# Monitoring of Evolving Laser Induced Periodic Surface Structures
## 5 From Fringe Patterns in the Far Field to the Experimentally Observed Fringes  
So far, we have investigated how the far field distribution at a given distance from the sample surface is affected by the presence of a surface nanostructure. This does not directly relate to the experimental observation of the fringe pattern since an imaging system is involved. To further elucidate the origin of the fringe pattern, we investigate the far-field distribution as function of distance from the sample surface. Figure 7 shows a two-dimensional map of the far field distribution \(E^{2}(x,y=0,z)\). The color scale is chosen such that the modulations in the Airy rings are easily visible, that is, the central part of the beam profile is saturated.  
The 2D-map shows that the modulations already observed in the farfield distribution at a given distance actually originate from new, discrete angular contributions to the outgoing wave. Some examples are highlighted by thick, coloured lines. These new angular components are generated by the surface nanostructure. It seems that they originate from a virtual source around \((65\ldots 75)\) um behind the sample. The angles with respect to the sample normal are \(\alpha=\sim\)5\({}^{\circ}\) (cyan), \(\sim\)9\({}^{\circ}\) (black), \(\sim\)10\({}^{\circ}\) (red) and \(\sim\)11\({}^{\circ}\). In addition, we not only observe single beams outgoing at different angles but bundles of beams as exemplarily shown for the cyan lines. Those contributions outgoing under sufficiently small angles can be imaged by our imaging system. The (half) acceptance angle of the parabola is 14\({}^{\circ}\), that is, the highlighted beams will be transported by the imaging system. Since the virtual source lies behind the sample plane, these newly generated discrete beams will be focused in front of the original image plane. This is sketched in Figure 8. The original beam from the unstructured surface(red) is focused in the image plane and exhibits an intensity profile similar to the beam incident on the sample surface with the Airy diffraction pattern of the aperture(thick red curve). Since the reflectivity in the structured region is strongly reduced, the intensity ratio between the individual intensity maxima of the reflected light may however be different from the incident light. In the image plane, the wave vector (thick, red arrow) of this beam is parallel to the optical axis. The blue line shows the LIPSS generated beam with its focal plane in front of the original image plane, that is, its wave vector (thick blue arrow) includes a finite angle \(\beta\) with the original beam (reflected from the surface) which leads to interference in the image plane.  
Figure 7: Simulated far-field distribution map \(E^{2}(x,y=0,z)\) as function of distance from the sample surface which is located at \(z=0\).  
\(\beta\) is given by \(\alpha\) and the magnification factor of the imaging system \(M=10\) via \(M\tan\beta=\tan\alpha\), that is, for new angular components of \(\sim\)5\({}^{\circ}\) on the sample side, the intersection angle of the two wavefronts at the image plane is \(\sim\)0.5\({}^{\circ}\). Interference between these two tilted beams give rise to fringes with a distance of \(\Delta L=\frac{\lambda}{\sin\beta}\sim 90\,\upmu\)m, which is in very good agreement with our experimental observation of \((75\dots 100)\,\upmu\)m.  
So far, we have concentrated on the smallest angles observed in the farfield distribution. The larger angles will lead to smaller fringe distances, for example, for the 9\({}^{\circ}\) angle on the sample side, we expect fringe periods of 50 \(\upmu\)m in the image plane, which corresponds to only 2 pixels of our CCD camera. This is the Nyquist limit to reproduce the fringe pattern. Given the assumption made in the numerical model and the precision in determining the angular contributions it is conceivable that those fringe patterns are not resolved with our setup.