# Full Explicit Numerical Modeling in Time-Domain for Nonlinear Electromagnetics Simulations in Ultrafast Laser Nanostructuring

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Modeling Electromagnetic Fields in Nonlinear Media

## 3. Algorithms

## 4. Performance and Framework

## 5. Results

- Low power regime: ${E}_{0}={10}^{4}\frac{\mathrm{V}}{\mathrm{m}}$ and $I=2.46\times {10}^{14}\frac{\mathrm{W}}{{\mathrm{m}}^{2}}$.
- Critical power regime: ${E}_{0}=3.5\times {10}^{5}\frac{\mathrm{V}}{\mathrm{m}}$ and $I=8.61\times {10}^{16}\frac{\mathrm{W}}{{\mathrm{m}}^{2}}$.
- High power regime: ${E}_{0}={10}^{6}\frac{\mathrm{V}}{\mathrm{m}}$ and $I=2.45\times {10}^{18}\frac{\mathrm{W}}{{\mathrm{m}}^{2}}$.

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Stability

#### Appendix A.1. Stability Condition on Metal

#### Appendix A.2. Stability Condition on Kerr Effect

#### Appendix A.3. Stability Condition on Raman Effect

#### Appendix A.4. Stability Condition on Photoionization Effect

## Appendix B. Computational Resources

Computational Resources: |
---|

CPU: Intel Core i9-10980XE |

MB: GIGABYTE X299X |

RAM: Crucial CT32G4RF D4293 DDR4-2933 |

ROM: 2x Crucial P5 1TB PCIe M.2 2280SS SSD |

GPU: GeForce GTX 1080 Ti |

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**Figure 1.**Scheme of the calculation box, illustrating the delay $n{t}_{0}\left(k\right)$ that depends on the refractive index of the media ${n}_{i}$ as well as the location of the electromagnetic field in the media.

**Figure 3.**The figure shows the time per step in the simulation updating in a computational domain with dynamics limits (index k is a dynamical magnitude). Furthermore, the number of threads in the palatalization process is dynamics.

**Figure 5.**Propagation of the pulsed beam given by a time sequence of instants which has been plotted before reaching the theoretical critical power, with an energy per pulse of 2.8 $\mathsf{\mu}$J allowing to reach an electron density of ${10}^{27}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{m}}^{-3}$, the beam splits, enclosing the charge and defocusing the beam. (

**a**) Several instants of the pulsed square electromagnetic field when the power is beyond the critical power (High power regime). (

**b**) Pulse propagation sequences during the travel inside the fused silica bulk for low power regime. (

**c**) Sequence of frames that sketches the pulse along during its travel inside the fused silica bulk from the instance ${t}_{1}=6290{\Delta}_{t}$ to ${t}_{18}=9620{\Delta}_{t}$ (where ${\Delta}_{t}=2.65765\times {10}^{-16}$ s and Courant number 0.69). The difference between two time-steps is 200${\Delta}_{t}$ in the sequence. The power of this sequence is in the threshold between low and high power regimes (critical power).

**Figure 6.**(

**a**) Subsample taken from the computational domain in which we plot the pulsed density of power and the history of the power density. (

**b**) Power density power plotted by means of iso-slices for a pulsed Bessel beam in power regime close to the critical value. the selected time corresponds to the Bessel beam entrance inside the fused silica block. We consider all nonlinear effects in this simulation. (

**c**) History of the power density for a pulsed Bessel beam. At the interface between the fused silica bulk and the air a modulation is observed in the history of power density.

**Figure 7.**(

**a**) Computational domain including clusters of spherical nanovoids, randomly located inside the fused silica bulk. (

**b**) Power history of the pulsed Bessel beam propagating inside the nanoporous block. (

**c**) History of the photo-ionized electrons density in the complete domain.

**Figure 8.**(

**a**) Power history of the pulsed Bessel beam propagating inside the fused silica block in which we place a cluster of spherical metal particles. (

**b**) History of the photo-ionized electrons density in the complete domain.

**Figure 9.**Pattern observed in the history of the photo-ionized electron history for (

**a**) voids and (

**b**) metal particles.

**Table 1.**The list of the new explicit algorithms added to the classical FDTD scheme in order to deal with the nonlinear effects. The current list of algorithms is depicted in the form of a flow chart and it is complemented by the global integration view shown in Figure 2.

Step 0: | Update 6D-Discrete Incident Electric Field Array |

↓ | |

Step 1: | Total field / Scattering field algorithm on $\overrightarrow{E}$ |

↓ | |

Step 2: | $|{\overrightarrow{Q}}_{a}^{s+\frac{1}{2}}|=|{\overrightarrow{Q}}_{a}^{s-\frac{1}{2}}|+\overline{\overline{g}}\left(s{\Delta}_{t}-\frac{(k-{k}_{0}){\Delta}_{z}}{{H}_{c}{c}_{0}}\right){|{\overrightarrow{E}}^{s-\frac{1}{2}}|}^{2}$ |

↓ | |

Step 3: | ${\overrightarrow{E}}^{s+\frac{1}{2}}={\overrightarrow{E}}^{s-\frac{1}{2}}+{C}_{B}^{s}\left(\nabla \wedge {\overrightarrow{H}}^{s}-{g}_{c}{\overrightarrow{J}}_{Metal}^{s-\frac{1}{2}}\right)$ |

↓ | |

Step 4: | ${\overrightarrow{J}}_{Metal}^{s+\frac{1}{2}}={g}_{a}{\overrightarrow{J}}_{Metal}^{s-\frac{1}{2}}+{g}_{b}\left({\overrightarrow{E}}^{s+\frac{1}{2}}-{\overrightarrow{E}}^{s-\frac{1}{2}}\right)$ |

↓ | |

Step 5: | $|{\overrightarrow{A}}^{s+\frac{1}{2}}|=|{\overrightarrow{A}}^{s+\frac{1}{2}}|+|{\overrightarrow{E}}^{s+\frac{1}{2}}|{\Delta}_{t}$ |

↓ | |

Step 6: | ${T}_{ci}^{{\Sigma}_{Time}^{s+\frac{1}{2}}}={T}_{ci}^{{\Sigma}_{Time}^{s+\frac{1}{2}}}+\frac{-{\Delta}_{t}\sqrt{{E}_{g}}|{\overrightarrow{E}}^{s+\frac{1}{2}}|{e}^{j{\Delta}_{t}\left(s+\frac{1}{2}\right)({E}_{g}+{\xi}_{ci})}}{({E}_{g}+{\xi}_{ci})\sqrt{2{m}_{vc}^{*}}\left(1-\frac{j}{\mathsf{\Omega}}|{\overrightarrow{A}}^{s+\frac{1}{2}}|\right)}$$\forall {\xi}_{ci}=0,...,{W}_{f}$ |

↓ | |

Step 7: | ${n}_{ph-{e}^{-}}^{s+\frac{1}{2}}={N}_{0}\sum _{{\xi}_{ci}=0}^{{W}_{f}}{\left|{T}_{ci}^{{\Sigma}_{Time}^{s+\frac{1}{2}}}\right|}^{2}$ |

↓ | |

Step 8: | ${n}_{f{e}^{-}}^{s+1}=\frac{2-{\Delta}_{t}\left({f}_{r}^{s+\frac{1}{2}}-{f}_{a}^{s+\frac{1}{2}}-{f}_{\eta}^{s+\frac{1}{2}}\right)}{2+{\Delta}_{t}\left({f}_{r}^{s+\frac{1}{2}}-{f}_{a}^{s+\frac{1}{2}}-{f}_{\eta}^{s+\frac{1}{2}}\right)}{n}_{f{e}^{-}}^{s}+\frac{2{f}_{d}^{s+\frac{1}{2}}{\Delta}_{t}}{2+{\Delta}_{t}\left({f}_{r}^{s+\frac{1}{2}}-{f}_{a}^{s+\frac{1}{2}}-{f}_{\eta}^{s+\frac{1}{2}}\right)}{n}_{ph-{e}^{-}}^{s+\frac{1}{2}}$ |

↓ | |

Step 9: | CPML on Magnetic Field |

↓ | |

Step 10: | Update 6D-Discrete Incident Magnetic Field Array |

↓ | |

Step 11: | Total field / Scattering field algorithm on $\overrightarrow{H}$ |

↓ | |

Step 12: | ${\overrightarrow{H}}^{s+1}={\overrightarrow{H}}^{n}-\frac{{\Delta}_{t}}{\mu}\nabla \wedge {\overrightarrow{E}}^{s+\frac{1}{2}}$ |

↓ | |

Step 13: | CPML on Magnetic Field |

↓ | |

Step 14: | ${\overrightarrow{E}}^{s-\frac{1}{2}}={\overrightarrow{E}}^{s+\frac{1}{2}}$; ${\overrightarrow{J}}_{Metal}^{s-\frac{1}{2}}={\overrightarrow{J}}_{Metal}^{s+\frac{1}{2}}$; |

${n}_{f{e}^{-}}^{s}={n}_{f{e}^{-}}^{s+1}$; ${\overrightarrow{H}}^{s}={\overrightarrow{H}}^{s+1}$; $|{\overrightarrow{Q}}_{a}^{s-\frac{1}{2}}|=|{\overrightarrow{Q}}_{a}^{s+\frac{1}{2}}|$; |

**Table 2.**List of stability conditions. Note that $Min\left[\right]$ selects the minimum value in a list.

Stability Conditions | ||
---|---|---|

All cases | ${\Delta}_{t}\le \frac{{n}_{i}}{{c}_{0}}\sqrt{\frac{1}{{\Delta}_{x}^{2}}+\frac{1}{{\Delta}_{y}^{2}}+\frac{1}{{\Delta}_{z}^{2}}}$ | |

Metal | $\mathcal{R}\left[a\right]\ge 0$ | $\mathcal{R}\left[b\right]\ge 0$ |

Kerr | $|{\tilde{\overrightarrow{E}}}_{0}|<\sqrt{\frac{{\u03f5}_{\infty}}{{D}_{f}{\chi}_{0}^{\left(3\right)}}}Min\left[\frac{1}{\sqrt{\alpha}},\frac{1}{\sqrt{1-\alpha}}\right]$ | |

Raman | ||

Photoionization | ${\sigma}_{f{e}^{-}}\ge 0$ |

Parameters | ||||
---|---|---|---|---|

Name | Symbol | Value | Unit | Ref. |

Gap energy | ${E}_{g}$ | 9 | eV | [28] |

Third-order elec-tric susceptibility | ${\chi}_{0}^{\left(3\right)}$ | ${10}^{-22}$ | $\frac{{\mathrm{m}}^{2}}{{\mathrm{V}}^{2}}$ | [28] |

Raman sinusoidal time | ${\tau}_{1}$ | 12.2 | fs | [29] |

Raman response fraction | $1-\alpha $ | 0.18 | - | [29] |

Electron recombi-nation time | ${\tau}_{r}$ | 150 | fs | [10] |

Mobility | ${\mu}_{f{e}^{-}}$ | $1.13\times {10}^{-22}$ | $\frac{{\mathrm{m}}^{2}}{\mathrm{Vs}}$ | [21] |

Density of valence electrons | ${N}_{0}$ | ${10}^{22}$ | ${\mathrm{cm}}^{-3}$ | [10] |

Electron mass reduced particle | ${m}_{vc}^{*}$ | 0.5 | ${\mathrm{m}}_{0}$ | [10] |

Work function | ${W}_{f}$ | 5 | eV | [28] |

Avalanche ioniza-tion coefficient | $\Theta $ | 10${}^{-3}$ | $\frac{{\mathrm{m}}^{2}}{\mathrm{Ws}}$ | [28] |

Raman decay time | ${\tau}_{2}$ | 32 | fs | [29] |

Kerr response fraction | $\alpha $ | 0.82 | - | [29] |

Laser characteristic time for electrons production | ${\tau}_{d}$ | 3 | ps | [10] |

Impact ionization efficiency | $\eta $ | 0.01 | - | [28] |

Fitting parameter for the spatial expansion of the valence wave function | $\mathsf{\Omega}$ | 0.8 | - | [10] |

Metal | a | $19.2+j2.7$ | PHz | |

parameters | b | $1.3-j3.9$ | PHz | [15] |

Axicon refractive index | ${n}_{a}$ | 1.45 | - | [30] |

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**MDPI and ACS Style**

Moreno, E.; Nguyen, H.D.; Stoian, R.; Colombier, J.-P. Full Explicit Numerical Modeling in Time-Domain for Nonlinear Electromagnetics Simulations in Ultrafast Laser Nanostructuring. *Appl. Sci.* **2021**, *11*, 7429.
https://doi.org/10.3390/app11167429

**AMA Style**

Moreno E, Nguyen HD, Stoian R, Colombier J-P. Full Explicit Numerical Modeling in Time-Domain for Nonlinear Electromagnetics Simulations in Ultrafast Laser Nanostructuring. *Applied Sciences*. 2021; 11(16):7429.
https://doi.org/10.3390/app11167429

**Chicago/Turabian Style**

Moreno, Enrique, Huu Dat Nguyen, Razvan Stoian, and Jean-Philippe Colombier. 2021. "Full Explicit Numerical Modeling in Time-Domain for Nonlinear Electromagnetics Simulations in Ultrafast Laser Nanostructuring" *Applied Sciences* 11, no. 16: 7429.
https://doi.org/10.3390/app11167429