Abstract

We describe a new class of propagation-invariant light beams with Fourier transform given by an eigenfunction of the quantum mechanical pendulum. These beams, whose spectra (restricted to a circle) are doubly periodic Mathieu functions in azimuth, depend on a field strength parameter. When the parameter is zero, pendulum beams are Bessel beams, and as the parameter approaches infinity, they resemble transversely propagating one-dimensional Gaussian wave packets (Gaussian beam-beams). Pendulum beams are the eigenfunctions of an operator that interpolates between the squared angular momentum operator and the linear momentum operator. The analysis reveals connections with Mathieu beams, and insight into the paraxial approximation.

© 2013 Optical Society of America

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    [CrossRef]

2012

M. R. Dennis and J. B. Götte, Phys. Rev. Lett. 109, 183903 (2012).
[CrossRef]

K. Lombardo and M. A. Alonso, Am. J. Phys. 80, 82 (2012).
[CrossRef]

J. D. Ring, J. J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, Opt. Express 20, 18955 (2012).
[CrossRef]

2011

2009

M. Mazilu, D. J. Stephenson, F. Gunn-Moore, and K. Dholakia, Laser Photon. Rev. 4, 529 (2009).
[CrossRef]

2008

S. Franke-Arnold, L. Allen, and M. Padgett, Laser Photon. Rev. 2, 299 (2008).
[CrossRef]

2007

2006

2004

2003

J. C. Gutiérrez-Vega, R. M. Rodríguez-Dagnino, M. A. Meneses-Nava, and S. Chávez-Cerda, Am. J. Phys. 71, 233 (2003).
[CrossRef]

G. W. Forbes, M. A. Alonso, and A. E. Siegman, J. Phys. A 36, 7027 (2003).
[CrossRef]

2000

1998

Z. Bouchal, J. Wagner, and M. Chlup, Opt. Commun. 151, 207 (1998).
[CrossRef]

1992

S. Danakas and P. K. Aravind, Phys. Rev. A 45, 1973 (1992).
[CrossRef]

L. Allen, M. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

1987

1979

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

1975

C. P. Boyer, E. G. Kalnins, and W. Miller, J. Math. Phys. 16, 512 (1975).
[CrossRef]

1928

E. U. Condon, Phys. Rev. 31, 891 (1928).
[CrossRef]

Abramochkin, E. G.

E. G. Abramochkin and V. G. Volostnikov, J. Opt. A 6, S157 (2004).
[CrossRef]

Allen, L.

S. Franke-Arnold, L. Allen, and M. Padgett, Laser Photon. Rev. 2, 299 (2008).
[CrossRef]

L. Allen, M. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

Alonso, M. A.

K. Lombardo and M. A. Alonso, Am. J. Phys. 80, 82 (2012).
[CrossRef]

M. R. Dennis, J. B. Götte, R. P. King, M. A. Morgan, and M. A. Alonso, Opt. Lett. 36, 4452 (2011).
[CrossRef]

G. W. Forbes, M. A. Alonso, and A. E. Siegman, J. Phys. A 36, 7027 (2003).
[CrossRef]

Aravind, P. K.

S. Danakas and P. K. Aravind, Phys. Rev. A 45, 1973 (1992).
[CrossRef]

Balazs, N. L.

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

Bandres, M. A.

Beijersbergen, M.

L. Allen, M. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

Berry, M. V.

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

Bouchal, Z.

Z. Bouchal, J. Wagner, and M. Chlup, Opt. Commun. 151, 207 (1998).
[CrossRef]

Boyer, C. P.

C. P. Boyer, E. G. Kalnins, and W. Miller, J. Math. Phys. 16, 512 (1975).
[CrossRef]

Chávez-Cerda, S.

Chlup, M.

Z. Bouchal, J. Wagner, and M. Chlup, Opt. Commun. 151, 207 (1998).
[CrossRef]

Christodoulides, D. N.

Condon, E. U.

E. U. Condon, Phys. Rev. 31, 891 (1928).
[CrossRef]

Danakas, S.

S. Danakas and P. K. Aravind, Phys. Rev. A 45, 1973 (1992).
[CrossRef]

Dennis, M. R.

Dholakia, K.

Durnin, J.

Forbes, G. W.

G. W. Forbes, M. A. Alonso, and A. E. Siegman, J. Phys. A 36, 7027 (2003).
[CrossRef]

Franke-Arnold, S.

S. Franke-Arnold, L. Allen, and M. Padgett, Laser Photon. Rev. 2, 299 (2008).
[CrossRef]

Götte, J. B.

Gunn-Moore, F.

M. Mazilu, D. J. Stephenson, F. Gunn-Moore, and K. Dholakia, Laser Photon. Rev. 4, 529 (2009).
[CrossRef]

Gutiérrez-Vega, J. C.

Gutíerrez-Vega, J. C.

Iturbe-Castillo, M. D.

Kalnins, E. G.

C. P. Boyer, E. G. Kalnins, and W. Miller, J. Math. Phys. 16, 512 (1975).
[CrossRef]

King, R. P.

Lindberg, J. J.

Lombardo, K.

K. Lombardo and M. A. Alonso, Am. J. Phys. 80, 82 (2012).
[CrossRef]

Lopez-Mariscal, C.

Mazilu, M.

J. D. Ring, J. J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, Opt. Express 20, 18955 (2012).
[CrossRef]

M. Mazilu, D. J. Stephenson, F. Gunn-Moore, and K. Dholakia, Laser Photon. Rev. 4, 529 (2009).
[CrossRef]

McLachlan, N. W.

N. W. McLachlan, Theory and Application of Mathieu Functions (Dover, 1964).

Meneses-Nava, M. A.

J. C. Gutiérrez-Vega, R. M. Rodríguez-Dagnino, M. A. Meneses-Nava, and S. Chávez-Cerda, Am. J. Phys. 71, 233 (2003).
[CrossRef]

Miller, W.

C. P. Boyer, E. G. Kalnins, and W. Miller, J. Math. Phys. 16, 512 (1975).
[CrossRef]

Milne, G.

Morgan, M. A.

Mourka, A.

Padgett, M.

S. Franke-Arnold, L. Allen, and M. Padgett, Laser Photon. Rev. 2, 299 (2008).
[CrossRef]

Ring, J. D.

Rodríguez-Dagnino, R. M.

J. C. Gutiérrez-Vega, R. M. Rodríguez-Dagnino, M. A. Meneses-Nava, and S. Chávez-Cerda, Am. J. Phys. 71, 233 (2003).
[CrossRef]

Rodríguez-Morales, G.

Siegman, A. E.

G. W. Forbes, M. A. Alonso, and A. E. Siegman, J. Phys. A 36, 7027 (2003).
[CrossRef]

A. E. Siegman, Lasers (University Science Books, 1990).

Siviloglou, G. A.

Spreeuw, R. J. C.

L. Allen, M. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

Stephenson, D. J.

M. Mazilu, D. J. Stephenson, F. Gunn-Moore, and K. Dholakia, Laser Photon. Rev. 4, 529 (2009).
[CrossRef]

Volostnikov, V. G.

E. G. Abramochkin and V. G. Volostnikov, J. Opt. A 6, S157 (2004).
[CrossRef]

Wagner, J.

Z. Bouchal, J. Wagner, and M. Chlup, Opt. Commun. 151, 207 (1998).
[CrossRef]

Woerdman, J. P.

L. Allen, M. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

Am. J. Phys.

J. C. Gutiérrez-Vega, R. M. Rodríguez-Dagnino, M. A. Meneses-Nava, and S. Chávez-Cerda, Am. J. Phys. 71, 233 (2003).
[CrossRef]

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

K. Lombardo and M. A. Alonso, Am. J. Phys. 80, 82 (2012).
[CrossRef]

J. Math. Phys.

C. P. Boyer, E. G. Kalnins, and W. Miller, J. Math. Phys. 16, 512 (1975).
[CrossRef]

J. Opt. A

E. G. Abramochkin and V. G. Volostnikov, J. Opt. A 6, S157 (2004).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. A

G. W. Forbes, M. A. Alonso, and A. E. Siegman, J. Phys. A 36, 7027 (2003).
[CrossRef]

Laser Photon. Rev.

S. Franke-Arnold, L. Allen, and M. Padgett, Laser Photon. Rev. 2, 299 (2008).
[CrossRef]

M. Mazilu, D. J. Stephenson, F. Gunn-Moore, and K. Dholakia, Laser Photon. Rev. 4, 529 (2009).
[CrossRef]

Opt. Commun.

Z. Bouchal, J. Wagner, and M. Chlup, Opt. Commun. 151, 207 (1998).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev.

E. U. Condon, Phys. Rev. 31, 891 (1928).
[CrossRef]

Phys. Rev. A

L. Allen, M. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, Phys. Rev. A 45, 8185 (1992).
[CrossRef]

S. Danakas and P. K. Aravind, Phys. Rev. A 45, 1973 (1992).
[CrossRef]

Phys. Rev. Lett.

M. R. Dennis and J. B. Götte, Phys. Rev. Lett. 109, 183903 (2012).
[CrossRef]

Other

A. E. Siegman, Lasers (University Science Books, 1990).

G. Wolf, NIST Digital Library of Mathematical Functions, Chap. 28, http://dlmf.nist.gov/ .

N. W. McLachlan, Theory and Application of Mathieu Functions (Dover, 1964).

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Figures (3)

Fig. 1.
Fig. 1.

Quantum pendulum eigenfunctions, given by Mathieu functions in azimuth as in the text. (a)–(c) n=0, (d)–(f) n=4, (a), (d) q=5, (b), (e) q=50, (c), (f) q=500. These are the spectra (with radius kr) of pendulum beams.

Fig. 2.
Fig. 2.

Pendulum beam intensities, with length scale kr1. Frames are Fourier transforms of those in Fig. (1): (a)–(c) Λ0, (d)–(f) Λ3, (a), (d) q=5, (b),(e) q=50, (c), (f) q=500. The profiles are propagation invariant, and interpolate between real Bessel beams (q=0) and “beam-beams” (q1), whose profiles resemble propagating 1D HG wave packets.

Fig. 3.
Fig. 3.

Amplitude distributions near the axis for q0. Thin lines are intensity contours, thick (dashed) lines real (imaginary) contours. (a) n=3, with amplitude 48(kr2xy/q2)8i(krx/q). (b) n=4 with amplitude 1+24(krx/q)224(kry/q)2+8i(kry/q). The functions globally behave like xy, and x2y2 but the nongeneric axial singularity is perturbed by small q, given by Eqs. (5) and (6).

Equations (9)

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Y(χ)+[α2qcos(2χ)]Y(χ)=0,
δ(|k|kr)cos(mθ)imcos(m[ϕπ/2])Jm(krr),
Λ2n(r;q)=m=0imA2m2n(q)cos(m[ϕπ2])Jm(krr),
Λ2n1(r;q)=m=0imB2m2n(q)sin(m[ϕπ2])Jm(krr),
Λ2n(r,ϕ)t=0n(n+t1)!(nt)!t!(2ikrrq)tcos(t[ϕπ2]),
Λ2n1(r,ϕ)t=1n(n+t1)!(nt)!t!(2ikrrq)tsin(t[ϕπ2]).
cem(χ;q),sem+1(χ;q)[πq22m+1(m!)2]1/4e(q1/4cosχ)2Hm(2q1/4cosχ),
ce2n(θ2,q)12[ce2n(θπ2;q16)+se2n1(θπ2;q16)],
Λ2n(r;q)Ce2n(ξ;Q)ce2n(η;Q)+iSe2n+1(ξ;Q)se2n+1(η;Q),

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