## 3 Model Violations
### Geometry  
#### 3.2.1 Restricted Beam Dimensions  
We return to our standard model (Figure 1) and turn to the discussion of selected model assumption violations. Within this section, we shortly discuss a phenomenon called the Goos-Hanchen shift [42] for historical reasons.  
In this context we consider a situation, that in contrast to model assumption V, the illumination area is _spatially restricted_. In particular, we assume some kind of illumination slit that is elongated along the \(y\)-axis while being spatially restricted along the \(x\)-axis (Figure 3).  
Clearly, even in this modified illumination geometry, the incident light beam may be reflected or transmitted at the sample surface. Once we have a spatially restricted illumination spot, it makes sense to ask where the light leaves the sample.  
Historically, this was first investigated in internal total reflection conditions at a single interface. As a matter of fact, the totally reflected light beam turns out to be laterally shifted along the \(x\)-axis with respect to the incident light beam for a characteristic distance \(l_{r}\). This is the so-called Goos-Hanchen shift [42; 43]. It turned out later that this phenomenon is however _not_ restricted to surfaces in total internal reflection conditions. In optical coatings, lateral shifts up to the sub-millimeter range have been experimentally established [44].  
In [43], [PERSON] was able to show that the value of the Goos-Hanchen shift depends on the first derivative of the phase \(\delta_{r}\) of the complex reflection coefficient with respect to the incidence angle \(\varphi\). The idea behind his model is that the spatially restricted illumination area is equivalent to relaxing the model requirement V. Hence, he assumed a certain distribution of incident plane waves with different incidence angles, and the lateral shift \(l_{r}\) is then obtained as follows:  
\[l_{r}=-\frac{d\delta_{r}}{dk_{x}}=-\frac{\lambda}{2\pi n_{1}\text{cos}\varphi }\frac{d\delta_{r}}{d\varphi} \tag{12}\]  
Here, \(n_{1}\) denotes the refractive index of the incidence medium.  
Again, the effect is not only relevant in total internal reflection conditions [10]. It has been found in the vicinity of the Brewster's angle with p-polarized light [45], at interfaces between transparent and absorbing media [45; 46; 47; 48; 49], as well as at metal surfaces [48; 49].  
Figure 3: Violation of model assumption V, resulting in a lateral shift (exaggerated for visibility) of the reflected light beam.  
[PERSON]'s argumentation is general enough to be applicable to thin film stacks as well [44; 50]. Correspondingly, lateral shifts in transmission (\(t\)) and reflection (\(r\)) are estimated in terms of Equation (13):  
\[\begin{array}{ll}r=|r|e^{i\delta_{r}}\Rightarrow&l_{r}=-\frac{\lambda}{2\pi} \frac{1}{n_{\text{l}}\text{cos}\frac{d\xi_{r}}{d\varphi}}\\ t=|t|e^{i\delta_{r}}\Rightarrow&l_{t}=-\frac{\lambda}{2\pi}\frac{1}{n_{ \text{l}}\text{cos}\frac{d\xi_{r}}{d\varphi}}\end{array} \tag{13}\]  
The lateral shift according to Equation (13) depends on the wavelength of the incident light. Potential applications therefore pursue wavelength demultiplexing tasks [51; 52].