A computational method for designing all-dielectric grating electromagnetic filters involves a combination of (1) numerical simulation of filter performance via integral-equation solution of Maxwell's equations and (2) genetic algorithms to find acceptable combinations of filter-design parameters. The method can be used to design filters of various types (e.g., band-pass, stop-band, or dichroic) for wavelengths from ultraviolet to microwave. One important potential application for the method might lie in the design of all-dielectric microwave dichroic filters for radomes; at present, all-dielectric versions are unknown and the filters are made of metals.

A typical all-dielectric grating filter can be a single- or multiple-layer plate with a periodic inhomogeneity in the dielectric material and/or a periodic variation in thickness (see figure). For the purpose of mathematical modeling, it is assumed that the filter lies in the *x × y* plane, the periodicity is in the *x* direction, and the filter is of infinite extent in the *y *direction and has a thickness *t* in the *z* direction. The material in each unit cell or subdivision of a unit cell is characterized by a complex permittivity and possibly by a complex permeability and/or a small conductivity.

The design problem is to find the permittivity, permeability, conductivity, and/or geometric parameters of the periodic structure (hereafter called "material parameters" for short) to obtain an acceptably close approximation to the desired reflectivity or transmissivity as a function of illumination angle and wavelength. This problem shares some of the formalism of the problem of identifying and locating dielectric objects, given the electromagnetic fields scattered from the object; both are nonlinear inverse source problems that involve the same integral-equation solutions of Maxwell's equations: The electromagnetic sources (electric and magnetic current densities) in a given material volume are related to outside electromagnetic fields by a linear integral equation derived from Maxwell's equations. The sources are related to the fields inside the volume by a constitutive equation that includes the material properties. Then the relationship linking the fields outside the source region to those inside is nonlinear in the material properties.

In this method, the nonlinear inverse problem is solved in two linear steps. For this purpose, the electromagnetic sources in the computational material volume are introduced as unknowns in addition to the material unknowns, making it possible to solve for source-volume fields and material parameters consistent with Maxwell's equations. Solutions are obtained iteratively by decoupling the two steps. First, one inverts for the material parameters only as part of a process of minimization of a cost function that is a measure of the deviation between the synthesized and desired reflectivity or transmissivity response. Next, given the material parameters, one computes the electromagnetic fields through direct solution of the integral equation in the source volume. The sources thus computed are used to compute the far fields and thus the synthesized transmissivity or reflectivity response of the filter.

The inversion to compute the material parameters is performed by use of a genetic-algorithm software package, called "PGAPACK," that incorporates capabilities for parallel processing. Genetic algorithms offer advantages over gradient-based and other local search methods for this and other electromagnetic-design problems, the solution spaces of which contain many extrema of cost functions. The ability of genetic algorithms to search parameter spaces globally not only makes it possible to avoid the common pitfall of converging on local (but not global) minima, but also holds promise for finding previously unknown solutions.

The possibility of finding previously unknown solutions is especially important in that it could enable one to design relatively simple or otherwise preferable filters and to investigate trends in material parameters.

The most expensive part of the computational cycle is solving for the electromagnetic fields. The computational burden is reduced by use of a matrix·vector formulation in which a set of frequency-dependent matrices need be filled only once. In addition, the number of frequencies for which the design equations are solved is reduced by using a transfer-function parameter-estimation technique in which the desired filter response is expressed as a quotient of frequency-dependent polynomials. The computation time is reduced further by taking full advantage of parallel-processing capabilities of PGAPACK on a Cray T3D computer. The parallelization scheme involves a simple master/slave configuration, with the expensive evaluation cycle distributed among the processors.

*This work was done by Cinzia Zuffada and Tom Cwik of Caltech for *NASA's Jet Propulsion Laboratory.* For further information, access the Technical Support Package (TSP) **free on-line at www.techbriefs.com **under the Physical Sciences category, *or circle no. 155 *on the TSP Order Card in this issue to receive a copy by mail ($5 charge).*

*NPO-20158*

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###### Genetic algorithms help design dielectric grating filters

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