Abstract

Conversion efficiencies and far-field profiles for third-order harmonic generation in an atomic medium irradiated by an intense Bessel–Gauss beam are calculated with an integral method. The calculation takes into account the nonperturbative variation of the atomic polarizabilities, target depletion by photoionization, and the effect of the free electrons. Numerical results are presented for a pump beam of 355-nm wavelength and up to 3×1013 W/cm2 intensity incident on hydrogen. They are compared with equivalent results for a pure Gaussian pump beam. Significant differences are found that originate from the different phase-matching properties and intensity profile of Bessel–Gauss beams.

© 1998 Optical Society of America

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References

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  1. B. Glushko, B. Kryzhanovsky, and D. Sarkisyan, “Self-phase-matching mechanism for efficient harmonic generation processes in a ring pump beam geometry,” Phys. Rev. Lett. 71, 243–246 (1993).
    [CrossRef] [PubMed]
  2. S. P. Tewari, H. Huang, and R. W. Boyd, “Theory of self-phase-matching,” Phys. Rev. A 51, 2707–2710 (1995); “Theory of third-harmonic generation using Bessel beams, and self-phase-matching,” Phys. Rev. A 54, 2314–2325 (1996).
    [CrossRef] [PubMed]
  3. J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  4. V. E. Peet, “Resonantly enhanced multiphoton ionization of xenon in Bessel beams,” Phys. Rev. A 53, 3679–3682 (1996).
    [CrossRef] [PubMed]
  5. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [CrossRef]
  6. J. K. Jabczynski, “A ‘diffraction-free’ resonator,” Opt. Commun. 77, 292–294 (1990).
    [CrossRef]
  7. A. L’Huillier, L. A. Lompré, G. Mainfray, and C. Manus, “High-order harmonic generation in rare gases,” in Atoms in Intense Laser Fields, M. Gavrila, ed., Advances in Atomic, Molecular and Optical Physics, Suppl. 1 (Academic, New York, 1992).
  8. The first argument of the complex exponential in Eq. (2), kz cos α, should be kz[1−(sin α)2/2] for EBG to be an exact solution of the paraxial-wave equation. The difference between kz cos α and kz[1−(sin α)2/2] is negligible here because it is of fourth order in α. Equation (2) has the advantage that EBG reduces to the exact Bessel form of Eq. (6) in the loose-focusing limit.
  9. V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).
  10. The cross section of the Gaussian beam of Fig. 1 would be larger, and for a fixed power its focal intensity would be smaller, if its confocal parameter were larger. One can optimize b and α so as to maximize the area of the focal plane irradiated above a given intensity for a given power. A numerical study showed that marginally larger areas can be achieved with an optimized Bessel-Gauss beam (α≠0) than with an optimized Gaussian beam (α=0); the difference between the two is a few percent at most.
  11. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).
  12. A novel method of graphically investigating phase matching, based on this idea, has been devised recently by Balcou et el. for a purely Gaussian beam geometry: P. Balcou, P. Salières, A. L’Huillier, and M. Lewenstein, “Generalized phase-matching conditions for high harmonics: the role of field-gradient forces,” Phys. Rev. A 55, 3204–3210 (1997).
    [CrossRef]
  13. This might be less important at high intensities, where the dipole moment usually saturates and ionization significantly reduces the number of atoms participating effectively in the harmonic generation process.
  14. R. M. Potvliege and R. Shakeshaft, “Multiphoton processes in an intense laser field: harmonic generation and total ionization rates for atomic hydrogen,” Phys. Rev. A 40, 3061–3079 (1989).
    [CrossRef] [PubMed]
  15. R. M. Potvliege and P. H. G. Smith, “Stabilization of excited states and harmonic generation: recent theoretical results in the Sturmian-Floquet approach,” in SuperIntense Laser-Atom Physics, B. Piraux, A. L’Huillier, and K. Rzazewski, eds., Vol. B316 of NATO ASI Series (Plenum, New York, 1993), pp. 173–184. In a few words, the ionization rate is −2 Im(E)/ħ, where E is the quasi energy of the dressed 1s state; dq is the term proportional to exp(−iqωt) in the dipole moment of the atom in the dressed 1s state; and, using SI units, χat(qω)ε0Eq/2 is the contribution linear in Eq to the dipole moment dq for a two-color field E1 cos ωt+ Eq cos qωt.
  16. A detailed account of these calculations will be presented elsewhere.
  17. Phase matching for the Gaussian beam is not more favorable at high intensity than in a weak field in the present case because |χ1−χ3| has the same magnitude in both limits, and the contributions of the geometric phase q tan−1(2z/b) (important for small confocal parameters) and of the electronic susceptibility χel(ω) have the same sign.
  18. The discussion above is consistent with the results of a recent experimental study of third-order harmonic generation in xenon comparing Gaussian and conical beams, published after completion of this manuscript [V. E. Peet and R. V. Tsubin, “Third-harmonic generation and multiphoton ionization in Bessel beams,” Phys. Rev. A 56, 1613–1620 (1997)]. Peet and Tsubin examined the variation with frequency of the harmonic yield in the vicinity of a three-photon resonance and at much weaker intensities than considered in this work for a focused Gaussian beam, an annular beam (α≈3°), and a Bessel beam (α=17°). The harmonic yield obtained with the annular beam exceeded that obtained with a Gaussian beam of same power in a small spectral range close to the resonance. No third-harmonic emission was detected for the Bessel beam, presumably because of the very low intensity of the beam that could be achieved in the experiment and the reabsorption of the emitted harmonic photons within the medium (with the cone half-angle α being fairly large, harmonic generation could be efficient only very close to the resonance where absorption is important; see Eq. 44).
    [CrossRef]
  19. See, for example, P. Antoine, A. L’Huillier, M. Lewenstein, P. Salières, and B. Carré, “Theory of high-order harmonic generation by an elliptically polarized laser field,” Phys. Rev. A 53, 1725–1745 (1996).
    [CrossRef] [PubMed]

1997 (2)

A novel method of graphically investigating phase matching, based on this idea, has been devised recently by Balcou et el. for a purely Gaussian beam geometry: P. Balcou, P. Salières, A. L’Huillier, and M. Lewenstein, “Generalized phase-matching conditions for high harmonics: the role of field-gradient forces,” Phys. Rev. A 55, 3204–3210 (1997).
[CrossRef]

The discussion above is consistent with the results of a recent experimental study of third-order harmonic generation in xenon comparing Gaussian and conical beams, published after completion of this manuscript [V. E. Peet and R. V. Tsubin, “Third-harmonic generation and multiphoton ionization in Bessel beams,” Phys. Rev. A 56, 1613–1620 (1997)]. Peet and Tsubin examined the variation with frequency of the harmonic yield in the vicinity of a three-photon resonance and at much weaker intensities than considered in this work for a focused Gaussian beam, an annular beam (α≈3°), and a Bessel beam (α=17°). The harmonic yield obtained with the annular beam exceeded that obtained with a Gaussian beam of same power in a small spectral range close to the resonance. No third-harmonic emission was detected for the Bessel beam, presumably because of the very low intensity of the beam that could be achieved in the experiment and the reabsorption of the emitted harmonic photons within the medium (with the cone half-angle α being fairly large, harmonic generation could be efficient only very close to the resonance where absorption is important; see Eq. 44).
[CrossRef]

1996 (3)

See, for example, P. Antoine, A. L’Huillier, M. Lewenstein, P. Salières, and B. Carré, “Theory of high-order harmonic generation by an elliptically polarized laser field,” Phys. Rev. A 53, 1725–1745 (1996).
[CrossRef] [PubMed]

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

V. E. Peet, “Resonantly enhanced multiphoton ionization of xenon in Bessel beams,” Phys. Rev. A 53, 3679–3682 (1996).
[CrossRef] [PubMed]

1993 (1)

B. Glushko, B. Kryzhanovsky, and D. Sarkisyan, “Self-phase-matching mechanism for efficient harmonic generation processes in a ring pump beam geometry,” Phys. Rev. Lett. 71, 243–246 (1993).
[CrossRef] [PubMed]

1990 (1)

J. K. Jabczynski, “A ‘diffraction-free’ resonator,” Opt. Commun. 77, 292–294 (1990).
[CrossRef]

1989 (1)

R. M. Potvliege and R. Shakeshaft, “Multiphoton processes in an intense laser field: harmonic generation and total ionization rates for atomic hydrogen,” Phys. Rev. A 40, 3061–3079 (1989).
[CrossRef] [PubMed]

1987 (2)

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Antoine, P.

See, for example, P. Antoine, A. L’Huillier, M. Lewenstein, P. Salières, and B. Carré, “Theory of high-order harmonic generation by an elliptically polarized laser field,” Phys. Rev. A 53, 1725–1745 (1996).
[CrossRef] [PubMed]

Bagini, V.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Balcou, P.

A novel method of graphically investigating phase matching, based on this idea, has been devised recently by Balcou et el. for a purely Gaussian beam geometry: P. Balcou, P. Salières, A. L’Huillier, and M. Lewenstein, “Generalized phase-matching conditions for high harmonics: the role of field-gradient forces,” Phys. Rev. A 55, 3204–3210 (1997).
[CrossRef]

Carré, B.

See, for example, P. Antoine, A. L’Huillier, M. Lewenstein, P. Salières, and B. Carré, “Theory of high-order harmonic generation by an elliptically polarized laser field,” Phys. Rev. A 53, 1725–1745 (1996).
[CrossRef] [PubMed]

Durnin, J.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Frezza, F.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Glushko, B.

B. Glushko, B. Kryzhanovsky, and D. Sarkisyan, “Self-phase-matching mechanism for efficient harmonic generation processes in a ring pump beam geometry,” Phys. Rev. Lett. 71, 243–246 (1993).
[CrossRef] [PubMed]

Gori, F.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Jabczynski, J. K.

J. K. Jabczynski, “A ‘diffraction-free’ resonator,” Opt. Commun. 77, 292–294 (1990).
[CrossRef]

Kryzhanovsky, B.

B. Glushko, B. Kryzhanovsky, and D. Sarkisyan, “Self-phase-matching mechanism for efficient harmonic generation processes in a ring pump beam geometry,” Phys. Rev. Lett. 71, 243–246 (1993).
[CrossRef] [PubMed]

L’Huillier, A.

A novel method of graphically investigating phase matching, based on this idea, has been devised recently by Balcou et el. for a purely Gaussian beam geometry: P. Balcou, P. Salières, A. L’Huillier, and M. Lewenstein, “Generalized phase-matching conditions for high harmonics: the role of field-gradient forces,” Phys. Rev. A 55, 3204–3210 (1997).
[CrossRef]

See, for example, P. Antoine, A. L’Huillier, M. Lewenstein, P. Salières, and B. Carré, “Theory of high-order harmonic generation by an elliptically polarized laser field,” Phys. Rev. A 53, 1725–1745 (1996).
[CrossRef] [PubMed]

Lewenstein, M.

A novel method of graphically investigating phase matching, based on this idea, has been devised recently by Balcou et el. for a purely Gaussian beam geometry: P. Balcou, P. Salières, A. L’Huillier, and M. Lewenstein, “Generalized phase-matching conditions for high harmonics: the role of field-gradient forces,” Phys. Rev. A 55, 3204–3210 (1997).
[CrossRef]

See, for example, P. Antoine, A. L’Huillier, M. Lewenstein, P. Salières, and B. Carré, “Theory of high-order harmonic generation by an elliptically polarized laser field,” Phys. Rev. A 53, 1725–1745 (1996).
[CrossRef] [PubMed]

Miceli Jr., J. J.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Peet, V. E.

The discussion above is consistent with the results of a recent experimental study of third-order harmonic generation in xenon comparing Gaussian and conical beams, published after completion of this manuscript [V. E. Peet and R. V. Tsubin, “Third-harmonic generation and multiphoton ionization in Bessel beams,” Phys. Rev. A 56, 1613–1620 (1997)]. Peet and Tsubin examined the variation with frequency of the harmonic yield in the vicinity of a three-photon resonance and at much weaker intensities than considered in this work for a focused Gaussian beam, an annular beam (α≈3°), and a Bessel beam (α=17°). The harmonic yield obtained with the annular beam exceeded that obtained with a Gaussian beam of same power in a small spectral range close to the resonance. No third-harmonic emission was detected for the Bessel beam, presumably because of the very low intensity of the beam that could be achieved in the experiment and the reabsorption of the emitted harmonic photons within the medium (with the cone half-angle α being fairly large, harmonic generation could be efficient only very close to the resonance where absorption is important; see Eq. 44).
[CrossRef]

V. E. Peet, “Resonantly enhanced multiphoton ionization of xenon in Bessel beams,” Phys. Rev. A 53, 3679–3682 (1996).
[CrossRef] [PubMed]

Potvliege, R. M.

R. M. Potvliege and R. Shakeshaft, “Multiphoton processes in an intense laser field: harmonic generation and total ionization rates for atomic hydrogen,” Phys. Rev. A 40, 3061–3079 (1989).
[CrossRef] [PubMed]

Salières, P.

A novel method of graphically investigating phase matching, based on this idea, has been devised recently by Balcou et el. for a purely Gaussian beam geometry: P. Balcou, P. Salières, A. L’Huillier, and M. Lewenstein, “Generalized phase-matching conditions for high harmonics: the role of field-gradient forces,” Phys. Rev. A 55, 3204–3210 (1997).
[CrossRef]

See, for example, P. Antoine, A. L’Huillier, M. Lewenstein, P. Salières, and B. Carré, “Theory of high-order harmonic generation by an elliptically polarized laser field,” Phys. Rev. A 53, 1725–1745 (1996).
[CrossRef] [PubMed]

Santarsiero, M.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Sarkisyan, D.

B. Glushko, B. Kryzhanovsky, and D. Sarkisyan, “Self-phase-matching mechanism for efficient harmonic generation processes in a ring pump beam geometry,” Phys. Rev. Lett. 71, 243–246 (1993).
[CrossRef] [PubMed]

Schettini, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Schirripa Spagnolo, G.

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Shakeshaft, R.

R. M. Potvliege and R. Shakeshaft, “Multiphoton processes in an intense laser field: harmonic generation and total ionization rates for atomic hydrogen,” Phys. Rev. A 40, 3061–3079 (1989).
[CrossRef] [PubMed]

Tsubin, R. V.

The discussion above is consistent with the results of a recent experimental study of third-order harmonic generation in xenon comparing Gaussian and conical beams, published after completion of this manuscript [V. E. Peet and R. V. Tsubin, “Third-harmonic generation and multiphoton ionization in Bessel beams,” Phys. Rev. A 56, 1613–1620 (1997)]. Peet and Tsubin examined the variation with frequency of the harmonic yield in the vicinity of a three-photon resonance and at much weaker intensities than considered in this work for a focused Gaussian beam, an annular beam (α≈3°), and a Bessel beam (α=17°). The harmonic yield obtained with the annular beam exceeded that obtained with a Gaussian beam of same power in a small spectral range close to the resonance. No third-harmonic emission was detected for the Bessel beam, presumably because of the very low intensity of the beam that could be achieved in the experiment and the reabsorption of the emitted harmonic photons within the medium (with the cone half-angle α being fairly large, harmonic generation could be efficient only very close to the resonance where absorption is important; see Eq. 44).
[CrossRef]

J. Mod. Opt. (1)

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Opt. Commun. (2)

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

J. K. Jabczynski, “A ‘diffraction-free’ resonator,” Opt. Commun. 77, 292–294 (1990).
[CrossRef]

Phys. Rev. A (5)

V. E. Peet, “Resonantly enhanced multiphoton ionization of xenon in Bessel beams,” Phys. Rev. A 53, 3679–3682 (1996).
[CrossRef] [PubMed]

A novel method of graphically investigating phase matching, based on this idea, has been devised recently by Balcou et el. for a purely Gaussian beam geometry: P. Balcou, P. Salières, A. L’Huillier, and M. Lewenstein, “Generalized phase-matching conditions for high harmonics: the role of field-gradient forces,” Phys. Rev. A 55, 3204–3210 (1997).
[CrossRef]

R. M. Potvliege and R. Shakeshaft, “Multiphoton processes in an intense laser field: harmonic generation and total ionization rates for atomic hydrogen,” Phys. Rev. A 40, 3061–3079 (1989).
[CrossRef] [PubMed]

The discussion above is consistent with the results of a recent experimental study of third-order harmonic generation in xenon comparing Gaussian and conical beams, published after completion of this manuscript [V. E. Peet and R. V. Tsubin, “Third-harmonic generation and multiphoton ionization in Bessel beams,” Phys. Rev. A 56, 1613–1620 (1997)]. Peet and Tsubin examined the variation with frequency of the harmonic yield in the vicinity of a three-photon resonance and at much weaker intensities than considered in this work for a focused Gaussian beam, an annular beam (α≈3°), and a Bessel beam (α=17°). The harmonic yield obtained with the annular beam exceeded that obtained with a Gaussian beam of same power in a small spectral range close to the resonance. No third-harmonic emission was detected for the Bessel beam, presumably because of the very low intensity of the beam that could be achieved in the experiment and the reabsorption of the emitted harmonic photons within the medium (with the cone half-angle α being fairly large, harmonic generation could be efficient only very close to the resonance where absorption is important; see Eq. 44).
[CrossRef]

See, for example, P. Antoine, A. L’Huillier, M. Lewenstein, P. Salières, and B. Carré, “Theory of high-order harmonic generation by an elliptically polarized laser field,” Phys. Rev. A 53, 1725–1745 (1996).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

B. Glushko, B. Kryzhanovsky, and D. Sarkisyan, “Self-phase-matching mechanism for efficient harmonic generation processes in a ring pump beam geometry,” Phys. Rev. Lett. 71, 243–246 (1993).
[CrossRef] [PubMed]

Other (9)

S. P. Tewari, H. Huang, and R. W. Boyd, “Theory of self-phase-matching,” Phys. Rev. A 51, 2707–2710 (1995); “Theory of third-harmonic generation using Bessel beams, and self-phase-matching,” Phys. Rev. A 54, 2314–2325 (1996).
[CrossRef] [PubMed]

A. L’Huillier, L. A. Lompré, G. Mainfray, and C. Manus, “High-order harmonic generation in rare gases,” in Atoms in Intense Laser Fields, M. Gavrila, ed., Advances in Atomic, Molecular and Optical Physics, Suppl. 1 (Academic, New York, 1992).

The first argument of the complex exponential in Eq. (2), kz cos α, should be kz[1−(sin α)2/2] for EBG to be an exact solution of the paraxial-wave equation. The difference between kz cos α and kz[1−(sin α)2/2] is negligible here because it is of fourth order in α. Equation (2) has the advantage that EBG reduces to the exact Bessel form of Eq. (6) in the loose-focusing limit.

R. M. Potvliege and P. H. G. Smith, “Stabilization of excited states and harmonic generation: recent theoretical results in the Sturmian-Floquet approach,” in SuperIntense Laser-Atom Physics, B. Piraux, A. L’Huillier, and K. Rzazewski, eds., Vol. B316 of NATO ASI Series (Plenum, New York, 1993), pp. 173–184. In a few words, the ionization rate is −2 Im(E)/ħ, where E is the quasi energy of the dressed 1s state; dq is the term proportional to exp(−iqωt) in the dipole moment of the atom in the dressed 1s state; and, using SI units, χat(qω)ε0Eq/2 is the contribution linear in Eq to the dipole moment dq for a two-color field E1 cos ωt+ Eq cos qωt.

A detailed account of these calculations will be presented elsewhere.

Phase matching for the Gaussian beam is not more favorable at high intensity than in a weak field in the present case because |χ1−χ3| has the same magnitude in both limits, and the contributions of the geometric phase q tan−1(2z/b) (important for small confocal parameters) and of the electronic susceptibility χel(ω) have the same sign.

This might be less important at high intensities, where the dipole moment usually saturates and ionization significantly reduces the number of atoms participating effectively in the harmonic generation process.

The cross section of the Gaussian beam of Fig. 1 would be larger, and for a fixed power its focal intensity would be smaller, if its confocal parameter were larger. One can optimize b and α so as to maximize the area of the focal plane irradiated above a given intensity for a given power. A numerical study showed that marginally larger areas can be achieved with an optimized Bessel-Gauss beam (α≠0) than with an optimized Gaussian beam (α=0); the difference between the two is a few percent at most.

J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1975).

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Figures (8)

Fig. 1
Fig. 1

(a) Radial intensity profile, I(z=0, r), and (b) radial power density, 2πrI(z=0, r), of a Gaussian beam (dashed curve, α=0) and a Bessel–Gauss beam (solid curve, α=1.2°) in the focal plane. The two beams have the same focal intensity, normalized to unity on axis, and the same total power.

Fig. 2
Fig. 2

Nonperturbative atomic quantities calculated in the Sturmian–Floquet approach, versus the incident intensity: (a) atomic dynamic dipole polarizabilities Re[αat(3ω)] (thick solid curve), Im[αat(3ω)] (thick dashed curve), Re[αat(ω)] (thin solid curve), and Im[αat(ω)] (thin dashed curve), in atomic units [multiply these results by 4πa03 to obtain the susceptibilities χat(3ω) and χat(ω) in SI units]; (b) phase of the atomic dipole moments d3 (for a fundamental wavelength λ=355 nm, thick solid curve) and d9 (for λ=1064 nm, thin solid curve); (c) modulus square of the atomic dipole moment d3 calculated nonperturbatively (thick solid curve) or in leading-order perturbation theory (thin solid curve) in atomic units; (d) probability that the atom is ionized by the pulse, as defined by Eq. (47) with t=+.

Fig. 3
Fig. 3

Optimum cone half-angle α, defined by Eq. (44), as a function of the intensity.

Fig. 4
Fig. 4

Conversion efficiency for a Gaussian beam (α=0) as a function of the peak focal intensity If0. The confocal parameter is bG=2 mm. The curves represent results of the quasi-stationary calculations. The markers represent results of time-dependent calculations. Solid curve and filled triangles, full results; dotted curve and open squares, photoionization neglected; open triangles, same as filled triangles but with absorption neglected; dashed curve, perturbation theory.

Fig. 5
Fig. 5

Same as Fig. 4 but for a Bessel–Gauss beam (α=1.2°). The confocal parameter is bBG=48.7 mm. Open circles: results represented by filled triangles in Fig. 4.

Fig. 6
Fig. 6

Conversion efficiency for an incident Bessel–Gauss beam (α=1.2°, bBG=48.7 mm) relative to that for an incident Gaussian beam of same power versus the confocal parameter of the latter, bG. The conversion efficiency is larger for the Bessel–Gauss beam when the ratio is larger than 1. The results were obtained with the time-dependent approach. The peak focal intensity If0 of the Bessel–Gauss beam is as follows: circles, 1×1013 W/cm2; squares, 2×1013 W/cm2; triangles, 3×1013W/cm2. The calculated points are joined by straight lines to guide the eye.

Fig. 7
Fig. 7

Far-field profile for a Bessel–Gauss beam (α=1.2°, bBG=48.7 mm) as a function of the far-field angle β for various peak focal intensities If0, obtained with the full time-dependent approach. Stars: corresponding results for a Gaussian beam (α=0, bG=2 mm) with If0=3×1013 W/cm2.

Fig. 8
Fig. 8

Far-field profiles for a Bessel–Gauss beam (α=1.2°, bBG=48.7 mm, If0=1×1013 W/cm2) as a function of the far-field angle β. Solid curve, same as in Fig. 7; dashed curve, same as the solid curve but with photoionization neglected; dot-dashed curve, same as the solid curve but with absorption neglected.

Tables (1)

Tables Icon

Table 1 Harmonic Yield for a Fixed Focal Intensity If of an Incident Bessel–Gauss Beam of Cone Half-Angle α and Confocal Parameter bBG; Results are Normalized to 1 at α=0

Equations (61)

Equations on this page are rendered with MathJax. Learn more.

3k1 cos α=k3 cos β,
EBG(r, z)=Ef1+(2z/b)2 J0kr sin α1+i(2z/b)×exp-k(r2+z2 sin2 α)/b1+(2z/b)2×expikz cos α-tan-1(2z/b)+(2z/b)×k(r2+z2 sin2 α)/b1+(2z/b)2,
EBG(r, z=0)=EfJ0(kr sin α)exp(-kr2/b),
EBG(r2z/kb sin α, zb/2)
-i Ef2π(2z/b)kr sin α× exp-kb4z2 (r-z sin α)2×exp{i[kz cos α+(k/2z)(r2+z2 sin2 α)]}.
EG(r, z)=Ef1+(2z/b)2 exp-kr2/b1+(2z/b)2×expikz-tan-1(2z/b)+(2z/b) kr2/b1+(2z/b)2,
EB(r, z)=EfJ0(kr sin α)exp(ikz cos α),
EBG(0)(r, z)=EfJ0(kr sin α)exp(ikz cos α-kr2/b),
IBG(0)(r)=IfJ02(kr sin α)exp(-2kr2/b),
PBG(α)=PG exp-bπ sin2 α2λI0bπ sin2 α2λ,
PG=PBG(α=0)=bλ4 If
IG,fbG=IBG,fbBG exp-bBGπ sin2 α2λI0bBGπ sin2 α2λ.
2Eq(x, t)-1q2ω2 2kq2Eq(x, t)t2=μ0 2PNL,q(x, t)t2.
Eq(x, t)=-μ04πmediumdx|x-x|-1×2PNL,q(x, t)t2t=t-(kq/qω)|x-x|.
2Eq(x)+kq2Eq(x)=-10 qωc2PNL,q(x),
Eq(x)=14π0 qωc2×medium exp(ikq|x-x|)|x-x|×PNL,q(x)dx.
|x-x||x|-1|x| [zz+rr cos(ϕ-ϕ)]+12|x|3 [r2r2 sin2(ϕ-ϕ)+r2z2+z2r2-2zzrr cos(ϕ-ϕ)],
|x-x|z-z+r2+r2-2rr cos(ϕ-ϕ)2(z-z),
|x-x|(z-z)cos β+r sin β-r sin β cos(ϕ-ϕ),
sin β=rz2+r2,
cos β=zz2+r2.
dq=|dq|exp(iΦq)exp[-iq(ωt-φE)],
PNL,q(x)=Nat(x)|dq(IBG)|×exp{i[Φq(IBG)+q arg(EBG)]},
Eq(x)Eqfar(x)=14π0|x| qωc2mediumNat(x)|dq[IBG(x)]|×exp[iΦtot(x, x)]dx,
Φtot(x, x)=qk1z cos α-tan-1(2z/b)+(2z/b)×k1(r2+z2 sin2 α)/b1+(2z/b)2+kq[(z-z)×cos β+r sin β-r sin β×cos(ϕ-ϕ)]+q argJ0k1r sin α1+i(2z/b)+Φq[IBG(r, z)].
Nat(x)=N0σ(z)[1-fion(x)],
kq(z-z)zzkq(ζ)dζ,
k1zzminzk1(ζ)dζ+k0, 1zmin,
kq(ζ)k0,q[1+N0σ(ζ)χq/2].
χel(qω)=-e2m0 1q2ω2=-8.970×10-34 λ2[nm]q2 m3.
Φtot(x, x)k0,qz2+r2+k0,q(cos α-cos β)z+N0k0,q(χ1 cos α-χq cos β)Σ(z)/2+k0,qσˆN0χq cos β/2+q-tan-1(2z/b)+(2z/b)k0,1(r2+z2 sin2 α)/b1+(2z/b)2+q argJ0k0,1r sin α1+i(2z/b)+Φq[IBG(r, z)]-k0,qr sin β cos(ϕ-ϕ).
Φtot(0)(x, x)k0,qz2+r2+k0,q(cos α-cos β)z+N0k0,q(χ1 cos α-χq cos β)Σ(z)/2+k0,qσˆN0χq cos β/2+q arg[J0(k0,1r sin α)]+Φq[IBG(0)(r)]-k0,qr sin β cos(ϕ-ϕ).
Eq(0) far(x)=2πN0σˆ exp[ik0,q(|x|)]4π0|x| qωc2× exp(iAumin)0rdr(1-fion)|dq[IBG(0)×(r)]|exp{iq arg[J0(k0,1r sin α)]}×exp{iΦq[IBG(0)(r)]}J0(k0,qr sin β)×exp(iD)Fz(A, B, C),
Fz(A, B, C)=exp(-iAumin)uminumaxK(u)×exp{i[Au+BK(u)]}exp[-CK(u)]du.
A=A(r, z)=2 qπλ σˆ(cos α-cos β),
B=B(r, r, z)=qπλ σˆN0 Re(χ1 cos α-χq cos β),
C=C(r, r, z)=qπλ σˆN0 Im(χ1 cos α-χq cos β),
D=D(r, r, z)=qπλ σˆN0χq cos β.
Fz=(S2+C2)-1{(C+iS)[1-cos S exp(-C)]+(S-iC)sin S exp(-C)},
cos β=1+Re(N0χ1/2)1+Re(N0χq/2) cos α=n1nq cos α,
cos β=1-N0|χel(ω)|/21-N0|χel(qω)|/2 cos α,
cos β=1+Re[N0χat(ω)/2]1+Re[N0χat(qω)/2] cos α.
cos α<1+Re[N0χat(qω)/2]1+Re[N0χat(ω)/2],
0drrJ0q(k0,1r sin α)J0(k0,qr sin β)exp(-qk0,1r2/b),
sin β=1q sin α,
sin αopt=n12-nq2n12-nq2/q2.
Ef=Ef0 exp[-(t-z cos α/c)2/2τ2],
If=If0 exp[-(t-z cos α/c)2/τ2],
f(Ip, t)=1-exp--t-z cos α/cΓ[Ip exp(-t2/τ2)]dt,
fion(x)fel(x)1-exp--+Γ[IBG(x)exp(-t2/τ2)]dt
Fz(A, B, C)=exp(-iAumin)×01 exp{i[AK-1(w)+Bw]}×exp(-Cw)dw,
K(u)=(1/2)[tan-1(2u)+1],
K(u)=1/(1+4u2),
K-1(w)=tan[2(w-1)]/2,
K(u)=1/2[u+(1/π)sin(πu)+1],
K(u)=cos2(πu/2),
Fz=iπ22 [exp(2iz)-1]N=-+JNB+iC2π×1[z-(N-1)π](z-Nπ)[z-(N+1)π],
K(u)=u+1/2,
K(u)=1,
K-1(w)=w-1/2,
Fz=[C+i(A+B)][1-cos(A+B)exp(-C)]+[(A+B)-iC]sin(A+B)exp(-C)(A+B)2+C2.

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