Simple approximations to transverse wiggler spectra integrated over finite apertures in the wiggle plane

Simple approximations to conventional transverse wiggler spectra integrated over finite apertures in the wiggle plane are derived. Account is taken of the departure of the wiggler's on-axis field profile from an idealized sinusoidal shape. By introducing a generalization of the conventional K parameter, it is shown that the approximations can be formulated from a Fourier decomposition of the wiggler's dominant field profile in the direction of particle motion without explicit reference to the particle beam trajectory. Use of the generalized K parameter for estimating the first harmonic output energies of undulators and the critical energies of wigglers is examined.