Abstract

Based on the separability of the Helmholtz equation into elliptical cylindrical coordinates, we present another class of invariant optical fields that may have a highly localized distribution along one of the transverse directions and a sharply peaked quasi-periodic structure along the other. These fields are described by the radial and angular Mathieu functions. We identify the corresponding function in the McCutchen sphere that produces this kind of beam and propose an experimental setup for the realization of an invariant optical field.

© 2000 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
  10. J. Turunen, A. Vasara, and A. T. Friberg, J. Opt. Soc. Am. A 8, 282 (1991).
    [CrossRef]
  11. N. W. McLachlan, Theory and Applications of Mathieu Functions (Oxford University, London, 1951).
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  13. J. C. Gutiérrez-Vega, “Formal analysis of the propagation of invariant optical fields with elliptical symmetries,” Ph.D. dissertation (Instituto Nacional de Astrofísica, Optica y Electroníca, Puebla, Mexico, 2000); jgutierr@campus.mty.itesm.mx.
  14. M. D. Feit and J. A. Fleck, J. Opt. Soc. Am. B 5, 633 (1988).
    [CrossRef]

2000

Z. Bouchal and J. Wagner, Opt. Commun. 176, 299 (2000).
[CrossRef]

1998

1991

1989

1988

Y. Y. Ananev, Opt. Spectrosc. (USSR) 64, 722 (1988).

M. D. Feit and J. A. Fleck, J. Opt. Soc. Am. B 5, 633 (1988).
[CrossRef]

1987

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

1976

E. G. Kalnins and W. Miller, J. Math. Phys. 17, 331 (1976).
[CrossRef]

1964

Ananev, Y. Y.

Y. Y. Ananev, Opt. Spectrosc. (USSR) 64, 722 (1988).

Bouchal, Z.

Z. Bouchal and J. Wagner, Opt. Commun. 176, 299 (2000).
[CrossRef]

Durnin, J. E.

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

Eberly, J. H.

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

Feit, M. D.

Fleck, J. A.

Friberg, A. T.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 5th ed. (Academic, London, 1994).

Gutiérrez-Vega, J. C.

J. C. Gutiérrez-Vega, “Formal analysis of the propagation of invariant optical fields with elliptical symmetries,” Ph.D. dissertation (Instituto Nacional de Astrofísica, Optica y Electroníca, Puebla, Mexico, 2000); jgutierr@campus.mty.itesm.mx.

Indebetouw, G.

Kalnins, E. G.

E. G. Kalnins and W. Miller, J. Math. Phys. 17, 331 (1976).
[CrossRef]

McCutchen, C. W.

McLachlan, N. W.

N. W. McLachlan, Theory and Applications of Mathieu Functions (Oxford University, London, 1951).

Miceli, J. J.

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

Miller, W.

E. G. Kalnins and W. Miller, J. Math. Phys. 17, 331 (1976).
[CrossRef]

Piestun, R.

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 5th ed. (Academic, London, 1994).

Shamir, J.

Stratton, J. A.

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

Turunen, J.

Vasara, A.

Wagner, J.

Z. Bouchal and J. Wagner, Opt. Commun. 176, 299 (2000).
[CrossRef]

J. Math. Phys.

E. G. Kalnins and W. Miller, J. Math. Phys. 17, 331 (1976).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Commun.

Z. Bouchal and J. Wagner, Opt. Commun. 176, 299 (2000).
[CrossRef]

Opt. Spectrosc. (USSR)

Y. Y. Ananev, Opt. Spectrosc. (USSR) 64, 722 (1988).

Phys. Rev. Lett.

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

Other

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

N. W. McLachlan, Theory and Applications of Mathieu Functions (Oxford University, London, 1951).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products, 5th ed. (Academic, London, 1994).

J. C. Gutiérrez-Vega, “Formal analysis of the propagation of invariant optical fields with elliptical symmetries,” Ph.D. dissertation (Instituto Nacional de Astrofísica, Optica y Electroníca, Puebla, Mexico, 2000); jgutierr@campus.mty.itesm.mx.

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Figures (3)

Fig. 1
Fig. 1

IOF’s are characterized by the ring formed by the intersection between the cone θ = θ 0 and the McCutchen sphere k = k 0 .

Fig. 2
Fig. 2

(a) Transverse intensity pattern of a truncated zero-order Mathieu beam. (b) Angular spectrum of the zero-order Mathieu beam.

Fig. 3
Fig. 3

Evolution of an apertured Mathieu beam on planes (a) x z and (b) y z . (c) Another view of the evolution in plane x z . One may observe the very well-defined quasi-invariant beam within the conic overlapping region.

Equations (4)

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u x , y , z = exp ik z z 0 2 π A φ exp ik t x   cos   φ + y   sin   φ d φ ,
2 u t ξ , η ξ 2 + 2 u t ξ , η η 2 + h 2 k t 2 2 cosh   2 ξ - cos   2 η u t ξ , η = 0 ,
u 1 ξ , η , z ; q = Ce 0 ξ ; q + iFey 0 ξ ; q ce 0 η ; q exp ik z z u 2 ξ , η , z ; q = Ce 0 ξ ; q - iFey 0 ξ ; q ce 0 η ; q exp ik z z ,
u ξ , η , z ; q = Ce 0 ξ ; q ce 0 η ; q exp ik z z ,

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