On the Geltman-Hartree model of multiple ionisation by intense laser pulses

It is shown that the Geltman-Hartree model of multiple ionisation by intense laser pulses can reproduce well the experimental data for Xe of Rhodes and co-workers, provided the laser intensity spatial distribution is taken into account. Estimates for the ionisation probabilities of 5p and 5 s electrons are obtained. One of the most challenging current problems in multiphoton physics is the explanation of a relatively large number of multiply charged ions observed in the ionisation of multielectron atoms by intense picosecond laser pulses (Lompre and Mainfray 1984, Rhodes 1984, Luk et a1 1985). The existing theoretical models (Crance 1984, Boyer and Rhodes 1985, Bialynicka-Birula and Bialynicki-Birula 1985, Lewenstein 1986, Szoke 1985) are too qualitative for direct comparison with experimental data. The only exception is the model proposed by Geltman (1985) based on the Hartree model of an atom. Since in the Hartree atom electrons move independently in the selfconsistent field, the probability that qi electrons are ejected (by the laser pulse) from the ith atomic shell is (Geltman 1985) where p i is a single-electron ionisation probability (SEIP) and n, is the number of electrons which belong to the ith shell. When it is reasonable to assume that electrons are being ejected from more than one shell the probability that an ion of charge q is produced is given by where the prime over the sum sign indicates that summation goes over all qi satisfying ql+. . .+ qN = q, and N is the total number of shells involved. Then the relative abundance R4 of ions of charge q may be calculated as Rq = P " / P ' . The SEIPP, for an electron belonging to the ith shell could, in principle, be calculated from a singleelectron strong-field ionisation model. Geltman, however, fitted the Rhodes-Luk (Rhodes 1984) data for Xe atoms, taking into account only the outermost shell (5p6), and obtained the estimates for the single-5p-electron ionisation probability for different laser intensities ( 10'5-1017 W cm-') used in the experiment. The fit agrees well with the experimental data for weakly charged (small q ) ions, but does not reproduce the data for larger q. This discrepancy may be partly reduced by simultaneously fitting 0022-3700/86/090315 + 05$02.50 @ 1986 The Institute of Physics L315 L316 Letter to the Editor pSp (the SEIP for 5p electrons) and pss with the help of equation (2), in other words assuming that 5p and 5s electrons may be ejected in an arbitrary order (such an assumption is validated by the fact that in the Rhodes-Luk experiment up to eight electrons were ejected, the atoms being stripped of all 5p and 5s electrons). Even then, however, the more highly charged ionic states are not reproduced satisfactorily (Zakrzewski 1985). This may lead to the conclusion that one must go beyond the Hartree atom approximation to obtain better agreement with the data. We intend, in this letter, to show that the Geltman model may reproduce the experimental data quite well, even for highly charged ions, provided the spatial distribution of the incident laser light intensity is taken into account. To achieve this we must determine the dependence of the SEIP on the laser intensity I. Geltman (1985) has suggested that this takes a form p z = 1 -exp(-a,I“~t), where n, is the number of photons needed to ionise the atom. This form has a correct weak-field limit, but its validity in the strong-field regime, in which we are interested, is doubtful. The existing theories of strong-field ionisation (Keldysh 1964, Brandi et a l 1981, Mittleman 1983, Reiss 1984) suggest a much weaker dependence of the SEIP on the intensity in the relevant range of intensities. The most convincing, probably, is the theory developed recently by Lewenstein and co-workers (Lewenstein et a l 1985, 1986); however, the formulae for ionisation rates obtained in this theory are rather complicated and depend on the number of atomic parameters which are not easy to approximate. As we want to obtain qualitative agreement with the experimental data we have chosen to utilise the well known quasiclassical results of Keldysh (1964). These are relatively simple and easy to parametrise in a convenient way. Thus we take the ionisation rate w to be of the form w = (+ exP[-(21El/fiwL)f(y)l (3a) f( y ) = ( 1 + 1/2y2) [ y + ( y 2 + 1) 1’2] ( y 2 + 1)1’2/2y. (3b) where In equation (3a) E is the energy of the bound electron state, wL is the laser frequency, (T is a proportionality factor which depends on the atomic structure and y is the tunnelling parameter of Keldysh. Y = (WL/ e)(2mlE1/I)”2 (3 c ) where I is the laser light intensity. We can write now the rate equation for the number of ions produced N = w ( 1 N ) p I = 1 exp( w, t ) . and obtain the SEIP in the form (4) To fit the Rhodes-Luk data we make the following assumptions. (i) The laser intensity spatial dependence is Gaussian, i.e. I(x, y ) = Io exp[-(x2+ y 2 ) / r 2 ] (the light propagates along z axis). (ii) The atom density is uniform in a region much larger than r (thus we can put r = 1 for convenience). (iii) Both 5p and 5s electrons may be ejected. Thus we shall fit four parameters: the proportionality factors up, us in (3a) for 5p and 5s electrons, and the corresonding bound energies E, , E, (or rather EP/hwL, E,/fiwL). Letter to the Editor L317 (iv) As no error bars are indicated for the Rhodes-Luk data, we assume that the ionic counts are independent. Therefore the mean square deviation uLq = Nq, where Nq denotes the number of ions of charge q. With the help of formulae (1)-(4) we fit the Rhodes-Luk data for all three laser intensities I,, = lo”, 10l6 and 10’’ W cm-’ simultaneously. For each set of dimensionless parameters u,t, ust, )E,I/hw,, IE,I/huL we calculate the number of ions of charge q (integrating over the intensity profile), then the corresponding relative abundances R q = N q / NI and compare them with the data. The result of the weighted least-squares fit (following from assumption (iv)) is presented in figure 1 . Note the big improvement compared with Geltman’s fit (open circles) for all values of the intensity. This improvement is due to our taking into account the inhomogeneous spatial intensity distribution (Geltman’s results are the best fit of the SEIP for a given fixed intensity). For each intensity Geltman has fitted one parameter, thus making a three-parameter fit to the data. We have introduced four parameters; this may lead to the conclusion that the improvement is due to the number of fitted parameters. This is not the case. Firstly, if we allow for 5p and 5s electrons to be ejected and fit pp and p s for each intensity separately, we fit six parameters altogether, but the obtained fit, although better than that of Geltman (as mentioned above), is much worse than that presented in figure 1. Secondly, if we assume, following Geltman, that 5p electrons are ejected first, then the result of the fit (based on equations (1)-(4) for q = 1, . . . , 6 ) is only slightly worse than that presented in figure 1 , although only two parameters, up and IEp(/huL, are fitted. The obtained values of fitted parameters are apt = 0.52, u,t = 14.12, IEp(/hwL= 1.23 and IE,l/uL=4.44. The last two parameters may be compared with the true values of \Ei)/huL for Xe atoms. They are IEpI/h~L= 1.88 and IE,J/hwL=3.65. Taking into account an assumed oversimplified expression for the ionisation rate (1)-(4), the fitted values seen to be quite close (within about 30%) to the expected values. With the help of formula (4) we are now able (taking into account the fitted values of the parameters) to estimate the SEIP for 5p and 5s electrons for different laser intensities-see table 1. Note that the SEIP for 5s electrons grows more rapidly with intensity (because more photons are needed to ionise a 5s electron) and for the highest intensities it reaches unity. On the other hand, for I < 1013 W cm-’, the SEIP for 5s electrons is negligible. Table 1. Estimated SEIP for 5p pp and 5s p . electrons.