Abstract

We derive the analytical expression of the M2 factor of a Bessel–Gauss beam of any order and discuss its behavior for high and low values of the width of the Gaussian profile. The case of unapertured Bessel beams can be treated as a limiting case of our analysis.

© 1997 Optical Society of America

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References

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  1. P. M. Mejías, H. Weber, R. Martínez-Herrero, and A. González-Ureña, eds., Proceedings of Workshop on Laser Beam Characterization (Sociedad Espanda de Optica, Madrid, Spain, 1993).
  2. A. Siegman, Proc. SPIE 1224, 2 (1990).
    [CrossRef]
  3. R. Simon, N. Mukunda, and E. C. G. Sudarshan, Opt. Commun. 65, 322 (1988).
    [CrossRef]
  4. P. A. Bélanger, Opt. Lett. 16, 196 (1991).
    [CrossRef]
  5. A. Siegman, IEEE J. Quantum Electron. 27, 1146 (1991).
    [CrossRef]
  6. M. W. Sasnett, in The Physics and Technology of Laser Resonators, D. R. Hall and P. E. Jackson, eds. (Hilger, London, 1989), p. 132.
  7. M. A. Porras and R. Medina, Appl. Opt. 34, 8247 (1995).
    [CrossRef] [PubMed]
  8. J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
    [CrossRef] [PubMed]
  9. C. Paré and P. A. Bélanger, Opt. Commun. 123, 679 (1991).
    [CrossRef]
  10. F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 491 (1987).
    [CrossRef]
  11. J. Lu, X. Xu, H. Zou, and J. F. Greenleaf, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42, 649 (1995).
    [CrossRef]
  12. P. L. Greene and D. G. Hall, J. Opt. Soc. Am. A 13, 962 (1996).
    [CrossRef]
  13. I. S. Gradshtein and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980).
  14. R. N. Bracewell, The Fourier Transform and Its Applications, Rev. 2nd ed. (McGraw-Hill, New York, 1986).
  15. F. Gori, S. Vicalvi, M. Santarsiero, and R. Borghi, Opt. Lett. 21, 1205 (1996).
    [CrossRef] [PubMed]
  16. J. Turunen, A. Vasara, and A. T. Friberg, J. Opt. Soc. Am. A 8, 282 (1991).
    [CrossRef]
  17. F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 311 (1987).
    [CrossRef]

1996 (2)

1995 (2)

J. Lu, X. Xu, H. Zou, and J. F. Greenleaf, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42, 649 (1995).
[CrossRef]

M. A. Porras and R. Medina, Appl. Opt. 34, 8247 (1995).
[CrossRef] [PubMed]

1991 (4)

C. Paré and P. A. Bélanger, Opt. Commun. 123, 679 (1991).
[CrossRef]

P. A. Bélanger, Opt. Lett. 16, 196 (1991).
[CrossRef]

A. Siegman, IEEE J. Quantum Electron. 27, 1146 (1991).
[CrossRef]

J. Turunen, A. Vasara, and A. T. Friberg, J. Opt. Soc. Am. A 8, 282 (1991).
[CrossRef]

1990 (1)

A. Siegman, Proc. SPIE 1224, 2 (1990).
[CrossRef]

1988 (1)

R. Simon, N. Mukunda, and E. C. G. Sudarshan, Opt. Commun. 65, 322 (1988).
[CrossRef]

1987 (3)

F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 491 (1987).
[CrossRef]

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 311 (1987).
[CrossRef]

Bélanger, P. A.

P. A. Bélanger, Opt. Lett. 16, 196 (1991).
[CrossRef]

C. Paré and P. A. Bélanger, Opt. Commun. 123, 679 (1991).
[CrossRef]

Borghi, R.

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications, Rev. 2nd ed. (McGraw-Hill, New York, 1986).

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Friberg, A. T.

Gori, F.

F. Gori, S. Vicalvi, M. Santarsiero, and R. Borghi, Opt. Lett. 21, 1205 (1996).
[CrossRef] [PubMed]

F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 311 (1987).
[CrossRef]

F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 491 (1987).
[CrossRef]

Gradshtein, I. S.

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

Greene, P. L.

Greenleaf, J. F.

J. Lu, X. Xu, H. Zou, and J. F. Greenleaf, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42, 649 (1995).
[CrossRef]

Guattari, G.

F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 311 (1987).
[CrossRef]

F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 491 (1987).
[CrossRef]

Hall, D. G.

Lu, J.

J. Lu, X. Xu, H. Zou, and J. F. Greenleaf, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42, 649 (1995).
[CrossRef]

Medina, R.

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Mukunda, N.

R. Simon, N. Mukunda, and E. C. G. Sudarshan, Opt. Commun. 65, 322 (1988).
[CrossRef]

Padovani, C.

F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 491 (1987).
[CrossRef]

F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 311 (1987).
[CrossRef]

Paré, C.

C. Paré and P. A. Bélanger, Opt. Commun. 123, 679 (1991).
[CrossRef]

Porras, M. A.

Ryzhik, I. M.

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

Santarsiero, M.

Sasnett, M. W.

M. W. Sasnett, in The Physics and Technology of Laser Resonators, D. R. Hall and P. E. Jackson, eds. (Hilger, London, 1989), p. 132.

Siegman, A.

A. Siegman, IEEE J. Quantum Electron. 27, 1146 (1991).
[CrossRef]

A. Siegman, Proc. SPIE 1224, 2 (1990).
[CrossRef]

Simon, R.

R. Simon, N. Mukunda, and E. C. G. Sudarshan, Opt. Commun. 65, 322 (1988).
[CrossRef]

Sudarshan, E. C. G.

R. Simon, N. Mukunda, and E. C. G. Sudarshan, Opt. Commun. 65, 322 (1988).
[CrossRef]

Turunen, J.

Vasara, A.

Vicalvi, S.

Xu, X.

J. Lu, X. Xu, H. Zou, and J. F. Greenleaf, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42, 649 (1995).
[CrossRef]

Zou, H.

J. Lu, X. Xu, H. Zou, and J. F. Greenleaf, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42, 649 (1995).
[CrossRef]

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

A. Siegman, IEEE J. Quantum Electron. 27, 1146 (1991).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

J. Lu, X. Xu, H. Zou, and J. F. Greenleaf, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 42, 649 (1995).
[CrossRef]

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

Opt. Commun. (4)

F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 311 (1987).
[CrossRef]

C. Paré and P. A. Bélanger, Opt. Commun. 123, 679 (1991).
[CrossRef]

F. Gori, G. Guattari, and C. Padovani, Opt. Commun. 64, 491 (1987).
[CrossRef]

R. Simon, N. Mukunda, and E. C. G. Sudarshan, Opt. Commun. 65, 322 (1988).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (1)

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef] [PubMed]

Proc. SPIE (1)

A. Siegman, Proc. SPIE 1224, 2 (1990).
[CrossRef]

Other (4)

P. M. Mejías, H. Weber, R. Martínez-Herrero, and A. González-Ureña, eds., Proceedings of Workshop on Laser Beam Characterization (Sociedad Espanda de Optica, Madrid, Spain, 1993).

M. W. Sasnett, in The Physics and Technology of Laser Resonators, D. R. Hall and P. E. Jackson, eds. (Hilger, London, 1989), p. 132.

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

R. N. Bracewell, The Fourier Transform and Its Applications, Rev. 2nd ed. (McGraw-Hill, New York, 1986).

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Figures (1)

Fig. 1
Fig. 1

M2 factor of a Bessel–Gauss beam of order n.

Equations (22)

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M2=2πσ0σ,
Unr,ϑ=AJnβrexp-r2/w02expinϑ,
U˜nν,φ=πw02-in exp-β2w024expinφ×exp-π2w02ν2Inπβw02ν,
σ02=0Un2r3dr0Un2rdr,
σ2=0U˜n2ν3dν0U˜n2νdν,
F1α=0Jn2βrexp-αr2rdr,
F2α=0In2δνexp-αν2νdν,
σ02=-F1αF1αα=2/w02,
σ2=-F2αF2αα=2π2w02,
F1α=12αInβ22αexp-β22α,
Inx=inJn-ix,
F1α=ddα12α exp-β22αInβ22α=1β2ddαβ22αddxx exp-xInxx=β2/2α,
xInx=xIn+1x+nInx,
ddxx exp-xInx=Inxexp-x×1+n+xIn+1xInx-1.
F1α=-12α2 exp-β22αInβ22α×1+n+β22αIn+1β22αInβ22α-1,
σ02=w0221+n+γIn+1γInγ-1,
σ2=12π2w021+n+γIn+1γInγ+1.
M2=1+n+γRnγ2-γ21/2,
Rnγ=In+1γ/Inγ.
Unr,ϑAβn2nn!rn exp-r2/w02expinϑ,
Inxexpx2πx1-4n2-18x,
σ0w0/2,  σβ/2π,

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