Erratum to: Continuum Models of Ultrashort Laser–Matter Interaction in Application to Wide-Bandgap Dielectrics

The online version of the original chapter can be found under DOI 10.1007/978-3-319-02898-9_5

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Erratum to: Chapter 5 in: M. Castillejo et al. (eds.), Lasers in Materials Science, DOI 10.1007/978-3-319-02898-9_5

On pages 107–108 in Chap. 5, the symbol ${\bar{{\rlap{-} C}}}$ was incorrectly printed and appears as “C”. The corrected text relevant to this change follows.

The models of laser light propagation in transparent media based on the non-linear Schrödinger equation (NLSE) are widely utilized for studying the processes of laser excitation of dielectrics in the regimes of modification. The NLSE is an asymptotic parabolic approximation of Maxwell’s equations [81] applicable for describing unidirectional propagation of slowly varying envelopes of laser pulses. This equation describes the self-focusing effect which manifests itself as a laser beam collapse at beam energies beyond a critical value particular for a Kerr medium with the positive non-linear refractive index n ₂. We note that for transparent crystals and glasses the n ₂ values are typically in the range of 10⁻¹⁶–10⁻¹⁴ cm²/W. To account for additional physical effects such as a small nonparaxiality, plasma defocusing, multiphoton ionization, etc., the additional terms are introduced to the scalar models based on the NLSE [18, 19, 82–84]. An important detailed review of NLSE application for various laser beam propagation conditions is given in [85].

A generalized NLSE which takes into account radiation losses for generation of electron plasma on the beam way and plasma-induced changing of the permittivity of the medium can be written in the cylindrically symmetric form as [18, 19, 82–85]:

$$ \begin{aligned} \frac{{\partial {\bar{{\rlap{-} C}}}}}{\partial z} = &\frac{i}{{2k_{0} }}T^{ - 1} \left( {\frac{{\partial^{2} }}{{\partial r^{2} }} + \frac{1}{r}\frac{\partial }{\partial r}} \right){\bar{{\rlap{-} C}}} - \frac{ik^{\prime \prime }}{2}\frac{{\partial^{2} {\bar{{\rlap{-} C}}}}}{{\partial t^{2} }} \\ &+ \frac{{ik_{0} n_{2} T}}{{n_{0} }}\left[ {(1 - f_{R} )\left| {{{\bar{{\rlap{-} C}}}}} \right|^{2} \;+\; f_{R} \int\limits_{ - \infty }^{t} {R(t - \tau )\left| {{{\bar{{\rlap{-} C}}}}} \right|^{2} d\tau } } \right]{{{\bar{{\rlap{-} C}}}}} \\ &- \frac{\sigma }{2}(1 + i\omega \tau_{c} )T^{ - 1} (n_{e} {{{\bar{{\rlap{-} C}}}}}) - \frac{1}{2}\frac{{W_{\text{PI}} (\left| {{{\bar{{\rlap{-} C}}}}} \right|)E_{g} }}{{\left| {{{\bar{{\rlap{-} C}}}}} \right|^{2} }}{{{\bar{{\rlap{-} C}}}}} \\ \end{aligned} $$

(5.3)

where ${{{\bar{{\rlap{-} C}}}}}$ is the complex envelope of the electric field strength of the light wave which is assumed to be slowly varying in time. For a Gaussian beam with cylindrical symmetry one has

$${{{\bar{{\rlap{-} C}}}}}(r,t,0) ={{{\bar{{\rlap{-} C}}}}}_{0} \exp \left( { - r^{2} /w^{2} - t^{2} /\tau_{\text{L}}^{2} - ik_{0} r^{2} /2f} \right). $$

(5.4)

Here ${{{\bar{{\rlap{-} C}}}}}_{0}^{2} = 2E_{\text{L}} /(\pi w^{2} \tau_{\text{L}} \sqrt {\pi /2} )$ is the input pulse intensity; E _L is the pulse energy; $w = w_{b} (1 + d^{2} /z_{f}^{2} )^{1/2}$ and w _b are the beam radius at the distance d from the geometric focus and the beam waist respectively; the curvature radius f and the focusing distance d are related as f = (d + z ²_f /d); z _f is the Rayleigh length; τ _L is the pulse duration (half-width determined by a decrease in the field envelope by 1/e times compared to the maximum value); k ₀ = n ₀ ω/c and ω are the wave number and the frequency of the carrier wave; n ₀ is the refractive index of the medium; c is speed of light; the parameter k″ describes the second-order group velocity dispersion; $E_{g} = E_{g0} + e^{ 2} {{{\bar{{\rlap{-} C}}}}}^{2} /\left( { 2cn_{0} \varepsilon_{0} m_{\text{r}} \omega^{ 2} } \right)$ is the effective ionization potential in the electromagnetic wave field expressed here via the electric field envelope [19]; m _r is the reduced mass of the electron and hole. Equation (5.3) takes into account the beam diffraction in the transverse direction, group velocity dispersion, the optical Kerr effect with a term corresponding to the delayed (Raman) response of the non-linear material (characterized by the parameter f _R), plasma defocusing, energy absorption due to photoionization and inverse bremsstrahlung. The operator T = 1 + (i/ω) × (∂/∂t) describes the self-steepening effects. The inverse bremsstrahlung process is described in the frames of the Drude model with the absorption cross section σ = k ₀ e ² ωτ _c /[n ²₀ ω ² ε ₀ m _e (1 + ω ² τ ² _c )]. The characteristic collisional time of electrons τ _c is a variable value dependent on electron energy and density (see comments in Sect. 5.3.2).

It should be noted that the linear term in (5.3) gives only an approximate estimation of the absorption efficiency when the free electron concentration considerably increases as the influence of the electron concentration on the absorption cross section is not taken into account. Additionally, the possibility of multiphoton absorption by free electrons is neglected which can be important at relatively high radiation intensities [86]. However, at laser beam focusing into the sample volume, the clamping effect limits the attainable intensities [57–59]. The rate equation describing generation and recombination kinetics of free electrons can be written as:

$$ \frac{{\partial n_{e} }}{\partial t} = \left[ {W_{\text{PI}} (\left| {{{{\bar{{\rlap{-} C}}}}}} \right|) + \frac{{\sigma n_{e} }}{{(1 + m_{\text{r}} /m_{e} )E_{g} }} \left| {{{{\bar{{\rlap{-} C}}}}}} \right|^{2} } \right]\frac{{n_{\text{at}} - n_{e} }}{{n_{\text{at}} }} - \frac{{n_{e} }}{{\tau_{\text{tr}} }}. $$

(5.5)

Here n _at is the atomic density in the undisturbed material matrix. Equation (5.5) takes into account free electron production in the processes of photoionization and avalanche as well as electron recombination in a trapping-like process associated with local deformations of the atomic lattice (see Sect. 5.2). The rate of photoionization W _PI can be described by the Keldysh formalism [53, 54] or in a simplified form for purely multiphoton ionization regimes when the clamping effect limits laser intensity levels to ${\gamma} \gtrsim 1$ [58].

Numerical investigations based on the NLSE allow elucidating important features of laser pulse propagation through transparent solids such as filamentation [83, 85], clamping [42, 83, 85], strong dependence of the laser energy deposition geometry on pulse duration [19, 42] for different irradiation conditions. Remarkable is the temporal dynamics of laser energy deposition into bulk dielectrics in the modification regimes [19, 42]. On an example of fused silica, it has been demonstrated that only a small fraction of the pulse leading edge, containing 10–15 % of the pulse energy, is absorbed with a high efficiency near and in front of the geometric focus. Due to strong defocusing scattering of the electron plasma generated by the pulse leading edge, the rest laser beam does not fall into the region near the geometric focus. However, as a result of the self-focusing effect, the later parts of the beam are absorbed before the geometric focus and, integrally, they generate the second region of efficient absorption (compare Figs. 11 and 12 in [42]). An important consequence of the complex correlation between self-focusing and plasma defocusing effects is that the local intensity over the whole pulse does not exceed app. 5 × 10¹³ W/cm², pointing once more to unavoidable intensity clamping. In the context of the clamping effect, the problem of the efficient delivery of laser energy into a local region inside transparent samples remains open. In particular, at high numerical apertures (${\text{NA}} \gtrsim 1$) the laser light may be concentrated to a small focal volume with consequences of strong material damage [23].

The validity of the NLSE for ultrashort laser beams focused inside transparent crystals and glasses can be broken down in many situations that is conditioned by neglecting some small terms upon its derivation from Maxwell’s equations. The condition of a slowly varying envelope limits applications of the NLSE to relatively long laser pulses. For pulse durations of order of 10 fs and shorter, either the NLSW has to be generalized with additional terms to accounting features of such extremely short pulses or, more appropriate, the complete set of Maxwell’s equations are to be used for describing light propagation through a non-linear medium. Another strong limitation imposed on using the NLSE is the requirement of unidirectionality of the light beam. This requirement makes impossible to apply the NLSE to describing tightly focused beams as well as to the cases when dense electron plasma is generated causing light scattering to large angles. Maxwell’s equations are free of the above limitations.