# Optimized Computation of Tight Focusing of Short Pulses Using Mapping to Periodic Space

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Spectral Solvers

#### 2.2. Problem Statement

#### 2.3. Mapping to and from the Computational Subregion

## 3. Results and Discussion

#### 3.1. Verification and Accuracy Determination

#### 3.2. Examples

#### 3.2.1. Focusing of a Gaussian Laser Pulse

#### 3.2.2. Focusing of a Laser Pulse with a Circular Flat-Top Transverse Profile

#### 3.2.3. Focusing of Realistic Laser Pulses

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**An example of simulation of the tight focusing problem. The electric field intensity is shown for (

**a**) $t=-{R}_{0}/c$, (

**b**) $t=-{R}_{0}/2c$, (

**c**) $t=0$ and (

**d**) $t={R}_{0}/2c$ with f-number $=0.3$ ($\theta \approx 1$ rad), $\lambda =1$ μm, ${R}_{0}=16\lambda $, $L=2\lambda $, $\epsilon =0.1$ rad, ${P}_{0}=1$ W. The pulse propagates towards positive x direction, the transverse directions are spanned by y and z axes (see Equation (8)).

**Figure 2.**Illustration of the idea of the proposed method. The pictures show the evolution of the electromagnetic field composed by the equidistant replication of the initial pulse along the x axis: the field intensity is shown for (

**a**) $t=-{R}_{0}/c$, (

**b**) $t=-{R}_{0}/2c$, (

**c**) $t=0$ and (

**d**) $t={R}_{0}/2c$ with f-number $=0.3$ ($\theta \approx 1$ rad), $\lambda =1$ μm, ${R}_{0}=16\lambda $, $L=2\lambda $, $\epsilon =0.1$ rad, ${P}_{0}=1$ W.

**Figure 3.**Schematic illustration of the proposed method in 3D space (

**a**) and the cross-section in the x-y plane at $z=0$ (

**b**). The restriction given by Equation (12) is derived from requiring that the initial (red) and replicated (blue) pulses do not overlap. The shaded region is the subregion where the simulation is performed.

**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 $f\text{-}\mathrm{number}=0.3$. (

**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 × 896.

**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 $f=1$.

**Figure 6.**The dependence of (

**a**) the peak value of cycle-averaged intensity on f-number and (

**b**) the coordinate x at which the peak is reached, at a fixed peak power ${P}_{0}=1$ W and wavelength $\lambda =1$ μm (see the explanation in the text).

**Figure 7.**Evolution of the peak amplitude ${a}_{0}$ at focus as a function of the standard deviation of the initial phase ($\sigma \left(\varphi \right)$ in radian, solid line) and standard deviation of ${a}_{0}$ over 20 runs (dashed lines). Inlets exhibit examples of the normalized transverse intensity profile obtained at focus for two different values of $\sigma \left(\varphi \right)$. Transverse directions are normalized to $\lambda $.

**Figure 8.**Example of focal spot calculated from realistic profile: (

**a**) typical transverse intensity profile (a. u.) and (

**b**) wavefront (in radians) without correction of extra aberrations during beam transport from the end of the laser to the experiment and (

**c**) normalized focal intensity obtained applying our method with a f-number $f=3$.

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**MDPI and ACS Style**

Panova, E.; Volokitin, V.; Efimenko, E.; Ferri, J.; Blackburn, T.; Marklund, M.; Muschet, A.; De Andres Gonzalez, A.; Fischer, P.; Veisz, L.;
et al. Optimized Computation of Tight Focusing of Short Pulses Using Mapping to Periodic Space. *Appl. Sci.* **2021**, *11*, 956.
https://doi.org/10.3390/app11030956

**AMA Style**

Panova E, Volokitin V, Efimenko E, Ferri J, Blackburn T, Marklund M, Muschet A, De Andres Gonzalez A, Fischer P, Veisz L,
et al. Optimized Computation of Tight Focusing of Short Pulses Using Mapping to Periodic Space. *Applied Sciences*. 2021; 11(3):956.
https://doi.org/10.3390/app11030956

**Chicago/Turabian Style**

Panova, Elena, Valentin Volokitin, Evgeny Efimenko, Julien Ferri, Thomas Blackburn, Mattias Marklund, Alexander Muschet, Aitor De Andres Gonzalez, Peter Fischer, Laszlo Veisz,
and et al. 2021. "Optimized Computation of Tight Focusing of Short Pulses Using Mapping to Periodic Space" *Applied Sciences* 11, no. 3: 956.
https://doi.org/10.3390/app11030956