Bandgap optimization of two-dimensional photonic crystals using semidefinite programming and subspace methods

Authors:
H. Men;N. C. Nguyen;R. M. Freund;P. A. Parrilo;J. Peraire
Affiliations:
National University of Singapore, Center for Singapore-MIT Alliance, Singapore 117576, Singapore;MIT Department of Aeronautics and Astronautics, 77 Massachusetts Ave., Cambridge, MA 02139, USA;MIT Sloan School of Management, 50 Memorial Drive, Cambridge, MA 02142, USA;MIT Department of Electrical Engineering and Computer Science, 77 Massachusetts Ave., Cambridge, MA 02139, USA;MIT Department of Aeronautics and Astronautics, 77 Massachusetts Ave., Cambridge, MA 02139, USA
Venue:
Journal of Computational Physics
Year:
2010

Citing 4
Cited 1

Semidefinite programming

SIAM Review
Band structure optimization of two-dimensional photonic crystals in H-polarization

Journal of Computational Physics
Primal-Dual Interior-Point Methods for Semidefinite Programming: Convergence Rates, Stability and Numerical Results

SIAM Journal on Optimization
Photonic Crystals: Molding the Flow of Light

Photonic Crystals: Molding the Flow of Light

Optimization of spectral functions of Dirichlet-Laplacian eigenvalues

Journal of Computational Physics

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Abstract

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.