Abstract

A new (to our knowledge) technique for the generation of a propagation-invariant elliptic hollow beam is reported. It avoids the use of the radial Mathieu function and hence is mathematically simpler. Bessel functions with their arguments having elliptic locus are used to generate the mask, which is then recorded using holographic technique. To generate such an elliptic beam, both the angular Mathieu function, i.e., elliptic vortex term, and the expression for the circular vortex are used separately. The resultant mask is illuminated with a plane beam, and the proper filtering of its Fourier transform generates the expected elliptic beam. Results with both vortex terms are satisfactory. It has been observed that even for higher ellipticity the vortices do not separate.

© 2006 Optical Society of America

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References

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  1. P. Coullet, L. Gil, and F. Rocca, "Optical vortices," Opt. Commun. 73, 403-408 (1989).
    [CrossRef]
  2. M. P. McDonald, L. Patterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of three-dimensional optically trapped structures," Science 296, 169-175 (2002).
  3. Y. Song, D. Milan, and W. T. Hill III, "Long, narrow all-light atom guide," Opt. Lett. 24, 1805-1807 (1999).
    [CrossRef]
  4. J. Yin, W. Gao, and Y. Zhu, "Generation of dark hollow beams and their applications," in Progress in Optics XLV, E.Wolf, ed. (Elsevier, 2003).
    [CrossRef]
  5. 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, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclassical Opt. 4, S52-S57 (2002).
    [CrossRef]
  6. J. C. Gutierrez Vega, M. D. Iturbe-Castillo, E. Tepichin, R. M. Rodriguez-Dagnino, S. Chavez Cerda, and G. H. C. New, "Experimental demonstration of optical Mathieu beams," Opt. Commun. 195, 35-40 (2001).
    [CrossRef]
  7. Z. Mei and D. Zhao, "Controllable dark-hollow beams and their propagation characteristics," J. Opt. Soc. Am. A 22, 1898-1902 (2005).
    [CrossRef]
  8. Z. Mei and D. Zhao, "Decentered controllable elliptical dark-hollow beams," Opt. Commun. 259, 415-423 (2006).
    [CrossRef]
  9. J. C. Gutiérrez Vega, "Formal analysis of the propagation of invariant optical fields in elliptic corrdinates," Ph.D. thesis (National Institute of Astrophysics and Optics, Mexico, 2000), Chaps. 3 and 4.
  10. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1964).
  11. J. Durnin, "Exact solutions for nondiffracting beams," J. Opt. Soc. Am. A 4, 651-654 (1987).
    [CrossRef]
  12. M. W. Maclachlan, Theory and Application of Mathieu Functions (Dover, 1964).
  13. N. Toyama and K. Shogen, "Computation of the value of the even and odd Mathieu functions of order N for a given parameter S and an argument X," IEEE Trans. Antennas Propag. AP-32, 537-539 (1984).
    [CrossRef]
  14. M. Schneider and J. Marquardt, "Fast computation of modified Mathieu functions applied to elliptical waveguide problems," IEEE Trans. Microwave Theory Tech. 47, 513-516 (1999).
    [CrossRef]
  15. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985).

2006 (1)

Z. Mei and D. Zhao, "Decentered controllable elliptical dark-hollow beams," Opt. Commun. 259, 415-423 (2006).
[CrossRef]

2005 (1)

2003 (1)

J. Yin, W. Gao, and Y. Zhu, "Generation of dark hollow beams and their applications," in Progress in Optics XLV, E.Wolf, ed. (Elsevier, 2003).
[CrossRef]

2002 (2)

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, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclassical Opt. 4, S52-S57 (2002).
[CrossRef]

M. P. McDonald, L. Patterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of three-dimensional optically trapped structures," Science 296, 169-175 (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, "Experimental demonstration of optical Mathieu beams," Opt. Commun. 195, 35-40 (2001).
[CrossRef]

2000 (1)

J. C. Gutiérrez Vega, "Formal analysis of the propagation of invariant optical fields in elliptic corrdinates," Ph.D. thesis (National Institute of Astrophysics and Optics, Mexico, 2000), Chaps. 3 and 4.

1999 (2)

M. Schneider and J. Marquardt, "Fast computation of modified Mathieu functions applied to elliptical waveguide problems," IEEE Trans. Microwave Theory Tech. 47, 513-516 (1999).
[CrossRef]

Y. Song, D. Milan, and W. T. Hill III, "Long, narrow all-light atom guide," Opt. Lett. 24, 1805-1807 (1999).
[CrossRef]

1989 (1)

P. Coullet, L. Gil, and F. Rocca, "Optical vortices," Opt. Commun. 73, 403-408 (1989).
[CrossRef]

1987 (1)

1985 (1)

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985).

1984 (1)

N. Toyama and K. Shogen, "Computation of the value of the even and odd Mathieu functions of order N for a given parameter S and an argument X," IEEE Trans. Antennas Propag. AP-32, 537-539 (1984).
[CrossRef]

1964 (2)

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1964).

M. W. Maclachlan, Theory and Application of Mathieu Functions (Dover, 1964).

Abramowitz, M.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1964).

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, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclassical Opt. 4, S52-S57 (2002).
[CrossRef]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985).

Arlt, J.

M. P. McDonald, L. Patterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of three-dimensional optically trapped structures," Science 296, 169-175 (2002).

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, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclassical Opt. 4, S52-S57 (2002).
[CrossRef]

J. C. Gutierrez Vega, M. D. Iturbe-Castillo, E. Tepichin, R. M. Rodriguez-Dagnino, S. Chavez Cerda, and G. H. C. New, "Experimental demonstration of optical Mathieu beams," Opt. Commun. 195, 35-40 (2001).
[CrossRef]

Coullet, P.

P. Coullet, L. Gil, and F. Rocca, "Optical vortices," Opt. Commun. 73, 403-408 (1989).
[CrossRef]

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, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclassical Opt. 4, S52-S57 (2002).
[CrossRef]

Dholakia, K.

M. P. McDonald, L. Patterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of three-dimensional optically trapped structures," Science 296, 169-175 (2002).

Durnin, J.

Gao, W.

J. Yin, W. Gao, and Y. Zhu, "Generation of dark hollow beams and their applications," in Progress in Optics XLV, E.Wolf, ed. (Elsevier, 2003).
[CrossRef]

Gil, L.

P. Coullet, L. Gil, and F. Rocca, "Optical vortices," Opt. Commun. 73, 403-408 (1989).
[CrossRef]

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, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclassical Opt. 4, S52-S57 (2002).
[CrossRef]

J. C. Gutierrez Vega, M. D. Iturbe-Castillo, E. Tepichin, R. M. Rodriguez-Dagnino, S. Chavez Cerda, and G. H. C. New, "Experimental demonstration of optical Mathieu beams," Opt. Commun. 195, 35-40 (2001).
[CrossRef]

Gutiérrez Vega, J. C.

J. C. Gutiérrez Vega, "Formal analysis of the propagation of invariant optical fields in elliptic corrdinates," Ph.D. thesis (National Institute of Astrophysics and Optics, Mexico, 2000), Chaps. 3 and 4.

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, "Experimental demonstration of optical Mathieu beams," Opt. Commun. 195, 35-40 (2001).
[CrossRef]

Maclachlan, M. W.

M. W. Maclachlan, Theory and Application of Mathieu Functions (Dover, 1964).

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, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclassical Opt. 4, S52-S57 (2002).
[CrossRef]

Marquardt, J.

M. Schneider and J. Marquardt, "Fast computation of modified Mathieu functions applied to elliptical waveguide problems," IEEE Trans. Microwave Theory Tech. 47, 513-516 (1999).
[CrossRef]

McDonald, M. P.

M. P. McDonald, L. Patterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of three-dimensional optically trapped structures," Science 296, 169-175 (2002).

Mei, Z.

Z. Mei and D. Zhao, "Decentered controllable elliptical dark-hollow beams," Opt. Commun. 259, 415-423 (2006).
[CrossRef]

Z. Mei and D. Zhao, "Controllable dark-hollow beams and their propagation characteristics," J. Opt. Soc. Am. A 22, 1898-1902 (2005).
[CrossRef]

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, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclassical Opt. 4, S52-S57 (2002).
[CrossRef]

J. C. Gutierrez Vega, M. D. Iturbe-Castillo, E. Tepichin, R. M. Rodriguez-Dagnino, S. Chavez Cerda, and G. H. C. New, "Experimental demonstration of optical Mathieu beams," Opt. Commun. 195, 35-40 (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, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclassical Opt. 4, S52-S57 (2002).
[CrossRef]

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, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclassical Opt. 4, S52-S57 (2002).
[CrossRef]

Patterson, L.

M. P. McDonald, L. Patterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of three-dimensional optically trapped structures," Science 296, 169-175 (2002).

Rocca, F.

P. Coullet, L. Gil, and F. Rocca, "Optical vortices," Opt. Commun. 73, 403-408 (1989).
[CrossRef]

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, "Experimental demonstration of optical Mathieu beams," Opt. Commun. 195, 35-40 (2001).
[CrossRef]

Schneider, M.

M. Schneider and J. Marquardt, "Fast computation of modified Mathieu functions applied to elliptical waveguide problems," IEEE Trans. Microwave Theory Tech. 47, 513-516 (1999).
[CrossRef]

Shogen, K.

N. Toyama and K. Shogen, "Computation of the value of the even and odd Mathieu functions of order N for a given parameter S and an argument X," IEEE Trans. Antennas Propag. AP-32, 537-539 (1984).
[CrossRef]

Sibbett, W.

M. P. McDonald, L. Patterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of three-dimensional optically trapped structures," Science 296, 169-175 (2002).

Song, Y.

Stegun, I.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1964).

Tepichin, E.

J. C. Gutierrez Vega, M. D. Iturbe-Castillo, E. Tepichin, R. M. Rodriguez-Dagnino, S. Chavez Cerda, and G. H. C. New, "Experimental demonstration of optical Mathieu beams," Opt. Commun. 195, 35-40 (2001).
[CrossRef]

Toyama, N.

N. Toyama and K. Shogen, "Computation of the value of the even and odd Mathieu functions of order N for a given parameter S and an argument X," IEEE Trans. Antennas Propag. AP-32, 537-539 (1984).
[CrossRef]

Volke-Sepulveda, K.

M. P. McDonald, L. Patterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of three-dimensional optically trapped structures," Science 296, 169-175 (2002).

Yin, J.

J. Yin, W. Gao, and Y. Zhu, "Generation of dark hollow beams and their applications," in Progress in Optics XLV, E.Wolf, ed. (Elsevier, 2003).
[CrossRef]

Zhao, D.

Z. Mei and D. Zhao, "Decentered controllable elliptical dark-hollow beams," Opt. Commun. 259, 415-423 (2006).
[CrossRef]

Z. Mei and D. Zhao, "Controllable dark-hollow beams and their propagation characteristics," J. Opt. Soc. Am. A 22, 1898-1902 (2005).
[CrossRef]

Zhu, Y.

J. Yin, W. Gao, and Y. Zhu, "Generation of dark hollow beams and their applications," in Progress in Optics XLV, E.Wolf, ed. (Elsevier, 2003).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

N. Toyama and K. Shogen, "Computation of the value of the even and odd Mathieu functions of order N for a given parameter S and an argument X," IEEE Trans. Antennas Propag. AP-32, 537-539 (1984).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

M. Schneider and J. Marquardt, "Fast computation of modified Mathieu functions applied to elliptical waveguide problems," IEEE Trans. Microwave Theory Tech. 47, 513-516 (1999).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt. (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, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclassical Opt. 4, S52-S57 (2002).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (3)

J. C. Gutierrez Vega, M. D. Iturbe-Castillo, E. Tepichin, R. M. Rodriguez-Dagnino, S. Chavez Cerda, and G. H. C. New, "Experimental demonstration of optical Mathieu beams," Opt. Commun. 195, 35-40 (2001).
[CrossRef]

Z. Mei and D. Zhao, "Decentered controllable elliptical dark-hollow beams," Opt. Commun. 259, 415-423 (2006).
[CrossRef]

P. Coullet, L. Gil, and F. Rocca, "Optical vortices," Opt. Commun. 73, 403-408 (1989).
[CrossRef]

Opt. Lett. (1)

Science (1)

M. P. McDonald, L. Patterson, K. Volke-Sepulveda, J. Arlt, W. Sibbett, and K. Dholakia, "Creation and manipulation of three-dimensional optically trapped structures," Science 296, 169-175 (2002).

Other (5)

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, 1985).

J. C. Gutiérrez Vega, "Formal analysis of the propagation of invariant optical fields in elliptic corrdinates," Ph.D. thesis (National Institute of Astrophysics and Optics, Mexico, 2000), Chaps. 3 and 4.

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1964).

M. W. Maclachlan, Theory and Application of Mathieu Functions (Dover, 1964).

J. Yin, W. Gao, and Y. Zhu, "Generation of dark hollow beams and their applications," in Progress in Optics XLV, E.Wolf, ed. (Elsevier, 2003).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

(a) Stable behavior of the Fourier coefficients of the AMF. (b) Bessel function of different orders.

Fig. 2
Fig. 2

Experimental setup used to generate an elliptic vortex.

Fig. 3
Fig. 3

(a) Simulated output of the elliptic beam generated using the AMF of m = 7 , q = 20 and Bessel function n = 7 , e = 0.8 ; (b) corresponding spectrum; (c) filtered output.

Fig. 4
Fig. 4

(a) Simulated output of the elliptic beam generated using the AMF of m = 7 , q = 8 and Bessel function n = 7 , e = 0.8 ; (b) corresponding spectrum; (c) filtered output.

Fig. 5
Fig. 5

(a) Simulated output of the elliptic beam generated using a circular vortex of order m = 3 and Bessel function n = 10 , e = 0.5 , (b) corresponding spectrum; (c) filtered output.

Fig. 6
Fig. 6

(a) Simulated output of the elliptic beam generated using a circular vortex of order m = 10 and Bessel function n = 10 , e = 0.5 ; (b) corresponding spectrum; (c) filtered output.

Fig. 7
Fig. 7

(a) Simulated output of the elliptic beam generated using a circular vortex of order m = 17 and Bessel function n = 10 , e = 0.5 ; (b) corresponding spectrum; (c) filtered output.

Tables (2)

Tables Icon

Table 1 Fourier Expansion of the AMF and Its Properties

Tables Icon

Table 2 Fourier Expansion of the RMF

Equations (12)

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

E ( r , t ) = R ( ξ ) Φ ( η ) exp ( ± i k z z ) exp ( ± i ω t ) .
2 R ( ξ ) ξ 2 ( a 2 q cosh 2 ξ ) R ( ξ ) = 0 ,
2 Φ ( η ) η 2 + ( a 2 q cos 2 η ) Φ ( η ) = 0 ,
q = ( h 2 k ρ 2 ) 4 ,
E ( ρ ) = exp ( i k z z ) 0 2 π A ( ϕ ) exp ( i k ρ ( x cos ϕ + y sin ϕ ) ) d ϕ ,
E ( ξ , η , z ; q ) = [ A m ( q ) J e m ( ξ ; q ) c e m ( η ; q ) + B m ( q ) J o m ( ξ ; q ) s e m ( η ; q ) ] exp ( i k z z ) .
2 A 0 2 + j = 1 ( A 2 j ) 2 = 1 .
J n ( x ) = s = 0 ( 1 ) s s ! ( n + s ) ! ( x 2 ) n + 2 s ,
( x a ( x , y ) ) 2 + ( y b ( x , y ) ) 2 = 1 ,
( b ( x , y ) ) 2 = ( 1 e 2 ) x 2 + y 2 .
t ( x , y ) = J n ( a ( x , y ) ) ( c e m ( x , y ; q ) + i s e m ( x , y ; q ) ) .
t ( x , y ) = J n [ a ( x , y ) ] exp ( ± i m ϕ ) , ϕ = tan 1 ( y x ) ,

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