# Optimized Computation of Tight Focusing of Short Pulses Using Mapping to Periodic Space
## 3 Results and Discussion
### Examples
#### 3.2.2 Focusing of a Laser Pulse with a Circular Flat-Top Transverse Profile  
In this subsection we apply the developed method for computing tight focusing of laser beams with more realistic structure. To account for the uniform amplification within the gain medium and the importance of not exceeding the breakdown intensity threshold, we consider the flat-top model of the pulse described in Section 2.2 with the field configuration parameters presented in Section 3.1. We then apply the described method to compute the peak field strength, as well as when and where it is reached. Figure 6 shows the dependency of the peak field amplitude and the coordinate at which it is reached on the f-number. The distance \(D\) between pulses was taken to be \(2L\), so that the error of the scheme does not exceed 1% of the peak amplitude (Section 3.1). The size of the computational grid was 128 \(\times\) 1024 \(\times\) 1024 nodes, and the instant of reaching peak power was determined with an accuracy of \(\Delta t=0.01\) s. The complete code by which these results were obtained is given in the Supplementary Materials.  
Note that for f-number \(\lesssim 0.6\) the peak field strength is achieved almost at the geometrical center of the initial spherical phase front. Nevertheless, for larger values of f-number the place of reaching the strongest field moves away from the geometrical centre with increase of f-number. This is the result of competition between geometrical propagation and diffraction. One can understand the nature of this mismatch through considering the limit of large \(f\)-numbers. When approaching to this limit, the pulse is limited by an increasingly small angle \(\theta\) and the phase front is increasingly close to plane under this limitation. In this case the pulse almost immediately starts transverse spreading due to diffraction, i.e., the field amplitude starts to decrease. That is why the peak field is achieved far away from the geometrical center of the sphere that defines the phase front. (If we fix the transverse size of the pulse, the center itself tends to be infinitely far from the initial location of the pulse in the limit of large \(f\)-numbers.) From this explanation we see that this mismatch is originated from the diffraction of the pulse in the initial state. This means that for larger values of \(f\)-number we would need to take a larger value of \(R_{0}\) to have the  
Figure 5: Comparison of numerical and analytical results for a tightly focused pulse with Gaussian temporal and angular profiles and fixed peak power \(P=1\) W: (**a**) The peak intensity at focus. The (**b**) transverse and (**c**) longitudinal fields in the plane \(z=0\), at focus, from simulations with an \(f\)-number of 1. (**d**) Comparison of theoretical (solid colors) and simulation (black, dashed) results for the transverse field along \(y=z=0\) (blue) and the longitudinal field along \(y=\lambda\) and \(z=0\), at focus, for \(f=1\).  
initial pulse in the zone of geometrical optics, i.e., so that it is well-described by a spherical wavefront. A simple estimate is to scale \(R_{0}\) with the Rayleigh length, i.e., \(R_{0}\sim 10\lambda f^{2}\). In brief, for large values of \(f\)-number, one should either account for the outlined shift (if approximate calculation is sufficient) or adjust the value of \(R_{0}\) accordingly (preferred).