Abstract

It is shown that three-dimensional nonparaxial beams are described by the oblate spheroidal exact solutions of the Helmholtz equation. For what is believed to be the first time, their beam behavior is investigated and their corresponding parameters are defined. Using the fact that the beam width of the family of paraxial Gaussian beams is described by a hyperbola, we formally establish the connection between the physical parameters of nonparaxial spheroidal beam solutions and those of paraxial beams. These results are also helpful for investigating exact vector nonparaxial beams.

© 2004 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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  16. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
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2002

S. R. Seshadri, J. Opt. Soc. Am. 19, 2134 (2002) and references therein.
[CrossRef]

2001

2000

Z. Ulanowski and I. K. Ludlow, Opt. Lett. 25, 1792 (2000).
[CrossRef]

A. Ciattoni, B. Crosignani, and P. Di Porto, Opt. Commun. 177, 9 (2000).
[CrossRef]

1999

S. Chavez-Cerda, J. Mod. Opt. 46, 923 (1999).

1998

Q. Cao and X. Deng, J. Opt. Soc. Am. A 15, 1144 (1998).
[CrossRef]

C. J. R. Sheppard and S. Saghafi, Phys. Rev. A 57, 2971 (1998).
[CrossRef]

1990

1988

1977

1975

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

1971

G. A. Deschamps, Electron. Lett. 7, 684 (1971).
[CrossRef]

Barrett, H. H.

Cao, Q.

Q. Cao and X. Deng, J. Opt. Soc. Am. A 15, 1144 (1998).
[CrossRef]

Chavez-Cerda, S.

S. Chavez-Cerda, J. Mod. Opt. 46, 923 (1999).

Ciattoni, A.

A. Ciattoni, B. Crosignani, and P. Di Porto, Opt. Commun. 177, 9 (2000).
[CrossRef]

Crosignani, B.

A. Ciattoni, B. Crosignani, and P. Di Porto, Opt. Commun. 177, 9 (2000).
[CrossRef]

Deng, X.

Q. Cao and X. Deng, J. Opt. Soc. Am. A 15, 1144 (1998).
[CrossRef]

Deschamps, G. A.

G. A. Deschamps, Electron. Lett. 7, 684 (1971).
[CrossRef]

Di Porto, P.

A. Ciattoni, B. Crosignani, and P. Di Porto, Opt. Commun. 177, 9 (2000).
[CrossRef]

Felsen, L. B.

Flammer, C.

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).

Jin, J.

S. Zhang and J. Jin, Computation of Special Functions (Wiley, New York, 1996), Chap. 15.

Kraus, H. G.

Landesman, B. T.

Lax, M.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Lekner, J.

J. Lekner, J. Opt. A Pure Appl. Opt. 3, 407 (2001).
[CrossRef]

Louisell, W. H.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Ludlow, I. K.

McKnight, W. B.

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

Saari, P.

Saghafi, S.

C. J. R. Sheppard and S. Saghafi, Phys. Rev. A 57, 2971 (1998).
[CrossRef]

Seshadri, S. R.

S. R. Seshadri, J. Opt. Soc. Am. 19, 2134 (2002) and references therein.
[CrossRef]

Sheppard, C. J. R.

C. J. R. Sheppard and S. Saghafi, Phys. Rev. A 57, 2971 (1998).
[CrossRef]

Shin, S. Y.

Siegman, A. E.

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

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Chap. VII.

Ulanowski, Z.

Zhang, S.

S. Zhang and J. Jin, Computation of Special Functions (Wiley, New York, 1996), Chap. 15.

Electron. Lett.

G. A. Deschamps, Electron. Lett. 7, 684 (1971).
[CrossRef]

J. Mod. Opt.

S. Chavez-Cerda, J. Mod. Opt. 46, 923 (1999).

J. Opt. A Pure Appl. Opt.

J. Lekner, J. Opt. A Pure Appl. Opt. 3, 407 (2001).
[CrossRef]

J. Opt. Soc. Am. A

Q. Cao and X. Deng, J. Opt. Soc. Am. A 15, 1144 (1998).
[CrossRef]

J. Opt. Soc. Am.

S. R. Seshadri, J. Opt. Soc. Am. 19, 2134 (2002) and references therein.
[CrossRef]

S. Y. Shin and L. B. Felsen, J. Opt. Soc. Am. 67, 699 (1977).

J. Opt. Soc. Am. A

Opt. Commun.

A. Ciattoni, B. Crosignani, and P. Di Porto, Opt. Commun. 177, 9 (2000).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

M. Lax, W. H. Louisell, and W. B. McKnight, Phys. Rev. A 11, 1365 (1975).
[CrossRef]

C. J. R. Sheppard and S. Saghafi, Phys. Rev. A 57, 2971 (1998).
[CrossRef]

Other

C. Flammer, Spheroidal Wave Functions (Stanford U. Press, Stanford, Calif., 1957).

S. Zhang and J. Jin, Computation of Special Functions (Wiley, New York, 1996), Chap. 15.

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

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Chap. VII.

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Figures (3)

Fig. 1
Fig. 1

Intensity and phase evolution (left) and transverse patterns (right) of oblate spheroidal modes.

Fig. 2
Fig. 2

Hyperbolic–elliptical geometry of a paraxial beam defined by the beam-width evolution.

Fig. 3
Fig. 3

(a) Comparison of on-axis intensity for oblate spheroidal nonparaxial and paraxial beams. Also shown is the normalized intensity for a highly nonparaxial beam (labeled c=0.4cp). (b) Corresponding transverse amplitude profiles at the plane z=0. Notice that, for the Laguerre paraxial mode, r can be obtained from r2=x2+y2=d21-η2.

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

d2Φdϕ2+m2Φ=0 ξ2+1d2Rdξ2+2ξdRdξ -amn-c2ξ2+m2ξ2+1R=0, 1-η2d2Sdη2-2ηdSdη+ amn-c2η2-m21-η2S=0,
Enmξ,η,ϕ=Rmnpξ;cSmnη;cexpimϕ,
R02=d2-w02
w0=2-1+1+c21/21/2k.

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