## Abstract

The nonlinear collective electron dynamics inside a large heated cluster irradiated by a strong linearly polarized short laser pulse are considered in the approximation of an incompressible medium. When the incident radiation frequency is near three-photon resonance with the Mie frequency, the field inside the cluster exhibits a third harmonic with an amplitude comparable with that of the fundamental. In the same parameter range, due to shielding, the field inside the cluster at the fundamental frequency is strongly reduced with respect to the incident field. The presence of the third harmonic can lead to a strong enhancement of the production of multiply charged ions. Third-harmonic generation by a cluster under the same conditions is analyzed, too.

©2003 Optical Society of America

Clusters have unique properties compared with atoms and molecules as well as with bulk matter, both individual clusters and cluster media. This shows, in particular, in their interaction with intense laser fields. In recent experiments, the interaction of short infrared (Ti:Sapphire, 800 nm) laser pulses with large van der Waals and metal clusters consisting of 10^{3}–10^{6} atoms was extensively investigated, with regard to energy absorption from the laser field [1], production of highly charged ions [2, 3], x-ray emission [4, 5], coherent high-order harmonic generation [6], and nuclear fusion [7]. In these experiments, typical laser intensities are in the range of 10^{15}–10^{17} W/cm^{2}, pulse durations are of the order of 100 fs, and typical cluster radii between 20 and 200 Å. These phenomena and the various theoretical approaches to their description are covered by two recent reviews [8, 9].

In bulk matter, laser energy is absorbed most efficiently when the laser frequency becomes comparable with the conduction-band plasma frequency *ω*_{p}
, corresponding to volume-plasmon excitation. In finite systems such as clusters, in addition the Mie resonance, corresponding to surface-plasmon excitation, plays a major role. For a spherical system, its frequency is *ω*_{M}
=*ω*_{p}
/√3. The Ti:Sa-laser frequency is several times smaller than *ω*_{M}
for typical electron densities of ~10^{22}–10^{23} cm^{-3} [2, 9]. Nevertheless, linear excitation of the Mie resonance may occur during Coulomb explosion of the cluster, when its density is reducing, provided this stage is reached while the laser pulse is still on [2, 10]. For clusters consisting of heavy atoms such as argon or xenon, an estimate of the time the cluster takes to double its radius yields around 100 fs if the cluster is fully ionized [11]. Due to the fact that for large clusters most of the electrons produced by ionization are trapped inside, the real time of radius doubling will be several times larger. Therefore, for large rare-gas clusters (>10^{4} atoms) irradiated by short (<100 fs) laser pulses, linear excitation of the Mie resonance is questionable.

In the present paper, we suggest that *nonlinear* (three-photon) excitation of the Mie resonance can be very efficient in the specified parameter range. To this end, considering sufficiently short pulses, we concentrate on the electron subsystem inside a large cluster and neglect the motion of the ions, assuming it is just only starting. The coherent collective motion of the electron subsystem is nonlinear and generates additional field components that oscillate at odd harmonics of the fundamental frequency. Under certain conditions, we find a very strong presence of the third harmonic in the inner electric field of the cluster, with an amplitude comparable with or even higher than that at the fundamental frequency inside the cluster. Notice, that inside the cluster the field component at the fundamental frequency due to shielding is weaker than the incident laser. In particular, the presence of the third harmonic may provide a simple model to explain, at least partially, the dramatic difference between clusters and atoms with respect to the production of multiply charged ions [12, 13].

In previous models, enhanced multiple ionization of cluster atoms was attributed to (i) impact ionization by the coherently moving electron cloud modelled as a quasi-particle with total charge -*eN*_{e}
[14], or (ii) avalanche inner ionization as a result of the combined action of the external laser field and the static electric field of the locally ionized cluster (the “ionization ignition model”) [15, 16], or (iii) binary collisions of the cluster atoms or ions with hot electrons heated by either inverse bremsstrahlung or linear excitation of the Mie resonance (the “nanoplasma model”) [1].

The model we here propose is based on the idea that in clusters (like in atoms) ions are produced by the action of the local self-consistent time-dependent electric field, while binary collisions only play a secondary role. This latter assumption has also been confirmed by numerical simulations of the laser-cluster interaction (see, for example, Ref. [17]). As remarked above, in clusters the internal field can differ substantially from the external laser field. In contrast to other self-consistent-field models [14, 15], the central point of our work is the essential role of high-harmonic components of the internal self-consistent field and, in particular, of the third harmonic. Owing to the homogeneous penetration of the laser field into the cluster, this mechanism produces a *volume* effect as opposed to the *surface* effect of the ionization-ignition model, where the ions are produced mostly near the cluster surface. The mechanism we propose is not likely to produce ions in the tail of the charge distribution, that is, with charges much higher than the average ion charge. It may, however, be relevant, if not dominant, for short laser pulses, which are over before the cluster has significantly expanded. In this case, first-order resonance with the Mie frequency cannot take place.

In a strong laser field, initial ionization inside the cluster, via tunnelling or above the barrier, occurs during the first few laser cycles and leaves a hot electron plasma with a mean energy close to the ponderomotive energy *U*_{p}
=*e*
^{2}
${\mathbf{E}}_{0}^{2}$/(4*m*_{e}*ω*
^{2}) of an electron in a laser field with electric field amplitude **E**
_{0} and frequency *ω*. For a linearly polarized Ti:Sa laser with frequency *h̅ω*≈1.55eV and intensity *I*=10^{16}W/cm^{2}, we have *U*_{p}
≈600eV. The electronelectron collision frequency is
${\nu}_{\mathrm{ee}}=\frac{4\sqrt{2\pi}{e}^{4}{n}_{e}{L}_{C}}{\left(3{T}_{e}^{\frac{3}{2}}{m}_{e}^{\frac{1}{2}}\right)}$
, where *n*_{e}
=3*N*_{e}
/(4*πR*
^{3}) is the mean electron density inside the cluster with radius *R* and *N*_{e}
free electrons, and *L*_{C}
≃10 is the Coulomb logarithm. For *T*_{e}
=1keV and *n*_{e}
=5×10^{22} cm^{-3}, the mean time between collisions is *τ*_{e}
≡1/ν_{ee}≃20fs. Therefore, during the action of a 100fs laser pulse the electrons inside the cluster with the temperature *T*≃*U*_{p}
will establish a quasi-equilibrium distribution.

Description of the electron dynamics inside a cluster under the conditions described above requires the solution of the self-consistent system of Maxwell and Vlasov equations. For large clusters, the problem may be simplified if one considers the electron subsystem as a continuous medium described by a density distribution *n*_{e}
(**r**, *t*) and a velocity distribution **v**(**r**, *t*). This liquid obeys the hydrodynamical equations

where **F**(**r**, *t*)=-*e*{**E**
_{L}
(*t*)-∇[*ϕ*_{e}
(**r**, *t*)+*ϕ*_{i}
(**r**, *t*)]}-${n}_{e}^{-1}$
∇*p*(**r**, *t*) is the total local force acting on the electron liquid (in the dipole approximation). It includes the forces exerted by the external laser field, by the scalar electric potentials of the electrons and the ions, and by the equilibrium (if established) electron-gas pressure. Nonlinear phenomena in cold metal clusters irradiated by a comparatively weak field (*I*<10^{14} W/cm^{2}) were considered on the basis of Eqs. (1) in Ref. [18].

We will consider the very simplest approximation to the above hydrodynamical model, which treats the electron fluid as incompressible with the initial quasi-equilibrium electron density ${n}_{e}^{0}$
(*r*). Hence, under the action of a strong laser pulse the electron cloud oscillates without deformation, and **v**(**r**, *t*)≡**v**(*t*)=**ẋ**(*t*) with **x**(*t*) the displacement of the cloud with respect to its equilibrium position. In general, this approximation is justified only if the oscillation amplitude of the electron cloud is much smaller than the cluster radius. In consequence, the electron density and potential are *n*_{e}
(**r**, *t*)=${n}_{e}^{\left(0\right)}$
(|**r**-**x**(*t*)|) and *ϕ*_{e}
(**r**, *t*)=${\varphi}_{e}^{\left(0\right)}$
(|**r**-**x**(*t*)|), respectively, while *ϕ*_{i}
(**r**, *t*)=${\varphi}_{i}^{\left(0\right)}$
(*r*). If we multiply the second of Eq. (1) with *m*_{e}*n*_{e}
(**r**, *t*) and integrate over space, we obtain a single-particle Newton equation for the cloud displacement

where

is the total force acting on the electron cloud from the ion core. Due to spherical symmetry, the pressure *p* and the electron potential in the force **F**(**r**, *t*) do not contribute to the force (3).

Even though the solution of Eq. (2) is simplest for a neutral cluster, we will consider the more realistic case of a heated cluster, which invariably is charged. We concentrate on the case when the outer cluster-ionization degree *η*=1-*N*_{e}
/(*z*_{i}*N*_{i}
) is small (*z*_{i}
and *N*_{i}
denote the average charge and the total number of ions), which is the case for our choice of parameters. Expanding the nonlinear Eq. (2) to third order in the electron cloud displacement amplitude, we obtain the cubic equation of motion

where *k*=(*R*/*n*
_{0})|${\mathit{\text{dn}}}_{e}^{0}$
(*r*)/d*r*|_{r=R} is defined by the gradient of the electron distribution (which in a heated cluster is continuous) at the cluster radius, and *ω*_{M}
is the Mie frequency defined with respect to the initial electron density *n*
_{0}≡*z*_{i}${n}_{i}^{0}$
(before the onset of outer ionization of the cluster). The corresponding equation for a neutral cluster was derived in Ref. [19]. We have introduced in Eq. (4) a phenomenological relaxation constant Γ, which is important near resonance. It summarizes the effects of both individual electron-ion collisions [20] and collisions of electrons with the inner cluster surface [21, 22], and can, in our parameter range, be estimated as *h̅*Γ~0.1 eV ^{1}. Details of the derivation of Eq. (4), with accurate justification of the incompressible-medium approximation, are given elsewhere [24]; for a cold cluster in a weak field, a derivation can be found in Ref. [25].

For the driving laser field with pulse duration *τ* (*ωτ*≫1), we take the linearly polarized pulse **E**
_{L}
(*t*)=**E**
_{0} sin^{2}(*t*/*τ*)cos*ωt*. In this case, the excursion amplitude **x**
_{0} of the linearized Eq. (4) can be written as

with *β*=*ω*_{M}
/*ω*. With our conditions (at *ω*
^{2}≪${\omega}_{M}^{2}$
), *x*
_{0}≈*eE*
_{0}/*m*_{e}${\omega}_{M}^{2}$
≪*R* is readily satisfied, as well as
${x}_{0}\ll {r}_{D}=\sqrt{\frac{3{T}_{e}}{\left(4\pi {e}^{2}{n}_{e}\right)}}$
, so that, on the scale of the Debye shielding length *r*_{D}
, the electron distribution ${n}_{e}^{0}$
(*r*) varies smoothly across the cluster radius.

The dimensionless derivative k, defined below Eq. (4), of the electron density at the cluster boundary can be estimated from the solution of the self-consistent system of the Poisson and Boltzmann equations inside the cluster,

with ${n}_{i}^{0}$
(*r*)=${n}_{i}^{0}$*θ*(*R*-*r*), and subject to the boundary conditions *ϕ*
_{0}(*R*)=*Q*/*R* and *ϕ*′_{0}(*R*)=-*Q*/*R*
^{2} with *Q*=*ηez*_{i}*N*_{i}
the cluster charge. For *R*=50 Å (~10^{4} rare-gas atoms), *z*_{i}${n}_{i}^{0}$
=10^{23} cm^{-3}, and *T*_{e}
=500 eV, and for 0.1≤*η*≤0.3 we obtain from Eqs. (6) 2.06≤*k*≤2.35. The time-dependent envelopes *E*_{ω}
(*t*) and *E*
_{3ω} (*t*) of the electric field inside the cluster at frequencies *ω* and 3*ω*, respectively, can then be found from a numerical solution of Eq. (4) by Fourier analysis of the cluster dipole moment **d**(*t*)=-*eN*_{e}
**x**(*t*) and the internal electric field **E**(*t*)=**E**
_{L}
(*t*)-**d**(*t*)/*R*
^{3}.

The calculated peak values ${E}_{\omega}^{0}$
and ${E}_{3\omega}^{0}$ of the envelopes *E*_{ω}
(*t*) and *E*
_{3ω}(*t*) of the internal electric field at the fundamental frequency and at the third harmonic, respectively, along the laser polarization axis are presented in Fig. 1. As can be seen from the figure, for laser frequencies near the three-photon resonance with the Mie frequency the third-harmonic amplitude inside the cluster can be comparable with the fundamental, or even a few times higher. The physical origin of such a high conversion efficiency from the fundamental to the third harmonic is the finite size of the cluster, which provides the nonlinear term in the restoring force of the electron-ion interaction, cf. Eq. (4). On the other hand, the amplitude of the field inside the cluster at the fundamental frequency is much lower than the amplitude of the incident laser field due to the shielding effect in a plasma with overcritical density. However, the third harmonic is only produced substantially if, in addition, the resonance frequency of the collective electron oscillations is close to three times the laser frequency, viz. *ω*_{M}
≈3*ω*. For a Ti:Sa laser, this condition can readily be satisfied. Note, also, the intensity-dependent shift of the resonance maximum of the third-harmonic amplitude presented in Fig. 1. The fact that this shift of the resonance maximum (which is proportional to intensity in the perturbation limit [26]) is towards *β* values larger than three can be traced to the negative sign of the nonlinear (cubic) term on the left-hand side of Eq. (4). In effect, this reduces the resonance frequency with respect to the reference Mie frequency.

At least two experimentally detectable manifestations of the nonlinear effects discussed above can be expected. First, the third harmonic inside the cluster gives rise to nonlinear light scattering. For a single cluster, in lowest-order perturbation theory and near the third-order resonance, the cross section for scattering into the third harmonic is

where *r*
_{0}=*e*
^{2}/*m*_{e}*c*
^{2} is the classical electron radius, *a*_{B}
is the Bohr radius, *I*_{at}
≈3.51×10^{16} W/cm^{2} is the reference atomic intensity, and *β*_{r}
denotes the intensity-dependent position (close to three) of the resonance maximum. Notice the distinctive dependence on the inverse fourth power of the cluster radius and the proportionality to ${N}_{e}^{2}$
, which reflects the coherent collective motion of the electron cloud. A qualitatively similar result was obtained for the third-order Kerr susceptibility of a cold metal cluster on the basis of a simple analytical model [27]. For the same parameters as above and at *I*≳10^{16} W/cm^{2}, in the resonant case (*β*=*β*_{r}
), we have *σ*(*ω*→3*ω*)~10 Mb^{2}. This corresponds to 10^{4} third-harmonic photons per cluster or about one photon per cluster ion during a 100-fs laser pulse^{3}. Off resonance (*β*
^{2}-${\beta}_{r}^{2}$
≃1), the cross section is reduced by one order of magnitude.

Second, owing to the presence of the inner electric field of frequency 3*ω* the probabilities of multiphoton processes, including excitation of bound-bound and bound-free transitions in multiply charged ions, may be much higher than in the field of the fundamental frequency alone, even if the amplitude of the third-harmonic field is smaller than the amplitude of the fundamental (but still comparable). As an illustration, we estimate the multiphoton-ionization probability by the third harmonic, *w*
_{3ω}=1-exp(-${\int}_{0}^{\tau}$
*W*
_{ion} [3*ω*, *E*
_{3ω}(*t*)]d*t*) [Fig. 2(a)], and compare it with the corresponding ionization probability *w*_{ω}
by the fundamental inside the cluster [Fig. 2(b)], for several charge states of argon, for an incident radiation intensity of 3×10^{16}W/cm^{2} (cf. Fig. 1). These estimates are based on the Keldysh-like ionization rate *W*
_{ion}(*ω*,*E*_{ω}
) with pre-exponential factor for hydrogenlike atoms, which applies both in the multiphoton and in the tunnelling regime, and on the calculated time-dependent envelopes of the internal electric field inside the cluster at the fundamental frequency and at the third harmonic. Near resonance, we observe a dramatic difference between *w*
_{3ω} and *w*_{ω}
. Even in absolute terms, the probability *w*
_{3ω} [Fig. 2(a)] for Ar^{4+} reaches unity near resonance. For all of the charge states of Fig. 2, according to the values of the Keldysh parameter *γ* the multiphoton regime applies, except near the resonance maximum where *γ*≲1. As expected, the enhancements increase with *γ*, i.e., for higher charge states. So, for our parameter range the third-harmonic field is more important for the ion production than the field at the fundamental frequency. These results may also explain, at least qualitatively, the enhancement of the yield of multiply charged ions that was observed in the laser-cluster experiments [12, 13].

In conclusion, we have shown that the electric field inside a laser-irradiated large heated cluster has a very substantial third-harmonic component if the tripled laser frequency is near-resonant with the Mie frequency. This can be readily achieved for a large cluster and a Ti:Sa laser. The presence of the third harmonic can enhance various processes inside the cluster, such as ionization into multiply charged states. This may be especially important for sufficiently short laser pulses. Third-harmonic generation is also strongly enhanced near resonance.

This work was supported by Deutsche Forschungsgemeinschaft (Project No. 436 RUS 113/676/0-1(R)), and by the Russian Foundation for Basic Researches (grants No. 02-02-04007NNIO-a, No. 02-02-16678 and No. 03-02-17112).

## Footnotes

^{1} | In linear approximation, for a cold metal cluster the proper system of the Vlasov and Maxwell equations may be solved almost analytically; cf. [23] and references therein. This yields the Mie resonance width Γ~ν/R, where ν is the typical electron velocity, and identifies Landau damping, properly modified for finite systems such as clusters, as the relevant damping mechanism. In this case, it can be treated as the result of collisions of electrons with the inner cluster surface, which leads to irreversible losses. |

^{2} | This corresponds to a third-order nonlinear susceptibility of the cluster substance (per unit volume) o${\mathrm{f}}_{\chi}^{\left(3\right)}$(3ω)~10^{-13} esu (in cgs units), which is in agreement with typical values for solids, which are in the range 10^{-12}-10^{-15} esu. |

^{3} | Nonlinear light scattering off cold metal clusters was considered in a phenomenological approach [28]. In contrast, in our approach the nonlinear susceptibility of the cluster substance is derived on the basis of simple physical model, which allows us to estimate the magnitude of the effect. |

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