6 research outputs found
Bandgap Optimization of Two-Dimensional Photonic Crystals Using Semidefinite Programming and Subspace Methods
In this paper, we consider the optimal design of photonic crystal structures for two-dimensional square lattices. The mathematical formulation of the bandgap optimization problem leads to an infinite-dimensional Hermitian eigenvalue optimization problem parametrized by the dielectric material and the wave vector. To make the problem tractable, the original eigenvalue problem is discretized using the finite element method into a series of finite-dimensional eigenvalue problems for multiple values of the wave vector parameter. The resulting optimization problem is large-scale and non-convex, with low regularity and non-differentiable objective. By restricting to appropriate eigenspaces, we reduce the large-scale non-convex optimization problem via reparametrization to a sequence of small-scale convex semidefinite programs (SDPs) for which modern SDP solvers can be efficiently applied. Numerical results are presented for both transverse magnetic (TM) and transverse electric (TE) polarizations at several frequency bands. The optimized structures exhibit patterns which go far beyond typical physical intuition on periodic media design
Robust topology optimization of three-dimensional photonic-crystal band-gap structures
We perform full 3D topology optimization (in which "every voxel" of the unit
cell is a degree of freedom) of photonic-crystal structures in order to find
optimal omnidirectional band gaps for various symmetry groups, including fcc
(including diamond), bcc, and simple-cubic lattices. Even without imposing the
constraints of any fabrication process, the resulting optimal gaps are only
slightly larger than previous hand designs, suggesting that current photonic
crystals are nearly optimal in this respect. However, optimization can discover
new structures, e.g. a new fcc structure with the same symmetry but slightly
larger gap than the well known inverse opal, which may offer new degrees of
freedom to future fabrication technologies. Furthermore, our band-gap
optimization is an illustration of a computational approach to 3D dispersion
engineering which is applicable to many other problems in optics, based on a
novel semidefinite-program formulation for nonconvex eigenvalue optimization
combined with other techniques such as a simple approach to impose symmetry
constraints. We also demonstrate a technique for \emph{robust} topology
optimization, in which some uncertainty is included in each voxel and we
optimize the worst-case gap, and we show that the resulting band gaps have
increased robustness to systematic fabrication errors.Comment: 17 pages, 9 figures, submitted to Optics Expres
A geometry optimization framework for photonic crystal design
AbstractThe performance of photonic crystal devices can depend strongly on their geometry. Alas, their fundamental physics offers relatively little by way of pointers in terms of optimum shapes, so numerical design search techniques must be used in an attempt to determine high performance layouts. We discuss strategies for solving this type of optimization problem, the main challenge of which is the conflict between the enormous size of the space of potentially useful designs and the relatively high computational cost of evaluating the performance of putative shapes. The optimization technique proposed here operates over increasing levels of fidelity, both in terms of the resolution of its non-parametric shape definition and in terms of the resolution of the numerical analysis of the performance of putative designs. This is a generic method, potentially applicable to any type of electromagnetic device shape design problem. We also consider a methodology for assessing the robustness of the optima generated through this process, investigating the impact of manufacturing errors on their performance. As an illustration, we apply this technology to the design of a two-dimensional photonic crystal structure; the result features a large complete band gap structure and a topology that is different from previously published designs
Extremal Spectral Gaps for Periodic Schr\"odinger Operators
The spectrum of a Schr\"odinger operator with periodic potential generally
consists of bands and gaps. In this paper, for fixed m, we consider the problem
of maximizing the gap-to-midgap ratio for the m-th spectral gap over the class
of potentials which have fixed periodicity and are pointwise bounded above and
below. We prove that the potential maximizing the m-th gap-to-midgap ratio
exists. In one dimension, we prove that the optimal potential attains the
pointwise bounds almost everywhere in the domain and is a step-function
attaining the imposed minimum and maximum values on exactly m intervals.
Optimal potentials are computed numerically using a rearrangement algorithm and
are observed to be periodic. In two dimensions, we develop an efficient
rearrangement method for this problem based on a semi-definite formulation and
apply it to study properties of extremal potentials. We show that, provided a
geometric assumption about the maximizer holds, a lattice of disks maximizes
the first gap-to-midgap ratio in the infinite contrast limit. Using an explicit
parametrization of two-dimensional Bravais lattices, we also consider how the
optimal value varies over all equal-volume lattices.Comment: 34 pages, 14 figure
Fabrication-Adaptive Optimization, with an Application to Photonic Crystal Design
It is often the case that the computed optimal solution of an optimization
problem cannot be implemented directly, irrespective of data accuracy, due to
either (i) technological limitations (such as physical tolerances of machines
or processes), (ii) the deliberate simplification of a model to keep it
tractable (by ignoring certain types of constraints that pose computational
difficulties), and/or (iii) human factors (getting people to "do" the optimal
solution). Motivated by this observation, we present a modeling paradigm called
"fabrication-adaptive optimization" for treating issues of
implementation/fabrication. We develop computationally-focused theory and
algorithms, and we present computational results for incorporating
considerations of implementation/fabrication into constrained optimization
problems that arise in photonic crystal design. The fabrication-adaptive
optimization framework stems from the robust regularization of a function. When
the feasible region is not a normed space (as typically encountered in
application settings), the fabrication-adaptive optimization framework
typically yields a non-convex optimization problem. (In the special case where
the feasible region is a finite-dimensional normed space, we show that
fabrication-adaptive optimization can be re-cast as an instance of modern
robust optimization.) We study a variety of problems with special structures on
functions, feasible regions, and norms, for which computation is tractable, and
develop an algorithmic scheme for solving these problems in spite of the
challenges of non-convexity. We apply our methodology to compute
fabrication-adaptive designs of two-dimensional photonic crystals with a
variety of prescribed features