  Inverse design of optical components based on adjoint sensitivity analysis
has the potential to address the most challenging photonic engineering
problems. However existing inverse design tools based on
finite-difference-time-domain (FDTD) models are poorly suited for optimizing
waveguide modes for adiabatic transformation or perturbative coupling, which
lies at the heart of many important photonic devices. Among these, dispersion
engineering of optical waveguides is especially challenging in ultrafast and
nonlinear optical applications involving broad optical bandwidths and
frequency-dependent anisotropic dielectric material response. In this work we
develop gradient back-propagation through a general purpose electromagnetic
eigenmode solver and use it to demonstrate waveguide dispersion optimization
for second harmonic generation with maximized phase-matching bandwidth. This
optimization of three design parameters converges in eight steps, reducing the
computational cost of optimization by ~100x compared to exhaustive search and
identifies new designs for broadband optical frequency doubling of laser
sources in the 1.3-1.4 micron wavelength range.Furthermore we demonstrate that
the computational cost of gradient back-propagation is independent of the
number of parameters, as required for optimization of complex geometries. This
technique enables practical inverse design for a broad range of previously
intractable photonic devices.
