What formalism is used to model thin film optical coatings that function across a range of angles of incident light?

Question

Thin film optical coatings [0] 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 [1], to increase its transmission of visible light.

The core mathematical model of thin film coating design is the Transfer-Matrix Method [2], 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 [3] 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?

[0] https://en.wikipedia.org/wiki/Thin-film_optics

[1] https://www.edmundoptics.com/knowledge-center/application-notes/lasers/anti-reflection-coatings

[2] https://en.wikipedia.org/wiki/Transfer-matrix_method_(optics)

[3] https://en.wikipedia.org/wiki/Riemann_sum

Answer 1

Typically thin film design is done on a computer. The design would use a number of specific wavelengths.

Sometimes you only need those specific wavelengths. E.G. you want the film to be transparent to one laser wavelength and reflective to another.

Other times you want good performance across a range, as a mirror that works across the visible spectrum. Or a band pass filter. In that case, you pick specific wavelengths to design with and check the performance in between. If you have a two layer film, there is no point to picking more than two wavelengths. If that isn't good enough, pick a $3^{rd}$ wavelength and add a layer. The expense will go up, but so will the performance.

The design process isn't guess and check. You start with an initial guess and calculate the reflectances or transmittances of interest.

You make a small step in thickness of each layer and recalculate. You can then numerically calculate the derivative of the reflectance/transmittance at each wavelength with respect to each layer's thickness. You can calculate a stepsize for each layer that will best change the outcome toward your desired reflectances/transmittances. You take that (or at least step toward that) as your new design.

You repeat until the design is close enough to your goal.

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The RP Photonics Encyclopedia and Buyer's Guide are usually good places to look for information about optics and where to buy them.

Dielectric Mirrors

Optical Thin-Film Design Services