# Optimized Computation of Tight Focusing of Short Pulses Using Mapping to Periodic Space
## 3 Results and Discussion
### Examples
#### 3.2.1 Focusing of a Gaussian Laser Pulse  
In this subsection we compare our simulation results with theoretical predictions for a focused Gaussian beam. In this case, a simulation is initialized with a linearly polarized, spherical pulse with transverse electric field \(E_{\perp}=E_{0}R_{0}\sin[k(r-R_{0})]u_{l}(r-R_{0})u_{ls}(\theta)/r\), where the longitudinal profile \(u_{l}(\zeta)=\exp[-\zeta^{2}\ln 4/(n\lambda)^{2}]\) and the transverse profile \(u_{ls}(\theta)=\exp(-\theta^{2}/\theta_{0}^{2})\). Here \(n\) is the number of wavelengths corresponding to the pulse duration (full width at half maximum) and the diffraction angle \(\theta_{0}\) is related to the f-number \(f\) by \(\tan\theta_{0}=1/(2f)\). The amplitude \(E_{0}\) is determined by requiring that the instantaneous power transmitted through the spherical surface \(r=R_{0}\) (the pulse peak) is equal to a fixed value \(P_{0}\), i.e., \(E_{0}=\sqrt{P_{0}/[\pi\epsilon_{0}R_{0}^{2}F(\theta_{0})]}\) where \(F(\theta_{0})=\int_{0}^{\pi}\sin(\theta)\exp(-2\theta^{2}/\theta_{0}^{2})\, \mathrm{d}\theta\). For \(\theta_{0}<2\) (equivalent to \(f>0.2\)), this auxiliary function is well approximated as \(F(\theta_{0})\simeq\theta_{0}^{2}/4-\theta_{0}^{4}/48+\theta_{0}^{6}/960\). We set the wavelength \(\lambda=1.0\) micron, the number of cycles \(n=4\), the initial position of the pulse \(R_{0}=20\lambda\), the peak power \(P_{0}=1\) W, and vary \(f\) from 0.25 to 2. The simulation covers the domain \(-30\lambda<x<10\lambda\) and \(-20\lambda<y,z<20\lambda\), which is represented by 512 grid points in each dimension.  
The peak intensity at focus, as obtained from simulations, is shown in Figure 5. If \(f<1\), there are substantial deviations from the theoretical prediction for a paraxial Gaussian beam \(I_{0}=2P_{0}/(\pi w_{0}^{2})\), where \(P_{0}\) is the instantaneous power passing through the focal plane \(z=0\) and the waist \(w_{0}=2f\lambda/\pi\) (see [12]). As the latter is calculated to lowest order  
Figure 4: (**a**) Relative error of the peak amplitude, comparing the computations in a subregion to that in the full domain, as a function of \(D\), the subregion size. Here f-number \(=0.3\). (**b**) Computational time of one iteration of the method (grid initialisation, forward FFT, spectral solver execution, and backward FFT), as a function of the parameter \(D/L\). All the simulations were performed using Hi-Chi. Computation time of sequential (all lines except the brown one) and multithreaded (brown line, 36 physical cores) versions of the code are presented. Size of the computational grid varies with the parameter \(D\) and is equal to \((56D/L)\times\) 896 \(\times\) 896.  
in the diffraction angle, or for large \(f\), this is to be expected. A more detailed comparison for the specific case \(f=1\) is shown in Figure 5b-d. While the transverse field along the laser axis is reasonably well predicted by theory, the longitudinal component is not. This supports the necessity of going beyond the paraxial approximation when considering tightly focused laser pulses. However, we also confirm that our simulations correctly reproduce the expected theoretical result in the limit that the focusing becomes weak.