  Disordered structures producing a non-iridescent color impression have been
shown to feature a spherically shaped Fourier transform of their
refractive-index distribution. We determine the direction and efficiency of
scattering from thin films made from such structures with the help of the Ewald
sphere construction which follows from first-order scattering approximation.
This way we present a simple geometrical argument why these structures are well
suited for creating short wavelength colors like blue but are hindered from
producing long wavelength colors like red. We also numerically synthesize a
model structure dedicated to produce a sharp spherical shell in reciprocal
space. The reflectivity of this structure as predicted by the first-order
approximation is compared to direct electromagnetic simulations. The results
indicate the Ewald sphere construction to constitute a simple geometrical tool
that can be used to describe and to explain important spectral and directional
features of the reflectivity. It is shown that total internal reflection in the
film in combination with directed scattering can be used to obtain long
wavelength structural colors.
