  Photonic bandgap engineering using aperiodic lattices (ALs) is systematically
studied. 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 real-space optimization of ALs tends to be
computationally demanding, and is also difficult to generalise beyond 1D.
However, via our Fourier space-based inverse optimization algorithm, we
efficiently tailor the relative strength of the AL Fourier components for
precise control of photonic band and localization properties.
