## 3 Model Violations
### Geometry
#### 3.2.2 Polarization Leakage  
In this section, we discuss the consequences of a violation of model requirement VI (no unambiguously defined incidence plane). Let us imagine a situation where the assumed plane sample surface is illuminated by light consisting of many different rays with different individual wavevectors. Let us further restrict on a simple model case, where all incident wavevectors are confined in a cone characterized by an apex-semiangle \(\Psi\) (Figure 4) [53]. The angle \(\varphi\) now denotes the angle between the surface normal and the symmetry axis of the cone (Figure 4, dotted line). The described situation may correspond to an oblique incidence of focused or defocused light.  
However, because of the assumed conical light incidence geometry, it is now no more possible to define an incidence plane in the strong sense. Indeed, each of the light rays that form the cone defines an individual incidence plane, and consequently, the termini of s- and p-polarization [10] use their assumed strong sense. Indeed, for example, light that is s-polarized with respect to the nominal incidence plane defined by the symmetry axis of the cone is not necessarily s-polarized for the individual incidence planes relevant for other \(\mathbf{k}\)-vectors; instead, it may contain p-polarized components. The result is an effect called polarization leakage. It is rather disturbing in any real experiment performed with linearly polarized light and, in particular, destroys the performance of thin film light polarizers.  
Quantitatively, the polarization leakage \(\Xi\) may be introduced in the following manner. Let the nominal incidence plane (as formed by the symmetry axis of the cone and the surface normal) coincide with the \(x\)-\(z\)-plane. Moreover, let the incident light be linearly polarized with the \(E\)-vector parallel to the \(y\)-axis. Hence, with respect to the nominal incidence plane, the light should be s-polarized. However, because of the spread in directions of the individual \(\mathbf{k}\)-vectors, this is not true for all light rays. Therefore, the transmittance relevant for light polarized along the \(y\)-direction (\(T_{y}\)) appears to be a biased superposition of the  
Figure 4: Assumed illumination geometry resulting in polarization leakage.  
s-polarized contribution (\(T_{s}\)) and the p-polarized contribution (\(T_{p}\)). Hence, we define \(\Xi\) through the following:  
\[T_{y}=\Xi T_{p}+(1-\Xi)T_{s} \tag{14}\]  
For a vanishing polarization leakage, i.e., \(\Xi=0\), the transmittance of \(y\)-polarized light naturally coincides with that of s-polarized light. Obviously, the polarization leakage will increase when \(\Psi\) increases. For other polarizations of the incident light, \(\Xi\) may be introduced in an analogous manner.  
While a general theory of the polarization leakage is developed in [54], a somewhat simpler but less general treatment restricted to a Lambertian intensity distribution results in the manageable expression Equation (15) (see [53]):  
\[\Xi\approx\frac{\Psi^{2}}{4\tan^{2}\varphi} \tag{15}\]  
In deriving Equation (15), \(\Psi\ll\varphi\) is assumed.  
Note that the leakage is determined by the illumination geometry only, the apex-semiangle is given in radians. Therefore, improvements of the coating design have no effect on the polarization leakage.  
As it turns out from Equation (15), the polarization leakage may be reduced by enlarging the angle \(\varphi\). In practice, that gave rise for the development of completely new types of thin film polarizers, working at incidence angles substantially larger than the usually applied incidence at \(45^{\circ}\)[55]. The working principle of these polarizers is based on attenuated total internal reflection at certain internal interfaces, and thus represents an example of the development of completely new optical components stimulated by the emergence of an originally disturbing effect like the polarization leakage.