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 lambda_u and wavenumber ku = 2pi/lambda exists. The excitation region consists of helically distributed material exhibiting a permanent dipole moment M and possibly permeable material of variable moment M = mu_0*h resulting in a field B = mu_0*(H+M) . In a current free region, Maxwell's equations are satisfied by a divergence free, div(H+M) = 0, irrotational, curl H = 0, field, and a scaler magnetic potential Phi_m exists, satisfying Poisson's equation D^2 Phi = -div M. Phi is 0 on axis, at infinity, and is continuous across boundaries. By representing the distribution of M, and consequentlyPhi, 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 M, the gap fields are:

Br = ..., B_phi = ... B_z = ...,
B0 = (2Br/pi)(ku*ri*K_{n-1}(n*ku*ri)+Kn(n*ku*ri)-....
Br is the magnets' residual field. r_i is gap radius, r_o is outer magnet radius, and n is harmonic number. In and Kn 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.