Topology-optimized multiple-disk resonators obtained using level set expression incorporating surface effects

Garuda Fujii; Tsuyoshi Ueta; Mamoru Mizuno; Masayuki Nakamura

doi:10.1364/OE.23.011312

1. Introduction

Recent developments in the fabrication of micro-nanostructures have resulted in advanced optical devices such as lasers [1–5] and waveguides [6]. Whispering-gallery-mode (WGM) lasers [7,8] are one of the advanced low-threshold lasers and WGMs in cylinders are exploited in optofluidic ring resonators consisting of thin-walled SiO_x cylinders [9], liquid-core optical ring-resonator sensors [10], and in lasing within micro/nanowire [11]. Multiple-disk resonators based on WGMs are used in various laser devices as coupled resonator optical waveguides (CROWs) [12], wavelength-selective switches [13], multiwavelength lasers [14], and multiplering-coupled microcavity lasers [15] because of their advantages such as narrow linewidth, high power, and low intensity noise. Because the structures of systems, composed of multiple disk resonators and waveguides, are simple, disk sizes and gap-widths between neighboring disks and waveguide have a large influence on the system performances of the multiple-disk resonators [13, 16]. When WGM behavior occurs, light waves propagate along the dielectric surfaces of the disk resonators by repeated total internal reflection. Hence, it is necessary to model clearly defined dielectric surfaces in simulations of the WGM oscillations to incorporate the effects of reflections precisely.

Structural optimizations are widely used in device design in engineering and are categorized into sizing, shape and topology optimizations. Topology optimization [17] is, in particular, a powerful numerical method for designing high-performance devices, because the topology optimization is a free-form design method and have more freedom to explore the design solutions than the other structural optimization methods. This design freedom of the topology optimization allows the creation of new holes and enable us to obtain high-performance device designs. The optimization method is used during product design in mechanical engineering in the form of stiffness maximization problems [17, 18], thermal problems [19–22], and vibration problems [23, 24]. Recently, the target fields of topology optimization was extended to acoustic problems [25, 26] and fluid dynamics problems [27, 28]. In the field of optics, several advanced designs of optical devices such as cloaks [29–32], superlenses [33], photonic crystal cavities [34, 35], waveguides [36–40], and metamaterials [41–43], have been proposed using topology optimization

Some structural expressions used in the topology optimizations have been proposed as homogenization method [17], density method [44], and level-set method [45]. The first two methods are widely used for the design of high-performance devices in optics. However, these methods provide optimal configurations that include intermediate materials between the dielectrics and air, the so-called grayscales [46]. The presence of these grayscales makes it difficult to fabricate optimally designed configurations. To remove them from the design configurations, discrete filtering schemes have been proposed [47–50]. These schemes succeed in removing them but they are not an ideal solution for this problem because grayscales still emerge during the optimization.

The original level-set method [51] was proposed as numerical algorithms for a variety of surface motion problems to represent evolutional interfaces implicitly. Level-set methods are applied to shape optimizations [52, 53], however, topological changes are not permitted during the shape optimizations because the outlines of the designed structures are evolved from an initial configuration by updating the level-set function. To create new holes during the shape optimization process, a variety of extended shape optimization methods based on level-set expressions have been proposed and are referred to as the bubble method [54], topological gradient method [55], and its coupling method [56]. These shape optimization methods can provide well-designed structures, but complicated parameter settings on the introduction of holes are necessary. To create new holes in optimization processes and fundamentally solve the grayscale problem, structural expressions based on the level-set methods were adopted during topology optimizations [57]. To make the evolution process robust and maintain numerical accuracy, the level-set function defined as the signed distance function must be reinitialized in the numerical process [58], however, the implementation of the reinitialization scheme is difficult. A piecewise level-set method [59,60] is introduced to be free from the reinitialization operation needed to maintain the signed distance characteristic of the function, but the optimal configurations obtained strongly depend on the initial configuration. To overcome the numerical problems mentioned above, an advanced topology optimization method using a level-set method incorporating a fictitious interface energy based on the phase field concept is proposed [45]. The method allows topological changes in the process of optimizing the topology and exhibits minimal dependency on initial configuration. Moreover, the method enables us to control the geometrical complexity of the optimal configurations and proposed the use of a reaction-diffusion equation for updating the level-set function. The level-set-based topology optimization can solve the grayscale problem inside optimal configurations whereas grayscale still exists slightly along the structural boundaries in the optimal configurations. To design optical devices that use the effect of dielectric surfaces, topology optimizations that consider clearly defined dielectric boundaries are needed.

In this work, we present topology-optimized multiple-disk resonators designed using a level-set-based topology optimization incorporating surface effects. With the formulation of the topology optimization based on the phase field concept [45], the modeling scheme of the clear boundaries is numerically implemented in the creation of finite element models, and clearly defined dielectric boundaries are obtained by linear interpolations of the level-set functions. The intensity of the electric field in a disk resonator is formulated as the objective functional to be maximized. The effect of the dielectric surfaces is incorporated into the process of optimizing the topology. We employ a finite element method (FEM) to analyze light behaviors. The method is used not only for the analyses but also for computations of adjoint field and updating the level-set functions [45].

2. Formulation and scheme

Figure 1(a) shows a schematic of an optimization problem for two-dimensional multiple-disk resonators. The dielectric disk resonators, Ω_dm, are designed and undergo a transformation in the design domain Ω_design. A fixed waveguide, Ω_wg, guides the light waves emitted from an oscillating dipole [3] located at x_d = (x_d, y_d). A perfectly matched layer-absorbing boundary condition (PML-ABC) [61] and an optimized-absorbing function [62] are used to simulate light scattering in the open regions. Figure 1(b) shows the level-set function defined on grid points near a dielectric boundary. The zero-points of the function are obtained by linearly interpolating the function, and the iso-surfaces of the level-set functions are interpreted as the dielectric boundaries. Finite elements are created based on the dielectric boundaries obtained and the grids. The numbers of nodes and elements are approximately 550000 and 1060000, respectively.

Fig. 1 (a) Structure of a multiple-disk resonators. The domain sizes are $L_{design}^{y} = 0.5 \times L_{design}^{x}$ , $L_{out}^{x} = 1.2 \times L_{design}^{x}$ , $L_{out}^{y} = 0.6 \times L_{design}^{x}$ , $L_{wg} = 0.12 \times L_{design}^{x}$ , $L_{grid} = L_{design}^{x} / 300$ , and $R_{disk} = 0.3 \times L_{design}^{x}$ . The position of the posing source is $(x_{d}, y_{d}) = (- L_{out}^{x}, R_{disk} + L_{wg} / 2)$ . (b) The clearly defined dielectric boundary and level-set function. ”+/−” symbols represent the signs of the level-set functions defined on the grid points. The symbol ”0” represents the location at which the level-set functions interpolated linearly in each grid become zero. Dielectric boundaries are modeled by the lines connecting the zero points of level-set functions in each grid.

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It is assumed that there is a transverse magnetic mode and the total electric field, E_z, satisfies the time-independent equation:

\nabla^{2} E_{z} + \frac{ω^{2}}{c^{2}} ε (x) E_{z} = - \frac{ω^{2}}{ε_{0} c^{2}} D_{d} δ (x - x_{d}),

where ω is the circular frequency of light, c the speed of light in a vacuum, ε₀ the permittivity in a vacuum, and D_d the z component of the polarization vector of the oscillating dipole [3]. ε(x) is a position-dependent relative permittivity defined as:

ε (x) = {\begin{array}{l} ε_{air} + χ (ε_{dm} - ε_{air}) & x \in Ω_{design} \\ ε_{air} & x \in Ω_{out} \\ ε_{dm} & x \in Ω_{wg}, \end{array}

where Ω_out represents the outer domain of Ω_design, ε_dm and ε_air are the relative permittivities of the dielectric material and air, respectively. χ is a characteristic function defined in Ω_design as:

χ (ϕ (x)) = {\begin{array}{l} 1 & if & x \in Ω_{dm} \\ 0 & if & x \in Ω_{design} \ Ω_{dm}, \end{array}

where ϕ(x) are the level-set functions [59, 60] having piecewise-constant values to the boundaries Γ_dm of the dielectric material such that:

{\begin{array}{l} - 1 \leq ϕ (x) < 0 & for & \forall x \in Ω_{design} \ Ω_{dm} \\ ϕ (x) = 0 & for & \forall x \in Γ_{dm} \\ 0 < ϕ (x) \leq 1 & for & \forall x \in Ω_{dm} \ Γ_{dm} . \end{array}

The values of the level-set functions are determined with respect to grid points, as indicated in Fig. 1(b). If the value of the level-set function on a grid point becomes positive (”+” symbols in Fig. 1(b)), the grid point is occupied by dielectric material, whereas if the value becomes negative (”−” symbol), the grid point is occupied by air. The positions of the ”0” level-set functions are obtained by the linear interpolation of the functions in each grid; the dielectric boundaries are modeled in the process of optimizing the topology.

Figure 2 shows the relations between the level-set functions ϕ(x) and its iso-surface at ϕ(x) = 0, and the structural boundaries Γ_dm. The dielectric structural configurations are implicitly represented using the iso-surface of the level-set function and the boundaries of dielectric structures are changed by updating the level-set function during the optimization process.

Fig. 2 Dielectric boundary and the iso-surface of level-set function.

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To design multiple-disk resonators, the total electric field E_z in the resonators, Ω_dm, must be maximized. Hence, the objective functional for maximizing the light intensity in the resonators is defined as:

\underset{ϕ}{maximize} F = \frac{1}{F_{0}} \int_{Ω_{dm}} E_{z} E_{z}^{*} d Ω,

where

E_{z}^{*}

is the complex conjugate of E_z and F₀ the integrated intensity of the total electric field for the initial configuration

F_{0} = \int_{Ω_{dm}} E_{z} E_{z}^{*} {d Ω |}_{Initial}

.

Topology optimizations need regularization to obtain optimal configurations because such optimizations are ill-posed problems. Based on the formulation of the level-set-based topology optimization [45], the objective functional (5) is regularized by adding a fictitious interface-energy term derived from the phase field model as follows:

\underset{ϕ}{minimize} F_{r} = - F + \int_{Ω_{design}} \frac{1}{2} τ {| \nabla ϕ |}^{2} d Ω,

where τ is a positive regularization parameter representing the ratio of the objective functional to the fictitious interface-energy term. The above regularization with the fictitious energy term also works as a perimeter control, that is, a type of geometrical constraint. For details on this geometrical constraint, we refer the reader to the literature [29, 45, 63].

The level-set functions representing the configurations of disk resonators are updated by solving the time-evolutional equation [45] as follows:

\frac{\partial ϕ}{\partial t} = - K (ϕ) ({\bar{F}}^{'} - τ \nabla^{2} ϕ),

where t is a fictitious time, K(ϕ) a positive coefficient of proportionality, F̄′ a topological derivatives [64–66], and the level-set functions ϕ become t-dependent functions. The fictitious time t is discretized by a time step Δt, used in the updating of ϕ from n step to n + 1 step, as follows:

\frac{ϕ (t + Δ t)}{Δ t} = \frac{ϕ (t)}{Δ t} - K (ϕ) [{\bar{F}}^{'} - τ \nabla^{2} ϕ (t)] .

The above time differential equation is solved using FEM and the level-set functions are updated from ϕ(t) to ϕ(t + Δt).

Figure 3 shows a flow chart of the proposed topology optimization. The details of the procedures are as follows:

(1) The initial level-set function is set in the design domain Ω_design. The value of level-set functions are specified for all grid points as defined in Eq. (4).
(2) The datum of the finite-element mesh is created based on the level-set function. In this process, finite element meshes are produced along the dielectric boundaries represented as the iso-surface of level-set function.
(3) Light propagation and trapping in the disk resonators are simulated by means of FEM.
(4) Based on the results obtained by the above analysis, the intensity of trapped light is computed as objective functional F defined in Eq. (5).
(5) The optimization process is finished if the objective functional has converged. We simply check that F_n, the first top peak of F obtained at the nth step, satisfies F_n−10, ··· ,F_n−1 < F_n, F_n > F_n+1, ··· ,F_n+10, with the desired criterion F_n ≥ 3 as giving an optimal performance. The optimization process is reiterated if F does not satisfy the above convergence requirements.
(6) The topological derivative [64–66] can be obtained using the adjoint variable method.
(7) Level-set functions are updated by solving the time-evolutional equation, Eq. (8), using FEM. The process returns to the creation of the finite element mesh for the updated configuration.

Fig. 3 Flowchart of topology optimization process.

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3. Results and discussions

Figures 4(a) and 4(b) show an initial configuration and the link structures between neighboring disks. We compute the transmittance of the waveguide in the initial configuration, T, defined as:

T = \frac{\int_{Γ_{out}} Re {\frac{E \times H^{*}}{2}} \cdot n_{out} d Γ}{\int_{Γ_{in}} Re {\frac{E \times H^{*}}{2}} \cdot n_{in} d Γ},

where Γ_in and Γ_out are the lines in the waveguide shown in Fig. 1(a), E and H^* are the total electric field and the complex conjugate of the magnetic field, Re{Z} represents the real part of the complex vector Z, n_in and n_out are the unit normal vectors on Γ_in and Γ_out, respectively, with the vectors directed in the positive x direction. The computed result of T, shown in Fig. 4(c), exhibits a steep dip at

ω L_{design}^{x} / 2 π c = 4.65362

. We plot a normalized electric field distribution,

E_{z} / | E_{i}^{0} |

, and its amplitude,

E_{z} / | E_{i}^{0} |

, at the frequency in Figs. 4(d) and 4(e).

E_{i}^{0}

denotes the incident electric field at the center of the Ω_design. The plotted field and amplitude represents a resonant state associated with a WGM in the initial configuration; its corresponding objective functional value becomes F = 1. The T-dip frequency

ω L_{design}^{x} / 2 π c = 4.65362

is fixed during the optimization procedure. The values of relative permittivities are set to ε_dm = 4 and ε_air = 1. Under the above physical parameters setting, the topology optimization is started from the initial configuration for enhancing the objective functional F.

Fig. 4 A resonance frequency $ω L_{design}^{x} / 2 π c = 4.65362$ and a WGM behavior in the initial configuration with ε_dm = 4 and ε_air = 1 at the resonance frequency.

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We also investigate eigenmodes of a system composed of a single disk and the waveguide (Fig. 5(a)) to investigate the effect of a resonant splitting [67]. The transmittance of the waveguide T when a single disk exists was computed (Fig. 5(b)). We observe that a resonant mode, shown in Figs. 5(c) and 5(d), at the dip frequency $ω L_{design}^{x} / 2 π c = 4.66074$ have the same distribution of electric field in a disk of of the initial configuration at $ω L_{design}^{x} / 2 π c = 4.65362$ . The dip of the single-disk system is split into dips corresponding to the three resonant modes of the initial configuration composed of three disks. We seek the other two modes and find them at $ω L_{design}^{x} / 2 π c = 4.5906$ and 4.6680 (Fig. 6).

Fig. 5 A resonance mode of a single disk at the dip frequency $ω L_{design}^{x} / 2 π c = 4.66074$ with ε_dm = 4 and ε_air = 1.

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Fig. 6 The other two resonant modes, in the initial configuration, originated in the mode in the single disk.

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Figure 7 shows the relationship between objective functional values and optimization step. For this computation, each iteration of this optimization takes about 30 seconds when using a computer configuration with two CPUs (3.3 GHz, 8 cores) and 128 GB memory. We define the optimal solution as the first peak of F satisfying F_n−10, ··· ,F_n−1 < F_n, F_n > F_n+1, ··· ,F_n+10, and the desired performance F_n ≥ 3. Structures obtained at after 8453, 28477, and 32805 steps were adopted as optimal structures when τ = 5 × 10⁻⁵, τ = 1 × 10⁻⁵ and τ = 8 × 10⁻⁶, respectively.

Fig. 7 Objective functional value F versus optimization step. Parameters are set to Δt = 1 × 10⁻⁸ and K(ϕ) = 1.

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Figures 8 and 9 show the optimal configurations obtained for these τ values. The distributions of the total electric field and their amplitudes at $ω L_{design}^{x} / 2 π c = 4.65362$ for the optimal configurations are given in Figs. 10 and 11. The Fs for the optimal configurations obtained for τ = 1 × 10⁻⁵ and τ = 8 × 10⁻⁶ reach values roughly four and a half times greater than that for the initial configuration. In Figs. 10 and 11, ”circuits” of light trajectories along the disk surfaces and the waveguides are observed and the amplitude of the light waves intensifies in the disk resonators.

Fig. 8 Obtained optimal configurations, the value of regularization parameter τ, optimization steps, and the perimeter of the optimal configurations $L_{p} / L_{design}^{x}$ .

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Fig. 9 Left and right link structures between neighboring disks for the obtained optimal configurations.

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Fig. 10 The distribution of total electric field, $E_{z} / | E_{i}^{0} |$ , where $| E_{i}^{0} |$ is the amplitude of incident electric field at the center of design domain Ω_design.

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Fig. 11 Amplitude distributions of total electric field, $E_{z} / | E_{i}^{0} |$ , where $| E_{i}^{0} |$ is the amplitude of the incident electric field at the center of design domain Ω_design.

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Large structural differences are not observed between the initial and the optimal configurations (i.e., Figs. 4(a) and 8). However, the most important difference between the configurations is observed between the neighboring dielectric disks as link structures, shown in close-up in Fig. 9. To estimate the width of the link structures, L_link in Fig. 9(b), the perimeters of the configurations, L_p, are computed and given in the subcaptions of Fig 8. The difference in the perimeters between the initial and the optimal configurations for τ = 1 × 10⁻⁵ become $Δ L_{p} / L_{design}^{x} = 5.654 - 4.990 = 0.664$ .

If the width of the link structures in the initial configuration are L_p = 0.00, and the left and right link structures in each optimal configuration are regarded as the same, the difference in the perimeters ΔL_p and L_link are related by ΔL_p ≈ 4L_link, and the widths of the link structures are estimated as:

L_{link} / L_{design}^{x} = {\begin{array}{l} 1.82 \times 10^{- 1} & τ = 5 \times 10^{- 5} \\ 1.66 \times 10^{- 1} & τ = 1 \times 10^{- 5} \\ 1.61 \times 10^{- 1} & τ = 8 \times 10^{- 6} . \end{array}

The wavelength, λ_dm, in the dielectric structures is obtained from:

\begin{array}{l} λ_{dm} / L_{design}^{x} & = 1 / [(ω L_{design}^{x} / 2 π c) \sqrt{ε_{dm}}], \\ = 1.00 \times 10^{- 1} . \end{array}

Hence λ_dm and widths L_link are roughly equal to each other. The link structures need to reflect the light and to confine the light waves in each dielectric disk to achieve disk resonance. Moreover, they need to transmit the light to the connecting circuit. To satisfy both competing functions, the widths of the link-structures need to be of the same length scale as the wavelength.

Figure 12 shows the transmittance T of the initial and the optimal configurations near the resonance frequency $ω L_{design}^{x} / 2 π c = 4.65362$ ; computed Q factors of the configurations are listed in Table 1. The values of the Q factors for the optimal configurations are enhanced roughly four-fold above that for the initial configuration. In the same comparison, the mode volumes of the resonant states for the optimal configurations increase only slightly. Resonance T-dip frequencies of the optimal configurations change slightly from that of the initial configurations as listed in Table 1. The little difference between the frequencies is caused by the changes of the link structures between neighboring disks.

Fig. 12 Transmission spectrum near the resonance frequency $ω L_{design}^{x} / 2 π c = 4.65362$ .

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Table 1. Q factor and mode volume.

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We investigated the propagating modes in the waveguide when light waves are emitted from an oscillating dipole. The dispersion relations for the even and odd modes of the light waves in the waveguide are plotted in Fig. 13. The light waves in the waveguide are confined in the y direction and propagate in the positive x direction. The fundamental and second propagating modes exist at the resonance frequency $ω L_{design}^{x} / 2 π c = 4.65362$ . Nevertheless, only the fundamental propagating mode emerges in the waveguide as incident waves because the oscillating dipole is located at the center of the waveguide. There is a possibility that the reflected and transmitted waves include the second propagating mode. In the waveguide neighboring disk resonators, some evanescent modes for the x direction exist in the waveguide.

Fig. 13 Dispersion relation in the waveguide. The resonance frequency $ω L_{design}^{x} / 2 π c = 4.65362$ is represented by black dashed line. Even and odd modes are plotted by red and blue dots, respectively.

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Figures 14 and 15 show the computed T and mode volume in the frequency range $3.0 \leq ω L_{design}^{x} / 2 π c \leq 7.0$ . Most of the dips in frequency in the mode volume in Fig. 15 do not correspond to those of T in Fig. 14. Steep periodic T-dips emerging like optical frequency combs [68] and aperiodic split dips near the periodic dips are observed in Fig. 14. The split dips arise from mode splittings [67] caused by multi-resonance in the multiple disks. It is important to classify the origin of all observed resonant modes emerging as dips in Fig. 14 and associate then with the original eigenmodes in a single disk. However, the task is difficult because there are a large number of eigenmodes in a single disk.

Fig. 14 Transmission spectrum in the waveguide T versus normalized frequency $ω L_{design}^{x} / 2 π c$ .

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Fig. 15 Mode volume versus normalized frequency $ω L_{design}^{x} / 2 π c$ .

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The circuit length, L_circuit, is considered to be the most important length for feedback in the circuit, becoming an integral multiple of the wavelength in the dielectric material, i.e., L_circuit = Nλ_dm, where N is an integer. The averaged period of the periodic dips appearing in Fig. 14 is $Δ (ω L_{design}^{x} / 2 π c) = 0.3126$ for τ = 1 × 10⁻⁵. When the difference in length for integers N + 1 and N is taken into consideration, the difference between the oscillating frequencies, the relative permittivity of the dielectric material, and L_circuit can be derived from the above relation as:

\frac{L_{circuit}^{x}}{L_{design}} = \frac{1}{Δ (\frac{ω L_{design}^{x}}{2 π c}) \sqrt{ε_{dm}}} .

The circuit length is calculated as

L_{circuit}^{x} / L_{design} = 1.602

when τ = 1 × 10⁻⁵. The perimeter of a disk is

L_{disk} / L_{design}^{x} = 2 π R_{disk} / L_{design}^{x} = 1.884

. Therefore, L_circuit is nearly the same as L_disk. Hence, resonance from the WGM within a disk is considered to be the most important feedback mechanism in the circuits.

4. Conclusion

This study demonstrated that topology optimization incorporating surface effects can provide optimal designs of multiple-disk resonators employing WGMs. The Q factors of the topology-optimized multiple-disk resonators are roughly three and a half times higher than that of the initial configuration. Clear dielectric boundaries are obtained by linear interpolations of level-set functions and influences from dielectric surfaces are taken into consideration during topology optimization. The link structures between neighboring disks are improved and the widths of the link structures are of order of the wavelength that provides an appropriate rate of reflection for resonance in the disk and transmission to connecting elements of the light circuit.

Acknowledgments

This work was supported by the CASIO SCIENCE PROMOTION FOUNDATION and partially supported by JSPS Grant-in-Aid for Young Scientists (B) (Grant No. 26870239), and a research grant from The Mazda Foundation.

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Structure	Dip frequency	Q factor	Mode volume

Initial configuration	4.65362	6.28 × 10²	8.32 × 10⁻²
τ = 5 × 10⁻⁵	4.65374	1.39 × 10³	8.61 × 10⁻²
τ = 1 × 10⁻⁵	4.65366	2.24 × 10³	8.60 × 10⁻²
τ = 8 × 10⁻⁶	4.65364	2.32 × 10³	8.62 × 10⁻²

Topology-optimized multiple-disk resonators obtained using level set expression incorporating surface effects

Abstract

1. Introduction

2. Formulation and scheme

3. Results and discussions

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (15)

Tables (1)

Equations (12)

Optics Express