Abstract

We examine a family of Bessel–Gauss beam solutions to the vector wave equation that allow a combination of azimuthal and radial polarization in the transverse electric field. Recently reported linear and azimuthal Bessel–Gauss beams may be identified as members of this set. Several free parameters determine the form and behavior of each beam; varying these parameters can produce distinctly different intensity patterns and beam behavior. We find a general diffraction integral for circularly symmetric disturbances and investigate two special cases, a thin lens and a circular aperture.

© 1998 Optical Society of America

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References

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  1. See, for example, A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), pp. 642–648.
  2. J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2348 (1970).
    [CrossRef]
  3. H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2349–2376 (1970).
    [CrossRef]
  4. L. W. Casperson, D. G. Hall, A. A. Tovar, “Sinusoidal–Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 14, 3341–3348 (1997).
    [CrossRef]
  5. L. W. Casperson, A. A. Tovar, “Hermite-sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 15, 954–961 (1998).
    [CrossRef]
  6. F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [CrossRef]
  7. R. H. Jordan, D. G. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel–Gauss beam solution,” Opt. Lett. 19, 427–429 (1994).
    [CrossRef] [PubMed]
  8. P. L. Greene, D. G. Hall, “Diffraction characteristics of the azimuthal Bessel–Gauss beam,” J. Opt. Soc. Am. A 13, 962–966 (1996).
    [CrossRef]
  9. T. Erdogan, D. G. Hall, “Circularly symmetric distributed feedback laser: coupled mode treatment of TE vector fields,” IEEE J. Quantum Electron. QE-28, 612–623 (1992).
    [CrossRef]
  10. R. H. Jordan, D. G. Hall, O. King, G. Wicks, S. Rishton, “Lasing behavior of circular grating surface-emitting semiconductor lasers,” J. Opt. Soc. Am. B 14, 449–453 (1997).
    [CrossRef]
  11. C. Palma, G. Cincotti, G. Guattari, M. Santarsiero, “Imaging of generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
    [CrossRef]
  12. B. Lü, W. Huang, “Three-dimensional intensity distribution of focused Bessel–Gauss beams,” J. Mod. Opt. 43, 509–515 (1996).
    [CrossRef]
  13. B. Lü, W. Huang, “Focal shift in unapertured Bessel–Gauss beams,” Opt. Commun. 109, 43–46 (1994).
    [CrossRef]
  14. R. Borghi, M. Santarsiero, “M2 factor of Bessel–Gauss beams,” Opt. Lett. 22, 262–264 (1997).
    [CrossRef] [PubMed]
  15. V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. S. Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).
  16. M. Santarsiero, “Propagation of generalized Bessel–Gauss beams through ABCD optical systems,” Opt. Commun. 132, 1–7 (1996).
    [CrossRef]
  17. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  18. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
    [CrossRef]
  19. Z. Bouchal, M. Olivı́k, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
    [CrossRef]
  20. D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. 21, 9–11 (1996).
    [CrossRef] [PubMed]
  21. T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
    [CrossRef]
  22. D. Pohl, “Operation of a ruby laser in the purely transverse electric mode TE01,” Appl. Phys. Lett. 20, 266–267 (1972).
    [CrossRef]
  23. J. Wynne, “Generation of the rotationally symmetric TE01 and TM01 modes from a wavelength-tunable laser,” IEEE J. Quantum Electron. QE-10, 125–127 (1974).
    [CrossRef]
  24. M. E. Marhic, E. Garmire, “Low-order TE0q operation of a CO2 laser for transmission through circular waveguides,” Appl. Phys. Lett. 38, 743–745 (1981).
    [CrossRef]
  25. 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]
  26. See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 59–60.
  27. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, San Diego, Calif., 1985), p. 585.
  28. W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd enlarged ed. (Springer-VerlagNew York, New York, 1966), Sec. 3.8.3, p. 93.
  29. Similar results for the m=0 mode, or ABG beam, have been presented in previous work (Ref. 8) for β≈10-6 k, though they are erroneously described therein as for β≈10-3 k.

1998 (2)

L. W. Casperson, A. A. Tovar, “Hermite-sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 15, 954–961 (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]

1997 (3)

1996 (6)

V. Bagini, F. Frezza, M. Santarsiero, G. Schettini, G. S. 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, G. Cincotti, G. Guattari, M. Santarsiero, “Imaging of generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
[CrossRef]

B. Lü, W. Huang, “Three-dimensional intensity distribution of focused Bessel–Gauss beams,” J. Mod. Opt. 43, 509–515 (1996).
[CrossRef]

P. L. Greene, D. G. Hall, “Diffraction characteristics of the azimuthal Bessel–Gauss beam,” J. Opt. Soc. Am. A 13, 962–966 (1996).
[CrossRef]

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

1995 (1)

Z. Bouchal, M. Olivı́k, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[CrossRef]

1994 (2)

1992 (2)

T. Erdogan, D. G. Hall, “Circularly symmetric distributed feedback laser: coupled mode treatment of TE vector fields,” IEEE J. Quantum Electron. QE-28, 612–623 (1992).
[CrossRef]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

1987 (3)

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

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

1981 (1)

M. E. Marhic, E. Garmire, “Low-order TE0q operation of a CO2 laser for transmission through circular waveguides,” Appl. Phys. Lett. 38, 743–745 (1981).
[CrossRef]

1974 (1)

J. Wynne, “Generation of the rotationally symmetric TE01 and TM01 modes from a wavelength-tunable laser,” IEEE J. Quantum Electron. QE-10, 125–127 (1974).
[CrossRef]

1972 (1)

D. Pohl, “Operation of a ruby laser in the purely transverse electric mode TE01,” Appl. Phys. Lett. 20, 266–267 (1972).
[CrossRef]

1970 (2)

J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2348 (1970).
[CrossRef]

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

Anderson, E. H.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, San Diego, Calif., 1985), p. 585.

Arnaud, J. A.

J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2348 (1970).
[CrossRef]

Bagini, V.

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

Borghi, R.

Bouchal, Z.

Z. Bouchal, M. Olivı́k, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[CrossRef]

Casperson, L. W.

Cincotti, G.

C. Palma, G. Cincotti, G. Guattari, M. Santarsiero, “Imaging of generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
[CrossRef]

Durnin, J.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[CrossRef]

Eberly, J. H.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Erdogan, T.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

T. Erdogan, D. G. Hall, “Circularly symmetric distributed feedback laser: coupled mode treatment of TE vector fields,” IEEE J. Quantum Electron. QE-28, 612–623 (1992).
[CrossRef]

Frezza, F.

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

Garmire, E.

M. E. Marhic, E. Garmire, “Low-order TE0q operation of a CO2 laser for transmission through circular waveguides,” Appl. Phys. Lett. 38, 743–745 (1981).
[CrossRef]

Goodman, J. W.

See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 59–60.

Gori, F.

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

Greene, P. L.

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, “Diffraction characteristics of the azimuthal Bessel–Gauss beam,” J. Opt. Soc. Am. A 13, 962–966 (1996).
[CrossRef]

Guattari, G.

C. Palma, G. Cincotti, G. Guattari, M. Santarsiero, “Imaging of generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
[CrossRef]

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

Hall, D. G.

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]

L. W. Casperson, D. G. Hall, A. A. Tovar, “Sinusoidal–Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 14, 3341–3348 (1997).
[CrossRef]

R. H. Jordan, D. G. Hall, O. King, G. Wicks, S. Rishton, “Lasing behavior of circular grating surface-emitting semiconductor lasers,” J. Opt. Soc. Am. B 14, 449–453 (1997).
[CrossRef]

P. L. Greene, D. G. Hall, “Diffraction characteristics of the azimuthal Bessel–Gauss beam,” J. Opt. Soc. Am. A 13, 962–966 (1996).
[CrossRef]

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

R. H. Jordan, D. G. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel–Gauss beam solution,” Opt. Lett. 19, 427–429 (1994).
[CrossRef] [PubMed]

T. Erdogan, D. G. Hall, “Circularly symmetric distributed feedback laser: coupled mode treatment of TE vector fields,” IEEE J. Quantum Electron. QE-28, 612–623 (1992).
[CrossRef]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Huang, W.

B. Lü, W. Huang, “Three-dimensional intensity distribution of focused Bessel–Gauss beams,” J. Mod. Opt. 43, 509–515 (1996).
[CrossRef]

B. Lü, W. Huang, “Focal shift in unapertured Bessel–Gauss beams,” Opt. Commun. 109, 43–46 (1994).
[CrossRef]

Jordan, R. H.

King, O.

R. H. Jordan, D. G. Hall, O. King, G. Wicks, S. Rishton, “Lasing behavior of circular grating surface-emitting semiconductor lasers,” J. Opt. Soc. Am. B 14, 449–453 (1997).
[CrossRef]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Lü, B.

B. Lü, W. Huang, “Three-dimensional intensity distribution of focused Bessel–Gauss beams,” J. Mod. Opt. 43, 509–515 (1996).
[CrossRef]

B. Lü, W. Huang, “Focal shift in unapertured Bessel–Gauss beams,” Opt. Commun. 109, 43–46 (1994).
[CrossRef]

Magnus, W.

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd enlarged ed. (Springer-VerlagNew York, New York, 1966), Sec. 3.8.3, p. 93.

Marhic, M. E.

M. E. Marhic, E. Garmire, “Low-order TE0q operation of a CO2 laser for transmission through circular waveguides,” Appl. Phys. Lett. 38, 743–745 (1981).
[CrossRef]

Miceli, J. J.

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Oberhettinger, F.

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd enlarged ed. (Springer-VerlagNew York, New York, 1966), Sec. 3.8.3, p. 93.

Olivi´k, M.

Z. Bouchal, M. Olivı́k, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[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]

Padovani, C.

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

Palma, C.

C. Palma, G. Cincotti, G. Guattari, M. Santarsiero, “Imaging of generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
[CrossRef]

Pohl, D.

D. Pohl, “Operation of a ruby laser in the purely transverse electric mode TE01,” Appl. Phys. Lett. 20, 266–267 (1972).
[CrossRef]

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]

R. H. Jordan, D. G. Hall, O. King, G. Wicks, S. Rishton, “Lasing behavior of circular grating surface-emitting semiconductor lasers,” J. Opt. Soc. Am. B 14, 449–453 (1997).
[CrossRef]

Rooks, M. J.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

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. S. Spagnolo, “Generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 1155–1166 (1996).

C. Palma, G. Cincotti, G. Guattari, M. Santarsiero, “Imaging of generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
[CrossRef]

Schettini, G.

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

Siegman, A. E.

See, for example, A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), pp. 642–648.

Soni, R. P.

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd enlarged ed. (Springer-VerlagNew York, New York, 1966), Sec. 3.8.3, p. 93.

Spagnolo, G. S.

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

Tovar, A. A.

Wicks, G.

Wicks, G. W.

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]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

Wynne, J.

J. Wynne, “Generation of the rotationally symmetric TE01 and TM01 modes from a wavelength-tunable laser,” IEEE J. Quantum Electron. QE-10, 125–127 (1974).
[CrossRef]

Zucker, H.

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

Appl. Phys. Lett. (4)

M. E. Marhic, E. Garmire, “Low-order TE0q operation of a CO2 laser for transmission through circular waveguides,” Appl. Phys. Lett. 38, 743–745 (1981).
[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]

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, “Circularly symmetric operation of a concentric-circle-grating, surface-emitting, AlGaAs/GaAs quantum-well semiconductor laser,” Appl. Phys. Lett. 60, 1921–1923 (1992).
[CrossRef]

D. Pohl, “Operation of a ruby laser in the purely transverse electric mode TE01,” Appl. Phys. Lett. 20, 266–267 (1972).
[CrossRef]

Bell Syst. Tech. J. (2)

J. A. Arnaud, “Nonorthogonal optical waveguides and resonators,” Bell Syst. Tech. J. 49, 2311–2348 (1970).
[CrossRef]

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

IEEE J. Quantum Electron. (2)

T. Erdogan, D. G. Hall, “Circularly symmetric distributed feedback laser: coupled mode treatment of TE vector fields,” IEEE J. Quantum Electron. QE-28, 612–623 (1992).
[CrossRef]

J. Wynne, “Generation of the rotationally symmetric TE01 and TM01 modes from a wavelength-tunable laser,” IEEE J. Quantum Electron. QE-10, 125–127 (1974).
[CrossRef]

J. Mod. Opt. (4)

C. Palma, G. Cincotti, G. Guattari, M. Santarsiero, “Imaging of generalized Bessel–Gauss beams,” J. Mod. Opt. 43, 2269–2277 (1996).
[CrossRef]

B. Lü, W. Huang, “Three-dimensional intensity distribution of focused Bessel–Gauss beams,” J. Mod. Opt. 43, 509–515 (1996).
[CrossRef]

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

Z. Bouchal, M. Olivı́k, “Non-diffractive vector Bessel beams,” J. Mod. Opt. 42, 1555–1566 (1995).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. Opt. Soc. Am. B (1)

Opt. Commun. (3)

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

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

B. Lü, W. Huang, “Focal shift in unapertured Bessel–Gauss beams,” Opt. Commun. 109, 43–46 (1994).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. Lett. (1)

J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Other (5)

See, for example, A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), pp. 642–648.

See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 59–60.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, San Diego, Calif., 1985), p. 585.

W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics, 3rd enlarged ed. (Springer-VerlagNew York, New York, 1966), Sec. 3.8.3, p. 93.

Similar results for the m=0 mode, or ABG beam, have been presented in previous work (Ref. 8) for β≈10-6 k, though they are erroneously described therein as for β≈10-3 k.

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Figures (10)

Fig. 1
Fig. 1

Calculated vector beam intensities at a propagation distance z=1 cm. Images (a), (b), (c), and (d) show modes m=0, 1, 2, and 8, respectively, each independently normalized. The parameters for each beam are λ=632.8 nm, w0=1 mm, and β=0.0005 k; the radial dimension is measured in millimeters.

Fig. 2
Fig. 2

Direction of polarization of the real part of each of the fields shown in Fig. 1. Images (a), (b), (c), and (d) show modes m=0, 1, 2, and 8, respectively. The m=0 mode, also called the ABG beam, is purely azimuthally polarized; higher-order modes contain both azimuthal and radial components.

Fig. 3
Fig. 3

Intensity of the m=2 beam at a radius ρ=1 mm at each azimuthal angle, for various values of the β parameter. Any value of β for which the intensity is constant for all ϕ is one for which the beam is circularly symmetric at this radius. It is circularly symmetric at all radii only for sufficiently small β.

Fig. 4
Fig. 4

Comparison of the absolute values of the Jm-1 (solid curve) and Jm+1 (dashed curve) terms of Eqs. (9) for m=2, plotted in arbitrary units against the β parameter at ρ=1 mm. For small β, Jm-1 clearly dominates.

Fig. 5
Fig. 5

Beam intensity patterns for the m=2 mode at z=1 cm. Images (a), (b), (c), and (d) show the patterns for β=0.0001 k, 0.0003 k, 0.0005 k, and 0.0008 k, respectively (k9.9 µm-1). Image (a) shows the approximate circular symmetry that occurs for very small β, image (b) shows the single azimuthally periodic ring that appears for intermediate values of β, and images (c) and (d) show the more complicated radial and azimuthal periodicity typical of larger β. Other beam parameters are as in Fig. 1, and the radial dimension is measured in millimeters.

Fig. 6
Fig. 6

Half-cone divergence angle on propagation plotted against the azimuthal mode number m, for several values of β. Other beam parameters are set as in Fig. 1. Only integer values of m are allowed, but the points for each β have been joined for clarity. For m>2 the angle increases with increasing β, but more interesting structure can be seen for small m. For β=0.0008 k, the maximum divergence angle (for m>0) occurs for m=6, at roughly 0.018°, and for large m the angle is near 0.011°.

Fig. 7
Fig. 7

Calculated vector beam intensities [column (a)] and corresponding measured far-field patterns emitted by CCGSE lasers [column (b)]. Each beam was calculated with λ=1 µm and β=0.0005 k; the mode numbers are 0, 2, 11, and 23, as numbered at the left, and the Gaussian waists w0 are 1.5, 1.5, 3.0, and 3.5 mm, respectively. Radial dimensions vary among the calculated and the measured patterns.

Fig. 8
Fig. 8

(a) Beam width, in micrometers, as a function of distance from the focal plane of a thin lens (f=3 mm, an f/1 lens for the m=0 mode), for several modes. The m=0 and m=2 modes are essentially identical and reach a focused 1/e2 width of 0.95 µm. The m=1 mode focuses to 0.65 µm. Beam parameters are as in Fig. 1, except that β=0.0001 k. (b) Spot size at the focal plane plotted against azimuthal mode number m for several values of β. Note that the modes do not all have the same initial width.

Fig. 9
Fig. 9

Intensities at a plane z=1 cm after a 25-µm-diameter pinhole. Images (a), (b), (c), and (d) show modes m=0, 1, 2, and 8, respectively, each independently normalized. Diffraction rings are present for all beams but are very faint for lower-order modes. Beam parameters are as in Fig. 1, except that β=0.0001 k. Radial dimensions are shown in millimeters.

Fig. 10
Fig. 10

Divergence angle of various azimuthal modes after passing through a 25-µm pinhole. The m=0 and m=2 modes diverge at 1.97°; the m=1 mode diverges at 1.19°. Beam parameters are as in Fig. 8, though the angle is nearly independent of β.

Equations (18)

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××E+0μ0 2Et2=0,
E=U(ρ, ϕ, z)exp[i(kz-ωt)].
U(ρ, ϕ, z)=g(ρ, z)f(ρ, ϕ, z).
g(ρ, z)=w0w(z) exp[-iΦ(z)]exp-ρ2/w021+iz/L,
f{ρ,ϕ}(ρ, ϕ, z)=Q(z)[amJm-1(u)±bmJm+1(u)]Θ{ρ,ϕ}(ϕ),
Q(z)=exp-iβ2z/(2k)1+iz/L,
u=βρ1+iz/L,
Θρ(ϕ)=cos(mϕ)or sin(mϕ),
Θϕ(ϕ)=-sin(mϕ)or cos(mϕ);
f{ρ,ϕ}(ρ, ϕ, z)=amQ(z)[Jm-1(u)±Jm+1(u)]Θ{ρ,ϕ}(ϕ).
Uξ(ρ, ϕ, z)=exp(ikz)iλz exp(ikρ2/2z)×02π0P(ρ)Uξ(ρ, θ, 0)exp[ikρ2/2z]×exp-i 2πρρλz cos(ϕ-θ)ρdρdθ
exp(iα cos β)=ν=-iνJν(α)exp(iνβ),
U{ρ,ϕ}(ρ, ϕ, z)=-amkimz expikz+ρ22zΘ{ρ,ϕ}(ϕ)×0P(ρ)Jm-1(βρ)Jm-1-kρρzJm+1(βρ)Jm+1-kρρz×g(ρ, 0)exp(ikρ2/2z)ρdρ,
P(ρ)=exp(-ikρ2/2f),
0Jν(at)Jν(bt)exp(-ct2)tdt=12c exp[-(a2+b2)/(4c)]Iνab2c,
U{ρ,ϕ}(ρ, ϕ, z)=amkim2Fzexpikz+ρ22z×exp14F β2+k2ρ2z2Im-1βkρ2FzIm+1βkρ2FzΘ{ρ,ϕ}(ϕ),
F=-1w02+ik2 1z-1f
P(ρ)=10ρa0ρ>a

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