Theoretical Studies on Multiphoton Absorption of Ultrashort Laser Pulses in Sapphire

We discuss in detail the multiphoton absorption theory for wide bandgap dielectrics that we have implemented in the framework of the time-dependent
$N$
th order perturbation theory. In particular, we have carried out calculations on the
$N$
-photon absorption coefficient
$K_{N} $
and interband transition rate
$W_{N} $
for a perfect sapphire crystal (
$\alpha $
-Al
₂
O
₃
). Furthermore, we derive an expression on the laser-pulse induced seed electron density
$n_{0} $
. Application of this theory on sapphire shows that
$n_{0} $
, induced by the ultrashort laser pulse with pulse duration of
$30\,\rm {ps}$
, wavelength of
$1\,\mu \rm {m}$
, and peak intensity of
$5\times 10^{11}\rm {W}/\rm {cm}^{2}$
at the focus point, cannot lead to multiphoton-based ablation. Indeed, our calculations show that the laser pulse in the
$N=7$
order absorption process generates a conduction electron density
$n_{0} \approx 6\times 10^{13}\rm {cm}^{-3}$
that is far too small compared with the required critical density
$n_{\rm {cr}} \approx 10^{19}-10^{21}\rm {cm}^{-3}$
. This is in agreement with the extremely small value of the probability (
$P_{\rm {ind}} \approx 2\times 10^{-9}$
) that we have estimated for a valence electron to be transferred to the conduction band under the influence of the abovementioned laser-pulse conditions. Therefore, we need to seek other mechanisms than multiphoton absorption, to explain ablation in sapphire. However, we show that by using larger photon energies and smaller order
$N$
values, it is possible to induce large enough multiphoton absorption excited electron densities, which will lead to ablation in sapphire, even for the abovementioned laser beam intensity. Finally, we discuss the Keldysh adiabatic parameter
$\gamma $
and its interpretation in the case of our experimental pulse-laser setup.