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Journal of the Optical Society of America B

Journal of the Optical Society of America B


  • Vol. 15, Iss. 3 — Mar. 1, 1998
  • pp: 1096–1106

Phase matching and harmonic generation in Bessel Gauss–beams

C. F. R. Caron and R. M. Potvliege  »View Author Affiliations

JOSA B, Vol. 15, Issue 3, pp. 1096-1106 (1998)

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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

OCIS Codes
(190.2620) Nonlinear optics : Harmonic generation and mixing
(190.4160) Nonlinear optics : Multiharmonic generation
(190.4180) Nonlinear optics : Multiphoton processes
(190.4410) Nonlinear optics : Nonlinear optics, parametric processes
(270.4180) Quantum optics : Multiphoton processes
(270.6620) Quantum optics : Strong-field processes

C. F. R. Caron and R. M. Potvliege, "Phase matching and harmonic generation in Bessel Gauss–beams," J. Opt. Soc. Am. B 15, 1096-1106 (1998)

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  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.
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  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.
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  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).
  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.
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  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).
  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).

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