Daniel Lesnic (Leeds Maths) – A study on the determination of the index of refraction of anti-reflective coatings

In this study, the real-world application is concerned with the determination of a space-dependent index of refraction of an optical anti-reflection coating. The model is based on solving an inverse coefficient identification problem for the 1D Helmholtz equation. The additional data necessary for the inversion can be the full complex reflection coefficient or its absolute value only, measured for many wavenumbers. The numerical method is based on a FDM

direct solver combined with a nonlinear Tikhonov regularization. It is shown that, in general, the knowledge of the full complex reflection coefficient is necessary to determine uniquely a spacewise continuous index of refraction. When only the absolute value of the reflection coefficient is used as input data, constraints need to be imposed, e.g., the knowledge of the full integrated refraction index, additional smoothness assumptions on the index of refraction, or more reflectance data measured for many wavelengths. Apart from this insight into the uniqueness of solution of the inverse problem, the principal conclusion is that a better fit of the reflectance measured data is obtained by using a continuously varying index of refraction than when this coefficient is sought as a piecewise constant function.