Recently, there has been plenty of work in designing and fabricatingmaterials with an effective negative refractive index. Veselago realized that aslab of material with a refractive index of -1 would act as a lens. Pendrysuggested that the Veselago lens would act as a superlens, providing a perfectimage of an object in contrast to conventional lenses which are only able tofocus a point source to an image having a diameter of the order of thewavelength of the incident field. Recent work has shown that similar focusing effects can be obtained withcertain slabs of ``conventional'' periodic composite materials: photoniccrystals. The present work seeks to answer the question of what periodicdielectric composite medium (described by dielectric coefficient with positivereal part) gives an optimal image of a point source. An optimization problem isformulated and it is shown that a solution exists provided the medium has smallabsorption. Solutions are characterized by an adjoint-state gradient condition,and several numerical examples illustrate both the plausibility of this designapproach, and the possibility of obtaining smaller image spot sizes than withtypical photonic crystals.
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