Abstract

We demonstrate the existence of parabolic beams that constitute the last member of the family of fundamental nondiffracting wave fields and determine their associated angular spectrum. Their transverse structure is described by parabolic cylinder functions, and contrary to Bessel or Mathieu beams their eigenvalue spectrum is continuous. Any nondiffracting beam can be constructed as a superposition of parabolic beams, since they form a complete orthogonal set of solutions of the Helmholtz equation. A novel class of traveling parabolic waves is also introduced for the first time.

© 2004 Optical Society of America

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References

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  1. J. Durnin, J. Opt. Soc. Am. A 4, 651 (1987).
    [CrossRef]
  2. J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, Phys. Rev. Lett. 83, 1171 (1999).
    [CrossRef]
  3. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1.
  4. E. G. Kalnins and W. Miller, J. Math. Phys. 18, 271 (1977), and references therein.
    [CrossRef]
  5. W. Miller, in Encyclopedia of Mathematics and Applications, G. C. Rota, ed. (Addison-Wesley, Reading, Mass., 1977).
  6. J. C. Gutierrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, Opt. Lett. 25, 1493 (2000).
    [CrossRef]
  7. S. Chávez-Cerda, J. C. Gutiérrez-Vega, and G. H. C. New, Opt. Lett. 26, 1803 (2001).
    [CrossRef]
  8. 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, Opt. Commun. 195, 35 (2001).
    [CrossRef]
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    [CrossRef]
  10. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, Cambridge, England, 1927).
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    [CrossRef]
  13. S. Chávez-Cerda, J. Mod. Opt. 46, 923 (1999).

2001 (2)

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, Opt. Commun. 195, 35 (2001).
[CrossRef]

S. Chávez-Cerda, J. C. Gutiérrez-Vega, and G. H. C. New, Opt. Lett. 26, 1803 (2001).
[CrossRef]

2000 (1)

1999 (2)

S. Chávez-Cerda, J. Mod. Opt. 46, 923 (1999).

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, Phys. Rev. Lett. 83, 1171 (1999).
[CrossRef]

1997 (1)

P. Hillion, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 57, 1702 (1997).
[CrossRef]

1989 (1)

1987 (1)

1977 (1)

E. G. Kalnins and W. Miller, J. Math. Phys. 18, 271 (1977), and references therein.
[CrossRef]

Chávez-Cerda, S.

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, Opt. Commun. 195, 35 (2001).
[CrossRef]

S. Chávez-Cerda, J. C. Gutiérrez-Vega, and G. H. C. New, Opt. Lett. 26, 1803 (2001).
[CrossRef]

J. C. Gutierrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, Opt. Lett. 25, 1493 (2000).
[CrossRef]

S. Chávez-Cerda, J. Mod. Opt. 46, 923 (1999).

Durnin, J.

Fagerholm, J.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, Phys. Rev. Lett. 83, 1171 (1999).
[CrossRef]

Feshbach, H.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1.

Friberg, A. T.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, Phys. Rev. Lett. 83, 1171 (1999).
[CrossRef]

Gutierrez-Vega, J. C.

Gutiérrez-Vega, J. C.

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, Opt. Commun. 195, 35 (2001).
[CrossRef]

S. Chávez-Cerda, J. C. Gutiérrez-Vega, and G. H. C. New, Opt. Lett. 26, 1803 (2001).
[CrossRef]

Hillion, P.

P. Hillion, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 57, 1702 (1997).
[CrossRef]

Indebetouw, G.

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, Opt. Commun. 195, 35 (2001).
[CrossRef]

J. C. Gutierrez-Vega, M. D. Iturbe-Castillo, and S. Chávez-Cerda, Opt. Lett. 25, 1493 (2000).
[CrossRef]

Kalnins, E. G.

E. G. Kalnins and W. Miller, J. Math. Phys. 18, 271 (1977), and references therein.
[CrossRef]

Miller, W.

E. G. Kalnins and W. Miller, J. Math. Phys. 18, 271 (1977), and references therein.
[CrossRef]

W. Miller, in Encyclopedia of Mathematics and Applications, G. C. Rota, ed. (Addison-Wesley, Reading, Mass., 1977).

Morse, P. M.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1.

New, G. H. C.

S. Chávez-Cerda, J. C. Gutiérrez-Vega, and G. H. C. New, Opt. Lett. 26, 1803 (2001).
[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, Opt. Commun. 195, 35 (2001).
[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, Opt. Commun. 195, 35 (2001).
[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, Opt. Commun. 195, 35 (2001).
[CrossRef]

Salo, J.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, Phys. Rev. Lett. 83, 1171 (1999).
[CrossRef]

Salomaa, M. M.

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, Phys. Rev. Lett. 83, 1171 (1999).
[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, Opt. Commun. 195, 35 (2001).
[CrossRef]

Watson, G. N.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, Cambridge, England, 1927).

Whittaker, E. T.

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, Cambridge, England, 1927).

J. Math. Phys. (1)

E. G. Kalnins and W. Miller, J. Math. Phys. 18, 271 (1977), and references therein.
[CrossRef]

J. Mod. Opt. (1)

S. Chávez-Cerda, J. Mod. Opt. 46, 923 (1999).

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

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, Opt. Commun. 195, 35 (2001).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (1)

J. Salo, J. Fagerholm, A. T. Friberg, and M. M. Salomaa, Phys. Rev. Lett. 83, 1171 (1999).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. (1)

P. Hillion, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 57, 1702 (1997).
[CrossRef]

Other (4)

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1.

W. Miller, in Encyclopedia of Mathematics and Applications, G. C. Rota, ed. (Addison-Wesley, Reading, Mass., 1977).

E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge U. Press, Cambridge, England, 1927).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1964), Chap. 19.

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Figures (4)

Fig. 1
Fig. 1

Transverse patterns of the stationary even and odd parabolic beams for a=0, 1.5, -4.

Fig. 2
Fig. 2

(a) Amplitude and (b) phase of the angular spectra of even (solid curve) and odd (dashed curves) parabolic beams with a=1.

Fig. 3
Fig. 3

Evolution of an apertured parabolic beam for a=1.5 along the plane x,z, which can be observed in the well-defined conical region where the field preserves invariance.

Fig. 4
Fig. 4

Transverse structure of (a) the amplitude U and (b) the phase of a traveling solution TU+η,ξ,a=101/2.

Equations (8)

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

Ur=exp-ikzz-ππAφ×exp-iktx cos φ+y sin φdφ,
d2Φηdη2+kt2η2+2ktaΦη=0,
d2Rξdξ2+kt2ξ2-2ktaRξ=0,
Pv,a=n=0cnvnn!,  cn+2=acn-nn-1cn-24.
Ueη,ξ;a=1π2Γ12Peσξ;aPeση;-a,
Uoη,ξ;a=2π2Γ32Poσξ;aPoση;-a,
Aeφ;a=12πsin φ1/2expia lntanφ2,
Aoφ;a=1i-Aeφ;a,φ-π,0Aeφ;a,φ0,π.

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