High-order harmonic generation in the presence of a static electric field

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.

Figure 1

(Color online) Notation $\alpha \beta m\eta$ used to label the saddle-point solutions in analogy with Fig. 2 of Ref. 22. The harmonic photon energy in units of the ponderomotive energy $U_{\mathrm{P}}$ is presented as a function of the real part of the ionization time $\mathrm{Re}{t}_0∕T=\mathrm{Re}{\phi }_0∕2\pi$ (left-hand part of the figure) and the real part of the recombination time $\mathrm{Re}t∕T=\mathrm{Re}\phi ∕2\pi$ (right-hand part). For $\eta =0$ (upper panel) we have $0<\mathrm{Re}\phi <2\pi$, while for $\eta =1$ (lower panel) it is $\pi <\mathrm{Re}\phi <3\pi$. The dashed curves correspond to the solution in the absence of the static field, while the solid curves are for ${\epsilon }_{\mathrm{S}}=0.02$. The shift of the maximal energy (cutoff) is denoted by horizontal lines. In the left-hand part of the figure, the counterpart of each right-hand-part curve identifies (with the same color) the corresponding ionization times. There are many further solutions that have ionization times $t_0$ beyond the left-hand margin of the figure. The curves have been calculated for laser-field intensity $1.2\times {10}^{14}\mathrm{W}{\mathrm{cm}}^{-2}$, photon energy $1.17\mathrm{eV}$, and for argon.

Figure 2

(Color online) Solutions $\beta m\eta =-100$ of the saddle-point equations, presented in the complex $\phi$ and ${\phi }_0$ planes, for the same laser and atomic parameters as in Fig. 1. The dashed lines correspond to the solutions in the absence of the static field while the solid lines are for ${\epsilon }_{\mathrm{S}}=0.1$ (upper panels) and ${\epsilon }_{\mathrm{S}}=0.2$ (lower panels). The corresponding values of the multi-index $\alpha \beta m\eta$ and the cutoff positions are denoted.

Figure 3

(Color online) Solutions $\beta m\eta =-101$ (upper panels) and $\beta m\eta =-111$ (lower panels) of the saddle-point equations for the same laser and atomic parameters as in Fig. 1, presented similarly as in Fig. 2, for three different values of ${\epsilon }_{\mathrm{S}}$. The corresponding values of the multi-index $\alpha \beta m\eta$ and the cutoff positions are denoted.

Figure 4

(Color online) Partial harmonic emission rates for $\beta m\eta =120$, ${\epsilon }_{\mathrm{S}}=0.1$, and for the same laser and atomic parameters as in Fig. 1. Partial contributions of the saddle-point solutions are presented by a dotted line for $\alpha =-1$ and by a dashed line for $\alpha =+1$. The coherent sum of the $\alpha =-1$ and $\alpha =+1$ contributions is presented by a blue solid line. The corresponding rate is multiplied by a factor of 0.1 in order to avoid an overlap with the uniform-approximation partial rate which is presented by a black solid line.

Figure 5

(Color online) Comparison of the results for the harmonic emission rates obtained by the uniform approximation (solid lines) with the results obtained by the numerical integration (dashed red lines), for three different values of the static electric field strength: ${\epsilon }_{\mathrm{S}}=0.1$, 0.15, and 0.25. The rates for ${\epsilon }_{\mathrm{S}}=0.1$ are multiplied by a factor of ${10}^4$, while the rates for ${\epsilon }_{\mathrm{S}}=0.25$ are multiplied by ${10}^{-4}$. The laser and atomic parameters are as in Fig. 1.

Figure 6

(Color online) The high-energy part of the harmonic emission rates obtained by the uniform approximation for the same parameters as in Fig. 5. The rates for three different values of the static electric field strength, ${\epsilon }_{\mathrm{S}}=0.1$ (solid line), 0.15 (dashed line), and 0.25 (dot-dashed line), are presented on the same scale.

Figure 7

(Color online) Comparison of the results for the harmonic emission rates obtained by the uniform approximation with the results obtained by the numerical integration (circles) for ${\epsilon }_{\mathrm{S}}=0.1$. Partial contributions of the various saddle-point solutions are presented separately and denoted by the index $\beta m\eta$ with the corresponding cutoff value given in parentheses. The laser and atomic parameters are as in Fig. 1.

Figure 8

(Color online) Same as in Fig. 7, but for ${\epsilon }_{\mathrm{S}}=0.15$.

Figure 9

(Color online) Same as in Fig. 7, but for ${\epsilon }_{\mathrm{S}}=0.25$.

Figure 10

(Color online) Quantum orbits for the harmonic order $n=60$ and for various saddle-point solutions with $\alpha =-1$ and $\eta =0$. For the upper panel we have $\beta m=-10$ (solid lines) and $\beta m=11$ (dashed lines), and for the middle panel it is $\beta m=12$ (solid lines) and $\beta m=-11$ (dashed lines), while for the lower panel it is $\beta m=-12$ (solid lines) and $\beta m=13$ (dashed lines). The static electric field strength is ${\epsilon }_{\mathrm{S}}=0.1$ (lower black curves), 0.15 (middle red curves), and 0.25 (upper green curves). The laser and atomic parameters are as in Fig. 1.

Figure 11

(Color online) Quantum orbits for ${\epsilon }_{\mathrm{S}}=0.1$ and for various saddle-point solutions with $\alpha =-1$ and $\eta =0$. For the upper panel we have $\beta m=-13$ (solid lines) and $\beta m=14$ (dashed lines), while for the lower panel it is $\beta m=15$ (solid lines) and $\beta m=-14$ (dashed lines). The harmonic order for the upper panel is $n=60$ (black curves) and $n=120$ [red (gray) curves], while for the lower panel it is $n=120$ (black curves) and $n=150$ [red (gray) curves]. The laser and atomic parameters are as in Fig. 1.