A theory of high-order harmonic generation by diatomic molecules is introduced. Various versions (with or without the dressing of the initial and/or final molecular state) of the molecular strong-field approximation are investigated. Using examples of homonuclear diatomic molecules such as ${\mathrm{H}}_2$, ${\mathrm{N}}_2$, and ${\mathrm{O}}_2$, it is shown that clear two-center interference minima in the harmonic spectra as a function of the molecular orientation appear only if the final molecular state is undressed. For ${\mathrm{H}}_2$ the positions of these minima are in agreement with the ab initio numerical results. Physically, the returned electron wave packet recombines into a molecular orbital, which is a linear combination of the atomic orbitals having different parities. The interference minima in the harmonic spectrum are caused by the destructive interference of the corresponding partial recombination amplitudes. In accordance with this, we have derived an interference minima condition which is valid for arbitrary homonuclear diatomic molecules.

We consider high-order harmonic generation by a linearly polarized laser field and a parallel static electric field. We first develop a modified saddle-point method which enables a quantitative analysis of the harmonic spectra even in the presence of Coulomb singularities. We introduce a classification of the saddle-point solutions and show that, in the presence of a static electric field which breaks the inversion symmetry, an additional classification number has to be introduced and that the usual saddle-point approximation and the uniform approximation in the case of the coalescing saddle points have to be modified. The theory developed offers a simple and accurate explanation of the static-field-induced multiplateau structure of the harmonic spectra. The longer quantum orbits are responsible for a long extension of the harmonic plateau, while the larger initial electron velocities are the reason of lower harmonic emission rates.

Electron-ion radiative recombination assisted by a bicircular laser field that consists of two circularly polarized fields counterrotating in the $xy$ plane and having the frequencies $r\omega$ and $s\omega$, which are integer multiples of the fundamental frequency $\omega$, is considered using the $S\text{-matrix}$ theory. The energy and polarization of soft x rays generated in this process are analyzed as functions of the incident electron energy and incident electron angle with respect to the $x$ axis. Numerical results for the process of direct recombination of electrons with ${\text{He}}^{+}$ ionic targets are presented. Abrupt cutoffs of the plateau structures in the emitted x-ray energy spectra are explained by classical analysis. Simpler or more complex oscillatory structures in the spectrum may appear as a result of the interference of a different number of classical orbits. Symmetry analysis and the numerical results show that the x-ray power spectrum and ellipticity are invariant with respect to a rotation of the incident electron momentum by the angle $2\pi /(r+s)$. We have visualized this by presenting the logarithm of the differential power spectrum and polarization of the emitted x rays in false colors as functions of the incident electron angle and the x-ray energy. We have also shown that the change of the relative phase of the bicircular field is equivalent to the change of the incident electron angle. By controlling this relative phase it is possible to control the polarization of the emitted soft x rays.

High-order harmonic generation by a bicircular field, which consists of two coplanar counter-rotating circularly polarized fields of frequency $r\omega$ and $s\omega$ ($r$ and $s$ are integers), is investigated for a polyatomic molecule. This field possesses dynamical symmetry, which can be adjusted to the symmetry of the molecular Hamiltonian and used to investigate the molecular symmetry. For polyatomic molecules having the $C_{r+s}$ symmetry, only the harmonics $n=q(r+s)\pm r,q=1,2,...,$ are emitted having the ellipticity ${\varepsilon }_n=\pm 1$. We illustrate this using the example of the planar molecules ${\mathrm{BH}}_3$ and ${\mathrm{BF}}_3$, which obey the $C_3$ symmetry. We show that for the ${\mathrm{BF}}_3$ molecule, similarly to atoms with a $p$ ground state, there is a strong asymmetry in the emission of high harmonics with opposite helicities. This asymmetry depends on the molecular orientation.

We investigate the laser-wavelength dependence of high-order above-threshold ionization and high-order harmonic generation rates. For both processes we have found pronounced enhancements in the spectra for those wavelengths for which the channel-closing condition is satisfied. These enhancements are explained in two complementary ways. One is in terms of the constructive interference of a large number of long quantum orbits, while the second interpretation is based on threshold anomalies. There are two types of these enhancements, which leave very different footprints in the spectra. When the ground-state parity of the atom changes, the two types are interchanged.

We investigate high-order harmonic generation of non-planar ammonia molecules by a bicircular field, which consists of two coplanar counter-rotating circularly polarised fields of frequencies ω and 2ω. This field possesses dynamical symmetries, which can be used together with the symmetry of the molecular Hamiltonian to obtain the selection rules for harmonic emission. We show analytically and confirm by numerical calculations that circularly polarised harmonics of order n = 3q ± 1, with q integer, as well as linearly polarised harmonics of order n = 3q are emitted. The presence of well-separated both linearly and circularly polarised harmonics in the same spectrum is unique to non-planar molecules.