# Optimized Computation of Tight Focusing of Short Pulses Using Mapping to Periodic Space
## 2 Materials and Methods
### Problem Statement  
We assume that, at the start of the simulation, the pulse that is to be advanced to focus is located within a spherical sector, defined by a polar angle \(\theta\) around the negative \(x\) axis that can be as large as \(\pi/2\). The opening angle \(\theta\) is related to the \(f\)-number, \(f\), as \(\theta=\arctan(1/(2f))\).  
The distance to the far-field zone is determined by the lowest characteristic spatial frequency component of the pulse. Since we are interested in the treatment of few-cycle pulses, we can take the central wavelength \(\lambda\) (the one that corresponds to the mean of the frequency band) as the typical scale for the lowest spatial frequency component. Then we require that the distance \(R_{0}\) from the initial location of the pulse to the geometrical center (focus position) is much larger than \(\lambda\). In practice we set \(R_{0}\approx 16\lambda\) but this can vary depending on the spectral bandwidth and on the required level of accuracy.  
The pulse can have arbitrary profiles along the longitudinal and transverse directions, i.e., perpendicular to, and along, the spherical surface respectively. The pulse can have arbitrary variation of profile, polarization and phase (wavefront) along the transverse directions, but the variations should not be too rapid, so that we still can assume that locally the pulse propagates predominantly towards the geometrical center of the sphere. The permitted magnitude of variations depends on the particular problem and on the required accuracy level. This is a matter of numerical verification.  
Although the problem statement is not restricted to any particular form, we consider a specific case of a short pulse under tight focusing. This facilitates the description of the method and provides a good demonstration of its capabilities. To mimic a laser beam being uniformly amplified within the amplifying medium, we consider a pulse with flat-top transverse profile. In this case we need to define the transverse shape \(u_{ts}\) so that the pulse amplitude is nonzero only within the angle \(\theta\). In order to provide a smooth transition near this angular limit, we introduce an edge smoothing angle \(\varepsilon\) and use \(\cos^{2}\) shape:  
\[u_{ts}(\alpha)=\omega\Big{(}\alpha,0,\theta-\frac{\varepsilon}{2}\Big{)}+ \left(\cos\frac{\pi\big{(}\alpha-\theta+\frac{\varepsilon}{2}\big{)}}{2 \varepsilon}\right)^{2}\cdot\omega\Big{(}\alpha,\theta-\frac{\varepsilon}{2}, \theta+\frac{\varepsilon}{2}\Big{)}, \tag{5}\]  
where we use the rectangular function  
\[\omega(x,x_{min},x_{max})=\begin{cases}1\text{ for }x\in[x_{min},x_{max}]\\ 0\text{ for }x\
otin[x_{min},x_{max}]\end{cases}. \tag{6}\]We consider a longitudinal shape of the form:  
\[u_{l}(x)=\sin\left(\frac{2\pi x}{\lambda}\right)\cdot\cos^{2}\left(\frac{\pi x}{L }\right)\cdot\omega\left(x,-\frac{L}{2},\frac{L}{2}\right), \tag{7}\]  
where \(L\) is the pulse length, \(\alpha\) is the angle between the \(x\) axis and the position vector \(\textbf{R}=(x,y,z)\).  
Finally, the spherical pulse can be defined as  
\[u(x,y,z) =u(\textbf{R})=\frac{A}{R}\cdot u_{l}(R-R_{0})\cdot u_{ls}\left( \arcsin\frac{\sqrt{y^{2}+z^{2}}}{R}\right) \tag{8}\] \[A=\sqrt{\frac{4P_{0}}{c(1-\cos\theta)}}, \tag{9}\]  
where \(P_{0}\) is the input power. We consider linear polarization and set the electromagnetic field as follows:  
\[\textbf{E}(\textbf{R}) =u(\textbf{R})\textbf{s}_{\textbf{0}} \tag{10}\] \[\textbf{B}(\textbf{R}) =u(\textbf{R})\textbf{s}_{\textbf{1}}, \tag{11}\]  
where \(\textbf{s}_{\textbf{0}}\) and \(\textbf{s}_{\textbf{1}}\) are, respectively, the normalized vectors \(\textbf{d}\times\textbf{d}\times\textbf{R}\) and \(\textbf{d}\times\textbf{R}\), and \(\textbf{d}\) is the polarization vector.  
An example of simulation of the tight focusing problem by the method described in Section 2.1 is demonstrated in Figure 1.