Abstract

The propagation of partially coherent flat-topped beams in uniaxial crystals orthogonal to the optical axis is investigated. Analytical propagation formulas for partially coherent flat-topped beams propagating through uniaxial crystals orthogonal to the optical axis are derived and some analyses are illustrated by numerical examples related to the propagation properties of partially coherent circular flat-topped beams. It is found that the propagation properties of partially coherent flat-topped beams in uniaxial crystals are closely related to the initial coherence and the ratio of extraordinary and ordinary refractive indices.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. Desilvestri, P. Laporta, V. Magni, and O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors--the super-Gaussian approach,” IEEE J. Quantum Electron. 26, 1172-1177 (1988).
    [CrossRef]
  2. M. S. Bowers, “Diffractive analysis of unstable optical resonator with super-Gaussian mirrors,” Opt. Lett. 17, 1319-1321 (1992).
    [CrossRef] [PubMed]
  3. M. R. Perrone and A. Piegari, “On the super-Gaussian unstable resonators for high-gain short pulse laser media,” IEEE J. Quantum Electron. 29, 1423-1427 (1993).
    [CrossRef]
  4. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335-341 (1994).
    [CrossRef]
  5. V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, and G. S. Spagnolo, “Propagation of axially symmetric flattened Gaussian beam,” J. Opt. Soc. Am. A 13, 1385-1394 (1996).
    [CrossRef]
  6. Y. Li, “Light beam with flat-topped profiles,” Opt. Lett. 27, 1007-1009 (2002).
    [CrossRef]
  7. Y. Li, “New expressions for flat-topped beams,” Opt. Commun. 206, 225-234 (2002).
    [CrossRef]
  8. Y. Li, “Flat-topped beam with non-circular cross-sections,” J. Mod. Opt. 50, 1957-1966 (2003).
    [CrossRef]
  9. Y. Cai, “Light beams with elliptical flat-topped profiles,” J. Opt. A, Pure Appl. Opt. 6, 390-395 (2004).
    [CrossRef]
  10. R. Borghi and M. Santarsiero, “Modal decomposition of partially coherent flat-topped beams produced by multimode lasers,” Opt. Lett. 23, 313-315 (1998).
    [CrossRef]
  11. Y. Cai and S. He, “Partially coherent flattened Gaussian beam and its paraxial propagation properties,” J. Opt. Soc. Am. A 23, 2623-2628 (2006).
    [CrossRef]
  12. Y. Cai and Q. Lin, “A partially coherent elliptical flattened Gaussian beam and its propagation,” J. Opt. A, Pure Appl. Opt. 6, 1061-1066 (2004).
    [CrossRef]
  13. D. Ge, Y. Cai, and Q. Lin, “Partially coherent flat-topped beam and its propagation,” Appl. Opt. 43, 4732-4738 (2004).
    [CrossRef] [PubMed]
  14. Y. Dan, B. Zhang, and P. Pan, “Propagation of partially coherent flat-topped beams through a turbulent atmosphere,” J. Opt. Soc. Am. A 25, 2223-2231 (2008).
    [CrossRef]
  15. Y. Cai, X. Lü, H. Eyyuboğlu, and Y. Baykal, “Paraxial propagation of a partially coherent flattened Gaussian beam through apertured ABCD optical systems,” Opt. Commun. 281, 3221-3229 (2008).
    [CrossRef]
  16. Y. Dan and B. Zhang, “Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Express 16, 15563-15575 (2008).
    [CrossRef] [PubMed]
  17. X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, “Influence of atmosphere turbulence on the spatial correlation properties of partially coherent flat-topped beams,” J. Opt. Soc. Am. A 24, 3554-3563 (2007).
    [CrossRef]
  18. M. Alavinejad, B. Ghafary, and D. Razzaghi, “Spectral changes of partially coherent flat topped beam in turbulent atmosphere,” Opt. Commun. 281, 2173-2178 (2008).
    [CrossRef]
  19. F. Wang and Y. Cai, “Experimental generation of a partially coherent flat-topped beam,” Opt. Lett. 33, 1795-1797 (2008).
    [CrossRef] [PubMed]
  20. J. Stamnes and G. Sherman, “Radiation of electromagnetic fields in uniaxially anisotropic medium,” J. Opt. Soc. Am. 66, 780-788 (1976).
    [CrossRef]
  21. A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A 18, 1656-1661 (2001).
    [CrossRef]
  22. A. Ciattoni, G. Cincotti, and C. Palma, “Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals,” Opt. Commun. 195, 55-61 (2001).
    [CrossRef]
  23. G. Cincotti, A. Ciattoni, and C. Palma, “Laguerre-Gauss and Bessel-Gauss beams in uniaxial crystals,” J. Opt. Soc. Am. A 19, 1680-1688 (2002).
    [CrossRef]
  24. A. Ciattoni, G. Cincotti, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A 19, 792-796 (2002).
    [CrossRef]
  25. B. Lü and S. Luo, “Propagation properties of three-dimensional flattened Gaussian beams in uniaxially anisotropic crystals,” Opt. Laser Technol. 36, 51-56 (2004).
    [CrossRef]
  26. D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281, 202-209 (2008).
    [CrossRef]
  27. D. Deng, J. Shen, Y. Tian, J. Shao, and Z. Fan, “Propagation properties of beams generated by Gaussian mirror resonator in uniaxial crystals,” Optik (Jena) 118, 547-551 (2007).
  28. A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A 20, 2163-2171 (2003).
    [CrossRef]
  29. A. Ciattoni and C. Palma, “Nondiffracting beams in uniaxial media propagating orthogonally to the optical axis,” Opt. Commun. 224, 175-183 (2003).
    [CrossRef]
  30. A. Ciattoni and C. Palma, “Anisotropic beam spreading in uniaxial crystals,” Opt. Commun. 231, 79-92 (2004).
    [CrossRef]
  31. D. Liu and Z. Zhou, “Various dark hollow beams propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. A, Pure Appl. Opt. 10, 095005 (2008).
    [CrossRef]
  32. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).
  33. W. H. Carter, “Spot size and divergence for Hermite Gaussian beams of any order,” Appl. Opt. 19, 1027-1029 (1980).
    [CrossRef] [PubMed]

2008 (7)

Y. Cai, X. Lü, H. Eyyuboğlu, and Y. Baykal, “Paraxial propagation of a partially coherent flattened Gaussian beam through apertured ABCD optical systems,” Opt. Commun. 281, 3221-3229 (2008).
[CrossRef]

M. Alavinejad, B. Ghafary, and D. Razzaghi, “Spectral changes of partially coherent flat topped beam in turbulent atmosphere,” Opt. Commun. 281, 2173-2178 (2008).
[CrossRef]

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281, 202-209 (2008).
[CrossRef]

D. Liu and Z. Zhou, “Various dark hollow beams propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. A, Pure Appl. Opt. 10, 095005 (2008).
[CrossRef]

F. Wang and Y. Cai, “Experimental generation of a partially coherent flat-topped beam,” Opt. Lett. 33, 1795-1797 (2008).
[CrossRef] [PubMed]

Y. Dan, B. Zhang, and P. Pan, “Propagation of partially coherent flat-topped beams through a turbulent atmosphere,” J. Opt. Soc. Am. A 25, 2223-2231 (2008).
[CrossRef]

Y. Dan and B. Zhang, “Beam propagation factor of partially coherent flat-topped beams in a turbulent atmosphere,” Opt. Express 16, 15563-15575 (2008).
[CrossRef] [PubMed]

2007 (2)

X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, “Influence of atmosphere turbulence on the spatial correlation properties of partially coherent flat-topped beams,” J. Opt. Soc. Am. A 24, 3554-3563 (2007).
[CrossRef]

D. Deng, J. Shen, Y. Tian, J. Shao, and Z. Fan, “Propagation properties of beams generated by Gaussian mirror resonator in uniaxial crystals,” Optik (Jena) 118, 547-551 (2007).

2006 (1)

2004 (5)

D. Ge, Y. Cai, and Q. Lin, “Partially coherent flat-topped beam and its propagation,” Appl. Opt. 43, 4732-4738 (2004).
[CrossRef] [PubMed]

A. Ciattoni and C. Palma, “Anisotropic beam spreading in uniaxial crystals,” Opt. Commun. 231, 79-92 (2004).
[CrossRef]

Y. Cai and Q. Lin, “A partially coherent elliptical flattened Gaussian beam and its propagation,” J. Opt. A, Pure Appl. Opt. 6, 1061-1066 (2004).
[CrossRef]

B. Lü and S. Luo, “Propagation properties of three-dimensional flattened Gaussian beams in uniaxially anisotropic crystals,” Opt. Laser Technol. 36, 51-56 (2004).
[CrossRef]

Y. Cai, “Light beams with elliptical flat-topped profiles,” J. Opt. A, Pure Appl. Opt. 6, 390-395 (2004).
[CrossRef]

2003 (3)

Y. Li, “Flat-topped beam with non-circular cross-sections,” J. Mod. Opt. 50, 1957-1966 (2003).
[CrossRef]

A. Ciattoni and C. Palma, “Nondiffracting beams in uniaxial media propagating orthogonally to the optical axis,” Opt. Commun. 224, 175-183 (2003).
[CrossRef]

A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A 20, 2163-2171 (2003).
[CrossRef]

2002 (4)

2001 (2)

A. Ciattoni, G. Cincotti, and C. Palma, “Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals,” Opt. Commun. 195, 55-61 (2001).
[CrossRef]

A. Ciattoni, B. Crosignani, and P. Di Porto, “Vectorial theory of propagation in uniaxially anisotropic media,” J. Opt. Soc. Am. A 18, 1656-1661 (2001).
[CrossRef]

1998 (1)

1996 (1)

1994 (1)

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335-341 (1994).
[CrossRef]

1993 (1)

M. R. Perrone and A. Piegari, “On the super-Gaussian unstable resonators for high-gain short pulse laser media,” IEEE J. Quantum Electron. 29, 1423-1427 (1993).
[CrossRef]

1992 (1)

1988 (1)

S. Desilvestri, P. Laporta, V. Magni, and O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors--the super-Gaussian approach,” IEEE J. Quantum Electron. 26, 1172-1177 (1988).
[CrossRef]

1980 (1)

1976 (1)

Alavinejad, M.

M. Alavinejad, B. Ghafary, and D. Razzaghi, “Spectral changes of partially coherent flat topped beam in turbulent atmosphere,” Opt. Commun. 281, 2173-2178 (2008).
[CrossRef]

Ambrosini, D.

Bagini, V.

Baykal, Y.

Y. Cai, X. Lü, H. Eyyuboğlu, and Y. Baykal, “Paraxial propagation of a partially coherent flattened Gaussian beam through apertured ABCD optical systems,” Opt. Commun. 281, 3221-3229 (2008).
[CrossRef]

Borghi, R.

Bowers, M. S.

Cai, Y.

Y. Cai, X. Lü, H. Eyyuboğlu, and Y. Baykal, “Paraxial propagation of a partially coherent flattened Gaussian beam through apertured ABCD optical systems,” Opt. Commun. 281, 3221-3229 (2008).
[CrossRef]

F. Wang and Y. Cai, “Experimental generation of a partially coherent flat-topped beam,” Opt. Lett. 33, 1795-1797 (2008).
[CrossRef] [PubMed]

Y. Cai and S. He, “Partially coherent flattened Gaussian beam and its paraxial propagation properties,” J. Opt. Soc. Am. A 23, 2623-2628 (2006).
[CrossRef]

D. Ge, Y. Cai, and Q. Lin, “Partially coherent flat-topped beam and its propagation,” Appl. Opt. 43, 4732-4738 (2004).
[CrossRef] [PubMed]

Y. Cai, “Light beams with elliptical flat-topped profiles,” J. Opt. A, Pure Appl. Opt. 6, 390-395 (2004).
[CrossRef]

Y. Cai and Q. Lin, “A partially coherent elliptical flattened Gaussian beam and its propagation,” J. Opt. A, Pure Appl. Opt. 6, 1061-1066 (2004).
[CrossRef]

Carter, W. H.

Chen, S.

Chen, X.

Ciattoni, A.

Cincotti, G.

Crosignani, B.

Dan, Y.

Deng, D.

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281, 202-209 (2008).
[CrossRef]

D. Deng, J. Shen, Y. Tian, J. Shao, and Z. Fan, “Propagation properties of beams generated by Gaussian mirror resonator in uniaxial crystals,” Optik (Jena) 118, 547-551 (2007).

Desilvestri, S.

S. Desilvestri, P. Laporta, V. Magni, and O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors--the super-Gaussian approach,” IEEE J. Quantum Electron. 26, 1172-1177 (1988).
[CrossRef]

Di Porto, P.

Eyyuboglu, H.

Y. Cai, X. Lü, H. Eyyuboğlu, and Y. Baykal, “Paraxial propagation of a partially coherent flattened Gaussian beam through apertured ABCD optical systems,” Opt. Commun. 281, 3221-3229 (2008).
[CrossRef]

Fan, Z.

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281, 202-209 (2008).
[CrossRef]

D. Deng, J. Shen, Y. Tian, J. Shao, and Z. Fan, “Propagation properties of beams generated by Gaussian mirror resonator in uniaxial crystals,” Optik (Jena) 118, 547-551 (2007).

Ge, D.

Ghafary, B.

M. Alavinejad, B. Ghafary, and D. Razzaghi, “Spectral changes of partially coherent flat topped beam in turbulent atmosphere,” Opt. Commun. 281, 2173-2178 (2008).
[CrossRef]

Gori, F.

He, S.

Ji, X.

Laporta, P.

S. Desilvestri, P. Laporta, V. Magni, and O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors--the super-Gaussian approach,” IEEE J. Quantum Electron. 26, 1172-1177 (1988).
[CrossRef]

Li, X.

Li, Y.

Y. Li, “Flat-topped beam with non-circular cross-sections,” J. Mod. Opt. 50, 1957-1966 (2003).
[CrossRef]

Y. Li, “New expressions for flat-topped beams,” Opt. Commun. 206, 225-234 (2002).
[CrossRef]

Y. Li, “Light beam with flat-topped profiles,” Opt. Lett. 27, 1007-1009 (2002).
[CrossRef]

Lin, Q.

D. Ge, Y. Cai, and Q. Lin, “Partially coherent flat-topped beam and its propagation,” Appl. Opt. 43, 4732-4738 (2004).
[CrossRef] [PubMed]

Y. Cai and Q. Lin, “A partially coherent elliptical flattened Gaussian beam and its propagation,” J. Opt. A, Pure Appl. Opt. 6, 1061-1066 (2004).
[CrossRef]

Liu, D.

D. Liu and Z. Zhou, “Various dark hollow beams propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. A, Pure Appl. Opt. 10, 095005 (2008).
[CrossRef]

Lü, B.

X. Ji, X. Chen, S. Chen, X. Li, and B. Lü, “Influence of atmosphere turbulence on the spatial correlation properties of partially coherent flat-topped beams,” J. Opt. Soc. Am. A 24, 3554-3563 (2007).
[CrossRef]

B. Lü and S. Luo, “Propagation properties of three-dimensional flattened Gaussian beams in uniaxially anisotropic crystals,” Opt. Laser Technol. 36, 51-56 (2004).
[CrossRef]

Lü, X.

Y. Cai, X. Lü, H. Eyyuboğlu, and Y. Baykal, “Paraxial propagation of a partially coherent flattened Gaussian beam through apertured ABCD optical systems,” Opt. Commun. 281, 3221-3229 (2008).
[CrossRef]

Luo, S.

B. Lü and S. Luo, “Propagation properties of three-dimensional flattened Gaussian beams in uniaxially anisotropic crystals,” Opt. Laser Technol. 36, 51-56 (2004).
[CrossRef]

Magni, V.

S. Desilvestri, P. Laporta, V. Magni, and O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors--the super-Gaussian approach,” IEEE J. Quantum Electron. 26, 1172-1177 (1988).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Pacileo, A. M.

Palma, C.

A. Ciattoni and C. Palma, “Anisotropic beam spreading in uniaxial crystals,” Opt. Commun. 231, 79-92 (2004).
[CrossRef]

A. Ciattoni and C. Palma, “Nondiffracting beams in uniaxial media propagating orthogonally to the optical axis,” Opt. Commun. 224, 175-183 (2003).
[CrossRef]

A. Ciattoni and C. Palma, “Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond,” J. Opt. Soc. Am. A 20, 2163-2171 (2003).
[CrossRef]

G. Cincotti, A. Ciattoni, and C. Palma, “Laguerre-Gauss and Bessel-Gauss beams in uniaxial crystals,” J. Opt. Soc. Am. A 19, 1680-1688 (2002).
[CrossRef]

A. Ciattoni, G. Cincotti, and C. Palma, “Propagation of cylindrically symmetric fields in uniaxial crystals,” J. Opt. Soc. Am. A 19, 792-796 (2002).
[CrossRef]

A. Ciattoni, G. Cincotti, and C. Palma, “Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals,” Opt. Commun. 195, 55-61 (2001).
[CrossRef]

Pan, P.

Perrone, M. R.

M. R. Perrone and A. Piegari, “On the super-Gaussian unstable resonators for high-gain short pulse laser media,” IEEE J. Quantum Electron. 29, 1423-1427 (1993).
[CrossRef]

Piegari, A.

M. R. Perrone and A. Piegari, “On the super-Gaussian unstable resonators for high-gain short pulse laser media,” IEEE J. Quantum Electron. 29, 1423-1427 (1993).
[CrossRef]

Razzaghi, D.

M. Alavinejad, B. Ghafary, and D. Razzaghi, “Spectral changes of partially coherent flat topped beam in turbulent atmosphere,” Opt. Commun. 281, 2173-2178 (2008).
[CrossRef]

Santarsiero, M.

Shao, J.

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281, 202-209 (2008).
[CrossRef]

D. Deng, J. Shen, Y. Tian, J. Shao, and Z. Fan, “Propagation properties of beams generated by Gaussian mirror resonator in uniaxial crystals,” Optik (Jena) 118, 547-551 (2007).

Shen, J.

D. Deng, J. Shen, Y. Tian, J. Shao, and Z. Fan, “Propagation properties of beams generated by Gaussian mirror resonator in uniaxial crystals,” Optik (Jena) 118, 547-551 (2007).

Sherman, G.

Spagnolo, G. S.

Stamnes, J.

Svelto, O.

S. Desilvestri, P. Laporta, V. Magni, and O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors--the super-Gaussian approach,” IEEE J. Quantum Electron. 26, 1172-1177 (1988).
[CrossRef]

Tian, Y.

D. Deng, J. Shen, Y. Tian, J. Shao, and Z. Fan, “Propagation properties of beams generated by Gaussian mirror resonator in uniaxial crystals,” Optik (Jena) 118, 547-551 (2007).

Wang, F.

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Xu, S.

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281, 202-209 (2008).
[CrossRef]

Yu, H.

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281, 202-209 (2008).
[CrossRef]

Zhang, B.

Zhou, Z.

D. Liu and Z. Zhou, “Various dark hollow beams propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. A, Pure Appl. Opt. 10, 095005 (2008).
[CrossRef]

Appl. Opt. (2)

IEEE J. Quantum Electron. (2)

S. Desilvestri, P. Laporta, V. Magni, and O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors--the super-Gaussian approach,” IEEE J. Quantum Electron. 26, 1172-1177 (1988).
[CrossRef]

M. R. Perrone and A. Piegari, “On the super-Gaussian unstable resonators for high-gain short pulse laser media,” IEEE J. Quantum Electron. 29, 1423-1427 (1993).
[CrossRef]

J. Mod. Opt. (1)

Y. Li, “Flat-topped beam with non-circular cross-sections,” J. Mod. Opt. 50, 1957-1966 (2003).
[CrossRef]

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

Y. Cai, “Light beams with elliptical flat-topped profiles,” J. Opt. A, Pure Appl. Opt. 6, 390-395 (2004).
[CrossRef]

Y. Cai and Q. Lin, “A partially coherent elliptical flattened Gaussian beam and its propagation,” J. Opt. A, Pure Appl. Opt. 6, 1061-1066 (2004).
[CrossRef]

D. Liu and Z. Zhou, “Various dark hollow beams propagating in uniaxial crystals orthogonal to the optical axis,” J. Opt. A, Pure Appl. Opt. 10, 095005 (2008).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

Opt. Commun. (8)

M. Alavinejad, B. Ghafary, and D. Razzaghi, “Spectral changes of partially coherent flat topped beam in turbulent atmosphere,” Opt. Commun. 281, 2173-2178 (2008).
[CrossRef]

Y. Li, “New expressions for flat-topped beams,” Opt. Commun. 206, 225-234 (2002).
[CrossRef]

Y. Cai, X. Lü, H. Eyyuboğlu, and Y. Baykal, “Paraxial propagation of a partially coherent flattened Gaussian beam through apertured ABCD optical systems,” Opt. Commun. 281, 3221-3229 (2008).
[CrossRef]

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335-341 (1994).
[CrossRef]

A. Ciattoni and C. Palma, “Nondiffracting beams in uniaxial media propagating orthogonally to the optical axis,” Opt. Commun. 224, 175-183 (2003).
[CrossRef]

A. Ciattoni and C. Palma, “Anisotropic beam spreading in uniaxial crystals,” Opt. Commun. 231, 79-92 (2004).
[CrossRef]

D. Deng, H. Yu, S. Xu, J. Shao, and Z. Fan, “Propagation and polarization properties of hollow Gaussian beams in uniaxial crystals,” Opt. Commun. 281, 202-209 (2008).
[CrossRef]

A. Ciattoni, G. Cincotti, and C. Palma, “Ordinary and extraordinary beams characterization in uniaxially anisotropic crystals,” Opt. Commun. 195, 55-61 (2001).
[CrossRef]

Opt. Express (1)

Opt. Laser Technol. (1)

B. Lü and S. Luo, “Propagation properties of three-dimensional flattened Gaussian beams in uniaxially anisotropic crystals,” Opt. Laser Technol. 36, 51-56 (2004).
[CrossRef]

Opt. Lett. (4)

Optik (Jena) (1)

D. Deng, J. Shen, Y. Tian, J. Shao, and Z. Fan, “Propagation properties of beams generated by Gaussian mirror resonator in uniaxial crystals,” Optik (Jena) 118, 547-551 (2007).

Other (1)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1
Fig. 1

Normalized average irradiance distribution I ( x , 0 ) I ( x , 0 ) max of a partially coherent circular flat-topped beam with different N at z = 0 .

Fig. 2
Fig. 2

3D-normalized irradiance distribution and contour graph of a partially coherent circular flat-topped beam in uniaxial crystals at several propagation distances z = ( a ) 1000 , (b) 3000, (c) 6000, and (d) 10,000 μ m .

Fig. 3
Fig. 3

Cross line ( y = 0 ) of the normalized irradiations of a partially coherent circular flat-topped beam in uniaxial crystals with different initial coherent width σ g and the different ratio of extraordinary and ordinary refractive indices at different propagation distances (a) z = 1000 μ m , n e n o = 1.1 , (b) z = 6000 μ m , n e n o = 1.1 , (c) z = 1000 μ m , n e n o = 1.5 , (d) z = 6000 μ m , n e n o = 1.5 .

Fig. 4
Fig. 4

Dependence of the effective beam spot sizes W x and W y of a partially coherent circular flat-topped beam in uniaxial crystals on the propagation distance for different values of the initial coherent width σ g .

Fig. 5
Fig. 5

Dependence of the effective beam spot sizes W x and W y of a partially coherent circular flat-topped beam in uniaxial crystals on the propagation distance for the different ratio of extraordinary and ordinary refractive indices.

Equations (23)

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

E N ( x , y ) = n = 1 N ( 1 ) n 1 N ( N n ) exp ( n ( x 2 w 0 x 2 + y 2 w 0 y 2 ) ) ,
Γ ( x 1 , y 1 , x 2 , y 2 , 0 ) = I ( x 1 , y 1 , 0 ) I ( x 2 , y 2 , 0 ) g ( x 1 x 2 , y 1 y 2 ) ,
g ( x 1 x 2 , y 1 y 2 ) = exp [ ( x 1 x 2 ) 2 2 σ g 2 ( y 1 y 2 ) 2 2 σ g 2 ] ,
Γ ( x 1 , y 1 , x 2 , y 2 , 0 ) = m = 1 N n = 1 N ( 1 ) m + n N 2 ( N m ) ( N n ) exp ( n ( x 1 2 w 0 x 2 + y 1 2 w 0 y 2 ) ) exp ( m ( x 2 2 w 0 x 2 + y 2 2 w 0 y 2 ) ) exp [ ( x 1 x 2 ) 2 2 σ g 2 ( y 1 y 2 ) 2 2 σ g 2 ] ,
E x ( r , z ) = exp ( i k 0 n e z ) d 2 k exp ( i k r i n e 2 k x 2 + n o 2 k y 2 2 k 0 n o 2 n e z ) E ̃ x ( k ) ,
E y ( r , z ) = exp ( i k 0 n o z ) d 2 k exp ( i k r i k x 2 + k y 2 2 k 0 n o z ) E ̃ y ( k ) ,
E ̃ ( k ) = 1 ( 2 π ) 2 d 2 k exp ( i k r ) E ( r , 0 )
E x ( ρ x , ρ y , z ) = k 0 n o 2 π i z exp ( i k 0 n e z ) d x d y exp { k 0 2 i z n e [ n o 2 ( ρ x x ) 2 + n e 2 ( ρ y y ) 2 ] } E x ( x , y , 0 ) ,
E y ( ρ x , ρ y , z ) = k 0 n o 2 π i z exp ( i k 0 n o z ) d x d y exp { k 0 n o 2 i z [ ( ρ x x ) 2 + ( ρ y y ) 2 ] } E y ( x , y , 0 ) .
Γ ( x 1 , y 1 , x 2 , y 2 , 0 ) = E ( x 1 , y 1 , 0 ) E * ( x 2 , y 2 , 0 ) ,
Γ ( ρ 1 x , ρ 1 y , ρ 2 x , ρ 2 y , z ) = E ( ρ 1 x , ρ 1 y , z ) E * ( ρ 2 x , ρ 2 y , z ) ,
Γ ( ρ 1 x , ρ 1 y , ρ 2 x , ρ 2 y , z ) = E ( ρ 1 x , ρ 1 y , z ) E * ( ρ 2 x , ρ 2 y , z ) = k 0 2 n o 2 4 π 2 z 2 Γ ( x 1 , y 1 , x 2 , y 2 , 0 ) exp { k 0 2 i z n e [ n o 2 ( ρ 1 x x 1 ) 2 + n e 2 ( ρ 1 y y 1 ) 2 ] } exp { k 0 2 i z n e [ n o 2 ( ρ 2 x x 2 ) 2 + n e 2 ( ρ 2 y y 2 ) 2 ] } d x 1 d y 1 d x 2 d y 2 .
Γ ( ρ 1 x , ρ 1 y , ρ 2 x , ρ 2 y , z ) = k 0 2 n o 2 4 z 2 m = 1 N n = 1 N ( 1 ) m + n N 2 ( N m ) ( N n ) exp ( i k 0 n o 2 2 z n e ρ 1 x 2 i k 0 n o 2 2 z n e ρ 2 x 2 + i k 0 n e 2 z ρ 1 y 2 i k 0 n e 2 z ρ 2 y 2 ) 1 a x b x a y b y exp ( k 0 2 n o 4 4 a x z 2 n e 2 ρ 1 x 2 k 0 2 n e 2 4 a y z 2 ρ 1 y 2 ) exp [ k 0 2 n o 4 4 b x z 2 n e 2 ( ρ 2 x ρ 1 x 2 a x σ g 2 ) 2 k 0 2 n e 2 4 b y z 2 ( ρ 2 y ρ 1 y 2 a y σ g 2 ) 2 ] ,
a x = n w 0 x 2 + 1 2 σ g 2 + k 0 n o 2 2 i z n e ,
b x = m w 0 x 2 + 1 2 σ g 2 k 0 n o 2 2 i z n e 1 4 a x σ g 4 ,
a y = n w 0 y 2 + 1 2 σ g 2 + k 0 n e 2 i z ,
b y = m w 0 y 2 + 1 2 σ g 2 k 0 n e 2 i z 1 4 a y σ g 4 .
W s = 2 s 2 I ( x , y , z ) d x d y I ( x , y , z ) d x d y ( s = x , y ) ,
F 0 = I ( x , y , z ) d x d y = I ( x , y , z = 0 ) d x d y = π m = 1 N n = 1 N ( 1 ) m + n N 2 ( N m ) ( N n ) w 0 x w 0 y m + n ,
F x = x 2 I ( x , y , z ) d x d y = k 0 2 n o 2 8 z 2 m = 1 N n = 1 N ( 1 ) m + n N 2 ( N m ) ( N n ) π A x 3 2 A y 1 2 a x b x a y b y ,
F y = y 2 I ( x , y , z ) d x d y = k 0 2 n o 2 8 z 2 m = 1 N n = 1 N ( 1 ) m + n N 2 ( N m ) ( N n ) π A x 1 2 A y 3 2 a x b x a y b y ,
A x = k 0 2 n o 4 4 a x z 2 n e 2 + k 0 2 n o 4 4 b x z 2 n e 2 ( 1 1 2 a x σ g 2 ) 2 ,
A y = k 0 2 n e 2 4 a y z 2 + k 0 2 n e 2 4 b y z 2 ( 1 1 2 a y σ g 2 ) 2 .

Metrics