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MULTIPLE IONIZATION OF ATOMS THROUGH MULTIPHOTON ABSORPTION
A. l’Huillier
To cite this version:
A. l’Huillier. MULTIPLE IONIZATION OF ATOMS THROUGH MULTIPHOTON ABSORPTION.
Journal de Physique Colloques, 1987, 48 (C9), pp.C9-415-C9-425. �10.1051/jphyscol:1987971�. �jpa-
00227392�
JOURNAL DE PHYSIQUE
Colloque C9, suppl6ment au n012, Tome 48, dgcembre 1987
MULTIPLE IONIZATION OF ATOMS THROUGH MULTIPHOTON ABSORPTION
A. L'HUILLIER
Service de Physique des Atomes et des Surfaces, CEN Saclay, F-91191 Gif-sur-Yvette Cedex, France
Resume: Nous presentons les principaux r6sultats de l'ionisatipp multsple des gaz rares en champ laser intense (superieur a 10 W.cm ) . Nous envisageons differents mecanismes responsables de l'ejection de plusieurs electrons : ionisation en touche externe ou interne, en une ou plusieurs &tapes. Nous montrons les differences entre l'ionisation multiple par absorption d'un photon et l'ionisation multiple par absorption de plusieurs photons.
Abstract: We review the main aspects of ppltiple, ionization of rare gases in strong laser fields (above 10 W.cm ) . We discuss the mechanisms responsible for the multi-electron ejection: inner-shell ionization or outer-shell ionization, one-step or multi-step. W e show the differences between one-photon and multi-photon multiple ionization.
Introduction
Figure 1 presents a schematic photoelectron spectrum which illustrates some inner-shell and outer-shell ionization processes observed in one-photon absorption [1,2]. When a 100 eV-energy photon interacts with a Xe atom, it can produce ionization into the outer 5s and 5p subshells or ionization into the.inner Id-shell [3-51. Besides the main photoelectron lines (4d,5s,5p) indicated in Fig.1, one can
I X e photoelectron spectrum hV = 100eV
0 20 4 0 6 0 80
~ i n e t i c enerqy feV
).Au9er4d 5s 5~
Figure 1: Schematic photoelectron spectrum obtained in one-photon absorption at 100 eV.
L
Satellites
I
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987971
C9-416 JOURNAL DE PHYSIQUE
observe shake-up satellites and a double ionization (shake-off) continuum. 4d-inner-shell ionization can decay via an Auger process, giving rise to Auger electrons, at fixed energies. For simplicity, we have not represented double Auger and shake off relaxation processes leading to triple ionization.
The corresponding i+on spec&rum consists mainly of xe2+ ions (about 80%), and also Xe and Xe ions [6]. Doubly charged ions are produced essentially through ionization in the 4d inner shell followed by Auger decay and also through direct double ionization in the outer shell (a much weaker process at this photon energy).
Suppose that the 100 eV-energy photon is replaced by a "bunch" of infra-red or visible photons (e.g. 100 photons of 1 eV-energy).
Multiphoton ionization (MPI) experiments 17-18 show_2 that an atom exposed to a high laser intensity (abgye 10 W-cm ) can indeed absorb a 100 eV- total energy, since Xe ions are readily observed.
What becomes of the physical picture represented in Fig.1 ? Do inner-shell ionization, shake-up or shake-off processes occur in multiphoton absorption as in one-photon absorption ? The purpose of the present paper is to describe the main results obtained in multiple MPI so far and to compare them with what is known in single-photon absorption, pointing out the specific properties of multiphoton absorption. In the first section, we discuss the problem of multiphoton inner-shell ionization. In Sec.2, we concentrate on multiple multiphoton ionization in the outer shell. We consider different mechanisms responsible for the multiple ionization. Finally, in Sec.3, we discuss ionization processes such that the ion is left in an excited state (analog to shake-up).
I- MULTIPHOTON INNER-SHELL IONIZATION Ion dectection experimental results
A MPI experiment consists in focussing an intense pulsed laser into an intepction chamber filled with a gas at low pressure (typically 10 torr) and in detecting the ions [7-121, the electrons [I41 or the photons 1151 produced. The main characteristics (wavelength, pulse duration, intensity) of the different lasers used in the experiments together with the maximum charge states observed in the rare gases are indicated in table 1. Some experiments have also been performed in alkaline-earths, producing up to triply charged ions
[I61
.
Reference [71 181 191 [lo] [Ill [I21
Laser ArF Kr F Nd-YAG Dye ~ d - Y A G C O ~ Wavelength 193 nm 248 nm 532 nm 586 nm 1.06 pm 10.6 pm Photon energy 6.4 eV 5 eV 2.4 eV 2.1 eV 1.2 eV 0.1 eV Pulse length 5 ps 0.5 ps 50 ps 2 ps 50 ps 1 ns Maximum inten 1016 3 1017 1013 1014 3 1012 l0l4 -sity (W.cm-2)
xe;: xe"' Xe 5+ 6 + Results Kr6+ ICr6+
Ar N e : :
He
Table 1: Summary of the laser characteristics and experimental results
The highest charge state reported is xel1* 181
.
All the outer 5s and 5p electrons plus three 4d electrons have been removed. It means that a total energy of 1 keV has been abf+orbeqy Is inner-&ell ionization responsible for the formation of Xe , Xe,
up to Xe , as it is mostly the case in one-photon absorption as soon as the energy of the incident photon is larger than the 4d-shell ionization energy ? The present understanding of this problem is that outer-shell stripping of the atom dominates over any inner-shell ionization mechanism. A 4d electron can be removed only when most of the outer shell has been stripped. We shall give two different arguments.Atomic response as function of the field frequency [17,181
Let us describe how the response of a Xe atom changes as the frequency of the photon field increases from zero to above the Id-shell ionization energy. The 5p and 4d shells of Xe are known to show strong collective properties, giving rise to strong dipole couplings and polarizabilities [2,19,20]. One way of describing these polarization effects is to introduce an effective field taking into account the dynamic screening of the radiation field due to the electrons [2,20]. This effective field physically represents the field to which each individual electron is exposed. The response of an atom to an electromagnetic field strongly depends on the field frequency.
Figure 2 shows the variation of the effective field r(o) as a function of r for different photon energies varying from 0 to 2 Ry (27.2 eV), i.e. below and above the 5p-shell ionization energy. r(@) becomes complex when the photon energy is larger than the 5p-ionization energy. r(w) is calculated by means of the random-phase approximation
(RPA) with a local density basis set (LDRPA).
Figure 2: Variation of the effec- Figure 3: Schematic representation tive field (real part and imagi- of the laser pulse shape (see nary part) as a function of r for text)
different photon energies (0 to 2 Ry).
At low photon energy, r(@) is much lower than the external field (r, materialized by the straight line in Fig.2), in the 5p-region (up to 6 a.u.) and even more in the 4d region (0-2 a.u.). The external field is screened out from the atom; it cannot penetrate. This
C9-4 18 JOURNAL DE PHYSIQUE
lowering of the effective field will make it difficult to ionize a 5p-electron and even more difficult to ionize an inner Id-electron [I71
.
In contrast, at photon energies above the 5p-ionization energy (see Fig.21, or above the Id-ionization energy (see Ref.20), the effective field is considerably enhanced. A well-known consequence of this collective behaviour is the Id-giant dipole resonance observed in the photoionization cross-section.In conclusion, the same many-electron effects (i.e. polarization effec,ts) which, in one-photon absorption, may lead to a strong absorption of the radiation field, have the opposite consequence in MPI. The coupling of the external shell of Xe with the external field is reduced (compared to an independent-particle picture) by the screening due to the electrons. The coupling of any inner-shell with the field is even more reduced.
Competition between outer-shell and inner-shell ionization:
influence of the laser rise time
The key idea for understanding multiple MPI experiments is that an atom does not experience a single intensity (I) (i-e. a square pulse) but a continuous distribution (I(t)) (i.e. a real laser pulse, with a slow rising time, as schematized in Fig.3) 1211
.
Let us think in terms of lowest-order perturbation theory (for the radiation field): a multiphoton ionization rate is written as 01 N , where N is the minimum number of photons required for ionization and 0 is an atomic factor, called N-photon ionization generalized cross-section.Let N , N be the numbers of photons required for ionization respec?iveiy, in the Id-shell and in the 5p-shell of Xe; to,tl the
_t N
corresponding ionization times defined '0 [ I (t)
1
dt = 1 (i=0,1). Ii=I(ti) are called saturation intensities (see Fig.3).Since the Id-ionization energy (67.5 eV) is much larger than the 5p-ionization energy (12.1 eV), the 4d-ionization non-linear order No, the lifetime to and the saturation intensity I, are much larger than respectively N , t , I , as shown in Fig.3. At low intensity (<I1), ionization in &he kp-shell is much more probable than ionization in the 4d-shell (a higher order process). At t= t, , the probability for ionization of a 5p-electron is equal to unity. A Xe (neutral) atom cannot survive at a time larger than t ; therefore, it will not be exposed to an intensity greater than
.
Any high-order ionization process from the ground-state of neutrai xenon, as ionization in the Id-shell, can be detected only if its probability at t<tl is not negligible compared to the probability for 5p-ionization. Since multiphoton inner-shell ionization requires the absorption of a large number of photons, outer-shell ionization occurs before the intensity become high enough for inner-shell ionization not to be negligible.In conclusion, inner-shell ionization of Xe (of the Id-shell or any deeper sheil) seems not to be very likely because inner-shell ionization energies are much larger than the 5p-shell ionization energy and because the coupling of inner shells with the radiation field is extremely reduced due to outer-shell screening. The more probable 5p ionization process occurs very rapidly, at the beginning of the laser pulse, depleting the neutral atom population.
However, as the atom becomes more ioni~ed, thze, 5p-ionization energy increases. Denoting by t and t the Xe and Xe lifetimes, We have t1 <t,<t (see Fig.3)
.
The ad-shel? ionization energy also rapidly increases bug the influence of the screening decreases. One could expect outer-shell ionization and inner-shell ionization to become more comparable as the charge state increases, so that there may' be some 5p or 5s electrons left when the first 4d electron is removed.11- MULTIPLE IONIZATION IN THE OUTER SHELL Direct or sequential ?
Let us now concentrate on multiple ionization in the outer shell.
A basic question is whether the electrons are removed together at the same time (a direct process) or one at a time in a sequential process (successive stripping [13,211). The same argument developed in the preceding section for inner-shell ionization can be applied for direct double (or multiple) ionization. Let t, be the characteristic time for direct double ionization. Since double ionization is of much higher order than single ionization, td>>ti (see Fig.3). Direct double ionization can be detected only if its probability at t<t, is not negligible compared to the single ionization probability
.
Sequential stripping is the dominant mechanism that governs multiple ionization, because of the "slow" rising time of the laser pulse (in the picosecond range, see table 1). Different phenomena may be obtained in the future at very short pulse durations, in the femtosecond range.The problem of direct versus sequential multiple ionization has been discussed by using statistical methods [22,23] and also by treating simplified problems involving a small number of photons (2 to 4) 124-261. Calculations of double ionization cross-sections show that a double ionization cross-section is of the same order of magnitude than an equivalent single ionization cross-section. In the different cases studied [24-261, direct double ionization is found to be negligible compared to the sequential mechanism, even when the latter is of higher order.
Experimental photoelectron spectrum
Xe photoelectron spectrum hv=6.4eV
I I I I I I
Electron energy (eV)
Figure 4: Photoelectron spectrum obtained in Xe at 1 0 ~ ~ ~ . c m - ~ (from Ref .lU)
In Fig.4 we prssent a photoelectron spectrum, obtained in xenon at 193 nm, lof6 W.cm [141; All,+the beaks observed can be attributed to 5p-ionization of Xe, Xe , Xe
.
T&e hatched peaks correspond to 2,4 , S2-photon ionization of Xe. Xe can be left in two configurations
P which explains the double-peak stsycture observed in 3, 4<3-phoP6g
ionization.
The pl@n ,pea$s repr~sent, 4 , 5, 6 , 7-photon ionization of Xe leading to Xe ( P 2 , Po or PI , D2).
Finally, the shaded peaks are due to 5, 6, 7-photon ionzzation of xe2+. InC9-420 JOURNAL DE PHYSIQUE
comparison with Fig.1, there are no 5s or 4d photoelectron lines, no Auger peaks, no double (or triple) ionization continua, no satellites.
On the other hand, a 5p-ionization photoelectron spectrum consists not only of a single peak but a series of peaks spaced by the photon energy. These peaks are due to ionization with more photons than the minimum required (an "above-threshpld-ionization" process). Finally, Fig.4 shows 5p-ionization in Xe and xe2+, which confirms that multiple MPI ionization is essentially a sequential process.
Photoelectron spectroscopy experiments performed at longer wavelength reveal interesting strong field effects, for example, disappearance of low-order peaks due to ponderomotive effects; shift of the photoelectron lines at short pulse duration 1271.
Calculation of single ionization probabilities
Multiple ionization being mostly a succession of single outer-shell ionization processes, the problem reduces to calculating the 5p-ionization probabilities for the different species. Although reliable calculations exist for one-electron atoms (alkalis 1281 and
H [ 2 9 1 ) , this is not yet the case as far as rare gases and other
complex atoms are concerned. Two limits are being investigated.
(i) Weak field- short wavelength
By this "weak field- short wavelength" limit, we refer to calculations of two- and three- photon ionization of rare gases, to lowest-order in the radiation field. In these approaches, the emphasis is put on the description of the atom, the aim is to understand the influence of many-electron effects (polarization, correlation) in multiphoton ionization 130,311
.
V@ i v : , - - yQ
m m
-- - - - - n
jQ
(a) (b) (c) (dl
Figure 5: Diagrammatic representation of many-electron effects in two-photon ionization (see text)
Fig.5 gives a diagrammatic representation of some polarization and correlation effects in two-photon ionization, to first order in the Coulomb interaction. The time goes upwards. The wavy line represents the field interaction, the dashed line the Coulomb interaction. Fig.5 (a-b) describes many-electron effects related to the absorption of the first photon : Fig.S(a) is a linear screening effect (intra- and inter-shell coupling), Fig.5(b) is a ground-state correlation effect. The second photon absorption introduces additional many-electron effects, for example, monopole and quadrupole polarization (Fig.5(c)) and non-linear screening (Fig.5(d)). By non-linear screening, we mean that several electron-hole pairs can be simultaneously excited by absorbing photons. Fig.S(d) describes a two-photon ionization process via a (virtual or real) doubly-excited intermediate state.
The polarization and correlation effects represented in Fig.5(a,b) belong to the linear response of the atom to the external perturbation. The description of a non-linear process such as two-photon ionization requires also the inclusion of the non-linear
response (Fig.5(c,d)).
Without going further in the formalism, described elsewhere [31], we show in Fig.6 numerical calculations of two-photon ionization of Xe within different approximations [311. The dashed line (HF) represents an independent-electron calculation based upon an average-of configuration Hartree-Fock type of potential. The solid line is an RPAE calculation (including linear and non-linear polarization effects, see Fig.5). The open circles refer to a calculation of McGuire [32], within an independent-electron approximation and the solid circles to a multichannel quantum defect theory (MQDT) calculation of Gangopadhyay et a1 [331.
Photon energy (Ry)
Figure 6: Two-photon ionization cross-section of Xe (see text)
Comparison between these different results shows that it is indeed essential to go beyond independent-electron type of calculations and to include linear and non-linear many-electron effects in order to get a quantitative description of two-photon ionization. Moreover, a simple extension of what has been done a few years ago in one-photon ionization [2,19,20,34] (i.e.linear response) to multiphoton ionization cannot properly describe the dynamics of the non-linear process.
(ii) High intensity- long wavelength
It can be doubted that calculations such as those presented above could ever be extended to high non-linear orders and high intensities.
Attempts have been made to estimate MPI cross-section by means of statistical considerations [221, scaling arguments [211 (with calculations in H as a starting point), or using the Keldysh-Reiss approximation, which consists in taking the final state as a free-electron Volkov state (dressed by the field) 1351. Reasonable agreement is obtained with experimental results at long wavelengths (1.06 pm). This could indicate that in the high intensity-short wavelength regime, details of the atomic structure, influence of resonances on discrete states, etc.., do not matter much. The atomic
C9-422 JOURNAL DE PHYSIQUE
levels are strongly perturbed by the field, they are shifted and broadened, and the ionization probability weakly depends on the wavelength.
A promising approach developed by Kulander 1361 and recently applied to He is to solve numerically the time-dependent Hartree-Fock (TDHF) equations. There is no approximation on the relative strength of the Coulomb field and the radiation field. This numerical approach could link both perturbative and non-perturbative limits.
111- MULTIPHOTON IONIZATION WITH EXCITATION
An important question is whether multiphoton absorption can lead to the production of ions in excited states. These excited ions could rapidly (i.e. during the laser pulse duration) absorb more photons and become further ionized, thus contributing to sequential multiple ionization. They could also decay (and this would take a much longer time) by emitting radiation. This question is of course particularly relevant in the X-ray laser research perspective : can coherent radiation in the (soft) X-ray range be generated by multiphoton absorption pumping ?
Let us point out the difference between one-photon and multi-photon ionization with excitation. In one-photon absorption (Fig.11, excited ions are produced by relaxation (Fig.7(a),shake-up) or correlation (Fig.7 (b) )
.
la)
Figure 7: Diagrammatic represen- Figure 8:Wavelength dependence of tation of ionization with exci- photoelectron peaks for a linear tation processes through the polarization of the laser.
absorption of one photon (a), (a) Sr (5s); ( b ) ~r+(4d);
(b) or two photons (c), (d)
.
(c) ~r+(5p); from 1161In two-photon (or multiphoton) ionization, shake-up processes, such as the one represented in Fig.7(c) only come as a correction to the basic process leading to the formation of an excited ion, shown in Fig.7(d). This process is such that one electron is ionized by absorbing a photon; a second electron is excited by absorbing a second photon. The Coulomb interaction is not needed for the multiple excitation (the Coulomb interaction between the hole lines in Fig.7(d)
is only involved in the self-consistent readjustment of the Coulomb field during the two-electron ejection, it does not explicitly appear in the transition amplitude). One could expect multi-excitation (collective) processes to occur more easily in multi-photon absorption than in one-photon absorption, because this multi-excited state can in principle be reached directly through photon absorption.
Rare gases are not good candidates for ionization with excitation, because excited ionic states lie relatively high in the energy spectrum. It is therefore more probable that the ion be produced in the ground state (see the discussion about the laser pulse shape). No evidence for ionization with excitation (i.e. satellites) has been found in photoelectron spectroscopy (Fig.4) [14].
Better candidates for ionization with excitation are alkaline- earths which present low-lying excited continua. Electron spectroscopy measurements 1161 performed +in Strontium show peaks corresponding to ionization of Sr into Sr (512, ~r'(4d) and ~ r + ( 5 p ) , (and also ionization of these ions to Sr ) . Fig.8 represents the variation of the amplitude of some electron peaks as a function of the laser wavelength in the 558 nm- 564 nm region. Fig.: (a,b,c) corresponds respectively to three-photon iqnization into Sr (5s) (ground state), foyr-photon ionization into Sr (4d) and four-photon ionization into Sr (5p). Except for the peak labelled I1 which sogresponds to a two-photon resonance on the bound doubly excited 5p ( P ) state, the other peaks in Fig.8 have been attributed, by means oP experimental one-photon absorption UV data 1371 and configuration interaction calcu&ations [38], to thrfe-photon resonances,on autoionizing statss : 4d4f ( D ) (peak I); 4d4$( F ) (peak IV);5p5d( D J (peak III);5p5d( D l ) (peak V? : finally 5p6s ( P ) (small peak b p i d e s peak I). These states may autoionize (Fig.8(a)
) f
leading to Sr in its ground state. They may also beintermediate
states to+ a four-photon ionization process, leading to Sr (4d) (Fig.l(b)) or Sr (5p) (Fig.B(c)).Most of the peaks are observed with nearly equal amplitudes in all spectra. It shows that there is a strong configuration mixing in the intermediate (or final) doubly excited state. Such experiments are extremely useful for the spectroscopy of doubly excited states in Sr.
Most resonances in Fig.8 involve J=3 states, which cannot be reached through one-photon absorption.
Fluorescence measurements
An experimental proof of the production of excited ionic species would be the detection of the fluorescence emitted from these excited ions. However, for experimental reasons, namely a poor photon collection e f J i c i ~ p c y , one needs to,, incisease_, the atomic density from typically 10 cm up to about 10 -10 cm
.
The laser light does not interact anymore with isolated atoms but with a collective assembly of atoms. Additional processee come into play. The emitted radiation may be due to fluorescence from excited states of atoms or ions produced either directly by the laser or by various plasma recombination mechanisms; it may also be due to harmonic generation or more generally any parametric processes. Experiments [I51 essentially show harmonic generation up to very-high order, especially in light rare gases (He, Ne ,Ar) plus fluorescence gniss3pn (not yet interpretated) from very ionized species (e.g. Xe , Xe ) .Conclusion
Multiple ionization through one-photon absorption and through multi-photon absorption seems to involve different physical phenomena.
In one-photon absorption. except at low photon energy or for light elements, inner-shell ionization followed by Auger cascades is the dominant process leading to the production of multiply charged ions.
C9-424 JOURNAL DE PHYSIQUE
In contrast, in multiphoton absorption, the interaction remains mostly in the outer shell and the production of multiply charged ions is probably due to sequential stripping of the outermost shell.
More information could be gained by studying the fluorescence light emitted from the excited species. However, recent experiments in this direction show phenomena, though interesting, that go beyond the scope of multiphoton ionization itself : high-order harmonic generation and fluorescence from excited ionic species which may be due to plasma recombination.
Acknowledgements
I would like to thank Drs.I.Nenner and P-Morin for very helpful discussions and Dr.G.Wendin for a careful reading of the manuscript.
REFERENCES
0
1 -T.Aberg in "Proceedings of the International Conference on Inner-shell Ionization Phenomena and Future Applications", Eds.
R.W.Fink, S.T.Manson, J.M.Palms and P-Venugopala Rao,U.S.AEC Report No. CONF-720404, p. 1509 (1972)
2 -G.Wendin in "New Trends In Atomic Physics", Les Houches, 1982, Session XXXVIII, eds.G.Grynberg and R-Stora, Elsevier Science Publishers, B.V.(1984), p 555 : Com.At.Mol.Phys. 17, 115 (1986) 3 -V.Schmidt, S.Krummacher, F.Wuilleumier and P-Dhez, Phys.Rev.A 24,
1803 (1981)
4 -U.Becker, T-Prescher, E.Schmidt, B.Sonntag and H.E.Wetze1 Phys.Rev.A 33, 3891 (1986)
5 -W.Eberhardt, G-Kalkoffen and C.Kunz Phys.Rev.Lett. 41, 156 (1978) 6 -D.M.P.Holland, K-Codling, J.B.West and G.V.Marr J.Phys.B 12, 2465
(1979)
7 -T.S.Luk, U.Johann,H.Egger, H.Pummer and C.K.Rhodes, Phys.Rev A 32, 214 (1985)
8 -G.A.Kyrala, D-E-Casperson, P.H.Y.Lee, L.A.Jones, A.J.Taylor and G.T.Schappert, in "Proceedings of the International Conference on Multiphoton Processes IV", Boulder, Colorado, 1987.
9 -A.L'Huillier,L.A.Lompr4,GGMainfray and C.Manus, Phys.Rev.A 27, 2503 (1983)
10-M.D.Perry, E-M-Campbell, 0.L.Landen and A-Szdke, in Ref.8
11-A.L'Huillier,L.A.Lompri,G.Mainfray and C-Manus, J.Phys. B16, 1363 (1983)
12-F-Yergeau, S.L.Chin and P.Lavigne J.Phys.B 20, 723 (1987);
S.L.Chin, W.Xiong and P.Lavigne J.Opt.Soc.Am. 4, 853 (1987) 13-M.Crance, Phys.Repts.144. 117 (1987).
14-U.Johann,T.S.Luk,H.Egger and C.K.Rhodes,Phys.Rev.A 34,1084 (1986).
15-A.McPherson, G.Gibson, H.Jara, U.Johann, T.S.Luk, 1-A-Mcfntyre, K.Boyer and C.K.Rhodes, J.Opt.Soc.Am B4, 595 (1987)
16-G-Petite and P-Agostini, J-Physique 47, 795 (1986)
17-G-Wendin, L.JOnsson and A.LIHuillier Phys.Rev.Lett 56, 1241 (1986) 18-A.LIHuillier, L.Jbnsson and G-Wendin, Phys.Rev.A 33, 3938 (1986) 19-M.Ya.Amusia, Adv.At.Mol.Phys. 17, 1 (1981)
20-A-Zangwill and P-Soven, Phys.Rev.A 21, 1561 (1980)
21-P.Lambropoulos Phys.Rev.Lett. 55, 2141 (1985); P.Lambropoulos and X.Tang J.Opt.Soc.Am, 4, 821 (1987)
22-M.Crance J.Phys.B 18, L155 (1985)
23-X.D.Mu, T.Aberg,A.Blomberg and B-Crasemann, Phys.Rev.Lett. 56, 1909 (1986)
24-M-Crance and M-Aymar J-Physique 46, 1887 (1985)
25-X.Tang and P.Lambropoulos, Phys-Rev-Lett. 58, 108 (1987)
26-A.LIHuillier and G.Wendin, Phys.Rev.A, in press; see also these proceedings (contributed paper).
27-G-Petite, P.Agostini and F.Yergeau, J.Opt.Soc.Am. 4, 765 (1987) and Refs. therein.
28-M-Aymar and M.Crance, J.Phys.B 14, 3585 (1981)
29-Y-Gontier, N.K.Rahman and M-Trahin Phys.Rev.A 14, 2109 (1976) 30-A.F.Starace, Phys.Scripta T17, 221 (1987)
31-A.LIHuillier and G.Wendin, J.Phys.B 20, L37 (1987); Phys.Rev.A, 36, xxx (1987).
32-E.J,McGuire Phys.Rev.A 24, 835 (1981)
33-P.Gangopadhyay, X.Tang, P.Lambropoulos and R.Shakeshaft, Phys.Rev.A 34, 2998 (1986)
34-H.P.Kelly, Phys-Scripta T17, 109 (1987)
35-see, e.g. H.R.Reiss, J.0pt.Soc.Am.B 4, 726 (1987).
36-K-Kulander, Phys.Rev.A 35, 445 (1987); Phys.Rev.A, in press.
37-J.P.Connerade, M.A.Baig, W.R.S.Garton and G.H.Newson, Pr0c.R.
S0c.Lond.A 31, 295 (1980)
38-X.Tang and P-Lambropoulos, private communication.
