Abstract

We determine the conical half-angle at which a weak, loosely focused Bessel–Gauss beam of fixed focal intensity and confocal parameter is most efficient at generating a given harmonic in a gaseous target of arbitrary density profile. Simple analytical results are compared with fully numerical calculations for hydrogen, xenon, and rubidium. The variation of the conversion efficiency with the conical half-angle is shown to depend on the properties of the medium only through the macroscopic dispersion.

© 1999 Optical Society of America

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  1. T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 70, 1401–1404 (1993); “Nonlinear optics of Bessel beams,” 71, 209 (1993).
    [CrossRef] [PubMed]
  2. 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]
  3. J. Peatross, J. L. Chaloupka, and D. D. Meyerhofer, “High-order harmonic generation with an annular laser beam,” Opt. Lett. 19, 942–944 (1994).
    [CrossRef] [PubMed]
  4. S. P. Tewari, H. Huang, and R. W. Boyd, “Theory of self-phase-matching,” Phys. Rev. A 51, R2707–R2710 (1995); “Theory of third-harmonic generation using Bessel beams, and self-phase-matching,” Phys. Rev. A 54, 2314–2325 (1996).
    [CrossRef] [PubMed]
  5. S. Klewitz, P. Leiderer, S. Herminghaus, and S. Sogomonian, “Tunable stimulated Raman scattering by pumping with Bessel beams,” Opt. Lett. 21, 248–250 (1996).
    [CrossRef] [PubMed]
  6. V. E. Peet, “Resonantly enhanced multiphoton ionization of xenon in Bessel beams,” Phys. Rev. A 53, 3679–3682 (1996).
    [CrossRef] [PubMed]
  7. V. E. Peet and R. V. Tsubin, “Third-harmonic generation and multiphoton ionization in Bessel beams,” Phys. Rev. A 56, 1613–1620 (1997).
    [CrossRef]
  8. K. Shinozaki, C. A. Xu, H. Sasaki, and T. Kamijoh, “A comparison of optical second-harmonic generation efficiency using Bessel and Gaussian beams in bulk crystals,” Opt. Commun. 133, 300–304 (1997).
    [CrossRef]
  9. L. Niggl and M. Maier, “Efficient conical emission of stimulated Raman Stokes light generated by a Bessel pump beam,” Opt. Lett. 22, 910–912 (1997).
    [CrossRef] [PubMed]
  10. A. P. Piskarsas, V. Smilgevic̆ius, and A. P. Stabinis, “Optical parametric oscillator pumped by a Bessel beam,” Appl. Opt. 36, 7779–7782 (1997).
    [CrossRef]
  11. V. N. Belyĭ, N. S. Kazak, and N. A. Khilo, “Characteristics of parametric frequency conversion making use of Bessel beams,” Quantum Electron. 28, 522–525 (1998).
    [CrossRef]
  12. S. Klewitz, S. Sogomonian, M. Woerner, and S. Herminghaus, “Stimulated Raman scattering of femtosecond Bessel pulses,” Opt. Commun. 154, 186–190 (1998).
    [CrossRef]
  13. 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).
    [CrossRef]
  14. C. F. R. Caron, “Harmonic generation in gases using Bessel–Gauss beams,” Ph.D. dissertation (University of Durham, Durham, UK 1998).
  15. F. Gori, G. Guattari, and C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [CrossRef]
  16. V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, and G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).
  17. D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. 21, 9–11 (1996).
    [CrossRef] [PubMed]
  18. P. Pääkkönen and J. Turunen, “Resonators with Bessel–Gauss modes,” Opt. Commun. 156, 359–366 (1998).
    [CrossRef]
  19. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987); 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, “Comparison of Bessel and Gaussian beams,” Opt. Lett. OPLEDP 13, 79–80 (1988).
    [CrossRef] [PubMed]
  20. Z. Bouchal and M. Olivik, “Nondiffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
    [CrossRef]
  21. A. L’Huillier, X. F. Li, and L. A. Lompré, “Propagation effects in high-order harmonic generation in rare gases,” J. Opt. Soc. Am. B 7, 527–536 (1990).
    [CrossRef]
  22. 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, Supplement 1 (Academic, New York, 1992), pp. 139–206.
  23. C. F. R. Caron and R. M. Potvliege, Comput. Phys. Commun. (to be published).
  24. See, e.g., J. F. Reintjes, Nonlinear Optical Parametric Processes in Liquids and Gases (Academic, New York, 1984).
  25. For example, γ=1.4 for λ=1064 nm, q=3, |z|≤2 mm, b=50 mm, α=1.2 deg, and β=0.4 deg. That γ is close to unity means that the integral multiplying (1+2iz/b) in integral (18) is essentially real. For z≪b the imaginary part of this integral is proportional to z/b and to an integral of a combination of products of J0 and J1 functions, a power, and an exponential, while the real part is nearly constant in z and can be approximated by integral (17) taken at z=0. It is not surprising that the imaginary part is dominated by the real part, and hence that γ≈1, since the (rapid) oscillations of the J1 functions are out of phase with those of the J0 functions, while the J0 functions oscillate in phase in the real part when β≈α/q.
  26. This result is, of course, well known for rectangular density profiles. (See Ref. 24.)
  27. Another system fulfilling these conditions is krypton at λ≈ 348 nm and λ≈369 nm. Interesting differences between Gaussian and noncollinear beams have been described for resonant harmonic generation and multiphoton ionization in xenon at λ≈440 nm. (See Ref. 7.)
  28. We verified this by evaluating the contribution of pressure broadening to the absorption coefficient, for hydrogen and xenon at 355 nm, in the simple approaches of W. R. Ferrell, M. G. Payne, and W. R. Garrett, “Resonance broadening and shifting of spectral lines in xenon and krypton,” Phys. Rev. A 36, 81–89 (1987); and of G. Peach, “Collisional broadening of spectral lines,” in Atomic, Molecular, and Optical Physics Handbook, G. W. F. Drake, ed. (American Institute of Physics, Melville, New York, 1996), p. 669, Eq. (57.27).
    [CrossRef] [PubMed]
  29. W. F. Chan, G. Cooper, X. Guo, G. R. Burton, and C. E. Brion, “Absolute optical oscillator strengths for the electronic excitation of atoms at high resolution. III. The photoabsorption of argon, krypton and xenon,” Phys. Rev. A 46, 149–171 (1992). We took Ei=E0+11.67 eV in Eq. (29), 0.46 eV below the true 2P3/2 ionization threshold, to allow for the fact that the contribution of individual bound states cannot be isolated from the data given in this reference above that energy.
    [CrossRef] [PubMed]
  30. A. H. Kung, “Third-harmonic generation in a pulsed supersonic jet of xenon,” Opt. Lett. 8, 24–26 (1983).
    [CrossRef] [PubMed]
  31. T. M. Miller and B. Bederson, “Atomic and molecular polarizabilities—a review of recent advances,” Adv. At. Mol. Phys. 13, 1–55 (1977).
    [CrossRef]
  32. L. J. Zych and J. F. Young, “Limitation of 3547 to 1182 Å conversion efficiency in Xe,” IEEE J. Quantum Electron. 14, 147–149 (1978).
    [CrossRef]
  33. H. Puell, K. Spanner, W. Falkenstein, W. Kaiser, and C. R. Vidal, “Third-harmonic generation of mode-locked Nd:glass laser pulses in phase-matched Rb-Xe mixtures,” Phys. Rev. A 14, 2240–2257 (1976).
    [CrossRef]

1998 (4)

V. N. Belyĭ, N. S. Kazak, and N. A. Khilo, “Characteristics of parametric frequency conversion making use of Bessel beams,” Quantum Electron. 28, 522–525 (1998).
[CrossRef]

S. Klewitz, S. Sogomonian, M. Woerner, and S. Herminghaus, “Stimulated Raman scattering of femtosecond Bessel pulses,” Opt. Commun. 154, 186–190 (1998).
[CrossRef]

P. Pääkkönen and J. Turunen, “Resonators with Bessel–Gauss modes,” Opt. Commun. 156, 359–366 (1998).
[CrossRef]

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).
[CrossRef]

1997 (4)

L. Niggl and M. Maier, “Efficient conical emission of stimulated Raman Stokes light generated by a Bessel pump beam,” Opt. Lett. 22, 910–912 (1997).
[CrossRef] [PubMed]

A. P. Piskarsas, V. Smilgevic̆ius, and A. P. Stabinis, “Optical parametric oscillator pumped by a Bessel beam,” Appl. Opt. 36, 7779–7782 (1997).
[CrossRef]

V. E. Peet and R. V. Tsubin, “Third-harmonic generation and multiphoton ionization in Bessel beams,” Phys. Rev. A 56, 1613–1620 (1997).
[CrossRef]

K. Shinozaki, C. A. Xu, H. Sasaki, and T. Kamijoh, “A comparison of optical second-harmonic generation efficiency using Bessel and Gaussian beams in bulk crystals,” Opt. Commun. 133, 300–304 (1997).
[CrossRef]

1996 (4)

D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. 21, 9–11 (1996).
[CrossRef] [PubMed]

S. Klewitz, P. Leiderer, S. Herminghaus, and S. Sogomonian, “Tunable stimulated Raman scattering by pumping with Bessel beams,” Opt. Lett. 21, 248–250 (1996).
[CrossRef] [PubMed]

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

1995 (1)

Z. Bouchal and M. Olivik, “Nondiffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[CrossRef]

1994 (1)

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]

1992 (1)

W. F. Chan, G. Cooper, X. Guo, G. R. Burton, and C. E. Brion, “Absolute optical oscillator strengths for the electronic excitation of atoms at high resolution. III. The photoabsorption of argon, krypton and xenon,” Phys. Rev. A 46, 149–171 (1992). We took Ei=E0+11.67 eV in Eq. (29), 0.46 eV below the true 2P3/2 ionization threshold, to allow for the fact that the contribution of individual bound states cannot be isolated from the data given in this reference above that energy.
[CrossRef] [PubMed]

1990 (1)

1987 (1)

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

1983 (1)

1978 (1)

L. J. Zych and J. F. Young, “Limitation of 3547 to 1182 Å conversion efficiency in Xe,” IEEE J. Quantum Electron. 14, 147–149 (1978).
[CrossRef]

1977 (1)

T. M. Miller and B. Bederson, “Atomic and molecular polarizabilities—a review of recent advances,” Adv. At. Mol. Phys. 13, 1–55 (1977).
[CrossRef]

1976 (1)

H. Puell, K. Spanner, W. Falkenstein, W. Kaiser, and C. R. Vidal, “Third-harmonic generation of mode-locked Nd:glass laser pulses in phase-matched Rb-Xe mixtures,” Phys. Rev. A 14, 2240–2257 (1976).
[CrossRef]

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).

Bederson, B.

T. M. Miller and B. Bederson, “Atomic and molecular polarizabilities—a review of recent advances,” Adv. At. Mol. Phys. 13, 1–55 (1977).
[CrossRef]

Belyi?, V. N.

V. N. Belyĭ, N. S. Kazak, and N. A. Khilo, “Characteristics of parametric frequency conversion making use of Bessel beams,” Quantum Electron. 28, 522–525 (1998).
[CrossRef]

Bouchal, Z.

Z. Bouchal and M. Olivik, “Nondiffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[CrossRef]

Brion, C. E.

W. F. Chan, G. Cooper, X. Guo, G. R. Burton, and C. E. Brion, “Absolute optical oscillator strengths for the electronic excitation of atoms at high resolution. III. The photoabsorption of argon, krypton and xenon,” Phys. Rev. A 46, 149–171 (1992). We took Ei=E0+11.67 eV in Eq. (29), 0.46 eV below the true 2P3/2 ionization threshold, to allow for the fact that the contribution of individual bound states cannot be isolated from the data given in this reference above that energy.
[CrossRef] [PubMed]

Burton, G. R.

W. F. Chan, G. Cooper, X. Guo, G. R. Burton, and C. E. Brion, “Absolute optical oscillator strengths for the electronic excitation of atoms at high resolution. III. The photoabsorption of argon, krypton and xenon,” Phys. Rev. A 46, 149–171 (1992). We took Ei=E0+11.67 eV in Eq. (29), 0.46 eV below the true 2P3/2 ionization threshold, to allow for the fact that the contribution of individual bound states cannot be isolated from the data given in this reference above that energy.
[CrossRef] [PubMed]

Caron, C. F. R.

Chaloupka, J. L.

Chan, W. F.

W. F. Chan, G. Cooper, X. Guo, G. R. Burton, and C. E. Brion, “Absolute optical oscillator strengths for the electronic excitation of atoms at high resolution. III. The photoabsorption of argon, krypton and xenon,” Phys. Rev. A 46, 149–171 (1992). We took Ei=E0+11.67 eV in Eq. (29), 0.46 eV below the true 2P3/2 ionization threshold, to allow for the fact that the contribution of individual bound states cannot be isolated from the data given in this reference above that energy.
[CrossRef] [PubMed]

Cooper, G.

W. F. Chan, G. Cooper, X. Guo, G. R. Burton, and C. E. Brion, “Absolute optical oscillator strengths for the electronic excitation of atoms at high resolution. III. The photoabsorption of argon, krypton and xenon,” Phys. Rev. A 46, 149–171 (1992). We took Ei=E0+11.67 eV in Eq. (29), 0.46 eV below the true 2P3/2 ionization threshold, to allow for the fact that the contribution of individual bound states cannot be isolated from the data given in this reference above that energy.
[CrossRef] [PubMed]

Falkenstein, W.

H. Puell, K. Spanner, W. Falkenstein, W. Kaiser, and C. R. Vidal, “Third-harmonic generation of mode-locked Nd:glass laser pulses in phase-matched Rb-Xe mixtures,” Phys. Rev. A 14, 2240–2257 (1976).
[CrossRef]

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]

Guo, X.

W. F. Chan, G. Cooper, X. Guo, G. R. Burton, and C. E. Brion, “Absolute optical oscillator strengths for the electronic excitation of atoms at high resolution. III. The photoabsorption of argon, krypton and xenon,” Phys. Rev. A 46, 149–171 (1992). We took Ei=E0+11.67 eV in Eq. (29), 0.46 eV below the true 2P3/2 ionization threshold, to allow for the fact that the contribution of individual bound states cannot be isolated from the data given in this reference above that energy.
[CrossRef] [PubMed]

Hall, D. G.

Herminghaus, S.

S. Klewitz, S. Sogomonian, M. Woerner, and S. Herminghaus, “Stimulated Raman scattering of femtosecond Bessel pulses,” Opt. Commun. 154, 186–190 (1998).
[CrossRef]

S. Klewitz, P. Leiderer, S. Herminghaus, and S. Sogomonian, “Tunable stimulated Raman scattering by pumping with Bessel beams,” Opt. Lett. 21, 248–250 (1996).
[CrossRef] [PubMed]

Kaiser, W.

H. Puell, K. Spanner, W. Falkenstein, W. Kaiser, and C. R. Vidal, “Third-harmonic generation of mode-locked Nd:glass laser pulses in phase-matched Rb-Xe mixtures,” Phys. Rev. A 14, 2240–2257 (1976).
[CrossRef]

Kamijoh, T.

K. Shinozaki, C. A. Xu, H. Sasaki, and T. Kamijoh, “A comparison of optical second-harmonic generation efficiency using Bessel and Gaussian beams in bulk crystals,” Opt. Commun. 133, 300–304 (1997).
[CrossRef]

Kazak, N. S.

V. N. Belyĭ, N. S. Kazak, and N. A. Khilo, “Characteristics of parametric frequency conversion making use of Bessel beams,” Quantum Electron. 28, 522–525 (1998).
[CrossRef]

Khilo, N. A.

V. N. Belyĭ, N. S. Kazak, and N. A. Khilo, “Characteristics of parametric frequency conversion making use of Bessel beams,” Quantum Electron. 28, 522–525 (1998).
[CrossRef]

Klewitz, S.

S. Klewitz, S. Sogomonian, M. Woerner, and S. Herminghaus, “Stimulated Raman scattering of femtosecond Bessel pulses,” Opt. Commun. 154, 186–190 (1998).
[CrossRef]

S. Klewitz, P. Leiderer, S. Herminghaus, and S. Sogomonian, “Tunable stimulated Raman scattering by pumping with Bessel beams,” Opt. Lett. 21, 248–250 (1996).
[CrossRef] [PubMed]

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]

Kung, A. H.

L’Huillier, A.

Leiderer, P.

Li, X. F.

Lompré, L. A.

Maier, M.

Meyerhofer, D. D.

Miller, T. M.

T. M. Miller and B. Bederson, “Atomic and molecular polarizabilities—a review of recent advances,” Adv. At. Mol. Phys. 13, 1–55 (1977).
[CrossRef]

Niggl, L.

Olivik, M.

Z. Bouchal and M. Olivik, “Nondiffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[CrossRef]

Pääkkönen, P.

P. Pääkkönen and J. Turunen, “Resonators with Bessel–Gauss modes,” Opt. Commun. 156, 359–366 (1998).
[CrossRef]

Padovani, C.

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

Peatross, J.

Peet, V. E.

V. E. Peet and R. V. Tsubin, “Third-harmonic generation and multiphoton ionization in Bessel beams,” Phys. Rev. A 56, 1613–1620 (1997).
[CrossRef]

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

Piskarsas, A. P.

Potvliege, R. M.

Puell, H.

H. Puell, K. Spanner, W. Falkenstein, W. Kaiser, and C. R. Vidal, “Third-harmonic generation of mode-locked Nd:glass laser pulses in phase-matched Rb-Xe mixtures,” Phys. Rev. A 14, 2240–2257 (1976).
[CrossRef]

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]

Sasaki, H.

K. Shinozaki, C. A. Xu, H. Sasaki, and T. Kamijoh, “A comparison of optical second-harmonic generation efficiency using Bessel and Gaussian beams in bulk crystals,” Opt. Commun. 133, 300–304 (1997).
[CrossRef]

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).

Shinozaki, K.

K. Shinozaki, C. A. Xu, H. Sasaki, and T. Kamijoh, “A comparison of optical second-harmonic generation efficiency using Bessel and Gaussian beams in bulk crystals,” Opt. Commun. 133, 300–304 (1997).
[CrossRef]

Smilgeviccius, V.

Sogomonian, S.

S. Klewitz, S. Sogomonian, M. Woerner, and S. Herminghaus, “Stimulated Raman scattering of femtosecond Bessel pulses,” Opt. Commun. 154, 186–190 (1998).
[CrossRef]

S. Klewitz, P. Leiderer, S. Herminghaus, and S. Sogomonian, “Tunable stimulated Raman scattering by pumping with Bessel beams,” Opt. Lett. 21, 248–250 (1996).
[CrossRef] [PubMed]

Spagnolo, G. Schirripa

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

Spanner, K.

H. Puell, K. Spanner, W. Falkenstein, W. Kaiser, and C. R. Vidal, “Third-harmonic generation of mode-locked Nd:glass laser pulses in phase-matched Rb-Xe mixtures,” Phys. Rev. A 14, 2240–2257 (1976).
[CrossRef]

Stabinis, A. P.

Tsubin, R. V.

V. E. Peet and R. V. Tsubin, “Third-harmonic generation and multiphoton ionization in Bessel beams,” Phys. Rev. A 56, 1613–1620 (1997).
[CrossRef]

Turunen, J.

P. Pääkkönen and J. Turunen, “Resonators with Bessel–Gauss modes,” Opt. Commun. 156, 359–366 (1998).
[CrossRef]

Vidal, C. R.

H. Puell, K. Spanner, W. Falkenstein, W. Kaiser, and C. R. Vidal, “Third-harmonic generation of mode-locked Nd:glass laser pulses in phase-matched Rb-Xe mixtures,” Phys. Rev. A 14, 2240–2257 (1976).
[CrossRef]

Woerner, M.

S. Klewitz, S. Sogomonian, M. Woerner, and S. Herminghaus, “Stimulated Raman scattering of femtosecond Bessel pulses,” Opt. Commun. 154, 186–190 (1998).
[CrossRef]

Xu, C. A.

K. Shinozaki, C. A. Xu, H. Sasaki, and T. Kamijoh, “A comparison of optical second-harmonic generation efficiency using Bessel and Gaussian beams in bulk crystals,” Opt. Commun. 133, 300–304 (1997).
[CrossRef]

Young, J. F.

L. J. Zych and J. F. Young, “Limitation of 3547 to 1182 Å conversion efficiency in Xe,” IEEE J. Quantum Electron. 14, 147–149 (1978).
[CrossRef]

Zych, L. J.

L. J. Zych and J. F. Young, “Limitation of 3547 to 1182 Å conversion efficiency in Xe,” IEEE J. Quantum Electron. 14, 147–149 (1978).
[CrossRef]

Adv. At. Mol. Phys. (1)

T. M. Miller and B. Bederson, “Atomic and molecular polarizabilities—a review of recent advances,” Adv. At. Mol. Phys. 13, 1–55 (1977).
[CrossRef]

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

L. J. Zych and J. F. Young, “Limitation of 3547 to 1182 Å conversion efficiency in Xe,” IEEE J. Quantum Electron. 14, 147–149 (1978).
[CrossRef]

J. Mod. Opt. (2)

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

Z. Bouchal and M. Olivik, “Nondiffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[CrossRef]

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

Opt. Commun. (4)

S. Klewitz, S. Sogomonian, M. Woerner, and S. Herminghaus, “Stimulated Raman scattering of femtosecond Bessel pulses,” Opt. Commun. 154, 186–190 (1998).
[CrossRef]

P. Pääkkönen and J. Turunen, “Resonators with Bessel–Gauss modes,” Opt. Commun. 156, 359–366 (1998).
[CrossRef]

K. Shinozaki, C. A. Xu, H. Sasaki, and T. Kamijoh, “A comparison of optical second-harmonic generation efficiency using Bessel and Gaussian beams in bulk crystals,” Opt. Commun. 133, 300–304 (1997).
[CrossRef]

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

Opt. Lett. (5)

Phys. Rev. A (4)

H. Puell, K. Spanner, W. Falkenstein, W. Kaiser, and C. R. Vidal, “Third-harmonic generation of mode-locked Nd:glass laser pulses in phase-matched Rb-Xe mixtures,” Phys. Rev. A 14, 2240–2257 (1976).
[CrossRef]

W. F. Chan, G. Cooper, X. Guo, G. R. Burton, and C. E. Brion, “Absolute optical oscillator strengths for the electronic excitation of atoms at high resolution. III. The photoabsorption of argon, krypton and xenon,” Phys. Rev. A 46, 149–171 (1992). We took Ei=E0+11.67 eV in Eq. (29), 0.46 eV below the true 2P3/2 ionization threshold, to allow for the fact that the contribution of individual bound states cannot be isolated from the data given in this reference above that energy.
[CrossRef] [PubMed]

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

V. E. Peet and R. V. Tsubin, “Third-harmonic generation and multiphoton ionization in Bessel beams,” Phys. Rev. A 56, 1613–1620 (1997).
[CrossRef]

Phys. Rev. Lett. (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]

Quantum Electron. (1)

V. N. Belyĭ, N. S. Kazak, and N. A. Khilo, “Characteristics of parametric frequency conversion making use of Bessel beams,” Quantum Electron. 28, 522–525 (1998).
[CrossRef]

Other (11)

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987); 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, “Comparison of Bessel and Gaussian beams,” Opt. Lett. OPLEDP 13, 79–80 (1988).
[CrossRef] [PubMed]

S. P. Tewari, H. Huang, and R. W. Boyd, “Theory of self-phase-matching,” Phys. Rev. A 51, R2707–R2710 (1995); “Theory of third-harmonic generation using Bessel beams, and self-phase-matching,” Phys. Rev. A 54, 2314–2325 (1996).
[CrossRef] [PubMed]

T. Wulle and S. Herminghaus, “Nonlinear optics of Bessel beams,” Phys. Rev. Lett. 70, 1401–1404 (1993); “Nonlinear optics of Bessel beams,” 71, 209 (1993).
[CrossRef] [PubMed]

C. F. R. Caron, “Harmonic generation in gases using Bessel–Gauss beams,” Ph.D. dissertation (University of Durham, Durham, UK 1998).

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, Supplement 1 (Academic, New York, 1992), pp. 139–206.

C. F. R. Caron and R. M. Potvliege, Comput. Phys. Commun. (to be published).

See, e.g., J. F. Reintjes, Nonlinear Optical Parametric Processes in Liquids and Gases (Academic, New York, 1984).

For example, γ=1.4 for λ=1064 nm, q=3, |z|≤2 mm, b=50 mm, α=1.2 deg, and β=0.4 deg. That γ is close to unity means that the integral multiplying (1+2iz/b) in integral (18) is essentially real. For z≪b the imaginary part of this integral is proportional to z/b and to an integral of a combination of products of J0 and J1 functions, a power, and an exponential, while the real part is nearly constant in z and can be approximated by integral (17) taken at z=0. It is not surprising that the imaginary part is dominated by the real part, and hence that γ≈1, since the (rapid) oscillations of the J1 functions are out of phase with those of the J0 functions, while the J0 functions oscillate in phase in the real part when β≈α/q.

This result is, of course, well known for rectangular density profiles. (See Ref. 24.)

Another system fulfilling these conditions is krypton at λ≈ 348 nm and λ≈369 nm. Interesting differences between Gaussian and noncollinear beams have been described for resonant harmonic generation and multiphoton ionization in xenon at λ≈440 nm. (See Ref. 7.)

We verified this by evaluating the contribution of pressure broadening to the absorption coefficient, for hydrogen and xenon at 355 nm, in the simple approaches of W. R. Ferrell, M. G. Payne, and W. R. Garrett, “Resonance broadening and shifting of spectral lines in xenon and krypton,” Phys. Rev. A 36, 81–89 (1987); and of G. Peach, “Collisional broadening of spectral lines,” in Atomic, Molecular, and Optical Physics Handbook, G. W. F. Drake, ed. (American Institute of Physics, Melville, New York, 1996), p. 669, Eq. (57.27).
[CrossRef] [PubMed]

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Figures (5)

Fig. 1
Fig. 1

Rectangular (solid curve), cosine-squared (dashed curve), and triangular (dot-dashed curve) density profiles considered in this paper. The curves represent N(z)/N0 for a full width at half-maximum of 1 mm.

Fig. 2
Fig. 2

Conversion efficiency versus the Bessel angle of the incident beam for third-order harmonic generation in xenon at a fundamental wavelength of 355 nm. The confocal parameter is (a) b=50 mm, (b) b=5 mm, or (c) b=1 mm. The peak atomic density N0 is 2.0×1018 atoms/cm3, and the full width at half-maximum of the density profile is 0.5 mm. Solid curves and squares: rectangular profile. Dot-dashed curves and circles: cosine-squared profile. Dashed curves and triangles: triangular profile.

Fig. 3
Fig. 3

Conversion efficiency versus the Bessel angle of the incident beam for third-order harmonic generation in rubidium at a fundamental wavelength of 1064 nm. The confocal parameter is 15 mm, and the target has a cosine-squared density profile with a full width at half-maximum of 1.5 mm. Solid curve and squares: N0=2.9×1017 atoms/cm3. Dashed curve and circles: N0=4.0×1017 atoms/cm3. Dot-dashed curve and triangles: N0=2.0×1018 atoms/cm3.

Fig. 4
Fig. 4

Conversion efficiency versus the Bessel angle of the incident beam for seventh-order harmonic generation in xenon at a fundamental wavelength of 812 nm, normalized to unity at α=0. The confocal parameter is 5 mm, and the target has a cosine-squared density profile with a full width at half-maximum of 1 mm. Solid curve: N0=1×1018 atoms/cm3. Short-dashed curve and circle: N0=2.0×1018 atoms/cm3. Long-dashed curve and triangle: N0=3.0×1018 atoms/cm3. Dot-dashed curve and diamond: N0=4.0×1018 atoms/cm3.

Fig. 5
Fig. 5

Comparison of the conversion efficiency for different atomic species with the same macroscopic dispersion, N0(χ1-χq): (a) absolute values, (b) values normalized to unity at α=0. The density profile is rectangular with a width equal to a tenth of the confocal parameter. Short-dashed curves: rubidium, λ=1064 nm, b=15 mm, and N0=2.95×1017 atoms/cm3. Long-dashed curves: xenon, λ=355 nm, b=5 mm, and N0=2.0×1018 atoms/cm3. Dot-dashed curves: hydrogen, λ=355 nm, b=5 mm, and N0=4.0×1018 atoms/cm3. The two thin solid curves give the conversion efficiency for rubidium at the same wavelength and the same confocal parameters as the short-dashed curve but for N0=2.90 or 3.00×1017 atoms/cm3.

Tables (3)

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Table 1 Atomic Susceptibilities Adopted in this Paper for Hydrogen, Xenon, and Rubidiuma

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Table 2 Gas Density Profilesa

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Table 3 Absolute Conversion Efficiency R3(α) at α=0 (Corresponding to an Incident Gaussian Beam) and at α=αmax as a Function of the Confocal Parameter ba

Equations (34)

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EBG(r, z)=Ef1+(2z/b)2J0kr 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,
EB(r, z)=Ef J0(kr sin α)exp(ikz cos α).
EG(r, z)=Ef1+(2z/b)2exp-kr2/b1+(2z/b)2×expikz-tan-1(2z/b)+(2z/b) kr2/b1+(2z/b)2.
PBG(α)=PG exp-πb sin2 α2λI0πb sin2 α2λ,
PG=PBG(α=0)=bλ4If
E(x, t)=E0(x, t)+q>0Re[2Eq(x, t)exp(-iqωt)],
PNL(x, t)=PNL,0(x, t)+q>0Re[2PNL,q(x, t)exp(-iqωt)].
2[Eq(x, t)exp(-iqωt)]-1q2ω22t2[kq2Eq(x, t)
×exp(-iqωt)]=μ0 2t2[PNL,q(x, t)exp(-iqωt)],
Eq(x, t)=-μ04πexp(iqωt)medium 1|x-x|2t2PNL,q(x, t)×exp(-iqωt)t=t-(kq/qω)|x-x|dx.
Eq(x)
=14π0qωc2medium exp(ikq|x-x|)|x-x|PNL,q(x)dx.
kp=pk[1+N(z)Re χp/2],
PNL,q(x)=N(z)DqEBGq(r, z).
Eqas(x)=14π0|x|q2k2medium exp{iqk[|x|+Zq(z)-z cos β-r sin β cos(ϕ-ϕ)]}×N(z)DqEBGq(r, z)dx.
Zq(z)=(z)Re(χ1/2)cos α-[(z)-(zmax)]Re(χq/2)cos β,
(z)=zminzN(z)dz.
Eqas(x)=exp(iqk|x|)20|x|q2k2Dqzminzmax exp[iqkZq(z)]N(z)×0J0(qkr sin β)EBGq(r, z)r drdz.
0J0(qkr sin β)J0qkr sin α1+2iz/bexp-qkr2/b1+2iz/br dr,
(1+2iz/b)0J0(qkr sin β1+2iz/b)J0qkr sin α1+2iz/b
×exp(-qkr2/b)r dr.
Eqas(x)zminzmax exp[iΦq(z)]N(z)dz,
Φq(z)=qk(cos α-cos β)z+qkZq(z)-(q-γ)×(2z/b).
q(2πz2/λb)sin2 α1
S(α)=1Lz1z2dΦqdz2dz,
dSdα=2q sin αz1z2-qk1 cos α+kq cos β+2b(q-γ)k1dz.
αopt=q2q2-11/2NavRe(χ1-χq)-2λπb1-1q1/2.
Nav=1Lz1z2N(z)dz.
αopt=q2q2-11/2λqπ1/2[-Δkq¯-2(q-1)/b]1/2,
Δkq¯=2πqλ1Lz1z2(nq-n1)dz.
αopt=Nav Re(χ1-χq)-2λπb1-1q1/2
χp=4π2c2r0n fn(En-E0)2-p22ω2+Ei (df/dE)(E-E0)2-p22ω2dE.
Einc(x, t)=EBG(r, z)exp(-t2/τ2).
Rq(α)=|2Eq(x, t)|2rdrdϕdt|Einc(x, t)|2rdrdϕdt.

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