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.

[1]  Steven J. Cox,et al.  Band Structure Optimization of Two-Dimensional Photonic Crystals in H-Polarization , 2000 .

[2]  Bernhard J. Hoenders,et al.  Photonic bandgap optimization in inverted fcc photonic crystals , 2000 .

[3]  Shanhui Fan,et al.  Stopping and storing light coherently , 2005 .

[4]  D. Larkman,et al.  Photonic crystals , 1999, International Conference on Transparent Optical Networks (Cat. No. 99EX350).

[5]  Kim-Chuan Toh,et al.  Solving semidefinite-quadratic-linear programs using SDPT3 , 2003, Math. Program..

[6]  Steven G. Johnson,et al.  Optimal bistable switching in nonlinear photonic crystals. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  S. Osher,et al.  Maximizing band gaps in two-dimensional photonic crystals by using level set methods , 2005 .

[8]  Abraham Charnes,et al.  Programming with linear fractional functionals , 1962 .

[9]  B. Craven,et al.  The Dual of a Fractional Linear Program , 1973 .

[10]  H. Haus,et al.  Channel drop filters in photonic crystals. , 1998, Optics express.

[11]  Shanhui Fan,et al.  Guided and defect modes in periodic dielectric waveguides , 1995 .

[12]  G. Floquet,et al.  Sur les équations différentielles linéaires à coefficients périodiques , 1883 .

[13]  Ole Sigmund,et al.  Systematic design of phononic band–gap materials and structures by topology optimization , 2003, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[14]  John,et al.  Strong localization of photons in certain disordered dielectric superlattices. , 1987, Physical review letters.

[15]  Farid Alizadeh,et al.  Interior Point Methods in Semidefinite Programming with Applications to Combinatorial Optimization , 1995, SIAM J. Optim..

[16]  Y. R. Wang,et al.  Optimization of band gap of photonic crystals fabricated by holographic lithography , 2007 .

[17]  George Shu Heng Pau,et al.  Feasibility and Competitiveness of a Reduced Basis Approach for Rapid Electronic Structure Calculations in Quantum Chemistry , 2006 .

[18]  E. Yablonovitch,et al.  Inhibited spontaneous emission in solid-state physics and electronics. , 1987, Physical review letters.

[19]  Eli Yablonovitch,et al.  Inverse Problem Techniques for the Design of Photonic Crystals (INVITED) , 2004 .

[20]  G. Pau,et al.  Reduced-basis method for band structure calculations. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[21]  R. Saigal,et al.  Handbook of semidefinite programming : theory, algorithms, and applications , 2000 .

[22]  Michael L. Overton,et al.  Primal-Dual Interior-Point Methods for Semidefinite Programming: Convergence Rates, Stability and Numerical Results , 1998, SIAM J. Optim..

[23]  L. Rayleigh,et al.  XVII. On the maintenance of vibrations by forces of double frequency, and on the propagation of waves through a medium endowed with a periodic structure , 1887 .

[24]  Yurii Nesterov,et al.  Interior-point polynomial algorithms in convex programming , 1994, Siam studies in applied mathematics.

[25]  Steven G. Johnson,et al.  Photonic Crystals: Molding the Flow of Light , 1995 .

[26]  Ole Sigmund,et al.  Geometric properties of optimal photonic crystals. , 2008, Physical review letters.

[27]  John Stillwell,et al.  Symmetry , 2000, Am. Math. Mon..

[28]  F. Bloch.  Über die Quantenmechanik der Elektronen in Kristallgittern , 1929 .

[29]  J. Strutt.  Scientific Papers: On the Maintenance of Vibrations by Forces of Double Frequency, and on the Propagation of Waves through a Medium endowed with a Periodic Structure , 2009 .

[30]  Stephen P. Boyd,et al.  Semidefinite Programming , 1996, SIAM Rev..