We present analytical results that shed new light on the properties of photonic-crystal fibers (optical fibers with
periodic structures in their cladding). First, we discuss a general theorem, applicable to any periodic cladding
structure, that gives rigorous conditions for the existence of cutoff-free guided modes-it lets you look at a
structure, in most cases without calculation, and by inspection give a rigorous guarantee that index-guiding
will occur. This theorem especially illuminates the long-wavelength limit, which has proved diffcult to study
numerically, to show that the index-guided modes in photonic-crystal fibers (like their step-index counterparts)
need not have any theoretical cutoff for guidance. Second, we look in the opposite regime, that of very short
wavelengths. As previously identified by other authors, there is a scalar approximation that becomes exact in
this limit, even for very high contrast fibers. We show that this "scalar" limit has consequences for practical
operation at finite wavelengths that do not seem to have been fully appreciated: it tells you when band gaps
arise and between which bands, reveals the symmetry and "LP" degeneracies of the modes, and predicts the
scaling of cladding-related losses (roughness, absorption, etc.) as the size of a hollow core is increased.
Finite-difference time-domain (FDTD) methods suffer from reduced accuracy when modeling
discontinuous dielectric materials, due to the inhererent discretization ("pixellization"). We show
that accuracy can be significantly improved by using a sub-pixel smoothing of the dielectric function,
but only if the smoothing scheme is properly designed. We develop such a scheme based on a
simple criterion taken from perturbation theory, and compare it to other published FDTD smoothing
methods. In addition to consistently achieving the smallest errors, our scheme is the only one
that attains quadratic convergence with resolution for arbitrarily sloped interfaces. Finally, we
discuss additional difficulties that arise for sharp dielectric corners.
The finite-difference time-domain (FDTD) approach is now widely used to simulate the expected performance of photonic crystal, plasmonic, and other nanophotonic devices. Unfortunately, given the computational demands of full 3-D simulations, researchers can seldom bring this modeling tool to bear on more than a few isolated design points. Thus 3-D FDTD -- as it stands now -- is merely a <i>verification</i> rather than a <i>design optimization</i> tool. Over the long term, continuing improvements in available computing power can be expected to bring structures of current interest within general reach. In the meantime, however, many researchers appear to be exploring alternative modeling techniques, trading off flexibility of approach in return for more rapid turnaround on the devices of specific interest to them. In contrast, we are trying to improve the efficiency of 3-D FDTD by reducing its computational expense without sacrificing accuracy. We believe that these two approaches are completely complementary because even with vast amounts of computational power, any real-world system will still require a modular approach to modeling, spanning from the nanometer to the millimeter scale or beyond.