Abstract

We investigate theoretically and experimentally the decomposition of high-order Bessel beams in terms of a new family of nondiffracting beams, referred as Hermite-Bessel beams, which are solutions of the Helmholtz equation in Cartesian coordinates. Based on this decomposition we develop a geometrical representation of first-order Bessel beams, equivalent to the Poincaré sphere for the polarization states of light and implement an unitary transformation within our geometrical representation using linear optical elements.

© 2006 Optical Society of America

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  1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 85, 8185-8189 (1992).
    [CrossRef]
  2. E. Abramochkin and V. Volostnikov, "Beam transformations and nontransformed beams," Opt. Commun. 83, 123-135 (1991).
    [CrossRef]
  3. M. J. Padgett and J. Courtial, "Poincaré-sphere equivalent for light beams containing orbital angular momentum," Opt. Lett. 24, 430-432 (1999).
    [CrossRef]
  4. G. S. Agarwal, "SU(2) structure of the Poincaré sphere for light beams with orbital angular momentum," J. Opt. Soc. Am. B 16, 2914-2916 (1999).
    [CrossRef]
  5. K. Volke-Sepulveda, V. Garcés-Chéz, S. Chávez-Cerda, J. Arlt, and K. Dholakia,"Orbital angular momentum of a high-order Bessel light beam," J. Opt. B: Quantum and Semiclass. Opt. 4, S82-S89 (2002).
    [CrossRef]
  6. J. Durnin, "Exact solutions for nondiffracting beams.I The scalar theory," J. Opt. Soc. Am. A 4, 651-654(1987);J. Durnin, J. J. MiceliJr., and J. H. Eberly, "Diffraction-free beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
    [CrossRef]
  7. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, "Alternative formulation for invariant optical fields: Mathieu beams," Opt. Lett. 25, 1493-1495 (2000);J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramírez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G.H.C. New, "Experimental demonstration of optical Mathieu beams," Opt. Commun. 195, 35-40 (2001);S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutiérrez-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 Semiclass. Opt. 4, 52-57 (2002).
    [CrossRef]
  8. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, "Parabolic nondiffracting optical wave fields," Opt. Lett. 29, 44-46 (2004);C. López-Mariscal, M. A. Bandres, and J. C. Gutiérrez-Vega, and Sabino Chávez-Cerda, "Observation of parabolic nondiffracting optical fields," Opt. Express 13, 2364-2369 (2005).
    [CrossRef] [PubMed]
  9. A. Vasara, J. Turunen, and A. T. Friberg, "Realization of general nondiffracting beams with computer-generated holograms," J. Opt. Soc. Am. A 6, 1748-1754 (1989).
    [CrossRef] [PubMed]
  10. H. Sasada and M. Okamoto, "Transverse-mode beam splitter of a light beam and its application to quantum cryptography," Phys. Rev. A 68, 012323 (2003).
    [CrossRef]
  11. M.W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123 (1993).
    [CrossRef]

2004

2003

H. Sasada and M. Okamoto, "Transverse-mode beam splitter of a light beam and its application to quantum cryptography," Phys. Rev. A 68, 012323 (2003).
[CrossRef]

2002

K. Volke-Sepulveda, V. Garcés-Chéz, S. Chávez-Cerda, J. Arlt, and K. Dholakia,"Orbital angular momentum of a high-order Bessel light beam," J. Opt. B: Quantum and Semiclass. Opt. 4, S82-S89 (2002).
[CrossRef]

2000

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, "Alternative formulation for invariant optical fields: Mathieu beams," Opt. Lett. 25, 1493-1495 (2000);J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramírez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G.H.C. New, "Experimental demonstration of optical Mathieu beams," Opt. Commun. 195, 35-40 (2001);S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutiérrez-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 Semiclass. Opt. 4, 52-57 (2002).
[CrossRef]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, "Alternative formulation for invariant optical fields: Mathieu beams," Opt. Lett. 25, 1493-1495 (2000);J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramírez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G.H.C. New, "Experimental demonstration of optical Mathieu beams," Opt. Commun. 195, 35-40 (2001);S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutiérrez-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 Semiclass. Opt. 4, 52-57 (2002).
[CrossRef]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, "Alternative formulation for invariant optical fields: Mathieu beams," Opt. Lett. 25, 1493-1495 (2000);J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramírez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G.H.C. New, "Experimental demonstration of optical Mathieu beams," Opt. Commun. 195, 35-40 (2001);S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutiérrez-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 Semiclass. Opt. 4, 52-57 (2002).
[CrossRef]

1999

M. J. Padgett and J. Courtial, "Poincaré-sphere equivalent for light beams containing orbital angular momentum," Opt. Lett. 24, 430-432 (1999).
[CrossRef]

G. S. Agarwal, "SU(2) structure of the Poincaré sphere for light beams with orbital angular momentum," J. Opt. Soc. Am. B 16, 2914-2916 (1999).
[CrossRef]

1993

M.W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123 (1993).
[CrossRef]

1992

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 85, 8185-8189 (1992).
[CrossRef]

1991

E. Abramochkin and V. Volostnikov, "Beam transformations and nontransformed beams," Opt. Commun. 83, 123-135 (1991).
[CrossRef]

1989

1987

Abramochkin, E.

E. Abramochkin and V. Volostnikov, "Beam transformations and nontransformed beams," Opt. Commun. 83, 123-135 (1991).
[CrossRef]

Agarwal, G. S.

G. S. Agarwal, "SU(2) structure of the Poincaré sphere for light beams with orbital angular momentum," J. Opt. Soc. Am. B 16, 2914-2916 (1999).
[CrossRef]

Allen, L.

M.W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 85, 8185-8189 (1992).
[CrossRef]

Allison, I.

Arlt, J.

K. Volke-Sepulveda, V. Garcés-Chéz, S. Chávez-Cerda, J. Arlt, and K. Dholakia,"Orbital angular momentum of a high-order Bessel light beam," J. Opt. B: Quantum and Semiclass. Opt. 4, S82-S89 (2002).
[CrossRef]

Bandres, M. A.

Beijersbergen, M. W.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 85, 8185-8189 (1992).
[CrossRef]

Beijersbergen, M.W.

M.W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123 (1993).
[CrossRef]

Chávez-Cerda, S.

M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, "Parabolic nondiffracting optical wave fields," Opt. Lett. 29, 44-46 (2004);C. López-Mariscal, M. A. Bandres, and J. C. Gutiérrez-Vega, and Sabino Chávez-Cerda, "Observation of parabolic nondiffracting optical fields," Opt. Express 13, 2364-2369 (2005).
[CrossRef] [PubMed]

K. Volke-Sepulveda, V. Garcés-Chéz, S. Chávez-Cerda, J. Arlt, and K. Dholakia,"Orbital angular momentum of a high-order Bessel light beam," J. Opt. B: Quantum and Semiclass. Opt. 4, S82-S89 (2002).
[CrossRef]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, "Alternative formulation for invariant optical fields: Mathieu beams," Opt. Lett. 25, 1493-1495 (2000);J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramírez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G.H.C. New, "Experimental demonstration of optical Mathieu beams," Opt. Commun. 195, 35-40 (2001);S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutiérrez-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 Semiclass. Opt. 4, 52-57 (2002).
[CrossRef]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, "Alternative formulation for invariant optical fields: Mathieu beams," Opt. Lett. 25, 1493-1495 (2000);J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramírez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G.H.C. New, "Experimental demonstration of optical Mathieu beams," Opt. Commun. 195, 35-40 (2001);S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutiérrez-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 Semiclass. Opt. 4, 52-57 (2002).
[CrossRef]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, "Alternative formulation for invariant optical fields: Mathieu beams," Opt. Lett. 25, 1493-1495 (2000);J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramírez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G.H.C. New, "Experimental demonstration of optical Mathieu beams," Opt. Commun. 195, 35-40 (2001);S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutiérrez-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 Semiclass. Opt. 4, 52-57 (2002).
[CrossRef]

Courtial, J.

Dholakia, K.

K. Volke-Sepulveda, V. Garcés-Chéz, S. Chávez-Cerda, J. Arlt, and K. Dholakia,"Orbital angular momentum of a high-order Bessel light beam," J. Opt. B: Quantum and Semiclass. Opt. 4, S82-S89 (2002).
[CrossRef]

Durnin, J.

Eberly, J. H.

Friberg, A. T.

Garcés-Chéz, V.

K. Volke-Sepulveda, V. Garcés-Chéz, S. Chávez-Cerda, J. Arlt, and K. Dholakia,"Orbital angular momentum of a high-order Bessel light beam," J. Opt. B: Quantum and Semiclass. Opt. 4, S82-S89 (2002).
[CrossRef]

Gutiérrez-Vega, J. C.

M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, "Parabolic nondiffracting optical wave fields," Opt. Lett. 29, 44-46 (2004);C. López-Mariscal, M. A. Bandres, and J. C. Gutiérrez-Vega, and Sabino Chávez-Cerda, "Observation of parabolic nondiffracting optical fields," Opt. Express 13, 2364-2369 (2005).
[CrossRef] [PubMed]

M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, "Parabolic nondiffracting optical wave fields," Opt. Lett. 29, 44-46 (2004);C. López-Mariscal, M. A. Bandres, and J. C. Gutiérrez-Vega, and Sabino Chávez-Cerda, "Observation of parabolic nondiffracting optical fields," Opt. Express 13, 2364-2369 (2005).
[CrossRef] [PubMed]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, "Alternative formulation for invariant optical fields: Mathieu beams," Opt. Lett. 25, 1493-1495 (2000);J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramírez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G.H.C. New, "Experimental demonstration of optical Mathieu beams," Opt. Commun. 195, 35-40 (2001);S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutiérrez-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 Semiclass. Opt. 4, 52-57 (2002).
[CrossRef]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, "Alternative formulation for invariant optical fields: Mathieu beams," Opt. Lett. 25, 1493-1495 (2000);J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramírez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G.H.C. New, "Experimental demonstration of optical Mathieu beams," Opt. Commun. 195, 35-40 (2001);S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutiérrez-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 Semiclass. Opt. 4, 52-57 (2002).
[CrossRef]

J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, "Alternative formulation for invariant optical fields: Mathieu beams," Opt. Lett. 25, 1493-1495 (2000);J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramírez, E. Tepichín, R. M. Rodríguez-Dagnino, S. Chávez-Cerda, and G.H.C. New, "Experimental demonstration of optical Mathieu beams," Opt. Commun. 195, 35-40 (2001);S. Chávez-Cerda, M. J. Padgett, I. Allison, G. H. C. New, J. C. Gutiérrez-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 Semiclass. Opt. 4, 52-57 (2002).
[CrossRef]

Iturbe-Castillo, M. D.

López-Mariscal, C.

MacVicar, I.

Miceli, J. J.

New, G. H. C.

New, G.H.C.

O’Neil, A. T.

Okamoto, M.

H. Sasada and M. Okamoto, "Transverse-mode beam splitter of a light beam and its application to quantum cryptography," Phys. Rev. A 68, 012323 (2003).
[CrossRef]

Padgett, M. J.

Ramírez, G. A.

Rodríguez-Dagnino, R. M.

Sasada, H.

H. Sasada and M. Okamoto, "Transverse-mode beam splitter of a light beam and its application to quantum cryptography," Phys. Rev. A 68, 012323 (2003).
[CrossRef]

Spreeuw, R. J. C.

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 85, 8185-8189 (1992).
[CrossRef]

Tepichín, E.

Turunen, J.

van der Veen, H. E. L. O.

M.W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123 (1993).
[CrossRef]

Vasara, A.

Volke-Sepulveda, K.

K. Volke-Sepulveda, V. Garcés-Chéz, S. Chávez-Cerda, J. Arlt, and K. Dholakia,"Orbital angular momentum of a high-order Bessel light beam," J. Opt. B: Quantum and Semiclass. Opt. 4, S82-S89 (2002).
[CrossRef]

Volostnikov, V.

E. Abramochkin and V. Volostnikov, "Beam transformations and nontransformed beams," Opt. Commun. 83, 123-135 (1991).
[CrossRef]

Woerdman, J. P.

M.W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123 (1993).
[CrossRef]

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 85, 8185-8189 (1992).
[CrossRef]

J. Opt. B: Quantum and Semiclass. Opt.

K. Volke-Sepulveda, V. Garcés-Chéz, S. Chávez-Cerda, J. Arlt, and K. Dholakia,"Orbital angular momentum of a high-order Bessel light beam," J. Opt. B: Quantum and Semiclass. Opt. 4, S82-S89 (2002).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

G. S. Agarwal, "SU(2) structure of the Poincaré sphere for light beams with orbital angular momentum," J. Opt. Soc. Am. B 16, 2914-2916 (1999).
[CrossRef]

Opt. Commun.

E. Abramochkin and V. Volostnikov, "Beam transformations and nontransformed beams," Opt. Commun. 83, 123-135 (1991).
[CrossRef]

M.W. Beijersbergen, L. Allen, H. E. L. O. van der Veen, and J. P. Woerdman, "Astigmatic laser mode converters and transfer of orbital angular momentum," Opt. Commun. 96, 123 (1993).
[CrossRef]

Opt. Lett.

Phys. Rev. A

L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, "Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes," Phys. Rev. A 85, 8185-8189 (1992).
[CrossRef]

H. Sasada and M. Okamoto, "Transverse-mode beam splitter of a light beam and its application to quantum cryptography," Phys. Rev. A 68, 012323 (2003).
[CrossRef]

Supplementary Material (1)

» Media 1: GIF (833 KB)     

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Figures (5)

Fig. 1.
Fig. 1.

The Poincaré-equivalent sphere for representing first order cylindrical and Cartesian nondifracting light beams: a) BB and HB modes are represented by a density plot of their intensity profiles; b) BB and HB modes are represented by a tridimensional plot of their intensity.

Fig. 2.
Fig. 2.

Experimental setup, where M1, M2, and M3 are mirrors; BS1, BS2, BS3 and BS4 are beam-splitters; and SF is a spatial filter.

Fig. 3.
Fig. 3.

a) Intensity profile of a first order Bessel beam with l=+1; b) Interference pattern of the first order Bessel beam with l=+1.

Fig. 4.
Fig. 4.

Intensity profile measured at the output ports of the DEFG interferometer. a) HB 10 beam; b) HB 01 beam. In the multimedia file it is shown a movie where the output ports of the DEFG interferometer are measured by rotating the glass plate.

Fig. 5.
Fig. 5.

a) Intensity profile of the BB 1 beam after the passage by the pair of cylindrical lens; b) Interference pattern of the of the BB 1 beam after the passage by the pair of cylindrical lens. [Media 1]

Equations (6)

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

B B l ( z , ρ , ϕ ) = k l 2 l e i k z z [ J l 1 ( k ρ ) + J l + 1 ( k ρ ) ] 1 ( k ρ ) l 1 [ x + i y ] l
B B ± 1 ( z , ρ , ϕ ) = ± k 2 e i k z z { [ J 0 ( k ρ ) + J 2 ( k ρ ) ] x ± i [ J 0 ( k ρ ) + J 2 ( k ρ ) ] y } .
H B 10 = [ J 0 ( k ρ ) + J 2 ( k ρ ) ] H 1 ( x ) H 0 ( y )
H B 01 = [ J 0 ( k ρ ) + J 2 ( k ρ ) ] H 0 ( x ) H 1 ( y ) ,
B B 1 = k e i k z z ( H B 10 + i H B 01 )
B B 1 = k e i k z z ( H B 10 i H B 01 ) .

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