### Inverse Design Problems in Electromagnetics and Nano-Photonics

Eli Yablonovitch

University of California, Los Angeles

Photonic crystals, are 2 and 3 dimensional structures, that are the analog
for electromagnetics of the semiconductor crystals that play such an important
role in electronics. Thus Maxwell’s equations are to photonic crystals,
what Schrodinger’s equation is to conventional crystals. Since photonic
crystals are purely products of our imagination, the question has always
been what is the exact structure that should be fabricated?

Engineering design is formally a type of mathematical Inverse Problem.
The design goal is a certain electromagnetic specification or desired electromagnetic
performance. It is necessary to work backward from that goal to the exact
design of the dielectric boundary that achieves the objective. For example,
in mathematics, the Level Set Method has emerged as an excellent tool that
can contribute to algorithms for the optimization of boundaries and edges.

In the Photonic Crystal field, the era of purely intuitive design may
be obsolete. We must now concentrate more on design software, rational design,
and the numerical solution of inverse problems. There are a number of inverse
algorithms, including genetic algorithms, the error-propagation method,
and simulated annealing that can contribute to future progress in photonic
crystal design. It is expected that the study of photonic crystals will
more and more be study and development of rational inverse design algorithms
and software.