Abstract

In an elliptic coordinate system the solution for a diffraction-free beam is given by a Mathieu beam, which is mathematically complicated and therefore takes considerable computational time and a large memory space. A technique is reported for the generation of elliptic hollow beams, which are actually scaled versions of higher-order Bessel beams. The analysis is based on the fact that stretching coordinates in the space domain results in a contraction of the coordinates in the frequency domain, along with a change in the overall amplitude of the spectrum. The beam, produced holographically, is structurally stable and retains its shape up to 1m.

© 2006 Optical Society of America

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References

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    [CrossRef]
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2003 (1)

2002 (2)

M. P. McDonald, L. Patterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, Science 296, 169 (2002).

S. Chavez Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutierrez Vega, A. T. O'Neil, I. MacVicar, and J. Courtial, Acoust. Hologr. 4, S52 (2002).

2001 (1)

J. C. Gutierrez Vega, M. D. Iturbe-Castillo, E. Tepichin, R. M. Rodriguez-Dagnino, S. Chavez Cerda, and G. H. C. New, Opt. Commun. 195, 35 (2001).
[CrossRef]

1999 (1)

1988 (1)

1987 (1)

Allison, I.

S. Chavez Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutierrez Vega, A. T. O'Neil, I. MacVicar, and J. Courtial, Acoust. Hologr. 4, S52 (2002).

Arlt, J.

M. P. McDonald, L. Patterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, Science 296, 169 (2002).

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Chattrapiban, N.

Chavez Cerda, S.

S. Chavez Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutierrez Vega, A. T. O'Neil, I. MacVicar, and J. Courtial, Acoust. Hologr. 4, S52 (2002).

J. C. Gutierrez Vega, M. D. Iturbe-Castillo, E. Tepichin, R. M. Rodriguez-Dagnino, S. Chavez Cerda, and G. H. C. New, Opt. Commun. 195, 35 (2001).
[CrossRef]

J. C. Gutierrez Vega and S. Chavez Cerda, "Theory and numerical analysis of the Mathieu functions," presented at Instituto Technologico y de Estudios Superiores de Monterrey, Instituto Nacional de Astrofisica, Optica y Electronica Centro de Investigaciones en Optica, Monterrey, N.L., Mexico, March 2003.

Cofield, D.

Courtial, J.

S. Chavez Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutierrez Vega, A. T. O'Neil, I. MacVicar, and J. Courtial, Acoust. Hologr. 4, S52 (2002).

Dholakia, K.

M. P. McDonald, L. Patterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, Science 296, 169 (2002).

Durnin, J.

Friberg, A. T.

Gao, W.

J. Yin, W. Gao, and Y. Zhu, in Progress in Optics, E.Wolf, ed. (Elsevier, 2003), Vol. 45, p. 121.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Gutierrez Vega, J. C.

S. Chavez Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutierrez Vega, A. T. O'Neil, I. MacVicar, and J. Courtial, Acoust. Hologr. 4, S52 (2002).

J. C. Gutierrez Vega, M. D. Iturbe-Castillo, E. Tepichin, R. M. Rodriguez-Dagnino, S. Chavez Cerda, and G. H. C. New, Opt. Commun. 195, 35 (2001).
[CrossRef]

J. C. Gutierrez Vega and S. Chavez Cerda, "Theory and numerical analysis of the Mathieu functions," presented at Instituto Technologico y de Estudios Superiores de Monterrey, Instituto Nacional de Astrofisica, Optica y Electronica Centro de Investigaciones en Optica, Monterrey, N.L., Mexico, March 2003.

Hill, W. T.

Iturbe-Castillo, M. D.

J. C. Gutierrez Vega, M. D. Iturbe-Castillo, E. Tepichin, R. M. Rodriguez-Dagnino, S. Chavez Cerda, and G. H. C. New, Opt. Commun. 195, 35 (2001).
[CrossRef]

MacVicar, I.

S. Chavez Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutierrez Vega, A. T. O'Neil, I. MacVicar, and J. Courtial, Acoust. Hologr. 4, S52 (2002).

McDonald, M. P.

M. P. McDonald, L. Patterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, Science 296, 169 (2002).

Milan, D.

New, G. H. C.

S. Chavez Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutierrez Vega, A. T. O'Neil, I. MacVicar, and J. Courtial, Acoust. Hologr. 4, S52 (2002).

J. C. Gutierrez Vega, M. D. Iturbe-Castillo, E. Tepichin, R. M. Rodriguez-Dagnino, S. Chavez Cerda, and G. H. C. New, Opt. Commun. 195, 35 (2001).
[CrossRef]

O'Neil, A. T.

S. Chavez Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutierrez Vega, A. T. O'Neil, I. MacVicar, and J. Courtial, Acoust. Hologr. 4, S52 (2002).

Padgett, M. J.

S. Chavez Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutierrez Vega, A. T. O'Neil, I. MacVicar, and J. Courtial, Acoust. Hologr. 4, S52 (2002).

Patterson, L.

M. P. McDonald, L. Patterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, Science 296, 169 (2002).

Rodriguez-Dagnino, R. M.

J. C. Gutierrez Vega, M. D. Iturbe-Castillo, E. Tepichin, R. M. Rodriguez-Dagnino, S. Chavez Cerda, and G. H. C. New, Opt. Commun. 195, 35 (2001).
[CrossRef]

Rogers, E. A.

Roy, R.

Sibbett, W.

M. P. McDonald, L. Patterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, Science 296, 169 (2002).

Song, Y.

Tepichin, E.

J. C. Gutierrez Vega, M. D. Iturbe-Castillo, E. Tepichin, R. M. Rodriguez-Dagnino, S. Chavez Cerda, and G. H. C. New, Opt. Commun. 195, 35 (2001).
[CrossRef]

Turunen, J.

Vasara, A.

Volke-Sepulveda, K.

M. P. McDonald, L. Patterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, Science 296, 169 (2002).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Yin, J.

J. Yin, W. Gao, and Y. Zhu, in Progress in Optics, E.Wolf, ed. (Elsevier, 2003), Vol. 45, p. 121.

Zhu, Y.

J. Yin, W. Gao, and Y. Zhu, in Progress in Optics, E.Wolf, ed. (Elsevier, 2003), Vol. 45, p. 121.

Acoust. Hologr. (1)

S. Chavez Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutierrez Vega, A. T. O'Neil, I. MacVicar, and J. Courtial, Acoust. Hologr. 4, S52 (2002).

Appl. Opt. (1)

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

Opt. Commun. (1)

J. C. Gutierrez Vega, M. D. Iturbe-Castillo, E. Tepichin, R. M. Rodriguez-Dagnino, S. Chavez Cerda, and G. H. C. New, Opt. Commun. 195, 35 (2001).
[CrossRef]

Opt. Lett. (2)

Science (1)

M. P. McDonald, L. Patterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, Science 296, 169 (2002).

Other (4)

J. C. Gutierrez Vega and S. Chavez Cerda, "Theory and numerical analysis of the Mathieu functions," presented at Instituto Technologico y de Estudios Superiores de Monterrey, Instituto Nacional de Astrofisica, Optica y Electronica Centro de Investigaciones en Optica, Monterrey, N.L., Mexico, March 2003.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

J. Yin, W. Gao, and Y. Zhu, in Progress in Optics, E.Wolf, ed. (Elsevier, 2003), Vol. 45, p. 121.

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Figures (3)

Fig. 1
Fig. 1

(a) Experimental setup for fabrication of the DOE along with successive coordinate conversion and scaling. (b) Experimental setup for holographic reconstruction of the elliptical Bessel beam.

Fig. 2
Fig. 2

(a) Mask used for the production of elliptic Bessel beam of order m = 15 , with scale factor μ = 6 . (b) Corresponding simulated output produced by using MATLAB. Experimentally obtained (c) light distribution on spectrum plane; (d) beam pattern at z = 25 cm , 2 a = 7 mm , and 2 b = 3 mm ; (e) beam pattern at z = 100 cm . Observed dimensions of the ellipse are 2 a = 22 mm and 2 b = 9 mm .

Fig. 3
Fig. 3

Same as Fig. 2 except that the order number m = 15 scale factor μ = 8 . The observed beam dimensions at z = 25 cm and z = 100 cm are 2.5 mm × 7 mm and 8 mm × 22 mm , respectively.

Equations (5)

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F T { g ( x , y ) } = G ( u , v ) ,
F T { g ( x , μ y ) } = 1 μ G ( u , v μ ) ,
t ( ρ , ϕ ) = exp [ i ψ ( ρ , ϕ ) ] ,
k r = 2 π λ ρ sin γ cos ϕ ,
ψ ( ρ , ϕ ) = ( m ϕ 2 π ρ ρ 0 + 2 π λ ρ sin γ cos ϕ ) ,

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