Competition between ultraviolet and infrared nanosecond laser pulses during the optical breakdown of KH₂PO₄ crystals

Abstract

This study addresses the initiation of laser-induced breakdown of dielectric materials in the nanosecond regime under multi-wavelength conditions. In particular, the competition between multi-photon absorption and electronic avalanche as ionization mechanisms in KDP crystal is studied. Since they are both dependent on the laser frequency and intensity of incident radiations, we carried out two experiments: in mono-wavelength configuration at 1,064 nm and in multi-wavelengths configuration applying the simultaneous mixing of 1,064 and 355 nm radiations with various fluence ratios. To interpret experimental data, a model based on heat transfer and which includes ionization processes has been developed for both configurations. The comparison between experiments and modeling results first indicates that avalanche can be responsible for optical breakdown at 1,064 nm. Then, the study underlines the existence of a coupling effect in the multi-wavelength configuration where multi-photon absorption and electronic avalanche both contribute to the breakdown. From a general point of view, the model accounts for the experimental trends and particularly reveals that the electronic recombination timescale may have an important role in the scenario of nanosecond laser-induced breakdown.

Appendices

Appendix 1: DMT model: general presentation

The DMT approach has been developed at 3ω by Dyan et al. [21]. KDP is supposed to contain nanometer-scale defects that may initiate laser damage when temperature exceeds a given critical temperature. The model solves the 3D heat equation of these nanometer plasma balls whose absorption efficiency is described through the Mie theory [38], introducing both wavelength and size dependence. The plasma optical indices mandatory to Mie theory equations are then evaluated within the Drude model [30]. The resolution of the Fourier’s equation [39, 40] gives the defect temperature evolution as a function of its radius a. Thus in the mono-wavelength case, the temperature elevation θ_(ω) of the defect reads:

$$ \theta_{(\omega)} = Q_{\rm abs}^{(\omega)} I_{(\omega)} \frac{ \xi \sqrt{4 \kappa_{\rm KDP} \tau_{L}}} { 4 \lambda_{\rm KDP}} $$

(8)

where Q ^(ω)_abs , I _(ω) and τ_L are the absorption efficiency, the laser intensity and the pulse duration for the ω frequency. κ_KDP and λ_KDP are the diffusivity and the thermal conductivity of the KDP. ξ is a function taking into account the material properties [21]. The critical fluence F _c necessary to reach the critical temperature T _c for which a first damage site occurs can be written as [21]:

$$ F_{\rm c} = \gamma \frac{T_{\rm c} - T_{0}}{Q_{\rm abs}(a_{\rm c})} \sqrt{\tau_{\rm L}} $$

(9)

where γ is a factor depending on the material properties, T ₀ is the room temperature and a _c is the defect critical radius which is associated with T _c, usually close to 100 nm. Assuming that a damage site appears as soon as the critical temperature T _c is reached, and then considering that any fluence F _(ω) activates the precursor defects in a defined size range [17, 20, 21, 26, 41], the damage density ρ_dam as a function of the fluence F _(ω) can be finally extracted from the following expression:

$$ \rho_{\rm dam}(F) = \int\limits_{a_{-}(F)}^{a_{+}(F)} D_{\rm def}(a).{\rm d}a $$

(10)

where [a ₋(F), a ₊(F)] is the range of defects size for which T = T _c, D _def(a) is the density size distribution of absorbers assumed to be:

$$ D_{\rm def}(a) = \frac{C}{a^{p+1}} $$

(11)

where C and p are adjusting parameters. This distribution is consistent with the fact that the more numerous the precursors, the higher is the damage probability. By integrating Eq. (10), we thus obtain:

$$ \rho_{\rm dam}(F) = \frac{C}{p} \left( \frac{1}{a_{-}^{p}(F)} - \frac{1}{a_{+}^{p}(F)} \right). $$

(12)

As $\frac{1}{a_{+}}^{p}(F)$ is smaller than $\frac{1}{a_{-}}^{p}(F),$ only absorbers whose size is smaller than the critical size a _c contribute significantly to ρ_dam(F). Also, it can be shown that in the neighborhood of F _c and for a < a _c, the damage fluence can be written as $F \propto \frac{1}{a}.$ It results that the damage density ρ_dam evolves as a function of the laser fluence F as:

$$ \rho_{\rm dam}(F) \simeq \frac{C}{p} \frac{1}{a_{-}^{p}(F)} \propto \frac{C}{p} F^{p}. $$

(13)

Thus, ρ_dam evolves as a power law function of the fluence, where C and p can be determined to fit the experimental data.

Appendix 2: Derivation of t _heat

2.1 Mono-wavelength case

The evolution of the electronic density in the conduction band is given by the following rate equation:

$$ \frac{\partial{n_{\rm cb}}}{\partial{t}} = \sigma_{{\rm m}} \Upphi_{\rm ph}^{\rm m} n_{\rm vb} - \frac{n_{\rm cb}}{\tau_{\rm r}} $$

(14)

where σ_m is the m-photon absorption cross section, $\Upphi_{\rm ph}$ is the photon flux at 1ω (= $\frac{F}{\hbar \omega \tau_{\rm L}}$) and τ_r is the recombination time. Note that σ_m may account for assisted transitions with states located in the bandgap as suggested by Carr et al. [6] in their previous studies.

By integrating Eq. (14), assuming that for t = 0 n _cb = 0, the electronic density n _cb reads as:

$$ n_{\rm cb}(t) = n_{\rm vb} \times (\sigma_{m} \Upphi_{\rm ph}^{\rm m}) \tau_{\rm r} \left[ 1- {\rm e}^{-\frac{t}{\tau_{\rm r}}} \right] .$$

(15)

Then, for t = t _aval, which is the criterion for the avalanche to trigger, associated with the minimum density n _aval (one electron in the defect volume), Eq. (15) becomes:

$$ t_{\rm prod} = - \tau_{\rm r} \times ln\left( 1 - \frac{n_{\rm aval}}{n_{\rm vb}} \frac{(\hbar \omega \tau_{\rm L})^{\rm m}}{\sigma_{m} F^{\rm m} \tau_{\rm r}} \right). $$

(16)

Also, the heating time t _heat is defined by the following relation:

$$ t_{\rm heat} = \tau_{\rm L} - t_{\rm prod}. $$

(17)

To simplify, we assume $K_{1} = n_{\rm vb} \frac{\sigma_{m} F^{\rm m}}{(\hbar \omega \tau_{\rm L})^{\rm m}},$ which results in:

$$ t_{\rm heat} = \tau_{\rm L} + \tau_{\rm r} \times ln\left( 1 - \frac{n_{\rm aval}}{K_{1} \tau_{\rm r}} \right). $$

(18)

Thus, according to Eq. (18), there exists a heating time such as 0 < t _heat < 6.5 ns if the condition $ln\left( 1 - \frac{n_{\rm aval}}{K_{1} \tau_{\rm r}} \right) < 0$ is verified. At the same time, it is possible to define the fluence, referred to as F _aval for which t _heat > 0. F _aval is then defined by Eq. (19):

$$ F_{{\rm aval}} > \frac{1}{{\left[ {\left( {1 - {\rm e}^{{ - \frac{{\tau _{\rm L} }}{{\tau _{\rm r} }}}} } \right)\frac{{n_{{\rm vb}} }}{{n_{{\rm aval}} }}\tau _{\rm r} } \right]^{{\frac{1}{\rm m}}} }} \times \frac{1}{{\hbar \omega \tau _{\rm L} }} .$$

(19)

2.2 Multi-wavelength case

Since 1ω and 3ω are assumed to both produce the electronic density in the conduction band, it results in the following rate equation:

$$ \frac{\partial{n_{\rm cb}}}{\partial{\rm t}} = (\sigma_{m} \Upphi_{\rm ph}^{\rm m} + \sigma_{3} \Upphi_{\rm ph}^{3} ) n_{\rm vb} - \frac{n_{\rm cb}}{\tau_{\rm r}}. $$

(20)

The determination of t _heat has been previously proposed in Appendix 2.1. When now considering two radiations at the same time, with a mean pulse duration $\overline{\tau_{\rm L}}$ corresponding to $\overline{{\tau _{\rm L} }} = \frac{{\tau _{\rm L}^{{1\omega }} + \tau _{\rm L}^{{3\omega }} }}{2} ,$ the mathematical derivation remains the same and t _heat becomes:

$$ t_{\rm heat} = \overline{\tau_{\rm L}} + \tau_{\rm r} \times ln\left( 1 - \frac{n_{\rm aval}}{K_{1}^{'} \tau_{\rm r}} \right) $$

(21)

where K ^′₁ is defined by:

$$ K_{1}^{'} = n_{vb} \left( \frac{\sigma_{m} F_{1\omega}^{m}}{(\hbar \omega_{1} \tau_{\rm L}^{1\omega})^{\rm m}} + \frac{\sigma_{3} F_{3\omega}^{3}}{(\hbar \omega_{3} \tau_{\rm L}^{3\omega})^{3}} \right) .$$

(22)