Role of resonance-enhanced multiphoton excitation in high-harmonic generation of N${}_2$: A time-dependent density-functional-theory study

A minimum at $\sim$39 eV is observed in the high-harmonic-generation spectra of N${}_2$ for several laser intensities and frequencies. This minimum appears to be invariant for different molecular orientations. We reproduce this minimum for a set of laser parameters and orientations in time-dependent density-functional-theory calculations, which also render orientation-dependent maxima at 23–26 eV. Photon energies of these maxima overlap with ionization potentials of excited states observed in photoelectron spectra. Time profile analysis shows that these maxima are caused by resonance-enhanced multiphoton excitation. We propose a four-step mechanism, in which an additional excitation step is added to the well-accepted three-step model. Excitation to a linear combination of Rydberg states $c_4^{\prime }{}^1{\Sigma }_u^{+}$ and $c_3{{}^1\Pi }_u$ gives rise to an orientation-invariant minimum analogous to the “Cooper minimum” in argon. When the molecular axis is parallel to the polarization direction of the field, a radial node goes through the atomic centers, and hence the Cooper-like minimum coincides with the minimum predicted by a modified two-center interference model that considers the de-excitation of the ion and symmetry of the Rydberg orbital.

Figure 1

HHG of N${}_2$ at the equilibrium geometry calculated with the TDDFT method. The unit for laser intensity $I$ is 10${}^{14}$ W/cm${}^2$ and the pulse length is 20 optical cycles. Wavelengths are $\lambda =800$ nm [dashed (black) line and dot-dashed (green) line] and $\lambda =1064$ nm [solid (red) line].

Figure 2

HHG of N${}_2$ calculated with the TDDFT method. The laser intensity is $2\times {10}^{14}$ W/cm${}^2$ and the pulse length is 20 optical cycles. Orientation angles are $\theta =0^{\circ }$ [solid (black) line], $\theta ={45}^{\circ }$ [dashed (green) line], and $\theta ={90}^{\circ }$ [solid (red) line].

Figure 3

Ionization potential of N${}_2$ as a function of $R$ calculated with the Rydberg-Klein-Rees method [47] with spectroscopic data for N${}_2$ and N${}_2^{+}$ from Refs. [48, 49], respectively (dashed line), and ground-state vibrational wave function ${\chi }_0\left(R\right)$ computed with the sinc-function discrete variable representation method [50, 51] (solid line).

Figure 4

Ratio of the HHG intensity calculated with Eq. (13) for the ground vibrational state to that calculated by fixing the internuclear distance at $R_e$. The laser intensity is $2\times {10}^{14}$ W/cm${}^2$ and $\theta =0^{\circ }$.

Figure 5

Time profiles of the 9th, 11th, 15th, and 25th harmonics for $R=R_e=2.07a_0$. The laser intensity is $2\times {10}^{14}$ W/cm${}^2$ and $\theta =0^{\circ }$.

Figure 6

Contribution of individual initially occupied orbitals to HHG, obtained by freezing all other occupied orbitals. The laser intensity is $2\times {10}^{14}$ W/cm${}^2$ and $\theta =0^{\circ }$.

Figure 7

Effect of shifting the energy of the unoccupied 1${\pi }_g$ orbital up by 0.25$E_h$ (6.8 eV). The laser intensity is $2\times {10}^{14}$ W/cm${}^2$ and $\theta =0^{\circ }$.

Figure 8

HHG of N${}_2$ at a series of internuclear distances calculated with the TDDFT method. Distances are labeled for each spectrum. The laser intensity is $2\times {10}^{14}$ W/cm${}^2$ and the pulse length is 20 optical cycles. Wavelengths are $\lambda =800$ nm [solid (black) line]. The molecular axis is parallel to the polarization direction of the field. Positions of the minima predicted by the modified two-center interference model are shown by solid vertical (red) lines.