  Semi-analytical methods, such as rigorous coupled wave analysis, have been
pivotal for numerical analysis of photonic structures. In comparison to other
methods, they offer much faster computation, especially for structures with
constant cross-sectional shapes (such as metasurface units). However, when the
cross-sectional shape varies even mildly (such as a taper), existing
semi-analytical methods suffer from high computational cost. We show that the
existing methods can be viewed as a zeroth-order approximation with respect to
the structure's cross-sectional variation. We instead derive a high-order
perturbative expansion with respect to the cross-sectional variation. Based on
this expansion, we propose a new semi-analytical method that is fast to compute
even in presence of large cross-sectional shape variation. Furthermore, we
design an algorithm that automatically discretizes the structure in a way that
achieves a user specified accuracy level while at the same time reducing the
computational cost.
