Reexamination of the improved strong-field approximation: Low-energy structures in the above-threshold-ionization spectra for short-range potentials

The improved strong-field approximation (ISFA) is a version of the strong-field approximation which takes into account an additional interaction of the ionized electron with the parent ion within the first Born approximation. The ISFA describes well the middle- and high-energy parts of the electron spectra in the above-threshold ionization process. We show, using an example of a short-range potential, that the ISFA is able to describe the low-energy structure in the energy spectra if it is calculated without additional approximations. We introduce two different forms of the $T$-matrix element which are appropriate for application of two widely used approximations: the saddle-point approximation [i.e., its more advanced version, the uniform approximation (UA)] and the pole approximation (PA). We show that both the PA and UA are not able to describe the low-energy structure. Furthermore, the UA describes better the plateau of the spectrum than the PA. We also identify the origin of a very-low-energy structure; it is connected to the laser-free (i.e., without exchange of the laser photons) electron forward scattering.

Figure 1

The differential detachment rates of F${}^{-}$ as functions of the electron energy in units of $U_{\mathrm{P}}$. The electron emission angle is $\theta =0^{\circ }$ and the wavelength and intensity of the linearly polarized laser field are 1800 nm and $1.3\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$, respectively. Comparison of the exact result obtained calculating the five-dimensional integral (5D) with the result obtained using the uniform approximation (UA).

Figure 2

Same as in Fig. 1 but for the comparison of the result obtained using the pole approximation (PA) and the corresponding “exact” result (Ex), as described in the text.

Figure 3

Same as in Fig. 2 but for the comparison of the result obtained solving the principal value integral for all channels with $n<n_0$ (PV$<$) and the “exact” result.

Figure 4

Same as in Fig. 2 but for the comparison of the result obtained solving the principal value integral for all channels with $n\ge n_0$ (PV$\ge$) and the “exact” result.

Figure 5

Same as in Fig. 2 but for the comparison of the result obtained solving the principal value integral for all channels (PV) and the “exact” result.

Figure 6

Same as in Fig. 1 but for the comparison of the results obtained using two different “exact” methods of calculations: EX, summation over all intermediate photon channels and integration over all ionization times and the intermediate electron momenta; 5D, five-dimensional integral over the ionization and travel times and over the intermediate electron momenta. The low-energy part of the spectrum ($E_{\mathbf{p}}\le 4U_{\mathrm{P}}$) is presented in order to better show the contribution of the $n^{\prime }=n$ term of the $T$-matrix element, which is described by the amplitude $F_n$ (dashed green line with squares). See the text for explanation.

Figure 7

Comparison of the results obtained using the ISFA calculated using the five-dimensional integral (5D, black solid curve) with the results obtained with the direct SFA alone (green dashed curve) and the coherent sum of the SFA and ISFA (5D+ISFA, red dotted curve with circles). The negative ion and laser parameters are as in Fig. 1.