Abstract

Ince-Gaussian beams are introduced to describe the natural resonating modes produced by stable resonators, and they form the third completely orthogonal family of exact solutions of the paraxial wave equation. The fractional Fourier transform (FRFT) is applied to treat the propagation of Ince-Gaussian beams, and an analytical expression for an Ince-Gaussian beam passing through a FRFT system is derived. The normalized intensity distribution of an Ince-Gaussian beam in the FRFT plane is graphically illustrated with numerical examples, and the influences of the different parameters on the normalized intensity distribution are discussed in detail.

© 2009 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]
  12. Y. Cai and Q. Lin, “Fractional Fourier transform for elliptical Gaussian beam,” Opt. Commun. 217, 7-13 (2003).
    [CrossRef]
  13. Y. Cai and Q. Lin, “Transformation and spectrum properties of partially coherent beams in the fractional Fourier transform plane,” J. Opt. Soc. Am. A 20, 1528-1536 (2003).
    [CrossRef]
  14. Y. Cai and Q. Lin, “Properties of flattened Gaussian beam in the fractional Fourier transform plane,” J. Opt. A, Pure Appl. Opt. 5, 272-275 (2003).
    [CrossRef]
  15. Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5, 453-459 (2003).
    [CrossRef]
  16. Y. Cai and Q. Lin, “The fractional Fourier transform for a partially coherent pulse,” J. Opt. A, Pure Appl. Opt. 6, 307-311 (2004).
    [CrossRef]
  17. D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236, 225-235 (2004).
    [CrossRef]
  18. X. Du and D. Zhao, “Fractional Fourier transform of truncated elliptical Gaussian beams,” Appl. Opt. 45, 9049-9052 (2006).
    [CrossRef]
  19. X. Du and D. Zhao, “Fractional Fourier transform of off-axial elliptical cosh-Gaussian beams,” Optik (Stuttgart) 119, 379-382 (2008).
  20. X. Du and D. Zhao, “Fractional Fourier transforms of elliptical Hermite-cosh-Gaussian beams,” Phys. Lett. A 366, 271-275 (2007).
    [CrossRef]
  21. G. Zhou, “Fractional Fourier transform of Lorentz-Gauss beams,” J. Opt. Soc. Am. A 26, 350-355 (2009).
    [CrossRef]
  22. G. Zhou, “Fractional Fourier transform of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 56, 886-892 (2009).
    [CrossRef]
  23. M. A. Bandres and J. C. Gutiérrez-Vega, “Ince-Gaussian series representation of the two-dimensional fractional Fourier transform,” Opt. Lett. 30, 540-542 (2005).
    [CrossRef]
  24. F. M. Arscott, Periodic Differential Equations (Pergamon, 1964).
  25. F. M. Arscott, “The Whittaker-Hill equation and the wave equation in paraboloidal coordinates,” Proc. R. Soc. Edinburgh, Sect. A: Math. 67, 265-276 (1967).

2009 (2)

G. Zhou, “Fractional Fourier transform of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 56, 886-892 (2009).
[CrossRef]

G. Zhou, “Fractional Fourier transform of Lorentz-Gauss beams,” J. Opt. Soc. Am. A 26, 350-355 (2009).
[CrossRef]

2008 (3)

D. Deng and Q. Guo, “Ince-Gaussian beams in strongly nonlocal nonlinear media,” J. Phys. B 41, 145401-145407 (2008).
[CrossRef]

G. Zhou, K. Zhu, and F. Liu, “Vectorial structure of Ince-Gaussian beam in the far field,” J. Mod. Opt. 54, 2807-2817 (2008).
[CrossRef]

X. Du and D. Zhao, “Fractional Fourier transform of off-axial elliptical cosh-Gaussian beams,” Optik (Stuttgart) 119, 379-382 (2008).

2007 (2)

X. Du and D. Zhao, “Fractional Fourier transforms of elliptical Hermite-cosh-Gaussian beams,” Phys. Lett. A 366, 271-275 (2007).
[CrossRef]

S. Chu and K. Otsuka, “Numerical study for selective excitation of Ince-Gaussian modes in end-pumped solid-state lasers,” Opt. Express 15, 16506-16519 (2007).
[CrossRef]

2006 (3)

2005 (2)

2004 (6)

2003 (4)

Y. Cai and Q. Lin, “Fractional Fourier transform for elliptical Gaussian beam,” Opt. Commun. 217, 7-13 (2003).
[CrossRef]

Y. Cai and Q. Lin, “Properties of flattened Gaussian beam in the fractional Fourier transform plane,” J. Opt. A, Pure Appl. Opt. 5, 272-275 (2003).
[CrossRef]

Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5, 453-459 (2003).
[CrossRef]

Y. Cai and Q. Lin, “Transformation and spectrum properties of partially coherent beams in the fractional Fourier transform plane,” J. Opt. Soc. Am. A 20, 1528-1536 (2003).
[CrossRef]

2002 (1)

1967 (1)

F. M. Arscott, “The Whittaker-Hill equation and the wave equation in paraboloidal coordinates,” Proc. R. Soc. Edinburgh, Sect. A: Math. 67, 265-276 (1967).

Arscott, F. M.

F. M. Arscott, “The Whittaker-Hill equation and the wave equation in paraboloidal coordinates,” Proc. R. Soc. Edinburgh, Sect. A: Math. 67, 265-276 (1967).

F. M. Arscott, Periodic Differential Equations (Pergamon, 1964).

Bandres, M. A.

Bentley, J. B.

Cai, Y.

Y. Cai and Q. Lin, “The fractional Fourier transform for a partially coherent pulse,” J. Opt. A, Pure Appl. Opt. 6, 307-311 (2004).
[CrossRef]

Y. Cai and Q. Lin, “Properties of flattened Gaussian beam in the fractional Fourier transform plane,” J. Opt. A, Pure Appl. Opt. 5, 272-275 (2003).
[CrossRef]

Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5, 453-459 (2003).
[CrossRef]

Y. Cai and Q. Lin, “Fractional Fourier transform for elliptical Gaussian beam,” Opt. Commun. 217, 7-13 (2003).
[CrossRef]

Y. Cai and Q. Lin, “Transformation and spectrum properties of partially coherent beams in the fractional Fourier transform plane,” J. Opt. Soc. Am. A 20, 1528-1536 (2003).
[CrossRef]

Q. Lin and Y. Cai, “Fractional Fourier transform for partially coherent Gaussian-Schell model beams,” Opt. Lett. 27, 1672-1674 (2002).
[CrossRef]

Chu, S.

Davis, J. A.

Deng, D.

D. Deng and Q. Guo, “Ince-Gaussian beams in strongly nonlocal nonlinear media,” J. Phys. B 41, 145401-145407 (2008).
[CrossRef]

Du, X.

X. Du and D. Zhao, “Fractional Fourier transform of off-axial elliptical cosh-Gaussian beams,” Optik (Stuttgart) 119, 379-382 (2008).

X. Du and D. Zhao, “Fractional Fourier transforms of elliptical Hermite-cosh-Gaussian beams,” Phys. Lett. A 366, 271-275 (2007).
[CrossRef]

X. Du and D. Zhao, “Fractional Fourier transform of truncated elliptical Gaussian beams,” Appl. Opt. 45, 9049-9052 (2006).
[CrossRef]

Ge, D.

Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5, 453-459 (2003).
[CrossRef]

Guo, Q.

D. Deng and Q. Guo, “Ince-Gaussian beams in strongly nonlocal nonlinear media,” J. Phys. B 41, 145401-145407 (2008).
[CrossRef]

Gutiérrez-Vega, J. C.

Jing, F.

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236, 225-235 (2004).
[CrossRef]

Lin, Q.

Y. Cai and Q. Lin, “The fractional Fourier transform for a partially coherent pulse,” J. Opt. A, Pure Appl. Opt. 6, 307-311 (2004).
[CrossRef]

Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5, 453-459 (2003).
[CrossRef]

Y. Cai and Q. Lin, “Properties of flattened Gaussian beam in the fractional Fourier transform plane,” J. Opt. A, Pure Appl. Opt. 5, 272-275 (2003).
[CrossRef]

Y. Cai and Q. Lin, “Transformation and spectrum properties of partially coherent beams in the fractional Fourier transform plane,” J. Opt. Soc. Am. A 20, 1528-1536 (2003).
[CrossRef]

Y. Cai and Q. Lin, “Fractional Fourier transform for elliptical Gaussian beam,” Opt. Commun. 217, 7-13 (2003).
[CrossRef]

Q. Lin and Y. Cai, “Fractional Fourier transform for partially coherent Gaussian-Schell model beams,” Opt. Lett. 27, 1672-1674 (2002).
[CrossRef]

Liu, F.

G. Zhou, K. Zhu, and F. Liu, “Vectorial structure of Ince-Gaussian beam in the far field,” J. Mod. Opt. 54, 2807-2817 (2008).
[CrossRef]

Liu, H.

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236, 225-235 (2004).
[CrossRef]

Mao, H.

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236, 225-235 (2004).
[CrossRef]

Otsuka, K.

Schwarz, U. T.

Wang, S.

S. Wang and T. Xu, “Propagation of Ince-Gaussian beams in a thermal lens medium,” Opt. Commun. 265, 1-5 (2006).
[CrossRef]

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236, 225-235 (2004).
[CrossRef]

Wei, X.

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236, 225-235 (2004).
[CrossRef]

Xu, T.

S. Wang and T. Xu, “Propagation of Ince-Gaussian beams in a thermal lens medium,” Opt. Commun. 265, 1-5 (2006).
[CrossRef]

Zhao, D.

X. Du and D. Zhao, “Fractional Fourier transform of off-axial elliptical cosh-Gaussian beams,” Optik (Stuttgart) 119, 379-382 (2008).

X. Du and D. Zhao, “Fractional Fourier transforms of elliptical Hermite-cosh-Gaussian beams,” Phys. Lett. A 366, 271-275 (2007).
[CrossRef]

X. Du and D. Zhao, “Fractional Fourier transform of truncated elliptical Gaussian beams,” Appl. Opt. 45, 9049-9052 (2006).
[CrossRef]

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236, 225-235 (2004).
[CrossRef]

Zhou, G.

G. Zhou, “Fractional Fourier transform of Lorentz-Gauss beams,” J. Opt. Soc. Am. A 26, 350-355 (2009).
[CrossRef]

G. Zhou, “Fractional Fourier transform of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 56, 886-892 (2009).
[CrossRef]

G. Zhou, K. Zhu, and F. Liu, “Vectorial structure of Ince-Gaussian beam in the far field,” J. Mod. Opt. 54, 2807-2817 (2008).
[CrossRef]

Zhu, K.

G. Zhou, K. Zhu, and F. Liu, “Vectorial structure of Ince-Gaussian beam in the far field,” J. Mod. Opt. 54, 2807-2817 (2008).
[CrossRef]

Appl. Opt. (1)

J. Mod. Opt. (2)

G. Zhou, “Fractional Fourier transform of a higher-order cosh-Gaussian beam,” J. Mod. Opt. 56, 886-892 (2009).
[CrossRef]

G. Zhou, K. Zhu, and F. Liu, “Vectorial structure of Ince-Gaussian beam in the far field,” J. Mod. Opt. 54, 2807-2817 (2008).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (3)

Y. Cai and Q. Lin, “Properties of flattened Gaussian beam in the fractional Fourier transform plane,” J. Opt. A, Pure Appl. Opt. 5, 272-275 (2003).
[CrossRef]

Y. Cai, D. Ge, and Q. Lin, “Fractional Fourier transform for partially coherent and partially polarized Gaussian Schell-model beams,” J. Opt. A, Pure Appl. Opt. 5, 453-459 (2003).
[CrossRef]

Y. Cai and Q. Lin, “The fractional Fourier transform for a partially coherent pulse,” J. Opt. A, Pure Appl. Opt. 6, 307-311 (2004).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. Phys. B (1)

D. Deng and Q. Guo, “Ince-Gaussian beams in strongly nonlocal nonlinear media,” J. Phys. B 41, 145401-145407 (2008).
[CrossRef]

Opt. Commun. (3)

Y. Cai and Q. Lin, “Fractional Fourier transform for elliptical Gaussian beam,” Opt. Commun. 217, 7-13 (2003).
[CrossRef]

S. Wang and T. Xu, “Propagation of Ince-Gaussian beams in a thermal lens medium,” Opt. Commun. 265, 1-5 (2006).
[CrossRef]

D. Zhao, H. Mao, H. Liu, S. Wang, F. Jing, and X. Wei, “Propagation of Hermite-cosh-Gaussian beams in apertured fractional Fourier transforming systems,” Opt. Commun. 236, 225-235 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lett. (6)

Optik (Stuttgart) (1)

X. Du and D. Zhao, “Fractional Fourier transform of off-axial elliptical cosh-Gaussian beams,” Optik (Stuttgart) 119, 379-382 (2008).

Phys. Lett. A (1)

X. Du and D. Zhao, “Fractional Fourier transforms of elliptical Hermite-cosh-Gaussian beams,” Phys. Lett. A 366, 271-275 (2007).
[CrossRef]

Proc. R. Soc. Edinburgh, Sect. A: Math. (1)

F. M. Arscott, “The Whittaker-Hill equation and the wave equation in paraboloidal coordinates,” Proc. R. Soc. Edinburgh, Sect. A: Math. 67, 265-276 (1967).

Other (1)

F. M. Arscott, Periodic Differential Equations (Pergamon, 1964).

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