Abstract

Analytical expressions are obtained for energy and angular distributions of ejected electrons at the barrier-suppression and tunneling ionization of complex atoms and atomic ions by low-frequency strong electromagnetic radiation. The results reduce to previously known expressions in the case of the ground state of the hydrogen atom. Both linear and circular polarizations of the electromagnetic field are considered. The ionization rates are found by integration over angles and energies of the ejected electron in the case of barrier-suppression ionization of complex atoms and atomic ions. The barrier-suppression results reduce correctly to the tunneling results of the Ammosov–Delone–Krainov approach in the limit of weak fields compared with the barrier-suppression fields.

© 1997 Optical Society of America

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  1. M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and atomic ions by an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191–1194 (1986).
  2. P. B. Corkum, N. H. Burnett, and F. Brunel, “Above-threshold ionization in the long-wavelength limit,” Phys. Rev. Lett. 62, 1259–1262 (1989).
    [CrossRef] [PubMed]
  3. V. P. Krainov, W. Xiong, and S. L. Chin, “An introductory overview of tunnel ionization of atoms by intense laser,” Laser Phys. 2, 467–475 (1992).
  4. V. P. Krainov and B. Shokri, “Energy and angular distributions of electrons in above-barrier ionization of atoms by strong low-frequency radiation,” Sov. Phys. JETP 80, 657–661 (1995).
  5. S. Augst, D. D. Meyerhofer, D. Strickland, and S. L. Chin, “Laser ionization of noble gases by Coulomb-barrier suppression,” J. Opt. Soc. Am. B 8, 858–867 (1991).
    [CrossRef]
  6. N. B. Delone and V. P. Krainov, Multiphoton Processes in Atoms (Springer-Verlag, Berlin, 1994).
  7. V. P. Krainov and H. R. Reiss, “Coulomb–Volkov correction for a strong-field approximation,” in Proceedings of the XIVth International Conference on Coherent and Nonlinear Optics, V. N. Koroteev, ed. (Russian Academy of Sciences, Saint Petersburg, 1995), p. 523.
  8. H. R. Reiss, “Theoretical methods in quantum optics: S-matrix and Keldysh techniques for strong-field processes,” Prog. Quantum Electron. 16, 1–71 (1992).
    [CrossRef]
  9. H. R. Reiss and V. P. Krainov, “Approximation for a Coulomb–Volkov solution in strong fields,” Phys. Rev. A 50, R910–R912 (1994).
    [CrossRef]
  10. V. P. Krainov, “Theory of barrier-suppression ionization of atoms,” J. Nonlin. Opt. Phys. 4, 775–798 (1995).
    [CrossRef]
  11. F. A. Ilkov, J. E. Decker, and S. L. Chin, “Tunnel ionization of molecules by an intense CO2 laser,” Laser Phys. 3, 298–306 (1993).
  12. H. R. Reiss, “Effect of an intense electromagnetic field on a weakly bound system,” Phys. Rev. A 22, 1786–1798 (1980).
    [CrossRef]
  13. L. D. Landau and E. M. Lifshitz, Quantum Mechanics–Non-Relativistic Theory, 3rd ed. (Pergamon, Oxford, 1977).
  14. N. B. Delone and V. P. Krainov, Atoms in Strong Light Fields (Springer-Verlag, Berlin, 1985).
  15. N. B. Delone and V. P. Krainov, “Energy and angular electron spectra for the tunnel ionization of atoms by strong low-frequency radiation,” J. Opt. Soc. Am. B 8, 1207–1211 (1991).
    [CrossRef]
  16. S. P. Goreslavsky, “Tunnel ionization in a light field without Fourier expansion,” Sov. Phys. JETP 81, 564–572 (1995).
  17. S. P. Goreslavsky and S. V. Popruzhenko, “Momentum distribution of photoelectrons in a strong low-frequency elliptically polarized laser field,” Laser Phys. 6, 326–330 (1996).

1995 (2)

V. P. Krainov and B. Shokri, “Energy and angular distributions of electrons in above-barrier ionization of atoms by strong low-frequency radiation,” Sov. Phys. JETP 80, 657–661 (1995).

S. P. Goreslavsky, “Tunnel ionization in a light field without Fourier expansion,” Sov. Phys. JETP 81, 564–572 (1995).

1994 (1)

H. R. Reiss and V. P. Krainov, “Approximation for a Coulomb–Volkov solution in strong fields,” Phys. Rev. A 50, R910–R912 (1994).
[CrossRef]

1992 (1)

H. R. Reiss, “Theoretical methods in quantum optics: S-matrix and Keldysh techniques for strong-field processes,” Prog. Quantum Electron. 16, 1–71 (1992).
[CrossRef]

1991 (2)

1989 (1)

P. B. Corkum, N. H. Burnett, and F. Brunel, “Above-threshold ionization in the long-wavelength limit,” Phys. Rev. Lett. 62, 1259–1262 (1989).
[CrossRef] [PubMed]

1986 (1)

M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and atomic ions by an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191–1194 (1986).

1980 (1)

H. R. Reiss, “Effect of an intense electromagnetic field on a weakly bound system,” Phys. Rev. A 22, 1786–1798 (1980).
[CrossRef]

Ammosov, M. V.

M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and atomic ions by an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191–1194 (1986).

Augst, S.

Brunel, F.

P. B. Corkum, N. H. Burnett, and F. Brunel, “Above-threshold ionization in the long-wavelength limit,” Phys. Rev. Lett. 62, 1259–1262 (1989).
[CrossRef] [PubMed]

Burnett, N. H.

P. B. Corkum, N. H. Burnett, and F. Brunel, “Above-threshold ionization in the long-wavelength limit,” Phys. Rev. Lett. 62, 1259–1262 (1989).
[CrossRef] [PubMed]

Chin, S. L.

Corkum, P. B.

P. B. Corkum, N. H. Burnett, and F. Brunel, “Above-threshold ionization in the long-wavelength limit,” Phys. Rev. Lett. 62, 1259–1262 (1989).
[CrossRef] [PubMed]

Delone, N. B.

N. B. Delone and V. P. Krainov, “Energy and angular electron spectra for the tunnel ionization of atoms by strong low-frequency radiation,” J. Opt. Soc. Am. B 8, 1207–1211 (1991).
[CrossRef]

M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and atomic ions by an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191–1194 (1986).

Goreslavsky, S. P.

S. P. Goreslavsky, “Tunnel ionization in a light field without Fourier expansion,” Sov. Phys. JETP 81, 564–572 (1995).

Krainov, V. P.

V. P. Krainov and B. Shokri, “Energy and angular distributions of electrons in above-barrier ionization of atoms by strong low-frequency radiation,” Sov. Phys. JETP 80, 657–661 (1995).

H. R. Reiss and V. P. Krainov, “Approximation for a Coulomb–Volkov solution in strong fields,” Phys. Rev. A 50, R910–R912 (1994).
[CrossRef]

N. B. Delone and V. P. Krainov, “Energy and angular electron spectra for the tunnel ionization of atoms by strong low-frequency radiation,” J. Opt. Soc. Am. B 8, 1207–1211 (1991).
[CrossRef]

M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and atomic ions by an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191–1194 (1986).

Meyerhofer, D. D.

Reiss, H. R.

H. R. Reiss and V. P. Krainov, “Approximation for a Coulomb–Volkov solution in strong fields,” Phys. Rev. A 50, R910–R912 (1994).
[CrossRef]

H. R. Reiss, “Theoretical methods in quantum optics: S-matrix and Keldysh techniques for strong-field processes,” Prog. Quantum Electron. 16, 1–71 (1992).
[CrossRef]

H. R. Reiss, “Effect of an intense electromagnetic field on a weakly bound system,” Phys. Rev. A 22, 1786–1798 (1980).
[CrossRef]

Shokri, B.

V. P. Krainov and B. Shokri, “Energy and angular distributions of electrons in above-barrier ionization of atoms by strong low-frequency radiation,” Sov. Phys. JETP 80, 657–661 (1995).

Strickland, D.

J. Opt. Soc. Am. B (2)

Phys. Rev. A (2)

H. R. Reiss and V. P. Krainov, “Approximation for a Coulomb–Volkov solution in strong fields,” Phys. Rev. A 50, R910–R912 (1994).
[CrossRef]

H. R. Reiss, “Effect of an intense electromagnetic field on a weakly bound system,” Phys. Rev. A 22, 1786–1798 (1980).
[CrossRef]

Phys. Rev. Lett. (1)

P. B. Corkum, N. H. Burnett, and F. Brunel, “Above-threshold ionization in the long-wavelength limit,” Phys. Rev. Lett. 62, 1259–1262 (1989).
[CrossRef] [PubMed]

Prog. Quantum Electron. (1)

H. R. Reiss, “Theoretical methods in quantum optics: S-matrix and Keldysh techniques for strong-field processes,” Prog. Quantum Electron. 16, 1–71 (1992).
[CrossRef]

Sov. Phys. JETP (3)

M. V. Ammosov, N. B. Delone, and V. P. Krainov, “Tunnel ionization of complex atoms and atomic ions by an alternating electromagnetic field,” Sov. Phys. JETP 64, 1191–1194 (1986).

V. P. Krainov and B. Shokri, “Energy and angular distributions of electrons in above-barrier ionization of atoms by strong low-frequency radiation,” Sov. Phys. JETP 80, 657–661 (1995).

S. P. Goreslavsky, “Tunnel ionization in a light field without Fourier expansion,” Sov. Phys. JETP 81, 564–572 (1995).

Other (8)

S. P. Goreslavsky and S. V. Popruzhenko, “Momentum distribution of photoelectrons in a strong low-frequency elliptically polarized laser field,” Laser Phys. 6, 326–330 (1996).

V. P. Krainov, W. Xiong, and S. L. Chin, “An introductory overview of tunnel ionization of atoms by intense laser,” Laser Phys. 2, 467–475 (1992).

L. D. Landau and E. M. Lifshitz, Quantum Mechanics–Non-Relativistic Theory, 3rd ed. (Pergamon, Oxford, 1977).

N. B. Delone and V. P. Krainov, Atoms in Strong Light Fields (Springer-Verlag, Berlin, 1985).

V. P. Krainov, “Theory of barrier-suppression ionization of atoms,” J. Nonlin. Opt. Phys. 4, 775–798 (1995).
[CrossRef]

F. A. Ilkov, J. E. Decker, and S. L. Chin, “Tunnel ionization of molecules by an intense CO2 laser,” Laser Phys. 3, 298–306 (1993).

N. B. Delone and V. P. Krainov, Multiphoton Processes in Atoms (Springer-Verlag, Berlin, 1994).

V. P. Krainov and H. R. Reiss, “Coulomb–Volkov correction for a strong-field approximation,” in Proceedings of the XIVth International Conference on Coherent and Nonlinear Optics, V. N. Koroteev, ed. (Russian Academy of Sciences, Saint Petersburg, 1995), p. 523.

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Figures (2)

Fig. 1
Fig. 1

Ratio of the barrier-suppression ionization rate [Eq. (15)] wBSI to the tunneling ADK ionization rate [Eq. (10)] wADK by a circularly polarized field as a function of the dimensionless parameter k=2Ei/(2F)2/3=(Z/n)2/(2F)2/3 (Ei is the energy of the initial s state). The value of kBSI corresponds to the barrier-suppression field strength for the ground state of the hydrogen atom.

Fig. 2
Fig. 2

Ratio of the barrier-suppression ionization rate [Eq. (23)] wBSI to the tunneling ADK ionization rate [Eq. (21)] wADK by a linearly polarized field as a function of the dimensionless parameter k=2Ei/(2F)2/3=(Z/n)2/(2F)2/3 (Ei is the energy of the initial s state). The value of kBSI corresponds to the barrier-suppression field strength for the ground state of the hydrogen atom.

Equations (69)

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I=exp(-i  Udt).
dt=dr/p=(Z2/n2-2Fr)-1/2dr,
I=2Z2n2Frn.
Anf=-iΨf(V)I|V(r, t)|Ψn(0)dt,
V(r, t)=pA/c+A2/2c2,
A=(cF/ω)(iˆx cos ωt+iˆy sin ωt),
Anf=i(p2/2+Z2/2n2)Ψf(V)|I|Ψn(0)dt,
Ψf(V)=exp-ipr+(i/2)(p+A/c)2dt.
Ψn(0)=C(Z3/4π)1/2(Zr)n-1 exp(-Zr/n),
C=(2e)2n(2π)-n-1/2.
Anf=ZDπ2nN=N0JNpF sin θω2 exp(ift)-1if,
f=p2/2+Z2/2n2+F2/2ω2-Nω=0
D=4eZ3Fn4n.
dwdΩ=ω/2ZD2(2πn)2N=N0(N-F2/2ω3-Z2/2ωn2)1/2×JN2pF sin θω2,
JN(N/cosh α)=(2πNα)-1/2 exp[-N(tanh α-α)].
N/cosh α=(F/ω2)(1-ψ2/2)×(2Nω-Z2/n2-F2/ω2)1/2,
δN=N-F2/ω3-Z2/2ωn2,
(1-ψ2/2)(1+ω3δN/F2-ω6δN2/2F4)
=(1+ω2Z2/2F2n2+ω3δN/F2)(1-α2/2+5α4/24).
α=(ωZ/Fn)[1-(1/24)(ωZ/Fn)2+(1/2)(ω2δNn/FZ)2+(1/2)(ψFn/ωZ)2-ω3δN/2F2].
2N(tanh α-α)=2N(-α3/3+2α5/15),
2N(tanh α-α)=(-2Z3/3Fn3)[1+(3/2)(ψFn/ωZ)2-ω3δN/2F2+(3/2)(ω2δNn/FZ)2-(1/16)(ωZ/Fn)2].
exp[(-2Z3/3Fn3)(1-γ2/15)].
exp(-Zψ2F/nω2).
exp(-Zω4δN2/F3n),
δN=δN-F2/6ωn2.
dwdΩ=ωD216π3nδN exp-2Z33Fn3(1-γ2/15)-FZnω2ψ2-Zω4nF3δN2.
dwdΩ=(πnF3/Z)1/216π3nωD2×exp-2Z33Fn3(1-γ2/15)-FZnω2ψ2.
wADK=FD28πZexp-2Z33Fn3(1-γ2/15).
w=ω2D28π3nFZexp-2Z33Fn3(1-γ2/15)×N exp-Zω4F3n(N-Nmax)2.
Nmax=F2/ω3+2Z2/3ωn2.
ω(N-Nmax)=p2/2-F2/2ω2-Z2/6n2.
JN(N+N1/3t)=(2/N)1/3Ai(-21/3t),
t=(F2/ω3)2/3(-ψ2/2-ω6δN2/2F4-Z2ω2/2F2n2).
dwdΩ=ZωD2(2F)1/34π2n2×δNAi2Z2/n2+(Fψ/ω)2+(ω2δN/F)2(2F)2/3.
ωδN=p2/2-F2/2ω2-Z2/6n2.
dwdΩ=ZFD22π2n2ω0Ai2(x2+u)dx.
u=(Z/n)2(2F)-2/3+(F2/2)2/3(ψ/ω)2.
w(δN)=Zω2D2πn2F0Ai2(x2+v)dx,
v=(Z/n)2(2F)-2/3+4ω4δN2(2F)-8/3
w=Z(2F)1/3D22πn2-Ai2x2+y2+Z2n2(2F)2/3dxdy.
wBSI=Z(2F)1/3D22n2[Ai2(k)-kAi2(k)].
k=(Z/n)2(2F)-2/3.
w(np):w(ns)=[(n-1):n](3π2/4).
S=(1/2)0t(p+A/c)2dt=(p2/2+F2/4ω2)t+(pF/ω2)cos ωt-(F2/8ω3)sin 2ωt.
Anf=DZ/2n0t exp[ig(t)]dt,
g(t)=(p2/2+Z2/2n2+F2/4ω2)t+(pF/ω2)cos ωt-(F2/8ω3)sin 2ωt.
Anf(0)=DZ/2n02π/ω exp[ig(t)]dt.
Anf(1)=Anf(0) exp(iS0),
S0=(π/ω)(p2+Z2/n2+F2/2ω2).
Anf=Anf(0){1+exp(iS0)+exp(2iS0)+exp(3iS0)++exp[(K-1)iS0]}=Anf(0)[1-exp(KiS0)]:[1-exp(iS0)].
Wnf=|Anf(0)|2 sin2(KS0/2): sin2(S0/2).
Wnf=|Anf(0)|2(ω2t/2π)δ(p2/2+Z2/2n2+F2/4ω2-Nω).
dwdΩ=pω2FD24π(2π)3n202π/ω exp[ig(t)]dt2.
sin ωt0=(ω/F)[p+(p2+Z2/n2)1/2]=α1.
ωt0=arcsin α=α+α3/6+3α5/401.
g(t0)=(1/ω)[(p2/2+Z2/2n2+F2/4ω2)(α+α3/6+3α5/40)+(pF/ω)(-α2/2-α4/8)-(F2/4ω2)(α-α3/2-α5/8)].
exp[-(2/3F)(p2+Z2/n2)3/2].
exp[-(2/3F)(p2+Z2/n2)3/2(1-γ2/10)-p2γ3/3ω].
γ=ωZ/Fn
dwdΩ=pωD28π3nFexp-2Z33Fn3-p2ZnF-p2γ33ω.
w=2ω2D28π2Zpexp-2Z33Fn3-p2γ33ω.
Nw=wdN=(w/ω)d(Nω)=(w/ω)pdp.
wADK=FD28πZ3Fn3πZ31/2 exp-2Z33Fn3.
wl=3Fn3πZ31/2wc.
dwdΩ=const.×D2Ai2Z2/n2+p2+(Fn/Zw)p2γ3/3(2F)2/3.
const.=p2ω2Zπ2(2F)4/3n2.
dwdΩ=p2ω2ZD2π2(2F)4/3n2×Ai2Z2/n2+p2+(Fn/Zw)p2γ3/3(2F)2/3dN,
wBSI=43Fπn(2F)1/34eZ3Fn42n×0Ai2Z2n2(2F)2/3+x2x2dx.

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