# Monitoring of Evolving Laser Induced Periodic Surface Structures
## 3 Computational  
In order to investigate where the observed secondary fringe pattern is originating from, we carried out numerical simulations with Lumerical FDTD Solutions. Finite difference time domain (FDTD) method is widely applied for solving different electromagnetic problems by discretizing Maxwell's equations, both in time and space with central difference approximations. Typically a Cartesian volume element having \(\Delta x\), \(\Delta y\), \(\Delta z\) dimensions is used for the space discretization (the most common discretization technique is the Yee scheme [33; 34]) and a time step \(\Delta t\) for the time discretization [35]. In the FDTD method electromagnetic radiation (e.g., a plane wave) is injected to the simulation volume at some initial time and new field components are computed from differences based on the field components of the previous time interval. This process is continued iteratively until the transient solution for the fields has converged to a steady-state solution. The near-field solution that one obtains for the fields inside the computational domain is then Fourier transformed into the frequency domain and subsequently propagated into the far-field. This computational approach is extensively used for studying scattering problems of nanostructured surfaces [36].  
Our goal was to model the far-field response of the nanostructured surface illuminated by laser pulses. We applied a linearly polarized light source with central wavelength of 800 nm, with Airy-disk beam profile having 25 \(\upmu\)m FWHM diameter. The simulation volume had dimensions of 60 \(\times\) 60 \(\times\) 1.2 \(\upmu\)m\({}^{3}\). Perfectly matched layer boundary conditions were used on all sides of the calculation domain. The model system consisted of a substrate with a structured central part of 30 \(\upmu\)m diameter in the same geometrical arrangement as for the laser irradiated sample. The nanostructure is modeled by a one-dimensional sinusoidal height modulation. The direction of the periodic structures was perpendicular to the polarization direction of the illumination. We applied two substrates, Ti and Si\({}_{3}\)N\({}_{4}\). Their dielectric properties were taken from references [37; 38]. The whole structure was surrounded by vacuum (index of refraction \(n=1\)). To ensure an accurate calculation of the reflected beam from the surface structures, mesh/grid size of 24 nm was chosen. A 2D frequency domain field profile monitor at 500 nm above the surface was used to visualize the field distribution of the  
Figure 3: Si\({}_{3}\)N\({}_{4}\) sample surface imaged during nanostructuring by the back reflection diagnostic. (a) first pulse, (b) back reflection after the structure has been formed. Scales refer to the image plane. Magnification factor: \(M=10\times\).  
reflected light components. The data of this monitor was later applied to calculate far-field response of the structured surface by letting the field components propagate to a given distance in space. This way, we could calculate the static response of the structured sample surface in the far-field upon laser illumination.