  In this paper we present a systematic study of photonic bandgap engineering
using aperiodic lattices (ALs). Up to now ALs have tended to be defined by
specific formulae (e.g. Fibonacci, Cantor), and theories have neglected other
useful ALs along with the vast majority of non-useful (random) ALs. Here, we
present a practical and efficient Fourier space-based general theory to
identify all those ALs having useful band properties, which are characterized
by well-defined Fourier (i.e. lattice momentum) components. Direct control of
field localization comes via control of the Parseval strength competition
between the different Fourier components characterizing a lattice. Real-space
optimization of ALs tends to be computationally demanding. However, via our
Fourier space-based simulated annealing inverse optimization algorithm, we
efficiently tailor the relative strength of the AL Fourier components for
precise control of photonic band and localization properties.
