Shape optimization for the strong directional scattering of dielectric nanorods

Abstract

In this project, we consider the shape optimization of a dielectric scatterer aiming at efficient directional routing of light. In the studied setting, light interacts with a penetrable scatterer with dimension comparable to the wavelength of an incoming planar wave. The design objective is to maximize the scattering efficiency inside a target angle window. For this, a Helmholtz problem with a piecewise constant refractive index medium models the wave propagation, and an accurate Dirichlet-to-Neumann map models an exterior domain. The strategy consists of using a high-order finite element (FE) discretization combined with gradient-based numerical optimization. The latter consists of a quasi-Newton (BFGS) with backtracking line search. A discrete adjoint method is used to compute the sensitivities with respect to the design variables. Particularly, for the FE representation of the curved shape, we use a bilinear transfinite interpolation formula, which admits explicit differentiation with respect to the design variables. We exploit this fact and show in detail how sensitivities are obtained in the discrete setting. We test our strategy for a variety of target angles, different wave frequencies, and refractive indexes. In all cases, we efficiently reach designs featuring high scattering efficiencies that satisfy the required criteria.

Shape optimization for the strong routing of light in periodic diffraction gratings

“…6.2.1]. We remark that the computation of sensitivities for the present case is more challenging than what was presented in our previous work [63] since we need to include the influence of the quasi-periodic conditions in the adjoint equation.…”

“…The state v(α) solves the system S(α)v(α) = b(α), from where u(α) is recovered and used to evaluate F (α) using expression (39). The sensitivities of J k (α) are obtained [63] by employing the chain rule for complex valued problems:…”

“…In this section, we have derived all the necessary information required in the formulas for the sensitivities (70) of the Jacobian matrix (61), which in turn, give formulas for the explicit computation of the sensitivities of the local system matrix and local load vector as described in Section 3.7.1. The reader is referred to our previous work, Araújo et al [63], for further implementation details.…”