Thin film optical coatings  are atomically/molecularly thin layers of material applied to a substrate with the intent of affecting the optical properties of the substrate. For example, magnesium fluoride can be applied to glass as an anti-reflective coating , to increase its transmission of visible light.
The core mathematical model of thin film coating design is the Transfer-Matrix Method , which describes the propagation of electromagnetic radiation through a stack of multiple media, accounting for the transition at the interface between each layer.
Several parameters of this formalism can be varied to determine the net effect of a coating: material used in each layer; order of the layers; each layer's thickness; wavelength of incoming light; angle of incidence of incoming light; polarization of incoming light.
What formalism is used when the parameters describing incoming light are a range rather than a single value? For example, how does calculating the effect of a thin film on 650 nm, S-polarized light at a 45 degree angle of incidence, differ from calculating light with an equal power distribution from 550 to 650 nm? Or, 650 nm S-polarized light, from 45 degrees to normal with the film?
Is a sort of Riemannian discretization  performed, or are there closed form methods to determine the effect of the thin film across ranges of light parameter values?
Another perspective on this question would be to assume that one were designing a thin film coating to produce a particular effect (e.g. reflection) across a range of angles of incidence - is there a method for determining the requisite material properties of the coating without numerical guess-and-check?