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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Vol. 21, Iss. 5 — May. 1, 2004
  • pp: 873–880

Ince–Gaussian modes of the paraxial wave equation and stable resonators

Miguel A. Bandres and Julio C. Gutiérrez-Vega  »View Author Affiliations


JOSA A, Vol. 21, Issue 5, pp. 873-880 (2004)
http://dx.doi.org/10.1364/JOSAA.21.000873


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Abstract

We present the Ince–Gaussian modes that constitute the third complete family of exact and orthogonal solutions of the paraxial wave equation in elliptic coordinates and that are transverse eigenmodes of stable resonators. The transverse shape of these modes is described by the Ince polynomials and is structurally stable under propagation. Ince–Gaussian modes constitute the exact and continuous transition modes between Laguerre– and Hermite–Gaussian modes. The expansions between the three families are derived and discussed. As with Laguerre–Gaussian modes, it is possible to construct helical Ince–Gaussian modes that exhibit rotating phase features whose intensity pattern is formed by elliptic rings and whose phase rotates elliptically.

© 2004 Optical Society of America

OCIS Codes
(050.1960) Diffraction and gratings : Diffraction theory
(140.3300) Lasers and laser optics : Laser beam shaping
(140.3410) Lasers and laser optics : Laser resonators
(260.1960) Physical optics : Diffraction theory
(350.5500) Other areas of optics : Propagation

History
Original Manuscript: August 18, 2003
Revised Manuscript: December 5, 2003
Manuscript Accepted: December 5, 2003
Published: May 1, 2004

Citation
Miguel A. Bandres and Julio C. Gutiérrez-Vega, "Ince–Gaussian modes of the paraxial wave equation and stable resonators," J. Opt. Soc. Am. A 21, 873-880 (2004)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-21-5-873


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References

  1. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  2. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966). [CrossRef]
  3. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 96, 8185–8194 (1992). [CrossRef]
  4. G. Nienhuis, L. Allen, “Paraxial wave optics and harmonic oscillators,” Phys. Rev. A 48, 656–665 (1993). [CrossRef] [PubMed]
  5. E. G. Ince, “A linear differential equation with periodic coefficients,” Proc. London Math. Soc. 23, 56–74 (1923).
  6. F. M. Arscott, Periodic Differential Equations (Pergamon, Oxford, UK, 1964).
  7. F. M. Arscott, “The Whittaker-Hill equation and the wave equation in paraboloidal coordinates,” Proc. R. Soc. Edinburgh Sect. A 67, 265–276 (1967).
  8. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964), Chap. 19.
  9. A. G. Fox, T. Li, “Resonant modes in maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961). [CrossRef]
  10. I. Kimel, L. R. Elı́as, “Relations between Hermite and Laguerre Gaussian modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993). [CrossRef]
  11. A. T. O’Neil, J. Courtial, “Mode transformations in terms of the constituent Hermite–Gaussian or Laguerre–Gaussian modes and the variable-phase converter” Opt. Commun. 181, 35–45 (2000). [CrossRef]
  12. C. P. Boyer, E. G. Kalnins, W. Miller, “Lie theory and separation of variables. 7. The harmonic oscillator in ellipic coordinates and the Ince polynomials,” J. Math. Phys. 16, 512–523 (1975). [CrossRef]
  13. L. Allen, M. J. Padgett, M. Babiker, “The orbital momentum of light,” Prog. Opt. 39, 291–371 (1999). [CrossRef]
  14. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25, 1493–1495 (2000). [CrossRef]
  15. J. C. Gutiérrez-Vega, M. D. Iturbe-Castillo, G. A. Ramı́rez, E. Tepichı́n, R. M. Rodrı́guez-Dagnino, S. Chávez-Cerda, G. H. C. New, “Experimental demonstration of optical Mathieu beams,” Opt. Commun. 195, 35–40 (2001). [CrossRef]
  16. S. Chávez-Cerda, J. C. Gutiérrez-Vega, G. H. C. New, “Elliptic vortices of electromagnetic wave fields,” Opt. Lett. 26, 1803–1805 (2001). [CrossRef]
  17. J. Arlt, M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity,” Opt. Lett. 25, 191–193 (2000). [CrossRef]
  18. K. T. Gahagan, G. A. Swartzlander, “Optical vortex trapping of particles,” Opt. Lett. 21, 827–829 (1997). [CrossRef]

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