PHYSICAL REVIEW A VOLUME 48, NUMBER 3 SEPTEMBER 1993 Multiphoton ejection of strongly bound relativistic electrons in very intense laser fields T. Radozycki Centrum Fizyki Teoretycznej PAN, al. Lotnikom 82/$6, 02 668-Warszawa, Poland F. H. M. Faisal Fakultat fur Physik, Universitat Bielefeld, $800 Bielefeld, Federal Republic of Germany (Received 20 January 1993) We investigate multiphoton ejection probability of strongly bound electrons in relativistically intense laser fields. A solvable model of a Klein-Gordon electron bound in a finite-range separable potential and interacting with a circularly polarized plane-wave field is used for the analysis. For binding energies of the order of several keV the rates of electron ejection for m=100 eV are found to be significant at relativistic intensities but are extremely small for m=10 eV. For lower binding energies spectra are obtained for the available COz laser frequency (m=0. 117 eV) and Nd laser frequency (~=1. 169 eV) Numerical results show the stabilization efFect for both relativistic and nonrelativistic intensities and subthreshold frequencies. PACS number(s): 32.80. Wr, 32.80.Fb, 32.80.Rm, 42. 50. Hz I. INTRODUCTION Due to rapid developments [1] of laser intensity it is expected to be possible to experimentally investigate the interaction of deeply bound electrons (like those bound in highly charged ions) with very intense lasers. In view of the formidable mathematical difhculty of the neces- sary nonperturbative analysis involving real systems, it is of much interest to obtain a qualitative understand- ing of such problems from solvable models. An exactly solvable model for investigations of nonperturbative be- havior of a bound electron interacting with a strong cir- cularly polarized electromagnetic field is the well-known b-potential model, which has been introduced by Berson [2] and Manakov and Rapoport [3]. This model, however, may not be used for intensities that are so high that the energy of oscillation of the electron in the field (quiver en- ergy) becomes comparable to the rest mass energy, mc2 of the electron. In other words, in the intensity domain Equiver mc it becomes necessary to analyze the problem both rel- ativistically and nonperturbatively. For very strongly bound objects we need relativistic formulas also for the binding force itself. Recently, we have introduced [4] a relativistic model of a bound Klein-Gordon (KG) electron interacting with a circularly polarized electromagnetic field (with full mul- tipolar interaction) and obtained the exact solution of the corresponding KG equation. Since the KG equation involves the squaring of the potential, the zero-range b potential, used in the nonrelativistic model [2, 3], can- not be used here. We have, therefore, chosen a separa- ble finite-range potential that supports a discrete bound state (and the full continuum) as does the b potential in the nonrelativistic model. Note that, originally, sep- arable potentials have been introduced to study nuclear reactions [5 — 7] and more recently they have been used for the laser-atom interaction problem, in the nonrelativistic domain [8 — 12]. In this paper we investigate the total rate of ejection of deeply bound electrons, as a function of the binding energy and the field intensity, within the framework of the KG equation. II. THE BOUND-STATE MODEL POTENTIAL The KG equation of the model system is [('~t — &pl&)(&l)' — p' — ~']l@(t)) = o (2) & = &pl&)(41 and the potential functions P are taken in this work in the Gaussian form P(x) = jVpe (4) 2A where %p — — ( " )s~4 is chosen such that (&I&) = 1. (5) Note that the parameters A and Vo are arbitrary and can be chosen to imitate the bound state of interest. The eigenstate @(x,t) = e '+t4@(x) of Eq. (2) can be readily seen to satisfy the integral equation where the separable potential is defined in the form of a projection operator 1050-2947/93/48(3)/2407(6)/$06. 00 48 1993 The American Physical Society