Optimization of transmission spectra through periodic aperture arrays

We investigate the problem of designing periodic aperture arrays in a thin, perfect electric conductor, such that transmission energy of time-harmonic electromagnetic plane waves through the apertures attains a specified profile with respect to frequency. We first formulate a mathematical model for the electromagnetic transmission problem, and describe a regularized numerical discretization. We then formulate the design problem as a mathematical optimization in which an objective is minimized with respect to a discrete characteristic function describing aperture shape and topology. A level-set method combined with a filtering technique and a gradient-based minimization algorithm numerically solves the problem. Several numerical examples are presented which show that although it is possible to obtain improved transmission spectra, the problem is underposed and subject to numerical instability.

[1]  N. V. van Hulst,et al.  Strong influence of hole shape on extraordinary transmission through periodic arrays of subwavelength holes. , 2004, Physical review letters.

[2]  K. Svanberg.  The method of moving asymptotes—a new method for structural optimization , 1987 .

[3]  F. Santosa.  A Level-set Approach Inverse Problems Involving Obstacles , 1995 .

[4]  G. Buttazzo,et al.  An optimal design problem with perimeter penalization , 1993 .

[5]  H. Bethe.  Theory of Diffraction by Small Holes , 1944 .

[6]  Michael Sarrazin,et al.  Role of Wood anomalies in optical properties of thin metallic films with a bidimensional array of subwavelength holes , 2003, physics/0311013.

[7]  Ajay Nahata,et al.  Transmission resonances through aperiodic arrays of subwavelength apertures , 2007, Nature.

[8]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[9]  J. Petersson.  Some convergence results in perimeter-controlled topology optimization , 1999 .

[10]  Ajay Nahata,et al.  Influence of aperture shape on the transmission properties of a periodic array of subwavelength apertures. , 2004, Optics express.

[11]  C. S. Jog,et al.  A new approach to variable-topology shape design using a constraint on perimeter , 1996 .

[12]  Stanley Osher,et al.  A survey on level set methods for inverse problems and optimal design , 2005, European Journal of Applied Mathematics.

[13]  T. E. Bruns,et al.  Topology optimization of non-linear elastic structures and compliant mechanisms , 2001 .