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).

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).

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).

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).

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).

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

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).

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

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

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).

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

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).

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).

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

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

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

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).

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

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

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

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).

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).

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).

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).

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).

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

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

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).

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

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).

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).

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).

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

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).

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

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

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).

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).

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

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

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

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

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).

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).

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).

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).

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

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).

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).

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 E_{BG} 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 E_{BG} 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; d_{q} is the term proportional to exp(−iqωt) in the dipole moment of the atom in the dressed 1s state; and, using SI units, χ_{at}(qω)ε_{0}E_{q}/2 is the contribution linear in E_{q} to the dipole moment d_{q} for a two-color field E_{1} cos ωt+ E_{q} 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 |χ_{1}−χ_{3}| 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 χ_{el}(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).