PHYSICAL REVIEW A
VOLUME 56, NUMBER 5
NOVEMBER 1997
Dynamics of an intense laser-driven multiwell system: A model of ionized clusters S. X. Hu1,2* and Z. Z. Xu2 1
CCAST World Laboratory, P.O. Box 8730, Beijing 100080, People’s Republic of China Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, P.O. Box 800-211, Shanghai 201800, People’s Republic of China ~Received 9 December 1996!
2
We have presented a two-dimensional multiwell model to simulate the inertially confined ionized clusters of which unscreened ions contribute attractive potential wells to other nearby atoms. Dynamics of the intense laser-driven multiwell system has been explored. When the multiwell system is irradiated by an intense laser pulse, it will emit enhanced high-order harmonic radiation in the process of avalanche multiphoton ionization. Three facts, namely, the combined potential configuration lowering potential barrier, intermediate states resonance, and collision heating of electron with ions, are responsible for the avalanche ionization of clusters. And the enhanced high-order harmonic generation is attributed to a higher ionization potential and greater polarizability of the active electron in the clustering multiwell system. We can conclude that the cluster is a highly efficient nonlinear medium for coherent short-wavelength radiation, especially when an ultrashort (,100 fs), medium intensity laser pulse is used. @S1050-2947~97!00211-4# PACS number~s!: 42.50.Hz, 32.80.Wr, 36.40.Gk
*Mailing address: Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences, P. O. Box 800-211, Shanghai 201800, P. R. China. Electronic address:
[email protected]
et al.’s studies @9# showed that long-lived strong x-ray emission can be produced from the laser-cluster interaction. But the x-ray emission does not come from the clusters themselves; the clusters serve only to absorb the laser energy, with the bulk of the x-ray emission occurring after cluster has expanded. They attributed the strong x-ray emission to the efficient coupling of the laser energy into the cluster. The energy deposition process is dependent on the collisional heating of electrons. Efficient heating within the cluster results in rapid production of highly charged ions, followed by long-lived x-ray emission from the hot, underdense plasma that forms after the clusters have expanded. Highly charged atomic species @10# such as Xe201, Kr181, and multi-keV electron generation @11# have been observed in the recent experiments of laser-cluster interaction. The two recent models, i.e., the coherent electron motion model ~CEMM! @12# and the ionization ignition model ~IIM! @13#, were proposed to explain the productions of high charged state ions and highly energetic electron when clusters are irradiated by intense laser pulses. In the CEMM, the fieldcluster interaction can come into a regime of strong coupling in which the rate of multiple electron ejection can become comparable to the removal of a single electron. The enhanced coupling arises from the coherent motion of the fieldionization electrons behaving as a quasiparticle with a charge Ze and mass Zm e ; where e and m e are the charge and mass of the electron, respectively. The subsequent impact ionization dominates the ionization of other multiple electrons. The clustering environment provides a source of many electrons to participate in the coherent motions essential to the ionization process. But in the IIM, the electrons are treated classically in the combined fields of the ion cores and the laser. After the initial ionization events, the parent ion cores are inertially confined to the cluster since the much lighter electrons depart quickly, leaving the ion field unscreened. This results in a very large and inhomogeneous electric field in clusters. And the large field lowers the ionization barrier and enables subsequent ionization events to occur, which in turn further increases the field and lowers the ionization barrier.
1050-2947/97/56~5!/3916~7!/$10.00
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I. INTRODUCTION
Recently, much effort has gone into the understanding of the interaction of ultrashort intense laser pulses with matter @1#. Many experiments have been conducted to study the superstrong laser pulses with a low-density gas or a highdensity solid target. With the gas targets, the short-pulse coherent x ray can be generated through high-order harmonic generation ~HHG! @2–4# or through the creation of a plasma channel favored for x-ray laser gain @5#. However, gases usually exhibit poor conversion efficiency of the laser energy into x rays. Solid targets have successfully produced both photons and particles with energies up to the MeV range @6#. Due to a variety of strong absorption mechanisms in a solid density plasma, a significant fraction of the laser pulse energy can be deposited into the plasma. The hot plasma will result in production of amounts of long-lived (;nanoseconds) strong x rays. Clusters are a unique combination of both gas- and solidphase components. Solid density clusters can be formed in high-pressure gas jets, resulting from the cooling associated with the adiabatic expansion of the gas into vacuum @7#. The cooling causes the gas to supersaturate and nucleate. Under proper conditions, when the gas jet backing pressure exceeds a few atmospheres, the clusters formed in the expanding jet can be very large ~greater than 104 atoms per cluster! for gases such as Ar, Kr, and Xe. In recent years the intense laser interactions with atomic clusters have attracted much attention. Mcpherson et al. have studied the interaction of small size clusters of Kr and Xe with high-intensity (;1017 W/cm2), 248-nm laser pulses @8#. They observed anomalous prompt x-ray emission from the high charged state ions. It is assigned to the inner shell vacancies owing to the collisions of laser driven electrons with atoms in small clusters. For large size clusters, Ditmire
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© 1997 The American Physical Society
DYNAMICS ON AN INTENSE LASER-DRIVEN . . .
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Thus, the fields created by the initial ionization events ‘‘ignite’’ the cluster to undergo further ionization. These results from laser-cluster interaction indicate that the cluster is a highly efficient source of incoherent shortwavelength radiation when irradiated by an intense laser pulse. However, the high-order harmonic experiment with Ar clusters @14# and the numerical simulation @15# have shown that clusters are also a source of coherent short-wavelength radiation, especially when an ultrashort (,100 fs), medium intensity (;1014 W/cm2) laser pulse is used. This paper is an expansion of Ref. @15# and will explain the results of Ref. @14# with the suggestion of the ionization ignition model. In this paper, we present a two-dimensional multiple-potentialwell system to simulate the response of ionized clusters to an ultrashort intense laser pulse. Firstly we consider that an atom of a cluster is singly ionized with the multiphotonionization mechanism. The ionized electron departs from the atom quickly, leaving an unscreened ion. The unscreened Coulomb field of the ion will contribute an attractive potential well to its atomic neighbors. Inversely, when a sole atom is surrounded by multiple ions, it seems to fall into a combined multiwell potential field. The external laser field will drive a single active electron of the sole atom to oscillate in the combined potential field. The dynamics of the intense laser-driven multiwell system is investigated. We find that the enhanced ionization and harmonic emission can attribute to the clustering environment of the multiwell system. The ionization contribution of the electron collision with ions is checked by using a slightly elliptically polarized intense laser field. II. MULTIPLE-POTENTIAL-WELL MODEL
When an atom is embedded in an ionized cluster environment, an active electron of the atom will be subject to ~1! the Coulomb field of its parent core, ~2! the combined Coulomb field of all unscreened ions, ~3! partial shielding of the laser field by the free charges, ~4! scattering in the ionized cluster during its oscillatory motion. The ionized cluster with the size of 103 ions per cluster is held together inertially for time scales on the order of 100 fs @9# Therefore, we can reasonably assume that ions in the ionized cluster are clamped for the ultrashort (,100 fs) laser pulse considered. Namely, the ionized cluster has not expanded during the interaction. In the two-dimensional multiwell model, attractive potential wells contributed by ions distribute on these crossing points of the square grid. The atom locates on the center of the square grid. The interwell separation r can be evaluated by the solid density cluster condition. We consider the argon cluster throughout r57 a.u. The schematic diagram of the model is shown as Fig. 1~a!. Figure 1~b! is the static combined potential experienced by the active electron. It is the superposition of the unscreened Coulomb potential of the ions and that of the parent core. With the usual soft-core potential model @16#, we can formulate the combined potential as Ng
V ~ x,y ! 52
(
Ng
1
( Aq 2 1 ~ x2ir ! 2 1 ~ y2 jr ! e
i52N g j52N g
, 2 ~1!
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FIG. 1. Schematic diagram ~a! and static potential ~b! of multiwell system N g 53, r515 a.u.
where q e 51 is the soft-core potential parameter, it eliminates the singularity of the potential when x5ir and y 5 jr. The number of ions surrounding the atom is equal to (2N g 11) 2 21. The value of symbol N g means the size of the ionized cluster. When ions appear around an atom, the energy level of the atom will be affected. In the ab initio calculations we will firstly solve the eigenvalue problem of the electron motion in the static combined multiwell potential. Namely,
F
2
G
1 ]2 1 ]2 2 1V ~ x,y ! F n ~ x,y ! 5E n F n ~ x,y ! . ~2! 2 ]x2 2 ]y2
With the effective ‘‘spectral method’’ @17# for solving twodimensional ~2D! Schro¨dinger equation, we can calculate the eigenenergy and eigenstate of the electron motion in the multiwell system. Compared with that of an isolated atom, the ionization potential of the electron shifts actually to a higher value. This corresponds to the ground state shifting toward a deeper energy level, and the number of bound states of the space-confined system increases due to modification of the potential configuration. If we assume dipole coupling between the electron and the elliptically polarized laser field, the evolution of the electron wave packet from the initial ground state, i.e., C(x,y,t50)5F 1 (x,y), is described by the following time-dependent Schro¨dinger equation in atomic units (m5\5e51):
i
F
1 ]2 1 ]2 ] C ~ x,y,t ! 5 2 1V ~ x,y ! 22 ]t 2 ]x 2 ]y2 1xE 0 f ~ t ! cos v t
G
1y a E 0 f ~ t ! sinv t C ~ x,y,t ! ,
~3!
where E 0 and v are respectively the peak amplitude and frequency of the laser field, and a is the ellipticity of the driving field. The envelope of the laser pulse is sine squared, namely,
S. X. HU AND Z. Z. XU
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H
sin2 ~ p t/2T on! , 0
