## 2 Basic Model Assumptions in Conventional Thin Film Optics
### Material Description
#### 2.2.3 Kramers-Kronig Consistency  
Any physically reasonable dielectric function must suffice the Kramers-Kronig relations [10; 12; 13]:  
\[\text{Re}\varepsilon(\omega)=1+\frac{1}{\pi}VP\int_{-\infty}^{\infty}\frac{ \text{Im}\varepsilon(\omega_{2})d\omega_{2}}{\omega_{2}-\omega}=1+\frac{2}{ \pi}VP\int_{0}^{\infty}\frac{\text{Im}\varepsilon(\omega_{2})\omega_{2}d \omega_{2}}{\omega_{2}^{2}-\omega^{2}} \tag{10}\]  
\[\begin{split}\text{Im}\varepsilon(\omega)=-\frac{1}{\pi}VP\int_{- \infty}^{\infty}\frac{\text{Im}\varepsilon(\omega_{2})-1]d\omega_{2}}{\omega_ {2}-\omega^{2}}+\frac{\varepsilon_{1 st}}{\varepsilon_{0}d\omega}\\ =-\frac{2\omega}{\pi}VP\int_{0}^{\infty}\frac{\text{Im}\varepsilon (\omega_{2})-1]}{\omega_{2}^{2}-\omega^{2}}d\omega_{2}+\frac{\varepsilon_{1 st} }{\varepsilon_{0}d\omega}\end{split} \tag{11}\]  
With VP--Cauchys principal value of the improper integral, and \(\omega_{2}\)--integration variable. The term \(\frac{\sigma_{1 st}}{\varepsilon_{0}d\omega}=\frac{\omega_{2}^{2}}{\pi \gamma\omega}\) in Equation (11) arises from the singularity in the Drude function at \(\omega=0\) and is only relevant in electric conductors with a static electric conductivity \(\sigma_{stat}\
eq 0\). The Drude,Debye, and oscillator models are consistent with Equations (10) and (11), and so are their linear superpositions. As a special case, the \(\beta\_\)do model is Kramers-Kronig consistent as well. The situation is a bit more complicated with the Brendel model [36] because the dielectric function according to Equation (5) contains a contribution with a vanishing resonance frequency, which becomes equivalent to a Drude-like term and therefore requires an addendum to Equation (11) similar to what we have in the case of conductors. The problem may be overcome when using an appropriately apodised Gaussian function in Equation (5) instead.  
The Campi-Coriasso model [37; 38] seems to be the first Kramers-Kronig consistent model combining the oscillator model with [PERSON]'s law [39]. It uses the same number of parameters with the same meaning as the Tauc-Lorentz model, but its parametrization is different and it is far less common. In the original work, no analytical expression for the real part of the dielectric function is provided but can be found in [39], where a good overview on models combining [PERSON]'s law and the [PERSON] model is also given. The author of [39] concludes on the Kramers-Kronig consistency of both the Tauc-Lorentz and Cody-Lorentz approaches, while the Forouhi-Bloomer model is claimed as Kramers-Kronig inconsistent.  
In [40], a different physically consistent model combining the oscillator model with [PERSON]'s law is presented. It is called Advanced Dispersion Model and is a precursor of the Universal Dispersion Model [24]. In contrast to the latter, it requires a priori information about the physical and chemical structures of the films since the model contains physical parameters specific for the material, such as atomic fractions.  
In concluding this paragraph, let us mention that the Kramers-Kronig relations represent a rather general quantitative formulation of the physical connection between light refraction and absorption phenomena. In fact, those connections are intuitively used by any coating practitioner. It is well known from coating practice that materials with a large refractive index tend to have smaller optical gaps than materials with a smaller refractive index. These correlations are formulated in semi-empirical rules like the Moss and Ravindra rules [41], and represent a useful guide in any realistic coating design procedure.