Optimizing Finite Structures to Suppress the Photonic Density of States

TL;DR

The paper addresses suppressing the photonic density of states (DOS) in finite 2D structures to emulate photonic band-gap behavior. It develops a hybrid topology optimization framework that couples density-based material descriptions with level-set region boundaries within an isogeometric optimization context, optimizing the DOS objective across regular polygons and cavities. Key findings show that the polygonal design suppressing DOS best corresponds to hexagonal unit-cell tiling when the structure size exceeds the Bragg length, and that introducing hexagonal cavities further enhances suppression versus circular cavities, with implications for fiber-optic cables. The approach enables recovery of finite hexagonal supercells and discovery of photonic-crystal primitive-cell symmetries for given material parameters without exhaustive space-group searches, offering a practical route to tailored DOS suppression in photonic crystals.

Abstract

We propose a topology-optimization framework for optimizing finite structures of arbitrary shape by combining density-based methods with level-set approaches. We first optimize regular polygonal structures to suppress the photonic density of states and find that the best performing polygon is consistent with a tiling of space with hexagonal unit cells. We next show that introducing cavities into hexagonal structures further suppresses the photonic density of states, particularly when the cavity is also hexagonal. Such a result would find application in the design of fiber-optic cables. We then describe an approach for optimizing arbitrary x-simple or y-simple designs that can recover finite supercells of a hexagonal unit cell. Our approach can therefore discover the symmetry of photonic-crystal primitive unit cells that significantly suppress the photonic density of states for a given set of material parameters within a single optimization.

Figures (5)

• Figure 1: Regular polygons of area $81a^2$ optimized for TM-polarized sources with a grid-point resolution (GPR) value of 100. We employed $\omega_0 = 0.8\cdot 2\pi c/a$ and $\Delta \omega = \omega_0/10$ in generating the designs for a square (a), a pentagon (b), a hexagon (c), a heptagon (d), and an octagon (e). The convergence of the photonic density of states figure of merit normalized by the corresponding quantities for vacuum is shown for the square in (f), the pentagon in (g), the hexagon in (h), the heptagon in (i), and the octagon in (j). The permittivity values were confined to the range $\frac{\epsilon(\mathbf{r})}{\epsilon_0} \in [1,8.9]$ and each grid point in a design was randomly initialized. The horizontal axis of the plots (f)-(j) shows the interval from 0 to 500 iterations and the vertical axis of the plots (f)-(j) the range from $10^{-6}$ to 1. The minimum value obtained by the normalized objective is indicated in (f)-(j) using a dashed line and with the numerical value.
• Figure 2: Regular polygons of area $81a^2$ optimized for TE-polarized sources with a GPR value of 100. We employed $\omega_0 = 0.8\cdot 2\pi c/a$ and $\Delta \omega = \omega_0/10$ in generating the designs for a square (a), a pentagon (b), a hexagon (c), a heptagon (d), and an octagon (e). The convergence of the photonic density of states figure of merit normalized by the corresponding quantities for vacuum is shown for the square in (f), the pentagon in (g), the hexagon in (h), the heptagon in (i), and the octagon in (j). The permittivity values were confined to the range $\frac{\epsilon(\mathbf{r})}{\epsilon_0} \in [1,8.9]$ and each grid point in a design was randomly initialized. The horizontal axis of the plots (f)-(j) shows the interval from 0 to 500 iterations and the vertical axis of the plots (f)-(j) the range from $10^{-6}$ to 1. In (f)-(j), the minimum value obtained by the normalized objective is indicated both by a dashed line and with the numerical value.
• Figure 3: Hexagons with cavities optimized for TM- and TE-polarized sources with a GPR value of 100. We employed $\omega_0 = 0.8\cdot 2\pi c/a$ and $\Delta \omega = \omega_0/10$ in generating the designs using a TM-polarized source for a hexagon with a hexagonal cavity (a) and a hexagon with a circular cavity (b), and using a TE-polarized source for a hexagon with a hexagonal cavity (c) and a hexagon with a circular cavity (d). The convergence of the photonic density of states figure of merit normalized by the corresponding quantity for vacuum is shown in (e) for (a), (f) for (b), (g) for (c), and (h) for (d). The permittivity values were confined to the range $\frac{\epsilon(\mathbf{r})}{\epsilon_0} \in [1,8.9]$ and each grid point in a design was randomly initialized (see the color bar in Fig. \ref{['fig:TM_regularshapes']}). The horizontal axis of the plots in (e)-(h) show the interval from 0 to 500 iterations and the vertical axis of the plots in (e)-(h) the range from $10^{-6}$ to 1. In (e)-(h), the minimum value obtained by the normalized objective is indicated both with a dashed line and with the numerical value. The annular regions all had area $81a^2$.
• Figure 4: Optimization of designs with two control points for the upper boundary function and two for the lower boundary function using the continuous design mask proposed in Eq. (\ref{['eq:design_mask']}) with $\alpha(y) = \arctan(y)/\pi$. We employed a GPR of 100, $\omega_0 = 0.8\cdot 2\pi c/a$, and $\Delta \omega = \omega_0/10$ in optimizing the designs in an $8a\times8a$ design region. For $\kappa = 70$, the design, the convergence of the figure of merit, and the progression of the area are shown in (a), (b), and (c), respectively. The corresponding quantities for $\kappa = 80$ and $\kappa = 90$ are shown in (d), (e), and (f), and (g), (h), and (i), respectively. TM polarization was used. See the color bar in Fig. \ref{['fig:TM_regularshapes']} for the permittivity values. The optimizations terminated with a criteria of $10^{-8}$ for the relative tolerance of the optimization parameters johnson_nlopt_2007. A dashed line indicates the minimum of the normalized objective function and the numerical value is also displayed in (b), (e), and (h).
• Figure 5: Optimization of designs with six control points for the upper boundary function and six for the lower boundary function using the continuous design mask proposed in Eq. (\ref{['eq:design_mask']}) with $\alpha(y) = \arctan(y)/\pi$. We employed a GPR of 100, $\omega_0 = 0.8\cdot 2\pi c/a$, and $\Delta \omega = \omega_0/10$ in optimizing the designs in an $8a\times8a$ design region. As above, for $\kappa = 70$, the design, the convergence of the figure of merit, and the progression of the area are shown in (a), (b), and (c), respectively. The corresponding quantities for $\kappa = 80$ and $\kappa = 90$ are shown in (d), (e), and (f), and (g), (h), and (i), respectively. TM polarization was used. See the color bar in Fig. \ref{['fig:TM_regularshapes']} for the permittivity values. The optimizations terminated with a criteria of $10^{-8}$ for the relative tolerance of the optimization parameters johnson_nlopt_2007. The minimum value of the normalized optimization objective is indicated in (b), (e), and (h) with a dashed line and with the numerical value.