Abstract
Recently, there has been plenty of work in designing and fabricating materials with an effective negative refractive index. Veselago realized that a slab of material with a refractive index of −1 would act as a lens. Pendry suggested that the Veselago lens would act as a superlens, providing a perfect image of an object in contrast to conventional lenses which are only able to focus a point source to an image having a diameter of the order of the wavelength of the incident field.
Recent work has shown that similar focusing effects can be obtained with certain slabs of “conventional” periodic composite materials: photonic crystals. The present work seeks to answer the question of what periodic dielectric composite medium (described by dielectric coefficient with positive real part) gives an optimal image of a point source. An optimization problem is formulated and it is shown that a solution exists provided the medium has small absorption. Solutions are characterized by an adjoint-state gradient condition, and several numerical examples illustrate both the plausibility of this design approach, and the possibility of obtaining smaller image spot sizes than with typical photonic crystals.
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Communicated by Irena Lasiecka.
This work was partially supported by NSF grant DMS-0537015.
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Dobson, D.C., Simeonova, L.B. Optimization of Periodic Composite Structures for Sub-Wavelength Focusing. Appl Math Optim 60, 133–150 (2009). https://doi.org/10.1007/s00245-008-9063-8
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DOI: https://doi.org/10.1007/s00245-008-9063-8