  We present methods for analyzing and designing cylindrical electromagnetic
metasurfaces with non-circular cross sections based on conformal
transformations. It can be difficult to treat surfaces with non-canonical
geometries since they generally do not admit straightforward solutions to the
Helmholtz wave equation subject to the appropriate boundary conditions. This
leads to the reliance on full wave numerical techniques which are only suitable
for the analysis, but not the synthesis, of these surfaces. We address this
issue by employing conformal transformations to map the physical space into a
computational space in which the surface coincides with a circular cylinder.
The electromagnetic boundary conditions on the surface remain intact under the
transformations due to their angle-preserving nature. However, they are much
more easily enforced. As a result, analytical modal solutions for the scattered
fields are readily obtainable, which facilitate closed-form analysis and
synthesis equations for general non-circular cylindrical metasurfaces. One
important utility enabled by the proposed framework is the efficient
identification of electromagnetic field distributions that satisfy local power
conservation. This leads to passive and lossless surface designs, which are
highly desirable in practice as they do not require active and/or lossy
components.
