Transition from perturbative to nonperturbative interaction in low-order-harmonic generation

We present results of ab initio numerical calculations for low-order-harmonic generation as well as calculations of the higher-order terms in the respective perturbative power-series expansions of the susceptibilities for third- and fifth-order-harmonic generation. We find that the transition from perturbative to nonperturbative interaction in these low-order nonlinear processes occurs at about ${10}^{13}\mathrm{W}/{\mathrm{cm}}^2$. Our findings confirm previous results that any deviation from the predictions of lowest-order perturbation theory indicates that the perturbative series expansion is not applicable and, if required, needs to be replaced by a nonperturbative treatment of the interaction between the atom and the field. In particular, the results also show that the observation of low-order-harmonic yields cannot be considered as a test of higher-order Kerr effects.

Figure 1

Results of ab initio numerical calculations for a low-order-harmonic spectrum generated by a driver laser pulse at a central wavelength of 1600 nm, a peak intensity of $5\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$, and a pulse length of 10 cycles. The inset shows the relative error between calculations using radial box sizes of $R_{\max }=500$ and $R_{\max }=1000$.

Figure 2

Results for perturbative power-series coefficients ${\chi }_{n\omega }^{\left(N\right)}$ for $n=1$ (dashed-dotted lines), $n=3$ (solid lines), and $n=5$ (dotted lines) as a function of $n_{max}$.

Figure 3

Results of ab initio calculations for the integrated harmonic power (solid circles with solid lines) for the (a) first, (b) third, and (c) fifth harmonics as a function of the peak laser intensity of a laser pulse of 10 cycles at a wavelength of 1600 nm. The numerical results are compared to a perturbative $I^n$ power-law fit, which is matched to the ab initio results at the lowest intensity. The insets show the relative error between ab initio results and power-law predictions with respect to the ab initio results.

Figure 4

Results for the ratio of higher-order terms to the lowest-order nonlinear term in the perturbative series expansion for (a) ${\chi }_{\omega }$, (b) ${\chi }_{3\omega }$, and (c) ${\chi }_{5\omega }$. Also shown is the ratio of the sum of all higher-order terms calculated with respect to the lowest-order term (solid lines).