Abstract

In this work we compute the wavefronts and the caustics associated with the solutions to the scalar wave equation introduced by Durnin in elliptical cylindrical coordinates generated by the function A(ϕ)=ceν(ϕ,q)+iseν(ϕ,q), with ν being an integral or nonintegral number. We show that the wavefronts and the caustic are invariant under translations along the direction of evolution of the beam. We remark that the wavefronts of the separable Mathieu beams generated by A(ϕ)=ceν(ϕ,q) and A(ϕ)=seν(ϕ,q) are cones and their caustic is the z axis; thus, they are not structurally stable. However, in general, the Mathieu beam generated by A(ϕ)=ceν(ϕ,q)+iseν(ϕ,q) is stable because locally its caustic has singularities of the fold and cusp types. To show this property, we present the wavefronts and the caustics for the Mathieu beams with characteristic value aν=0 and q=0,0.2,0.3,0.5. For q=0, we obtain the Bessel beam of order zero; in this case, the wavefronts are cones and the caustic coincides with the z axis. For q0, the wavefronts are deformations of conical ones, and the caustic surface, for some values of q, has singularities of the cusp ridge type. Furthermore, we remark that the set of Mathieu beams with characteristic value aν=0 and 0q<1 has associated a caustic with singularities of the swallowtail type, which is structurally stable. Therefore, we conclude that this type of Mathieu beam is more stable than plane waves, Bessel beams, parabolic beams, and those generated by A(ϕ)=ceν(ϕ,q) and A(ϕ)=seν(ϕ,q). To support this conclusion, we present experimental results showing the pattern obtained after obstructing a plane wave, the Bessel beam of order m=5, and the Mathieu beam of order m=5 and q=50 with complex transversal amplitude given by Ce5(ξ,50)ce5(η,50)+iSe5(ξ,50)se5(η,50), where (ξ, η) are the elliptical coordinates on the plane.

© 2018 Optical Society of America

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  21. C. T. Sosa-Sánchez, G. Silva-Ortigoza, S. A. Juárez-Reyes, O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, I. Julián-Macías, and P. Ortega-Vidals, “Parabolic non-diffracting beams: geometrical approach,” J. Opt. 19, 085604 (2017).
    [Crossref]
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    [Crossref]
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    [Crossref]
  26. O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, S. A. Juárez-Reyes, I. Julián-Macías, P. Ortega-Vidals, C. Rickenstorff-Parrao, G. Silva-Ortigoza, R. Silva-Ortigoza, and C. T. Sosa-Sánchez, “Durnin-Whitney beams,” J. Opt. 19, 055606 (2017).
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2017 (3)

O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, S. A. Juárez-Reyes, I. Julián-Macías, P. Ortega-Vidals, G. Silva-Ortigoza, R. Silva-Ortigoza, and C. T. Sosa-Sánchez, “Wavefronts and caustic associated with Durnin’s beams,” J. Opt. 19, 015603 (2017).
[Crossref]

C. T. Sosa-Sánchez, G. Silva-Ortigoza, S. A. Juárez-Reyes, O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, I. Julián-Macías, and P. Ortega-Vidals, “Parabolic non-diffracting beams: geometrical approach,” J. Opt. 19, 085604 (2017).
[Crossref]

O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, S. A. Juárez-Reyes, I. Julián-Macías, P. Ortega-Vidals, C. Rickenstorff-Parrao, G. Silva-Ortigoza, R. Silva-Ortigoza, and C. T. Sosa-Sánchez, “Durnin-Whitney beams,” J. Opt. 19, 055606 (2017).
[Crossref]

2012 (2)

P. A. Sanchez-Serrano, D. Wong-Campos, S. Lopez-Aguayo, and J. C. Gutiérrez-Vega, “Engineering of nondiffracting beams with genetic algorithms,” Opt. Lett. 37, 5040–5042 (2012).
[Crossref]

P. Rose, M. Boguslawski, and C. Denz, “Nonlinear lattice structures based on families of complex nondiffracting beams,” New J. Phys. 14, 033018 (2012).
[Crossref]

2010 (1)

2008 (1)

2007 (2)

2006 (3)

2005 (2)

2004 (1)

2002 (1)

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 Semiclassical Opt. 4, S52–S57 (2002).
[Crossref]

2001 (1)

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).
[Crossref]

2000 (1)

1999 (1)

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, “Nondiffracting bulk-acoustic X waves in crystals,” Phys. Rev. Lett. 83, 1171–1174 (1999).
[Crossref]

1987 (2)

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[Crossref]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

1983 (2)

V. I. Arnold, “Singularities, bifurcations and catastrophes,” Usp. Fiz. Nauk 141, 569–590 (1983).
[Crossref]

Y. A. Krastov and Y. I. Orlov, “Caustics catastrophes, and wave fields,” Usp. Fiz. Nauk 141, 591–627 (1983).
[Crossref]

1976 (1)

C. P. Boyer, E. G. Kalnins, and W. Miller, “Symmetry and separation of variables for the Helmholtz and Laplace equations,” Nagoya Math. J. 60, 35–80 (1976).
[Crossref]

Allison, I.

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 Semiclassical Opt. 4, S52–S57 (2002).
[Crossref]

Alvarez-Elizondo, M. B.

Arnold, V. I.

V. I. Arnold, “Singularities, bifurcations and catastrophes,” Usp. Fiz. Nauk 141, 569–590 (1983).
[Crossref]

V. I. Arnold, S. M. Gussein-Zade, and A. N. Varchenko, Singularities of Differentiable Maps (Birkhauser, 1995), Vol. I.

Arrizón, V.

Bandres, M. A.

Boguslawski, M.

P. Rose, M. Boguslawski, and C. Denz, “Nonlinear lattice structures based on families of complex nondiffracting beams,” New J. Phys. 14, 033018 (2012).
[Crossref]

Boyer, C. P.

C. P. Boyer, E. G. Kalnins, and W. Miller, “Symmetry and separation of variables for the Helmholtz and Laplace equations,” Nagoya Math. J. 60, 35–80 (1976).
[Crossref]

Carrada, R.

Chávez-Cerda, S.

C. López-Mariscal, M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Observation of parabolic nondiffracting optical fields,” Opt. Express 13, 2364–2369 (2005).
[Crossref]

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 Semiclassical Opt. 4, S52–S57 (2002).
[Crossref]

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).
[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).
[Crossref]

Courtial, J.

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 Semiclassical Opt. 4, S52–S57 (2002).
[Crossref]

Dartora, C. A.

de Jesús Cabrera-Rosas, O.

C. T. Sosa-Sánchez, G. Silva-Ortigoza, S. A. Juárez-Reyes, O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, I. Julián-Macías, and P. Ortega-Vidals, “Parabolic non-diffracting beams: geometrical approach,” J. Opt. 19, 085604 (2017).
[Crossref]

O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, S. A. Juárez-Reyes, I. Julián-Macías, P. Ortega-Vidals, C. Rickenstorff-Parrao, G. Silva-Ortigoza, R. Silva-Ortigoza, and C. T. Sosa-Sánchez, “Durnin-Whitney beams,” J. Opt. 19, 055606 (2017).
[Crossref]

O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, S. A. Juárez-Reyes, I. Julián-Macías, P. Ortega-Vidals, G. Silva-Ortigoza, R. Silva-Ortigoza, and C. T. Sosa-Sánchez, “Wavefronts and caustic associated with Durnin’s beams,” J. Opt. 19, 015603 (2017).
[Crossref]

Denz, C.

P. Rose, M. Boguslawski, and C. Denz, “Nonlinear lattice structures based on families of complex nondiffracting beams,” New J. Phys. 14, 033018 (2012).
[Crossref]

Dholokia, K.

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4, 651–654 (1987).
[Crossref]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

Egorov, A. A.

Espíndola-Ramos, E.

O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, S. A. Juárez-Reyes, I. Julián-Macías, P. Ortega-Vidals, G. Silva-Ortigoza, R. Silva-Ortigoza, and C. T. Sosa-Sánchez, “Wavefronts and caustic associated with Durnin’s beams,” J. Opt. 19, 015603 (2017).
[Crossref]

O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, S. A. Juárez-Reyes, I. Julián-Macías, P. Ortega-Vidals, C. Rickenstorff-Parrao, G. Silva-Ortigoza, R. Silva-Ortigoza, and C. T. Sosa-Sánchez, “Durnin-Whitney beams,” J. Opt. 19, 055606 (2017).
[Crossref]

C. T. Sosa-Sánchez, G. Silva-Ortigoza, S. A. Juárez-Reyes, O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, I. Julián-Macías, and P. Ortega-Vidals, “Parabolic non-diffracting beams: geometrical approach,” J. Opt. 19, 085604 (2017).
[Crossref]

Fagerholm, J.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, “Nondiffracting bulk-acoustic X waves in crystals,” Phys. Rev. Lett. 83, 1171–1174 (1999).
[Crossref]

Friberg, A. T.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, “Nondiffracting bulk-acoustic X waves in crystals,” Phys. Rev. Lett. 83, 1171–1174 (1999).
[Crossref]

González, L. A.

Gussein-Zade, S. M.

V. I. Arnold, S. M. Gussein-Zade, and A. N. Varchenko, Singularities of Differentiable Maps (Birkhauser, 1995), Vol. I.

Gutiérrez-Vega, J. C.

P. A. Sanchez-Serrano, D. Wong-Campos, S. Lopez-Aguayo, and J. C. Gutiérrez-Vega, “Engineering of nondiffracting beams with genetic algorithms,” Opt. Lett. 37, 5040–5042 (2012).
[Crossref]

M. B. Alvarez-Elizondo, R. Rodríguez-Masegosa, and J. C. Gutiérrez-Vega, “Generation of Mathieu-Gauss modes with an axicon-based laser resonator,” Opt. Express 16, 18770–18775 (2008).
[Crossref]

J. C. Gutiérrez-Vega and M. A. Bandres, “Normalization of the Mathieu-Gauss optical beams,” J. Opt. Soc. Am. A 24, 215–220 (2007).
[Crossref]

C. López-Mariscal, J. C. Gutiérrez-Vega, G. Milne, and K. Dholokia, “Orbital angular momentum transfer in helical Mathieu beams,” Opt. Express 14, 4182–4187 (2006).
[Crossref]

C. López-Mariscal, M. A. Bandres, and J. C. Gutiérrez-Vega, “Observation of the experimental propagation properties of Helmholtz-Gauss beams,” Opt. Eng. 45, 068001 (2006).
[Crossref]

C. López-Mariscal, M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Observation of parabolic nondiffracting optical fields,” Opt. Express 13, 2364–2369 (2005).
[Crossref]

J. C. Gutiérrez-Vega and M. A. Bandres, “Helmholtz-Gauss waves,” J. Opt. Soc. Am. A 22, 289–298 (2005).
[Crossref]

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 Semiclassical Opt. 4, S52–S57 (2002).
[Crossref]

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).
[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).
[Crossref]

Hernández-Figueroa, H. E.

Hernández-Hernández, R. J.

Iturbe-Castillo, M. D.

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).
[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).
[Crossref]

Juárez-Reyes, S. A.

C. T. Sosa-Sánchez, G. Silva-Ortigoza, S. A. Juárez-Reyes, O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, I. Julián-Macías, and P. Ortega-Vidals, “Parabolic non-diffracting beams: geometrical approach,” J. Opt. 19, 085604 (2017).
[Crossref]

O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, S. A. Juárez-Reyes, I. Julián-Macías, P. Ortega-Vidals, C. Rickenstorff-Parrao, G. Silva-Ortigoza, R. Silva-Ortigoza, and C. T. Sosa-Sánchez, “Durnin-Whitney beams,” J. Opt. 19, 055606 (2017).
[Crossref]

O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, S. A. Juárez-Reyes, I. Julián-Macías, P. Ortega-Vidals, G. Silva-Ortigoza, R. Silva-Ortigoza, and C. T. Sosa-Sánchez, “Wavefronts and caustic associated with Durnin’s beams,” J. Opt. 19, 015603 (2017).
[Crossref]

Julián-Macías, I.

O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, S. A. Juárez-Reyes, I. Julián-Macías, P. Ortega-Vidals, G. Silva-Ortigoza, R. Silva-Ortigoza, and C. T. Sosa-Sánchez, “Wavefronts and caustic associated with Durnin’s beams,” J. Opt. 19, 015603 (2017).
[Crossref]

O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, S. A. Juárez-Reyes, I. Julián-Macías, P. Ortega-Vidals, C. Rickenstorff-Parrao, G. Silva-Ortigoza, R. Silva-Ortigoza, and C. T. Sosa-Sánchez, “Durnin-Whitney beams,” J. Opt. 19, 055606 (2017).
[Crossref]

C. T. Sosa-Sánchez, G. Silva-Ortigoza, S. A. Juárez-Reyes, O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, I. Julián-Macías, and P. Ortega-Vidals, “Parabolic non-diffracting beams: geometrical approach,” J. Opt. 19, 085604 (2017).
[Crossref]

Kalnins, E. G.

C. P. Boyer, E. G. Kalnins, and W. Miller, “Symmetry and separation of variables for the Helmholtz and Laplace equations,” Nagoya Math. J. 60, 35–80 (1976).
[Crossref]

Kartashov, Y. V.

Krastov, Y. A.

Y. A. Krastov and Y. I. Orlov, “Caustics catastrophes, and wave fields,” Usp. Fiz. Nauk 141, 591–627 (1983).
[Crossref]

Lopez-Aguayo, S.

López-Mariscal, C.

MacVicar, I.

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 Semiclassical Opt. 4, S52–S57 (2002).
[Crossref]

McLachlan, N. W.

N. W. McLachlan, Theory and Applications of Mathieu Functions (Clarendon, 1947).

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

Miller, W.

C. P. Boyer, E. G. Kalnins, and W. Miller, “Symmetry and separation of variables for the Helmholtz and Laplace equations,” Nagoya Math. J. 60, 35–80 (1976).
[Crossref]

Milne, G.

New, G. H. C.

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 Semiclassical Opt. 4, S52–S57 (2002).
[Crossref]

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).
[Crossref]

O’Neil, A. T.

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 Semiclassical Opt. 4, S52–S57 (2002).
[Crossref]

Orlov, Y. I.

Y. A. Krastov and Y. I. Orlov, “Caustics catastrophes, and wave fields,” Usp. Fiz. Nauk 141, 591–627 (1983).
[Crossref]

Ortega-Vidals, P.

C. T. Sosa-Sánchez, G. Silva-Ortigoza, S. A. Juárez-Reyes, O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, I. Julián-Macías, and P. Ortega-Vidals, “Parabolic non-diffracting beams: geometrical approach,” J. Opt. 19, 085604 (2017).
[Crossref]

O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, S. A. Juárez-Reyes, I. Julián-Macías, P. Ortega-Vidals, C. Rickenstorff-Parrao, G. Silva-Ortigoza, R. Silva-Ortigoza, and C. T. Sosa-Sánchez, “Durnin-Whitney beams,” J. Opt. 19, 055606 (2017).
[Crossref]

O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, S. A. Juárez-Reyes, I. Julián-Macías, P. Ortega-Vidals, G. Silva-Ortigoza, R. Silva-Ortigoza, and C. T. Sosa-Sánchez, “Wavefronts and caustic associated with Durnin’s beams,” J. Opt. 19, 015603 (2017).
[Crossref]

Padgett, M. J.

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 Semiclassical Opt. 4, S52–S57 (2002).
[Crossref]

Ramírez, G. A.

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).
[Crossref]

Ricardez-Vargas, I.

Rickenstorff-Parrao, C.

O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, S. A. Juárez-Reyes, I. Julián-Macías, P. Ortega-Vidals, C. Rickenstorff-Parrao, G. Silva-Ortigoza, R. Silva-Ortigoza, and C. T. Sosa-Sánchez, “Durnin-Whitney beams,” J. Opt. 19, 055606 (2017).
[Crossref]

Rodríguez-Dagnino, R. M.

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).
[Crossref]

Rodríguez-Masegosa, R.

Rose, P.

P. Rose, M. Boguslawski, and C. Denz, “Nonlinear lattice structures based on families of complex nondiffracting beams,” New J. Phys. 14, 033018 (2012).
[Crossref]

Ruiz, U.

Salo, J.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, “Nondiffracting bulk-acoustic X waves in crystals,” Phys. Rev. Lett. 83, 1171–1174 (1999).
[Crossref]

Salomaa, M. M.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, “Nondiffracting bulk-acoustic X waves in crystals,” Phys. Rev. Lett. 83, 1171–1174 (1999).
[Crossref]

Sanchez-Serrano, P. A.

Silva-Ortigoza, G.

O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, S. A. Juárez-Reyes, I. Julián-Macías, P. Ortega-Vidals, G. Silva-Ortigoza, R. Silva-Ortigoza, and C. T. Sosa-Sánchez, “Wavefronts and caustic associated with Durnin’s beams,” J. Opt. 19, 015603 (2017).
[Crossref]

O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, S. A. Juárez-Reyes, I. Julián-Macías, P. Ortega-Vidals, C. Rickenstorff-Parrao, G. Silva-Ortigoza, R. Silva-Ortigoza, and C. T. Sosa-Sánchez, “Durnin-Whitney beams,” J. Opt. 19, 055606 (2017).
[Crossref]

C. T. Sosa-Sánchez, G. Silva-Ortigoza, S. A. Juárez-Reyes, O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, I. Julián-Macías, and P. Ortega-Vidals, “Parabolic non-diffracting beams: geometrical approach,” J. Opt. 19, 085604 (2017).
[Crossref]

Silva-Ortigoza, R.

O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, S. A. Juárez-Reyes, I. Julián-Macías, P. Ortega-Vidals, C. Rickenstorff-Parrao, G. Silva-Ortigoza, R. Silva-Ortigoza, and C. T. Sosa-Sánchez, “Durnin-Whitney beams,” J. Opt. 19, 055606 (2017).
[Crossref]

O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, S. A. Juárez-Reyes, I. Julián-Macías, P. Ortega-Vidals, G. Silva-Ortigoza, R. Silva-Ortigoza, and C. T. Sosa-Sánchez, “Wavefronts and caustic associated with Durnin’s beams,” J. Opt. 19, 015603 (2017).
[Crossref]

Sosa-Sánchez, C. T.

O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, S. A. Juárez-Reyes, I. Julián-Macías, P. Ortega-Vidals, G. Silva-Ortigoza, R. Silva-Ortigoza, and C. T. Sosa-Sánchez, “Wavefronts and caustic associated with Durnin’s beams,” J. Opt. 19, 015603 (2017).
[Crossref]

C. T. Sosa-Sánchez, G. Silva-Ortigoza, S. A. Juárez-Reyes, O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, I. Julián-Macías, and P. Ortega-Vidals, “Parabolic non-diffracting beams: geometrical approach,” J. Opt. 19, 085604 (2017).
[Crossref]

O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, S. A. Juárez-Reyes, I. Julián-Macías, P. Ortega-Vidals, C. Rickenstorff-Parrao, G. Silva-Ortigoza, R. Silva-Ortigoza, and C. T. Sosa-Sánchez, “Durnin-Whitney beams,” J. Opt. 19, 055606 (2017).
[Crossref]

Tepichín, E.

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).
[Crossref]

Terborg, R. A.

Torner, L.

Varchenko, A. N.

V. I. Arnold, S. M. Gussein-Zade, and A. N. Varchenko, Singularities of Differentiable Maps (Birkhauser, 1995), Vol. I.

Volke-Sepúlveda, V.

Vysloukh, V. A.

Watson, G. N.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge University, 1927).

Whittaker, E. T.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge University, 1927).

Wong-Campos, D.

Appl. Opt. (1)

J. Opt. (3)

O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, S. A. Juárez-Reyes, I. Julián-Macías, P. Ortega-Vidals, G. Silva-Ortigoza, R. Silva-Ortigoza, and C. T. Sosa-Sánchez, “Wavefronts and caustic associated with Durnin’s beams,” J. Opt. 19, 015603 (2017).
[Crossref]

C. T. Sosa-Sánchez, G. Silva-Ortigoza, S. A. Juárez-Reyes, O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, I. Julián-Macías, and P. Ortega-Vidals, “Parabolic non-diffracting beams: geometrical approach,” J. Opt. 19, 085604 (2017).
[Crossref]

O. de Jesús Cabrera-Rosas, E. Espíndola-Ramos, S. A. Juárez-Reyes, I. Julián-Macías, P. Ortega-Vidals, C. Rickenstorff-Parrao, G. Silva-Ortigoza, R. Silva-Ortigoza, and C. T. Sosa-Sánchez, “Durnin-Whitney beams,” J. Opt. 19, 055606 (2017).
[Crossref]

J. Opt. B: Quantum Semiclassical Opt. (1)

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 Semiclassical Opt. 4, S52–S57 (2002).
[Crossref]

J. Opt. Soc. Am. A (5)

Nagoya Math. J. (1)

C. P. Boyer, E. G. Kalnins, and W. Miller, “Symmetry and separation of variables for the Helmholtz and Laplace equations,” Nagoya Math. J. 60, 35–80 (1976).
[Crossref]

New J. Phys. (1)

P. Rose, M. Boguslawski, and C. Denz, “Nonlinear lattice structures based on families of complex nondiffracting beams,” New J. Phys. 14, 033018 (2012).
[Crossref]

Opt. Commun. (1)

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).
[Crossref]

Opt. Eng. (1)

C. López-Mariscal, M. A. Bandres, and J. C. Gutiérrez-Vega, “Observation of the experimental propagation properties of Helmholtz-Gauss beams,” Opt. Eng. 45, 068001 (2006).
[Crossref]

Opt. Express (3)

Opt. Lett. (3)

Phys. Rev. Lett. (2)

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, “Nondiffracting bulk-acoustic X waves in crystals,” Phys. Rev. Lett. 83, 1171–1174 (1999).
[Crossref]

J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[Crossref]

Usp. Fiz. Nauk (2)

V. I. Arnold, “Singularities, bifurcations and catastrophes,” Usp. Fiz. Nauk 141, 569–590 (1983).
[Crossref]

Y. A. Krastov and Y. I. Orlov, “Caustics catastrophes, and wave fields,” Usp. Fiz. Nauk 141, 591–627 (1983).
[Crossref]

Other (3)

N. W. McLachlan, Theory and Applications of Mathieu Functions (Clarendon, 1947).

V. I. Arnold, S. M. Gussein-Zade, and A. N. Varchenko, Singularities of Differentiable Maps (Birkhauser, 1995), Vol. I.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th ed. (Cambridge University, 1927).

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Figures (5)

Fig. 1.
Fig. 1. Wavefront τ = 0 (blue) and the caustic (red) for the cases a ν = 0 and (a)  q = 0 , (b)  q = 0.2 , (c)  q = 0.3 , (d)  q = 0.5 . To obtain these plots we used Eqs. (21)–(23) with ϑ 0 = π / 4 .
Fig. 2.
Fig. 2. Caustic on the plane Z = 0 for the cases a ν = 0 and q = 0 , 0.2 , 0.3 , 0.5 . To obtain these plots we used Eqs. (25)–(27) with ϑ 0 = π / 4 and Z = 0 .
Fig. 3.
Fig. 3. Caustic surface associated with the subset of Mathieu beams given by Eq. (15) with a ν = 0 and 0 q < 1 in the space with local coordinates X , Y , and q , which has two structural stable singularities of the swallowtail type. To obtain this plot we used Eqs. (25) and (26) and Z = q with ϑ 0 = π / 4 .
Fig. 4.
Fig. 4. Intensity patterns associated with the beams given by Eq. (15) with ϑ 0 = π / 4 , a ν = 0 , and (a)  q = 0 , (b)  q = 0.2 , (c)  q = 0.3 , (d)  q = 0.5 . The wavefronts and caustic associated to these beams are given in Fig. 1.
Fig. 5.
Fig. 5. In the first row we present the ideal intensities of a plane wave, the Bessel beam of order m = 5 , and the Mathieu beam of order m = 5 and q = 50 ; in the second one we present the experimental realizations of those beams; in the third and fourth rows we present the experimental results when a half of the beam is obstructed.

Equations (27)

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

ψ ( r , t ) = e i ( k z z ω t ) 2 π π π A ( ϕ ) e i k t ( x cos ϕ + y sin ϕ ) d ϕ ,
x = f cosh ξ cos η , y = f sinh ξ sin η , z = z ,
1 f 2 ( cosh 2 ξ cos 2 η ) ( 2 ψ ξ 2 + 2 ψ η 2 ) + 2 ψ z 2 = 1 c 2 2 ψ t 2 .
ψ ( ξ , η , z ) = R ( ξ ) ϕ ( η ) e i ( k z z ω t ) .
d 2 ϕ d η 2 + ( a 2 q cos ( 2 η ) ) ϕ = 0 ,
d 2 R d ξ 2 ( a 2 q cosh ( 2 ξ ) ) R = 0 ,
ψ c m ( ξ , η , z , t ) = C e m ( ξ , q ) c e m ( η , q ) e i ( k z z ω t ) ,
ψ s m ( ξ , η , z , t ) = S e m ( ξ , q ) s e m ( η , q ) e i ( k z z ω t ) ,
ψ c m ( ξ , η , z , t ) = ( ρ m π π c e m ( ϕ , q ) e i k t f ( cosh ξ cos η cos ϕ + sinh ξ sin η sin ϕ ) d ϕ ) e i ( k z z ω t ) ,
ψ s m ( ξ , η , z , t ) = ( σ m π π s e m ( ϕ , q ) e i k t f ( cosh ξ cos η cos ϕ + sinh ξ sin η sin ϕ ) d ϕ ) e i ( k z z ω t ) ,
a = ν 2 + r = 0 α r q r .
ψ ( r , t ) = e i ( k z z ω t ) 2 π π π A ( ϕ ) e i k t f ( cosh ξ cos η cos ϕ + sinh ξ sin η sin ϕ ) d ϕ .
J ( x , y , z ) ( ξ , η , z ) = 1 2 f 2 [ cosh ( 2 ξ ) cos ( 2 η ) ] .
A ( ϕ ) = c e ν ( ϕ , q ) + i s e ν ( ϕ , q ) ,
ψ ( r , t ) = e i ω t 2 π π π O ( ϕ ) e i k 0 S ( ξ , η , z , ϕ ) d ϕ ,
S = f sin ϑ 0 ( cosh ξ cos η cos ϕ + sinh ξ sin η sin ϕ ) + cos ϑ 0 z + g ( ϕ ) k 0 ,
g = arctan ( s e ν ( ϕ ; q ) c e ν ( ϕ , q ) ) ,
O = [ c e ν ( ϕ ; q ) ] 2 + [ s e ν ( ϕ ; q ) ] 2 .
f cosh ξ cos η cos ϕ + f sinh ξ sin η sin ϕ = C z cos ϑ 0 g k 0 sin ϑ 0 ,
f cosh ξ cos η sin ϕ + f sin ξ sin η cos ϕ = g ϕ k 0 sin ϑ 0 ,
X ( τ , ϕ , Z ) = [ τ Z cos ϑ 0 g ] cos ϕ + g ϕ sin ϕ sin ϑ 0 ,
Y ( τ , ϕ , Z ) = [ τ Z cos ϑ 0 g ] sin ϕ g ϕ cos ϕ sin ϑ 0 ,
Z ( τ , ϕ , z ˜ ) = Z ,
τ = Z cos ϑ 0 + g + g ϕ ϕ .
X c = g ϕ ϕ cos ϕ + g ϕ sin ϕ sin ϑ 0 ,
Y c = g ϕ ϕ sin ϕ g ϕ cos ϕ sin ϑ 0 ,
Z c = Z ,

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