Abstract

We show that the elegant Laguerre–Gauss light beams of high radial order n are asymptotically equal to Bessel–Gauss light beams. The Bessel–Gauss beam equivalent to each elegant Laguerre–Gauss beam is found and shown to have almost identical propagation factors M2. In the limit n, elegant Laguerre–Gauss beams can be identified with Durnin’s Bessel beam. Our results suggest a new experimental procedure for generating light beams with nondiffractinglike properties directly from the output of a stable resonator.

© 2001 Optical Society of America

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    [CrossRef]
  47. M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
    [CrossRef]
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2000 (3)

1999 (3)

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A 60, 2438–2441 (1999).
[CrossRef]

R. Borghi, “Superposition scheme for J0-correlated partially coherent sources,” IEEE J. Quantum Electron. 35, 849–856 (1999).
[CrossRef]

See, for example, M. J. Padgett, L. Allen, “The angular momentum of light: optical spanners and the rotational frequency shift,” Opt. Quantum Electron. 31, 1–12 (1999), and references therein.
[CrossRef]

1998 (6)

S. Saghafi, C. J. R. Sheppard, “The beam propagation factor for higher-order Gaussian beams,” Opt. Commun. 153, 207–210 (1998).
[CrossRef]

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

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

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, S. Rishton, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers,” Appl. Phys. Lett. 72, 1284–1286 (1998).
[CrossRef]

P. L. Greene, D. G. Hall, “Properties and diffraction of vector Bessel–Gauss beams,” J. Opt. Soc. Am. A 15, 3020–3027 (1998).
[CrossRef]

S. Saghafi, C. J. R. Sheppard, “Near field and far field of elegant Hermite–Gaussian and Laguerre–Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998).
[CrossRef]

1997 (1)

1996 (5)

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

M. Santarsiero, “Propagation of generalized Bessel–Gauss beams through ABCD optical systems,” Opt. Commun. 132, 1–7 (1996).
[CrossRef]

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

D. G. Hall, “Vector-beam solution of Maxwell’s wave equation,” Opt. Lett. 21, 9–12 (1996).
[CrossRef] [PubMed]

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

1995 (1)

1994 (1)

1993 (2)

B. Glushko, B. Kryzhanovsky, 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]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

1992 (1)

1991 (3)

1988 (2)

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

J. Durnin, J. J. Miceli, J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13, 79–80 (1988).
[CrossRef] [PubMed]

1987 (3)

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

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

J. Durnin, J. J. Miceli, J. H. Eberly, “Nondiffracting beams,” Phys. Rev. Lett. 58, 1499–1502 (1987).
[CrossRef] [PubMed]

1986 (1)

1985 (2)

T. Takenaka, M. Yokota, O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
[CrossRef]

M. Yokota, T. Takenaka, O. Fukumitsu, “Relations between conventional and complex beams,” Trans. Inst. Electron. Commun. Eng. Jpn. 68-C, 1130–1131 (1985).

1978 (1)

C. J. R. Sheppard, T. Wilson, “Gaussian-beam theory of lenses with annular aperture,” Microwaves Opt. Acoust. 2, 105–112 (1978).
[CrossRef]

1977 (1)

1976 (1)

1975 (1)

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

1973 (2)

1971 (1)

1970 (1)

H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2349–2376 (1970).
[CrossRef]

1965 (1)

N. G. Vakhimov, “Open resonators with mirrors having variable reflection coefficients,” Radio Eng. Electron. Phys. (USSR) 10, 1439–1446 (1965).

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).

Allen, L.

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A 60, 2438–2441 (1999).
[CrossRef]

See, for example, M. J. Padgett, L. Allen, “The angular momentum of light: optical spanners and the rotational frequency shift,” Opt. Quantum Electron. 31, 1–12 (1999), and references therein.
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Altucci, C.

Arlt, J.

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A 60, 2438–2441 (1999).
[CrossRef]

Arnaud, J. A.

J. A. Arnaud, “Hamiltonian theory of beam mode propagation,” Prog. Opt. 11, 247–304 (1973).
[CrossRef]

J. A. Arnaud, “Mode coupling in first-order optics,” J. Opt. Soc. Am. 61, 751–758 (1971).
[CrossRef]

Bagini, V.

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

Beijersbergen, M. W.

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

Bélanger, P. A.

Beracha, I.

Borghi, R.

R. Borghi, “Superposition scheme for J0-correlated partially coherent sources,” IEEE J. Quantum Electron. 35, 849–856 (1999).
[CrossRef]

R. Borghi, M. Santarsiero, “M2 factor of Bessel–Gauss beams,” Opt. Lett. 22, 262–264 (1997).
[CrossRef] [PubMed]

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

Boyd, R. W.

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

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

Bruzzese, R.

Cao, Q.

Caron, C. F. R.

Casperson, L. W.

Chi, S.

Cincotti, G.

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

D’Antuoni, D.

de Lisio, C.

Dholakia, K.

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A 60, 2438–2441 (1999).
[CrossRef]

Durnin, J.

J. Durnin, J. J. Miceli, J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13, 79–80 (1988).
[CrossRef] [PubMed]

J. Durnin, J. J. Miceli, J. H. Eberly, “Nondiffracting beams,” Phys. Rev. Lett. 58, 1499–1502 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13, 79–80 (1988).
[CrossRef] [PubMed]

J. Durnin, J. J. Miceli, J. H. Eberly, “Nondiffracting beams,” Phys. Rev. Lett. 58, 1499–1502 (1987).
[CrossRef] [PubMed]

Frezza, F.

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

Fukumitsu, O.

M. Yokota, T. Takenaka, O. Fukumitsu, “Relations between conventional and complex beams,” Trans. Inst. Electron. Commun. Eng. Jpn. 68-C, 1130–1131 (1985).

T. Takenaka, M. Yokota, O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
[CrossRef]

Glushko, B.

B. Glushko, B. Kryzhanovsky, 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, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

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

Gradshtein, I. S.

I. S. Gradshtein, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980).

Greene, P. L.

P. L. Greene, D. G. Hall, “Properties and diffraction of vector Bessel–Gauss beams,” J. Opt. Soc. Am. A 15, 3020–3027 (1998).
[CrossRef]

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, S. Rishton, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers,” Appl. Phys. Lett. 72, 1284–1286 (1998).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

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

Hafizi, B.

Hall, D. G.

Heyman, E.

Huang, H.

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

Jordan, R. H.

Kryzhanovsky, B.

B. Glushko, B. Kryzhanovsky, 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]

Lax, M.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Liu, H. K.

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

McKnight, W. B.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Miceli, J. J.

J. Durnin, J. J. Miceli, J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13, 79–80 (1988).
[CrossRef] [PubMed]

J. Durnin, J. J. Miceli, J. H. Eberly, “Nondiffracting beams,” Phys. Rev. Lett. 58, 1499–1502 (1987).
[CrossRef] [PubMed]

Mukunda, N.

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

Olson, C.

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, S. Rishton, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers,” Appl. Phys. Lett. 72, 1284–1286 (1998).
[CrossRef]

Pääkkönen, P.

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

Padgett, M. J.

J. Arlt, K. Dholakia, L. Allen, M. J. Padgett, “Efficiency of second-harmonic generation with Bessel beams,” Phys. Rev. A 60, 2438–2441 (1999).
[CrossRef]

See, for example, M. J. Padgett, L. Allen, “The angular momentum of light: optical spanners and the rotational frequency shift,” Opt. Quantum Electron. 31, 1–12 (1999), and references therein.
[CrossRef]

Padovani, C.

F. Gori, G. Guattari, C. Padovani, “Modal expansion for J0-correlated Schell-model sources,” Opt. Commun. 64, 311–316 (1987).
[CrossRef]

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

Palma, C.

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

Potvliege, R. M.

Pratesi, R.

Rishton, S.

C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, S. Rishton, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers,” Appl. Phys. Lett. 72, 1284–1286 (1998).
[CrossRef]

Ronchi, L.

Rosen, J.

Ryzhik, I. M.

I. S. Gradshtein, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980).

Saghafi, S.

S. Saghafi, C. J. R. Sheppard, “The beam propagation factor for higher-order Gaussian beams,” Opt. Commun. 153, 207–210 (1998).
[CrossRef]

S. Saghafi, C. J. R. Sheppard, “Near field and far field of elegant Hermite–Gaussian and Laguerre–Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998).
[CrossRef]

Salik, B.

Santarsiero, M.

R. Borghi, M. Santarsiero, “M2 factor of Bessel–Gauss beams,” Opt. Lett. 22, 262–264 (1997).
[CrossRef] [PubMed]

M. Santarsiero, “Propagation of generalized Bessel–Gauss beams through ABCD optical systems,” Opt. Commun. 132, 1–7 (1996).
[CrossRef]

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

Sarkisyan, D.

B. Glushko, B. Kryzhanovsky, 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, 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, G. Schirripa Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

Shchegrov, A. V.

Sheppard, C. J. R.

S. Saghafi, C. J. R. Sheppard, “Near field and far field of elegant Hermite–Gaussian and Laguerre–Gaussian modes,” J. Mod. Opt. 45, 1999–2009 (1998).
[CrossRef]

S. Saghafi, C. J. R. Sheppard, “The beam propagation factor for higher-order Gaussian beams,” Opt. Commun. 153, 207–210 (1998).
[CrossRef]

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A. Siegman, “Defining the effective radius of curvature of a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
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R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD law,” Opt. Commun. 65, 322–328 (1988).
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T. Takenaka, M. Yokota, O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
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M. Yokota, T. Takenaka, O. Fukumitsu, “Relations between conventional and complex beams,” Trans. Inst. Electron. Commun. Eng. Jpn. 68-C, 1130–1131 (1985).

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

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N. G. Vakhimov, “Open resonators with mirrors having variable reflection coefficients,” Radio Eng. Electron. Phys. (USSR) 10, 1439–1446 (1965).

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M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
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C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, S. Rishton, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers,” Appl. Phys. Lett. 72, 1284–1286 (1998).
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M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

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T. Takenaka, M. Yokota, O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
[CrossRef]

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[CrossRef]

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C. Olson, P. L. Greene, G. W. Wicks, D. G. Hall, S. Rishton, “High-order azimuthal spatial modes of concentric-circle-grating surface-emitting semiconductor lasers,” Appl. Phys. Lett. 72, 1284–1286 (1998).
[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

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[CrossRef]

Opt. Commun. (8)

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

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

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[CrossRef]

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[CrossRef]

R. Simon, N. Mukunda, E. C. G. Sudarshan, “Partially coherent beams and a generalized ABCD law,” Opt. Commun. 65, 322–328 (1988).
[CrossRef]

S. Saghafi, C. J. R. Sheppard, “The beam propagation factor for higher-order Gaussian beams,” Opt. Commun. 153, 207–210 (1998).
[CrossRef]

M. W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, J. P. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96, 123–132 (1993).
[CrossRef]

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Other (6)

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A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).

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[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Power spectrum |V˜ns|2 of the eLG beams. (a) s=0, n=1, w0=0.2 mm, (b) s=0, n=4, w0=0.4 mm, (c) s=0, n=9, w0=0.6 mm. The values of n and w0 are chosen so that 2n/w0=β=constant for the three eLG beams.

Fig. 2
Fig. 2

(a), (b), and (c) Transversal intensity distributions at the waist of eLG beams (solid curves) of increasing radial order n=1, 4, and 9, respectively, and the corrresponding equivalent BG beam’s (dotted curves); k=107 mm-1, s=0, and β=104 m-1. According to Eqs. (8), w0=0.2, 0.4, and 0.6 mm for the eLG beams, and v0=0.28, 0.56, and 0.85 mm for the equivalent BG beams. The dashed curves represent the intensity distribution of the limiting nondiffracting Bessel beam J0(βr). (d)–(f) Intensity profiles of the same beams after a propagation distance D/2, where D=kv0/β is the diffraction-free range of each equivalent BG beam. In (g)–(i) the propagation distance is D.

Fig. 3
Fig. 3

Comparison of the beam quality factor of eLG beams (circles) and BG beams (triangles) as functions of the order n [see Eqs. (23) and (25)]. Values of s are (a) 0, (b) 1, and (c) 2.

Equations (37)

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Vns(r, θ)=Arw0sLnsr2w02exp-r2w02exp(isθ),
V˜ns(ρ, ϕ)=Aisπw02n!(πw0ρ)2n+sexp(-π2w02ρ2)exp(isϕ),
x2n+sexp(-x2)n!ns-12π1/2exp[-2(x-n)2],
V˜ns(ρ, ϕ)Aisw02πns-121/2× exp-2π2w02ρ-nπw02exp(isϕ),
Us(r, θ)=BJs(βr)exp-r2v02exp(isθ),
U˜s(ρ, ϕ)=Bisπv02exp-β2v024exp(-π2v02ρ2)×Is(πβv02ρ)exp(isϕ),
U˜s(ρ, ϕ)Bisπv0βexp-π2v02ρ-β2π2exp(isϕ)
v0=2w0,β=2n/w0,B=nsA.
Lns(x)expx2nxs/2Js(2nx)
Vns(r, θ)Ans/2Js2nw0rexp-r22w02exp(isθ),
Vns(r, θ, z)=Aw02wC2n+s/2+1rwCs×Lnsr2wC2exp-r2wC2exp(isθ),
Us(r, θ, z)=Bv02vc2Jsv02vC2βr×exp-r2+z2β2/k2vC2exp(isθ),
 
M2=2πσ0σ,
σ02=0r2|Vns(r, θ)|2r dr0|Vns(r, θ)|2r dr,
σ2=0ρ2|V˜ns(ρ, ϕ)|2ρ dρ0|V˜ns(ρ, ϕ)|2ρ dρ.
σ02=w020ts+1exp(-2t)[Lns(t)]2dt0tsexp(-2t)[Lns(t)]2dt
=-w02limb2G(b)G(b),
G(b)=0tsexp(-bt)[Lns(t)]2dt.
G(b)=Γ(2n+s+1)(n!)2(b-1)2nb2n+s+1×F-n, -n; -(2n+s); b(b-2)(b-1)2,
G(b)=Γ(2n+s+1)(n!)2(b-1)2n-1b2n+s+2[2n-(b-1)×(s+1)]×F-n, -n; -(2n+s); b(b-2)(b-1)2+2(b-1)2n-3b2n+s+1×F-n, -n; -(2n+s); b(b-2)(b-1)2,
F[-n, -n; -(2n+s), 0]=1,
F[-n, -n; -(2n+s), 0]=-n22n+s,
σ02=w0222n+s+s22n+s.
σ2=12π2w020t2n+s+1exp(-t)dt0t2n+sexp(-t)dt
=2n+s+12π2w02,
MeLG2=(2n+s+1) 2n+s+s22n+s1/2.
MeLG2=(2n+1)1/2
MBG2=1+s+2nIs+1(2n)Is+1(2n)2-4n21/2.
Vns(r, θ)=Arw0sLnsr2w02exp-r2w02exp(isθ),
V˜(ρ, ϕ)=2πisexp(isϕ)0f(r)Js(2πrρ),
V˜ns(ρ, ϕ)=2πis A exp(isϕ)0rw0sLnsr2w02×exp-r2w02Js(2πrρ)r dr.
V˜ns(ρ, ϕ)=2πisA exp(isϕ)w020xs+1Lns(x2)×exp(-x2)Js(2πw0xρ)dx,
lim0(-)nLnsy24=y2n22nn!,
V˜ns(ρ, ϕ)=Aisπw02n!(πw0ρ)2n+sexp(-π2w02ρ2)exp(isϕ),
f(p)(t)ppexp(-p)p!exp-(t-p)22p
x2pexp(-x2)ppexp(-p)p!exp[-2(x-p)2].

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