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Optics Express

Optics Express

  • Editor: J. H. Eberly
  • Vol. 5, Iss. 7 — Sep. 27, 1999
  • pp: 144–148
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Magnetic-field effect in atomic ionization by intense laser fields

J.R. Vàzquez de Aldana and Luis Roso  »View Author Affiliations


Optics Express, Vol. 5, Issue 7, pp. 144-148 (1999)
http://dx.doi.org/10.1364/OE.5.000144


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Abstract

Numerical integrations of the two-dimensional Schrödinger equation that describes a flat atom interacting with an intense and linearly polarized laser field are presented. Simulations show the influence of the drift that is due to the magnetic field in situations in which a strong dichotomy of the wave function would otherwise have been expected.

© Optical Society of America

The absence of systematic nonperturbative models to describe intense atomic laser field ionization has stimulated the development of purely numerical methods [1

1. J. H. Eberly, R. Grobe, C. K. Law, and Q. Su, “Numerical experiments in strong and super-strong fields,” in Atoms in Intense Laser Fields, M. Gavrila, ed. (Academic, San Diego, Calif., 1992).

,2

2. K. C. Kulander, K. J. Schafer, and J. L. Krause, “Time-dependent studies of multiphoton processes,” in Atoms in Intense Laser Fields, M. Gavrila, ed. (Academic, San Diego, Calif., 1992).

,3

3. Q. Su, A. Sanpera, and L. Roso-Franco, “Atomic stabilization in the presence of intense laser pulses,” Int. J. Mod. Phys. B 8, 1655 (1994). [CrossRef]

]. For example, stabilization of atoms against ionization by an intense high-frequency laser beam has been widely studied over the past decade [4

4. M. Gavrila, “Atomic structure and decay in high frequency fields,” in Atoms in Intense Laser Fields, M. Gavrila, ed. (Academic, San Diego, Calif., 1992), and references therein.

,5

5. M. Protopapas, C. H. Keitel, and P. L. Knight, “Atomic physics with super-high intensity lasers,” Rep. Prog. Phys. 60, 389–486 (1997), and references therein. [CrossRef]

]. All the approaches to stabilization with a linearly poralized laser field (Floquet theory in the Kramers Henneberger frame, direct numerical integration of the wave equation, classic methods, etc.) have in common the presence of a highly distorted wave function that is spread along the polarization direction over a region whose width is given by the classic motion of the electron. This effect, sometimes referred as dichotomy when the wave function shows mostly a two-peaked structure, is the starting point for our understanding of the physical origin of the atomic stabilization. The existence of this effect is surprising because the electron is not ionized during this motion close to the nucleus. In elliptically polarized fields stabilization is also expected, but the shape of the stabilized wave packet changes significantly [6

6. A. Patel, N. J. Kylstra, and P. L. Knight, “Ellipticity and pulse shape dependence of localised wavepackets,” Opt. Express 4, 496–537 (1999). [CrossRef] [PubMed]

].

To our knowledge, no clear experiments to observe atomic stabilization have been reported. One of the main difficulties in designing such an experiment is the high frequency of the laser photons needed. To get proper Kramers Henneberger stabilization it is necessary to use a short-wavelength, intense laser field. It is widely believed that this effect will appear as soon as convenient light sources become available.

The problem with magnetic effects is that they are absent from the standard one-dimensional model, that has been widely used for study of atomic stabilization. Fortunately today it is feasible to solve two-dimensional models that are able to account for magnetic-field effects. Three-dimensional models are still a bit too impractical for use in ionization situations, for which large numerical grids have to be employed.

There is a range of laser parameters (frequency and intensity) at which the velocity is small enough to permit us to assume that the electron dynamics is given by the Schrödinger equation and large enough to include the first-order relativistic correction: The influence of the magnetic field. To study this region, we present numerical solutions of the time-dependent Schrödinger equation

itΨ(r,t)=[12(p+1cA(r,t))2+V(r)]Ψ(r,t),
(1)

which describes an atom interacting with an electromagnetic field. Atomic units (au; e=m=1au, c=137 au) are used throughout this paper. Because of computer limitations we work with a two-dimensional model, considering a flat atom in the xy plane with a soft-core Coulomb potential:

V(x,y)=1(x2+y2+a)12,
(2)

where a is a parameter that avoids singularity at the origin and was chosen to be a=1 au in our calculations. The spatial grid where the Schrödinger equation is solved has 1300×3000 equally separated points, with a step size of Δxy=0.2 au. In this grid and for the chosen value of a, the ground-state energy of the model is EB =-0.43 au, as determined by the standard imaginary-time propagation method.

To have the electron dynamics restricted to the xy plane we assume that the electric field is linearly polarized along the y axis and that the pulse propagates along the x axis. Then the magnetic field is parallel to the z axis. Therefore the classic Lorentz force will drive the electron (initially at rest) inside the z=0 plane. The electric field is

Ey(x,t)=E0f(x,t)sin(kxωLt),
(3)

where f(x, t) is the laser turn-on, which is chosen to be linear and to last four cycles of the field. The vector potential is defined by

Ay(x,t)=c0tEy(x,t)dt.
(4)

Therefore the two-dimensional time-dependent Schrödinger equation for such an atomic model interacting with a laser field is

itΨ(x,y,t)=[12(2x2+2y2)icAy(x,t)y+12c2Ay2(x,t)+V(x,y)]Ψ(x,y,t).
(5)

Observe that the space dependence of the fields has been retained. We are not working in the dipole approximation or, in other words, we are retaining the magnetic-field effect in the nonrelativistic Lorentz force.

Ψ(x,y,t+Δt)=exp[iΔtĤ(t+Δt2)]Ψ(x,y,t)
exp[iΔt2Ĥx(t+Δt2)]exp[iΔtĤy(t+Δt2)]
×exp[iΔt2Ĥx(t+Δt2)]Ψ(x,y,t),
(6)

where we have defined

Ĥx(t)=122x2+12V(x,y)+12c2Ay2(x,t),
Ĥy(t)=122y2+12V(x,y)icAy(x,t)y.
(7)

Now we express the exponentials in the Cayley unitary form (which preserves the norm of the wave function), and we solve the equations by using the Crank Nicholson finite-difference scheme. It is not possible to use Fourier-transform methods directly because the term Ay (x, t)(/∂y) mixes momentum and space coordinates.

Laser parameters at which stabilization is expected are chosen for our simulations. Figure 1 shows the probability density of the electron after 10 cycles of the field, comparing these kinds of (nondipolar) simulation with the corresponding two-dimensional calculation in the dipole approximation [Ey (x, t)≃Ey (t)=E 0 f(t) sin(ωLt)] with the same laser parameters. Figure 1 corresponds to a laser field of amplitude E 0=15 au (intensity, 7.9×1018 W=cm 2) and frequency ωL =1 au (photon energy, 27 eV). This is a set of parameters for which a nonrelativistic study is reasonable but the first order correction υ/c is not extremely small. We selected these parameters because they were also selected for a recent paper in this journal [6

6. A. Patel, N. J. Kylstra, and P. L. Knight, “Ellipticity and pulse shape dependence of localised wavepackets,” Opt. Express 4, 496–537 (1999). [CrossRef] [PubMed]

]; we use a similar two-dimensional model, but we take into account the space dependence of the fields.

Working in the dipole approximation, we find that the wave function shows the expected dichotomy along the classic excursion, that is showed in the contour plot of Figure 1(a). In this case the dipole approximation could seem to be reasonable because the laser wavelength is close to 45 nm (or 861 au), which is still much bigger than the atomic size. But the dipole approximation is not valid because of the influence of the magnetic field. Figure 1(b) shows the computation for the same laser parameters with the space dependence of the fields retained. The ionized population that reaches the boundaries of the integration surface in both cases is negligible (less than 10-6 anytime).

Figure 1. Probability density |Ψ(x, y, t)|2 of the electron after 10 cycles of the field for E 0=15 au (intensity, 7.9×1018 W/cm 2) and ωL =1 au (photon energy, 27 eV). A linear envelope four-cycles turn-on has been employed. (a) results obtained in the dipole approximation, (b) simulation including the space dependence of the fields [Eq. (5)]. In both cases, contour plot lines are set to the same linear scale.

The effect of the Lorentz force is clear: The spread of the wave function is observed not only along the polarization axis but also along the propagation axis. The ionized population is affected by the momentum transferred by incident photons, which push it toward the propagation direction and break the symmetry along the polarization axis. The dichotomy is not so relevant as in the dipole case. Because of the intense electric field, the electron reaches a high velocity (in fact, we are not too far from a breakdown of the validity of the nonrelativisitic wave equation, as has been shown in the context of atomic stabilization). At such high velocities, a small magnetic field is enough to explain a drift of the electron along the propagation axis. A trivial integration of the Lorentz force equations for a free electron in a squared pulse laser field (no turn-on time, for simplicity) gives a magnetic drift (or drift along the propagation direction) that is proportional to E02=(cωL2 ) (electron initially at rest) in addition to the well-known oscillation at 2ωL ; this is the origin of the drift shown in Fig. 1(b). Obviously, in the case of the numerical simulation presented here, the laser turn-on and the atomic potential reduce the magnitude of the drift.

Acknowledgments

Partial support from the Training and Mobility of Researchers Programme of the European Commission, from the Spanish Dirección General de Investigación Científica y Técnica (under contract PB95 0955), and from the Consejería de Educación y Cultura of the Junta de Castilla y León (Fondo Social Europeo) under grant SA56/99 is acknowledged.

References

1.

J. H. Eberly, R. Grobe, C. K. Law, and Q. Su, “Numerical experiments in strong and super-strong fields,” in Atoms in Intense Laser Fields, M. Gavrila, ed. (Academic, San Diego, Calif., 1992).

2.

K. C. Kulander, K. J. Schafer, and J. L. Krause, “Time-dependent studies of multiphoton processes,” in Atoms in Intense Laser Fields, M. Gavrila, ed. (Academic, San Diego, Calif., 1992).

3.

Q. Su, A. Sanpera, and L. Roso-Franco, “Atomic stabilization in the presence of intense laser pulses,” Int. J. Mod. Phys. B 8, 1655 (1994). [CrossRef]

4.

M. Gavrila, “Atomic structure and decay in high frequency fields,” in Atoms in Intense Laser Fields, M. Gavrila, ed. (Academic, San Diego, Calif., 1992), and references therein.

5.

M. Protopapas, C. H. Keitel, and P. L. Knight, “Atomic physics with super-high intensity lasers,” Rep. Prog. Phys. 60, 389–486 (1997), and references therein. [CrossRef]

6.

A. Patel, N. J. Kylstra, and P. L. Knight, “Ellipticity and pulse shape dependence of localised wavepackets,” Opt. Express 4, 496–537 (1999). [CrossRef] [PubMed]

7.

M. Protopapas, C. H. Keitel, and P. L. Knight, “Relativistic mass shift effects in adiabatic intense laser field stabilization of atoms,” J. Phys. B 29, L591–L598 (1996). [CrossRef]

8.

C. H. Keitel and P. L. Knight, “Monte Carlo classical simulations of ionization and harmonic generation in the relativistic domain,” Phys. Rev. A 51, 1420–1430 (1995). [CrossRef] [PubMed]

9.

P. Moreno, “Harmonic generation by H and H2+ in intense laser pulses,” Ph. D. dissertation, (Universidad de Salamanca, Salamanca, Spain, 1997).

10.

A. D. Bandrauk and H. Shen, “Exponential split operator methods for solving coupled time-dependent Schrödinger equations,” J. Chem. Phys. 99, 1185–1193 (1993). [CrossRef]

OCIS Codes
(020.4180) Atomic and molecular physics : Multiphoton processes
(300.6410) Spectroscopy : Spectroscopy, multiphoton

ToC Category:
Research Papers

History
Original Manuscript: July 12, 1999
Published: September 27, 1999

Citation
J. Vazquez de Aldana and Luis Roso, "Magnetic-field effect in atomic ionization by intense laser fields.," Opt. Express 5, 144-148 (1999)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-5-7-144


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References

  1. J. H. Eberly, R. Grobe, C. K. Law and Q. Su, "Numerical experiments in strong and super-strong fields," in Atoms in Intense Laser Fields, M. Gavrila, ed. (Academic, San Diego, Calif., 1992).
  2. K. C. Kulander, K. J. Schafer and J. L. Krause, "Time-dependent studies of multiphoton processes," in Atoms in Intense Laser Fields, M. Gavrila, ed. (Academic, San Diego, Calif., 1992).
  3. Q. Su, A. Sanpera and L. Roso-Franco, "Atomic stabilization in the presence of intense laser pulses," Int. J. Mod. Phys. B 8, 1655 (1994). [CrossRef]
  4. M. Gavrila, "Atomic structure and decay in high frequency fields," in Atoms in Intense Laser Fields, M. Gavrila, ed. (Academic, San Diego, Calif., 1992), and references therein.
  5. M. Protopapas, C. H. Keitel and P. L. Knight,"Atomic physics with super-high intensity lasers," Rep. Prog. Phys. 60, 389-486 (1997), and references therein. [CrossRef]
  6. A. Patel, N. J. Kylstra and P. L. Knight, "Ellipticity and pulse shape dependence of localised wavepackets," Opt. Express 4, 496-537 (1999), http://www.opticsexpress.org/oearchive/source/10164.htm [CrossRef] [PubMed]
  7. M. Protopapas, C. H. Keitel and P. L. Knight, "Relativistic mass shift effects in adiabatic intense laser field stabilization of atoms," J. Phys. B 29, L591-L598 (1996). [CrossRef]
  8. C. H. Keitel and P. L. Knight, "Monte Carlo classical simulations of ionization and harmonic generation in the relativistic domain," Phys. Rev. A 51, 1420-1430 (1995). [CrossRef] [PubMed]
  9. P. Moreno, "Harmonic generation by H and H + 2 in intense laser pulses," Ph. D. dissertation, (Universidad de Salamanca, Salamanca, Spain, 1997).
  10. A. D. Bandrauk and H. Shen, "Exponential split operator methods for solving coupled time-dependent Schr�dinger equations," J. Chem. Phys. 99, 1185-1193 (1993). [CrossRef]

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