  We provide a nonperturbative theory for photoionization of transparent
solids. By applying a particular steepest-descent method, we derive analytical
expressions for the photoionization rate within the two-band structure model,
which consistently account for the $selection$ $rules$ related to the parity of
the number of absorbed photons ($odd$ or $even$). We demonstrate the crucial
role of the interference of the transition amplitudes (saddle-points), which in
the semi-classical limit, can be interpreted in terms of interfering quantum
trajectories. Keldysh's foundational work of laser physics [Sov. Phys. JETP 20,
1307 (1965)] disregarded this interference, resulting in the violation of
$selection$ $rules$. We provide an improved Keldysh photoionization theory and
show its excellent agreement with measurements for the frequency dependence of
the two-photon absorption and nonlinear refractive index coefficients in
dielectrics.
