Why does ellipsometry aim to build a model of a sample that reproduces experimental data ($\psi$ and $\Delta$, which are often defined by the total reflection coefficients: $\frac{r_p}{r_s}=tan(\psi)exp(i\Delta)$) instead of calculating the optical constants and thicknesses constituting the sample from the experimental data directly? From my understanding, and I may be wrong, it is possible to calculate the optical constants and thicknesses of the various layers directly from experimental data, assuming everything is ideal and measurements are made at enough angles of incidence. For instance, for a simple air-substrate system, we have: $\epsilon=sin^2(\phi_0)(1+tan^2(\phi_0)(\frac{1-tan(\psi)exp(i\Delta)}{1+tan(\psi)exp(i\Delta)})^2)$, where $\epsilon$ is the complex dielectric function of the substrate and $\phi_0$ is the angle of incidence.
I was not able to find any information that directly addressed my question in any textbooks or online, so I would greatly appreciate your insights on this. If anything needs clarifying, please let me know.
EDIT:
A comprehensive answer to my question is given in section 4.5 of "Ellipsometry and Polarized Light" by Azzam and Bashara.
From my understanding, it is often possible to obtain the optical properties of a sample from ψ and Δ using numerical techniques. Obtaining an analytic solution to the set of simultaneous equations relating the optical properties of the sample to ψ and Δ given ψ and Δ is often not possible as the equations in the set are usually nonlinear and transcendental in nature. In cases where an analytic solution can be obtained, a least squares solution is often needed anyways due to possible errors in the experimental data/model. This is why, again, from my understanding, optical constants aren't directly extracted from ψ and Δ.
It is not explicitly stated why numerically solving the set of equations given ψ and Δ is not preferred over building a model that closely reproduces the experimental ψ and Δ (my understanding of computer science and numerical methods is poor). I assume that the latter is computationally cheaper than the former, hence why the latter is the more stand method in the field of ellipsometry.
A lot of this is outside of my circle of competence, so if there is anything wrong with my reasoning, please let me know.
The equation you present is also a model, it's just a very simple one, so that would be as "indirect" as every other model.
Ellipsometry does not directly measure optical constants because it requires knowledge of complex details of the light-matter interaction. Instead, ellipsometry measures the change in polarization and phase of the polarized light due to its refraction and absorption by the material being studied. Based on these measurements, mathematical models and wave theory are used to infer the optical constants of the material.
You can not measure the index of refraction for a 1 micron thick sample simply with a ray of light through this sample. The samples for elipsometry are too thin to measure optical constants directly.
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