Nonlinear photoionization of transparent solids: A nonperturbative theory obeying selection rules

We provide a nonperturbative theory for photoionization of transparent solids, which consistently accounts for the selection rules related to the parity of the number of absorbed photons (odd or even). We derive closed-form analytical expressions for the photoionization rate within the two-band structure model. Our model exhibits excellent agreement with measurements for the frequency dependence of the two-photon absorption and nonlinear refractive index coefficients in sapphire and silica, two highly relevant materials for industrial applications. We demonstrate the crucial role of the interference of the transition amplitudes, which in the semiclassical limit can be interpreted in terms of interfering quantum trajectories that were disregarded in Keldysh's foundational work of laser physics [Keldysh, Sov. Phys. JETP 20, 1307 (1965)], resulting in the violation of selection rules.

Figure 1

Integration contour applied in Eq. (4). The integral along the real axis is indicated by a dashed line and the blown-up contour $\mathcal{C}$ is the solid line. The surface and contour lines refer to the value of $M\left(u\right)$.

Figure 2

Integration contour for ${\mathcal{L}}_{cv}$ in the $u$ domain is depicted. The poles of ${\mathcal{V}}_{cv}$ lie at $u=u_{\pm }$ and there is a branch cut between $-1<u<+1$ on the real axis. The contours ${\mathcal{C}}_R^{\left(r\right)}$ and ${\mathcal{C}}_R^{\left(l\right)}$ are used to connect the contours ${\mathcal{C}}_{\pm }^{\left(r\right)}$ and ${\mathcal{C}}_{\pm }^{\left(l\right)}$ at infinity.

Figure 3

Left column—comparison between the original Keldysh and the corrected Keldysh formulas of photoionization rates in ${\mathrm{SiO}}_2$ as functions of laser intensity at the wavelength of 266 nm. Solid curves correspond to cKLD formula and dashed curves reproduce the KLD formula. (a) Total PI rate. (b) Relative contribution of two- (black), three- (red), and four-photon (blue) processes to photoionization. The channel-closure region at $I\approx 70$ TW ${\mathrm{cm}}^{-2}$ is indicated by the gray shaded area, (a) and (b), and horizontal arrow, (a). Right column—comparison between theory (curves) and experiment (markers) in ${\mathrm{SiO}}_2$ (blue) and ${\mathrm{Al}}_2{\mathrm{O}}_3$ (purple). Solid and dashed curves are numerical results via cKLD and KLD theory, respectively. (c) 2PA coefficient ${\beta }_2$. Fused silica (empty circles) and crystalline quartz (filled circles) [41, 42], ${\mathrm{Al}}_2{\mathrm{O}}_3$ [43] (empty squares), and [25] (filled square). (d) Experimental and numerical nonlinear refractive index $n_2$. Experimental data are taken from [25]: empty circles for ${\mathrm{SiO}}_2$ and empty squares for ${\mathrm{Al}}_2{\mathrm{O}}_3$. Notes: Refs. [44, 45].