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 Nth 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 (α-Al₂O₃). Furthermore, we derive an expression on the laser-pulse induced seed electron density n₀. Application of this theory on sapphire shows that n₀, induced by the ultrashort laser pulse with pulse duration of 30 ps, wavelength of 1 μm, and peak intensity of 5 × 10¹¹W/cm² 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 n₀ ≈ 6 × 10¹³ cm^-3 that is far too small compared with the required critical density n^cr ≈ 10¹⁹ - 10²¹ cm^-3. This is in agreement with the extremely small value of the probability (P_ind ≈ 2 × 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 y and its interpretation in the case of our experimental pulse-laser setup.