In abstract terms, a helical undulator consist of a cylindrical "gap" region surrounded by a cylindrical "excitation" region. Within the gap, a rotating dipole field with longitudinal and azimuthal period and wavenumber exists. The excitation region consists of helically distributed material exhibiting a permanent dipole moment and possibly permeable material of variable moment resulting in a field . In a current free region, Maxwell's equations are satisfied by a divergence free, , irrotational, , field, and a scaler magnetic potential exists, satisfying Poisson's equation . is 0 on axis, at , and is continuous across boundaries. By representing the distribution of , and consequently, as Fourier series, a boundary value problem solution of an inhomogeneous, modified Bessel differential equation describes the generated fields. In the very simple case of a double helix of ideal magnet material with a continuously rotating vector , the gap fields are:
is the magnets' residual field. is gap radius, is outer magnet radius, and is harmonic number. and are the modified Bessel functions. The added complication of introducing non-linear permeable material necessitates a numerical solution, and code for that purpose is under development. The basic form of the fields in the gap is expected to change little, however, just the absolute field strength and harmonic content.