in Fig. 10(b). It can be observed that the bandgap is opened, and the flatness of the band at both ends of PBG is improved, which achieves the optimization effect. This verifies the validity of the topology optimization proposed in this paper and provides a new idea for the design of novel PC devices.
4. Conclusion
In this paper, we use the proposed optimization method to optimize two-dimensional SPCs with different lattice structures. Firstly, we calculate the band diagrams by the basic units of the two structures, analyze and discuss the number and location of the band gaps. Then we adopt MMA method to topologically optimize transmission spectrum by minimizing the objective function based on gradient optimization. Finally, we simulate and analyze the energy band of the optimized structure. The broadening of PBG and the flattening of the energy band verify the accuracy of this method. These results show that 2D PCs can be designed based on this optimization method to achieve the desired wide band gap and other realistic requirements. At the same time, the method can be widely applied to PC devices, providing an important theoretical basis for designing novel optical devices.
Funding
National Key Research and Development Program of China (No.2021YFA0718801, No.2021YFB3601201).
Acknowledgments
This work is supported by the National Key Research and Development Program of China (No.2021YFA0718801, No.2021YFB3601201). We thank for the helpful discussion with Prof. Pengfei Guan and the computational support from the Beijing Computational Science Research Center (CSRC).
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.
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