Reflectance optimization of inhomogeneous coatings with continuous variation of the complex refractive index

A model is derived for the reflectance optimization of an inhomogeneous coating made of absorbing materials. The model is applicable mainly for spectral regions where no transparent materials are available, such as in the extreme ultraviolet. The complex refractive index is assumed to take values within a given continuous domain and in a given sequence. The coating design is generated through a series of layer elements with a small refractive-index contrast across interfaces; the thickness of the element is calculated in terms of the refractive-index increment at the interface. The coating is optimized element by element starting from the substrate. When the refractive index varies both continuously and smoothly, the thickness element is of first order in the refractive-index increment. Suggestions are given on how to optimize a more general coating that alternates continuous and smooth refractive-index domains along with discrete indices, which results in a succession of inhomogeneous coatings and finite layers. An example is given to illustrate the model. A new material selection rule is obtained to discriminate whether the addition of a material on top of a partly grown coating will increase or decrease the reflectance of the coating. As a consequence, the model, which is highlighted toward the maximization of reflectance, can be used analogously for reflectance minimization such as for anti-reflection coatings.