Abstract

We develop a paraxial wave equation for an azimuthally polarized field propagating in free space. The equation’s beamlike solution is composed of a plane-wave propagation factor multiplied by a Bessel function of the first kind, of order one, and a Gaussian factor, which describe the transverse characteristics of the beam. We compare the propagation characteristics of the azimuthal Bessel–Gauss beam solution with a solution of the more familiar scalar paraxial wave equation, the linearly polarized Bessel–Gauss beam.

© 1994 Optical Society of America

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References

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  1. T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, Appl. Phys. Lett. 60, 1921 (1992).
    [CrossRef]
  2. J. E. Harvey, Am. J. Phys. 52, 243 (1984).
    [CrossRef]
  3. F. Gori, G. Guattari, C. Padovani, Opt. Commun. 64, 491 (1987).
    [CrossRef]
  4. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, San Diego, Calif., 1985), p. 585.
  5. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, Corrected and Enlarged Edition (Academic, San Diego, Calif., 1980), p. 710, no. 6.615.
  6. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), p. 1283.
  7. A. Yariv, Optical Electronics, 3rd ed. (CBS, New York, 1985), p. 57.
  8. J. Durnin, J. J. J. Miceli, J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
    [CrossRef] [PubMed]
  9. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1965), p. 378, no. 9.7.7.

1992 (1)

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, Appl. Phys. Lett. 60, 1921 (1992).
[CrossRef]

1987 (2)

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

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

1984 (1)

J. E. Harvey, Am. J. Phys. 52, 243 (1984).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1965), p. 378, no. 9.7.7.

Anderson, E. H.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, Appl. Phys. Lett. 60, 1921 (1992).
[CrossRef]

Arfken, G.

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

Durnin, J.

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

Eberly, J. H.

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

Erdogan, T.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, Appl. Phys. Lett. 60, 1921 (1992).
[CrossRef]

Gori, F.

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

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, Corrected and Enlarged Edition (Academic, San Diego, Calif., 1980), p. 710, no. 6.615.

Guattari, G.

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

Hall, D. G.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, Appl. Phys. Lett. 60, 1921 (1992).
[CrossRef]

Harvey, J. E.

J. E. Harvey, Am. J. Phys. 52, 243 (1984).
[CrossRef]

King, O.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, Appl. Phys. Lett. 60, 1921 (1992).
[CrossRef]

Miceli, J. J. J.

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

Padovani, C.

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

Rooks, M. J.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, Appl. Phys. Lett. 60, 1921 (1992).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, Corrected and Enlarged Edition (Academic, San Diego, Calif., 1980), p. 710, no. 6.615.

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), p. 1283.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1965), p. 378, no. 9.7.7.

Wicks, G. W.

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, Appl. Phys. Lett. 60, 1921 (1992).
[CrossRef]

Yariv, A.

A. Yariv, Optical Electronics, 3rd ed. (CBS, New York, 1985), p. 57.

Am. J. Phys. (1)

J. E. Harvey, Am. J. Phys. 52, 243 (1984).
[CrossRef]

Appl. Phys. Lett. (1)

T. Erdogan, O. King, G. W. Wicks, D. G. Hall, E. H. Anderson, M. J. Rooks, Appl. Phys. Lett. 60, 1921 (1992).
[CrossRef]

Opt. Commun. (1)

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

Phys. Rev. Lett. (1)

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

Other (5)

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1965), p. 378, no. 9.7.7.

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

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, Corrected and Enlarged Edition (Academic, San Diego, Calif., 1980), p. 710, no. 6.615.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), p. 1283.

A. Yariv, Optical Electronics, 3rd ed. (CBS, New York, 1985), p. 57.

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Figures (1)

Fig. 1
Fig. 1

Radial intensity distributions, normalized to 1 at each plane, of (a) the linearly polarized BG beam and (b) the azimuthally polarized BG beam at the planes z = 0, 1, 1.5, 2.0, 2.5 m. The beam parameters are w0 = 1 mm, θ = 1 mrad, and λ = 632.8 nm.

Equations (9)

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E ( r , z ) = ϕ ^ Ψ ( r , z ) = ϕ ^ f ( r , z ) exp ( i k z ) ,
1 r r ( r Ψ r ) + 2 f z 2 exp ( i k z ) + 2 i k f z exp ( i k z ) - Ψ r 2 = 0.
1 r r ( r f r ) - f r 2 + 2 i k f z = 0.
E ( r , 0 ) = ϕ ^ Ψ ( r , 0 ) = ϕ ^ A J 1 ( β r ) exp [ - ( r / w 0 ) 2 ] .
E ( x , y , z ) = - i exp ( i k z ) λ z - d x d y E ( x , y , 0 ) × exp { i π λ z [ ( x - x ) 2 + ( y - y ) 2 ] } ,
Ψ ( r , z ) = - A k z exp ( i k z + i k r 2 / 2 z ) × 0 ρ d ρ J 1 ( β ρ ) exp [ - ( ρ w 0 ) 2 ] exp ( i k ρ 2 2 z ) J 1 ( k r ρ / z ) .
E ( r , z ) = ϕ ^ A w 0 w ( z ) exp [ i ( k - β 2 / 2 k ) z - i Φ ( z ) ] J 1 [ β r / ( 1 + i z / L ) exp { [ - 1 w 2 ( z ) + i k 2 R ( z ) ] ( r 2 + β 2 z 2 / k 2 ) } .
1 r r ( r f r ) + 1 r 2 2 f ϕ 2 + 2 i k f z = 0.
E ( r , z ) = u ^ exp ( - i γ ) A w 0 / ( 4 π β r z / k ) 1 / 2 × exp [ i ( k z + k r 2 / 2 z ) ] exp { - [ ( r - β z / k ) / w ] 2 } .

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