Appl Math Optim (2009) 60: 133–150
Optimization of Periodic Composite Structures
for Sub-Wavelength Focusing
David C. Dobson · Lyubima B. Simeonova
Published online: 24 December 2008
© Springer Science+Business Media, LLC 2008
Abstract Recently, there has been plenty of work in designing and fabricating ma-
terials 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 con-
trast 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.
Keywords Periodic composite materials · Optimization · Sub-wavelength focusing
Recently, there has been a renewed and avid interest in studying a class of materials
known as the left-handed materials (LHMs). These materials have simultaneously
Communicated by Irena Lasiecka.
This work was partially supported by NSF grant DMS-0537015.
D.C. Dobson · L.B. Simeonova (
Department of Mathematics, University of Utah, Salt Lake City, UT 84112-0090, USA