ISSN 1063-7850, Technical Physics Letters, 2009, Vol. 35, No. 3, pp. 271–274. © Pleiades Publishing, Ltd., 2009. Original Russian Text © P.A. Golovinski, E.M. Mikhailov, M.A. Preobrazhenskii, 2009, published in Pis’ma v Zhurnal Tekhnicheskoœ Fiziki, 2009, Vol. 35, No. 6, pp. 48–55.
Semiclassical Description of High Harmonic Generation during the Above-Threshold Ionization of Atoms P. A. Golovinski, E. M. Mikhailov*, and M. A. Preobrazhenskii Voronezh State University of Architecture and Civil Engineering, Voronezh, Russia *e-mail:
[email protected] Received September 22, 2008
Abstract—A simple analytical formula for evaluating the intensity of high-frequency emission from an atom is obtained proceeding from the classical notions about the motion of an electron in the field of the atomic core and an external laser radiation field. Calculations using this formula are performed for a hydrogen atom that occurs initially in the ground state and is exposed to the action of ultrashort radiation pulses involving various numbers of field oscillations. PACS numbers: 42.65.Ky DOI: 10.1134/S1063785009030225
The most convincing quantitative description of high harmonic generation during nonlinear (abovethreshold) ionization of atoms is based upon complete simulation of quantum dynamics [1]. However, this approach did not provide general pattern of the process and did not reveal some important details of the mechanism of harmonic generation. For this reason, partial analytical models of the process have been developed as well, which are most adequately suited to describe harmonic generation in the tunneling junction limit. Increase in a (laser) radiation field intensity leads to suppression of the potential barrier holding electron in an atom, which becomes the main mechanism of electron detachment from outer levels in the course of ionization. Nonlinear scattering of the laser radiation field (in the corresponding range of parameters) is still not studied in sufficient detail. The present study has been devoted to constructing a semiclassical model of harmonic generation during the above-threshold ionization of atoms. In this case, when the external field intensity reaches a level sufficient for the ionization, the electron motion is described by a continuous spectrum. Previously, this motion was described in a quasi-classical approximation [2, 3]. In addition, the classical character of motion of the centerof-mass of the wave packet and its diffusion smearing were taken into consideration [4, 5]. Unfortunately, the problem of determining the initial parameters of this wave packet is not yet completely solved, which pertly depreciates the value of further refinements in allowance of the wave packet smearing. In this Letter, the consideration is restricted to a classical description of the above-threshold ionization, while the quantum character of the process is taken into account only by selecting the initial energy of electrons in an atom and by averaging the electron potential in the atom.
Let us assume that the electric field in the wave of linearly polarized laser radiation is strong, so that the electron motion during the above-threshold ionization is close to one-dimensional and proceeds predominantly in the direction of the electric field (along the x axis). In order to take into account the motion along the transverse (y and z) coordinates not involved directly in the one-dimensional model, the interaction with the atomic residue is described by an effective one-dimensional Coulomb field averaged with respect to these coordinates with a potential of Ua(x) = −U0/ a + x [6], where U0 is the effective charge of the atomic core and a2 = 〈y2 + z2〉 is the cutoff parameter that characterizes the average value of the sum of square transverse coordinates at the moment of atomic decay. In what follows, we use the atomic system of units where e = m = = 1, e being the electron charge, m the electron mass, and the Planck constant. In these units, the effective atomic core charge for a hydrogen atom is U0 = 1 and the cutoff parameter is expressed through the principal (n) and the orbital (l) quantum numbers as follows [7]: 2
2
2
n 2 2 a = ----- [ 5n + 1 – 3l ( l + 1 ) ]. 2
(1)
Thus, the electron in an atom ionized under the action of laser radiation moves in the following total potential: U0 - – xF ( t ), U ( t ) = – ------------------2 2 a +x
(2)
where F(t) is the force of the alternating electric field. During the above-barrier decay of an atom, the electron released from the potential well has a characteristic lon-
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Fig. 1. (a–c) Electron displacement versus time upon detachment from the atomic core under the action of an ultrashort asymmetric pulse of laser radiation (dashed curves show the electric field variation in the laser pulse) and (d) the time series of electron acceleration in case (a).
gitudinal coordinate x0 that coincides with the position of the potential energy maximum. For a hydrogen atom with the principal quantum number n, this is possible provided that the potential energy U(t) at the point of maximum is below the initial (ground) state energy 1 E n = – --------2 . 2n
(3)
Let us assume tat the electron escape from the atom takes place at the moment of time t0 when the field strength in a laser radiation pulse reaches for the first time a level of Fmin at which the above-threshold ionization channel is opened. This moment determines the onset of classical motion of electron in the system. The value of t0 significantly depends both on the electromagnetic wave field amplitude F0 and on the shape of the incident pulse. Let us consider the process of nonlinear scattering for a pulse of the following characteristic shape: 2
t F ( t ) = F 0 exp ⎛ – ----2⎞ cos ( ωt + ϕ ), ⎝ τ⎠
(4)
where τ is the pulse duration, ω is the carrier frequency, and ϕ is the initial phase. In fact, the pulse shape prior to opening of the above-threshold ionization channel is
insignificant for the subsequent process. For this reason, the same pulse shape (of different symmetry) is used to analyze the scattering of pulses by setting various amplitudes F0 and, hence, ionization onset times. According to the potential field (2), the classical equation of electron motion upon the above-threshold ionization event is as follows: U0 x x˙˙ = – ------------------------- + F ( t ), 2 2 3/2 (a + x )
(5)
and the corresponding initial equations are x ( t0 ) = x0 ,
x˙( t 0 ) = 0.
(6)
As a particular example, we consider the action of a laser pulse on an electron occurring initially in the ground state of hydrogen atom, for which the cutoff parameter is a2 = 3 in accordance with (1). In order to evaluate the probability of ionization during the pulse rise time, we use the well-known Ammosov–Delone– Krainov formula for the probability of hydrogen ionization per unit time: 2 exp ( 2 ) F at 2 F at⎞ - exp ⎛ – --- ------ , W ( F ) = ω at -------------------- -----⎝ π F 3 F⎠ TECHNICAL PHYSICS LETTERS
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SEMICLASSICAL DESCRIPTION OF HIGH HARMONIC GENERATION F, at.u. 0.20
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Fig. 3. Spectral dependence of the intensity of a scattered pulse of laser radiation.
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initial state obeying condition (3). As can be seen, the electron can return to the atom, provided that the field makes several oscillations. However, the situation changes sharply if the pulse has an asymmetric shape such as that depicted in Fig. 2. In this case, the action of the laser pulse field alone is insufficient to compensate for the kinetic energy the electron possesses immediately upon detachment from the atom. Hence, the electron goes rather far away from the residual atom and then returns back under the action of the Coulomb field.
60 80 100 120 140 t, at.u.
Fig. 2. Electron displacement under the action of an ultrashort asymmetric pulse of laser radiation: (a) laser pulse shape F(t); (b) electron trajectory x(t) and acceleration.
where Fat = 6.1 × 109 V/cm is the atomic field strength and ωat = 4.1 × 1016 s–1 is the atomic frequency. Since the above-threshold ionization takes place at Fat/|Fmin| > 16, the characteristic ionization time is three orders of magnitude greater than the period of laser field oscillations in the visible spectral range. As was noted above, the pulse shape at t < t0 does not significantly contribute to ionization and can be ignored in description of the subsequent high-frequency generation. The character of electron motion after the ionization event depends not only on the pulse shape, but on the initial electron binding energy in the atom as well. As a result, the electron motion under the action of an initial symmetric pulse of the laser radiation field obeying the condition = 0
–∞
can be asymmetric, so that the electron in the final state would acquire a nonzero energy. Figure 1 shows the time series of electron motion for various moments of its detachment from an atom in the TECHNICAL PHYSICS LETTERS
The problem of determining the temporal variation of the scattered pulse in the adopted approximation has been solved in a rather simple manner by numerically integrating Eq. (5), after which the scattered radiation field is calculated from the known acceleration. For a number of stimulated quantum processes, it is important to know the spectral composition I(ω) of emission (characterizing the energy distribution of photons), which is calculated as follows:
∞
∫ F ( t ) dt
In the dipole approximation, the field of emission at large distances from the atom is proportional to the second derivative of the electron dipole moment with respect to the time: ε(t) ~ x˙˙. Equation (5) leads to the appearance of two components in the acceleration, the first of which describes nonlinear transformation of the electron momentum. The second component accounts for the coherent scattering of a laser pulse by the electron, for which the scattered radiation pulse does not change shape as compared to that in the incident pulse. In the nonlinear case, various components of the scattered field practically cannot be separated.
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∞
I (ω) = ε(ω) , 2
ε(ω) =
∫ ε ( t ) exp ( iωt ) dt.
–∞
(7)
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In the case of an ultrashort ionizing pulse such as depicted in Fig. 2, the electron acceleration is significant at the moment of detachment (where it is caused by the action of the propagating pulse on the particle) and at the moment of its return to the atom. Figure 3 shows the results of calculations for an ultrashort pulse of radiation with a wavelength of λ = 137 nm, a duration of τ = 170 fs, and a field amplitude of F0 = 0.92 × 1011 V/m. It was found that the nonsmooth character of the spectrum appears when the electron performs a reciprocating motion, which is possible at certain ratio of the duration and carrier frequency of the pulse [9]. If the electron does not return to the atom, the spectral density of radiation has a smooth shape similar to that of the initial radiation pulse. The pattern of emission obtained within the framework of the proposed model of nonlinear scattering of a laser radiation pulse shows that the use of a onedimensional screened Coulomb potential allows some correct features of the process of harmonic generation (inherent in the initial three-dimensional problem) to be retained. In cases where the laser fields are sufficiently strong to provide for the above-barrier ionization, the process of radiation scattering can be described in the classical approximation. This is confirmed by the comparison of the results of classical and direct quantum calculations of the electron dynamics in this range of parameters [9, 10]. In this limit, the laser pulse transformation is described by a simple formula applied to both high and low frequencies of the incident radiation, while the spectrum of harmonics is determined from the Fourier transformation of the time series of the electron acceleration. Using the results of this investigation,
it is possible to experimentally evaluate the possibility of both linear [11] and nonlinear transformation of high-intensity ultrashort pulsed laser fields propagating in simple atomic media. REFERENCES 1. K. C. Kulander, K. J. Shafer, and J. L. Krause, in Atoms in Intense Laser Fields, Ed. by M. Gavrila (Academic Press, Boston, 1992), pp. 248–875. 2. J. P. Connerade, K. Conen, K. Dietz, and J. Henkel, J. Phys. B 25, 3771 (1992). 3. M. Lewenstein, K. C. Kulander, K. J. Schafer, and P. H. Buksbaum, Phys. Rev. A 51, 1495 (1995). 4. R. Grobe and M. V. Fedorov, Phys. Rev. Lett. 68, 2592 (1992). 5. P. A. Golovinski, Laser Phys. 3, 280 (1993). 6. A. V. Kim, M. Yu. Ryabikin, and A. M. Sergeev, Usp. Fiz. Nauk 169, 58 (1999) [Phys. Usp. 42, 54 (1999)]. 7. L. D. Landau and E. M. Lifshits, Quantum Mechanics: Non-Relativistic Theory (Nauka, Moscow, 1989, 4th ed.; Pergamon Press, Oxford, 1977, 3rd ed.). 8. M. V. Ammosov, N. B. Delone, and V. P. Krainov, Zh. Éksp. Teor. Fiz. 91, 2008 (1986) [Sov. Phys. JETP 64, 1191 (1986)]. 9. Sz. Chelkowski and A. D. Bandrauk, Phys. Rev. A 65, 061802 (2002). 10. H. M. Nilsen, L. B. Madsen, and J. B. Hansen, Phys. Rev. A 66, 025402 (2002). 11. P. A. Golovinski and E. M. Mikhailov, Laser Phys. Lett. 3, 259 (2006)
Translated by P. Pozdeev
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