Abstract

Analytical expressions for the three components of nonparaxial propagation of a polarized Lorentz–Gauss beam in uniaxial crystal orthogonal to the optical axis are derived and used to investigate its propagation properties in uniaxial crystal. The influences of the initial beam parameters and the parameters of the uniaxial crystal on the evolution of the beam-intensity distribution in the uniaxial crystal are examined in detail. Results show that the statistical properties of a nonparaxial Lorentz–Gauss beam in a uniaxial crystal orthogonal to the optical axis are closely determined by the initial beam’s parameters and the parameters of the crystal: the beam waist sizes—w0, w0x, and w0y—not only affect the size and shape of the beam profile in uniaxial crystal but also determine the nonparaxial effect of a Lorentz–Gauss beam; the beam profile of a Lorentz–Gauss beam in uniaxial crystal is elongated in the x or y direction, which is determined by the ratio of the extraordinary refractive index to the ordinary refractive index; with increasing deviation of the ratio from unity, the extension of the beam profile augments. The results indicate that uniaxial crystal provides an effective and convenient method for modulating the Lorentz–Gauss beams. Our results may be valuable in some fields, such as optical trapping and nonlinear optics, where a light beam with a special profile and polarization is required.

© 2014 Optical Society of America

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References

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  1. O. E. Gawhary and S. Severini, “Lorentz beam and symmetry properties in paraxial optics,” J. Opt. A 8, 409–414 (2006).
    [CrossRef]
  2. W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. 11, 400–402 (1975).
    [CrossRef]
  3. A. Naqwi and F. Durst, “Focus of diode laser beams a simple mathematical model,” Appl. Opt. 29, 1780–1785 (1990).
    [CrossRef]
  4. Y. Jiang, K. Huang, and X. Lu, “Radiation force of highly focused Lorentz–Gauss beams on a Rayleigh particle,” Opt. Express 19, 9708–9713 (2011).
    [CrossRef]
  5. J. Li, Y. Chen, and S. Xu, “Propagation properties of Lorentz beam in uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 43, 506–514 (2011).
    [CrossRef]
  6. G. Zhou, “The beam propagation factors and the kurtosis parameters of a Lorentz–Gauss beam,” Opt. Laser Technol. 41, 953–955 (2009).
    [CrossRef]
  7. G. Zhou, “Super Lorentz–Gauss modes and their paraxial propagation properties,” J. Opt. Soc. Am. A 27, 563–571 (2010).
    [CrossRef]
  8. G. Zhou, “Propagation of a partially coherent Lorentz–Gauss beam through a paraxial ABCD optical system,” Opt. Express 18, 4637–4643 (2010).
    [CrossRef]
  9. G. Zhou, “Generalized M2 factors of truncated partially coherent Lorentz and Lorentz–Gauss beams,” J. Opt. 12, 1–6 (2010).
  10. G. Zhou, “Nonparaxial propagation of a Lorentz–Gauss beam,” J. Opt. Soc. Am. B 25, 2594–2599 (2008).
    [CrossRef]
  11. G. Zhou, “Average intensity and spreading of a Lorentz–Gauss beam in turbulent atmosphere,” Opt. Express 18, 726–731 (2010).
    [CrossRef]
  12. W. Du and C. Zhao, “Propagation of Lorentz and Lorentz–Gauss beams through an apertured fractional Fourier transform optical system,” Opt. Laser Eng. 49, 25–31 (2011).
    [CrossRef]
  13. G. Zhou, “Beam propagation factors of a Lorentz–Gauss beam,” Appl. Phys. B 96, 149–153 (2009).
    [CrossRef]
  14. G. Zhou, “Analytical vectorial structure of a Lorentz–Gauss beam in the far field,” Appl. Phys. B 93, 891–899 (2008).
    [CrossRef]
  15. G. Zhou, “Focal shift of focused truncated Lorentz–Gauss beam,” J. Opt. Soc. Am. A 25, 2594–2599 (2008).
    [CrossRef]
  16. G. Zhou, “Fractional Fourier transform of Lorentz–Gauss beams,” J. Opt. Soc. Am. A 26, 350–355 (2009).
    [CrossRef]
  17. G. Zhou and R. Chen, “Wigner distribution function of Lorentz and Lorentz–Gauss beams through a paraxial ABCD optical system,” Appl. Phys. B 107, 183–193 (2012).
    [CrossRef]
  18. 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]
  19. D. Liu and Z. Zhou, “Propagation of partially coherent flat-topped beams in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A 26, 924–930 (2009).
    [CrossRef]
  20. D. Liu and Z. Zhou, “Propagation of partially polarized, partially coherent beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 54, 95–101 (2009).
    [CrossRef]
  21. X. Du and D. Zhao, “Propagation of uniformly polarized stochastic electromagnetic beams in uniaxial crystals,” J. Electromagn. Waves Appl. 24, 971–981 (2010).
    [CrossRef]
  22. C. Zhao and Y. Cai, “Paraxial propagation of Lorentz and Lorentz–Gauss beam in uniaxial crystals orthogonal to the optical axis,” J. Mod. Opt. 57, 375–384 (2010).
    [CrossRef]
  23. L. Zhang and Y. Cai, “Statistical properties of a nonparaxial Gaussian Schell-model beam in a uniaxial crystal,” Opt. Express 19, 13312–13325 (2011).
    [CrossRef]
  24. J. Li and Y. Chen, “Propagation of confluent hypergeometric beam through uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 44, 1603–1610 (2012).
    [CrossRef]
  25. D. Deng, C. Chen, X. Zhao, and H. Li, “Propagation of an Airy vortex beam in uniaxial crystals,” Appl. Phys. B 110, 433–436 (2013).
    [CrossRef]
  26. Y. Zhou, X. Wang, C. Dai, X. Chu, and G. Zhou, “Nonparaxial analysis in the propagation of a cylindrical vector Laguerre–Gaussian beam in a uniaxial crystal orthogonal to the optical axis,” Opt. Commun. 305, 113–125 (2013).
    [CrossRef]

2013

D. Deng, C. Chen, X. Zhao, and H. Li, “Propagation of an Airy vortex beam in uniaxial crystals,” Appl. Phys. B 110, 433–436 (2013).
[CrossRef]

Y. Zhou, X. Wang, C. Dai, X. Chu, and G. Zhou, “Nonparaxial analysis in the propagation of a cylindrical vector Laguerre–Gaussian beam in a uniaxial crystal orthogonal to the optical axis,” Opt. Commun. 305, 113–125 (2013).
[CrossRef]

2012

G. Zhou and R. Chen, “Wigner distribution function of Lorentz and Lorentz–Gauss beams through a paraxial ABCD optical system,” Appl. Phys. B 107, 183–193 (2012).
[CrossRef]

J. Li and Y. Chen, “Propagation of confluent hypergeometric beam through uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 44, 1603–1610 (2012).
[CrossRef]

2011

J. Li, Y. Chen, and S. Xu, “Propagation properties of Lorentz beam in uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 43, 506–514 (2011).
[CrossRef]

W. Du and C. Zhao, “Propagation of Lorentz and Lorentz–Gauss beams through an apertured fractional Fourier transform optical system,” Opt. Laser Eng. 49, 25–31 (2011).
[CrossRef]

Y. Jiang, K. Huang, and X. Lu, “Radiation force of highly focused Lorentz–Gauss beams on a Rayleigh particle,” Opt. Express 19, 9708–9713 (2011).
[CrossRef]

L. Zhang and Y. Cai, “Statistical properties of a nonparaxial Gaussian Schell-model beam in a uniaxial crystal,” Opt. Express 19, 13312–13325 (2011).
[CrossRef]

2010

G. Zhou, “Average intensity and spreading of a Lorentz–Gauss beam in turbulent atmosphere,” Opt. Express 18, 726–731 (2010).
[CrossRef]

G. Zhou, “Propagation of a partially coherent Lorentz–Gauss beam through a paraxial ABCD optical system,” Opt. Express 18, 4637–4643 (2010).
[CrossRef]

G. Zhou, “Super Lorentz–Gauss modes and their paraxial propagation properties,” J. Opt. Soc. Am. A 27, 563–571 (2010).
[CrossRef]

G. Zhou, “Generalized M2 factors of truncated partially coherent Lorentz and Lorentz–Gauss beams,” J. Opt. 12, 1–6 (2010).

X. Du and D. Zhao, “Propagation of uniformly polarized stochastic electromagnetic beams in uniaxial crystals,” J. Electromagn. Waves Appl. 24, 971–981 (2010).
[CrossRef]

C. Zhao and Y. Cai, “Paraxial propagation of Lorentz and Lorentz–Gauss beam in uniaxial crystals orthogonal to the optical axis,” J. Mod. Opt. 57, 375–384 (2010).
[CrossRef]

2009

D. Liu and Z. Zhou, “Propagation of partially polarized, partially coherent beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 54, 95–101 (2009).
[CrossRef]

G. Zhou, “Beam propagation factors of a Lorentz–Gauss beam,” Appl. Phys. B 96, 149–153 (2009).
[CrossRef]

G. Zhou, “The beam propagation factors and the kurtosis parameters of a Lorentz–Gauss beam,” Opt. Laser Technol. 41, 953–955 (2009).
[CrossRef]

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

D. Liu and Z. Zhou, “Propagation of partially coherent flat-topped beams in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A 26, 924–930 (2009).
[CrossRef]

2008

G. Zhou, “Focal shift of focused truncated Lorentz–Gauss beam,” J. Opt. Soc. Am. A 25, 2594–2599 (2008).
[CrossRef]

G. Zhou, “Analytical vectorial structure of a Lorentz–Gauss beam in the far field,” Appl. Phys. B 93, 891–899 (2008).
[CrossRef]

G. Zhou, “Nonparaxial propagation of a Lorentz–Gauss beam,” J. Opt. Soc. Am. B 25, 2594–2599 (2008).
[CrossRef]

2006

O. E. Gawhary and S. Severini, “Lorentz beam and symmetry properties in paraxial optics,” J. Opt. A 8, 409–414 (2006).
[CrossRef]

2003

1990

1975

W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. 11, 400–402 (1975).
[CrossRef]

Cai, Y.

L. Zhang and Y. Cai, “Statistical properties of a nonparaxial Gaussian Schell-model beam in a uniaxial crystal,” Opt. Express 19, 13312–13325 (2011).
[CrossRef]

C. Zhao and Y. Cai, “Paraxial propagation of Lorentz and Lorentz–Gauss beam in uniaxial crystals orthogonal to the optical axis,” J. Mod. Opt. 57, 375–384 (2010).
[CrossRef]

Chen, C.

D. Deng, C. Chen, X. Zhao, and H. Li, “Propagation of an Airy vortex beam in uniaxial crystals,” Appl. Phys. B 110, 433–436 (2013).
[CrossRef]

Chen, R.

G. Zhou and R. Chen, “Wigner distribution function of Lorentz and Lorentz–Gauss beams through a paraxial ABCD optical system,” Appl. Phys. B 107, 183–193 (2012).
[CrossRef]

Chen, Y.

J. Li and Y. Chen, “Propagation of confluent hypergeometric beam through uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 44, 1603–1610 (2012).
[CrossRef]

J. Li, Y. Chen, and S. Xu, “Propagation properties of Lorentz beam in uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 43, 506–514 (2011).
[CrossRef]

Chu, X.

Y. Zhou, X. Wang, C. Dai, X. Chu, and G. Zhou, “Nonparaxial analysis in the propagation of a cylindrical vector Laguerre–Gaussian beam in a uniaxial crystal orthogonal to the optical axis,” Opt. Commun. 305, 113–125 (2013).
[CrossRef]

Ciattoni, A.

Dai, C.

Y. Zhou, X. Wang, C. Dai, X. Chu, and G. Zhou, “Nonparaxial analysis in the propagation of a cylindrical vector Laguerre–Gaussian beam in a uniaxial crystal orthogonal to the optical axis,” Opt. Commun. 305, 113–125 (2013).
[CrossRef]

Deng, D.

D. Deng, C. Chen, X. Zhao, and H. Li, “Propagation of an Airy vortex beam in uniaxial crystals,” Appl. Phys. B 110, 433–436 (2013).
[CrossRef]

Du, W.

W. Du and C. Zhao, “Propagation of Lorentz and Lorentz–Gauss beams through an apertured fractional Fourier transform optical system,” Opt. Laser Eng. 49, 25–31 (2011).
[CrossRef]

Du, X.

X. Du and D. Zhao, “Propagation of uniformly polarized stochastic electromagnetic beams in uniaxial crystals,” J. Electromagn. Waves Appl. 24, 971–981 (2010).
[CrossRef]

Dumke, W. P.

W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. 11, 400–402 (1975).
[CrossRef]

Durst, F.

Gawhary, O. E.

O. E. Gawhary and S. Severini, “Lorentz beam and symmetry properties in paraxial optics,” J. Opt. A 8, 409–414 (2006).
[CrossRef]

Huang, K.

Jiang, Y.

Li, H.

D. Deng, C. Chen, X. Zhao, and H. Li, “Propagation of an Airy vortex beam in uniaxial crystals,” Appl. Phys. B 110, 433–436 (2013).
[CrossRef]

Li, J.

J. Li and Y. Chen, “Propagation of confluent hypergeometric beam through uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 44, 1603–1610 (2012).
[CrossRef]

J. Li, Y. Chen, and S. Xu, “Propagation properties of Lorentz beam in uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 43, 506–514 (2011).
[CrossRef]

Liu, D.

D. Liu and Z. Zhou, “Propagation of partially coherent flat-topped beams in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A 26, 924–930 (2009).
[CrossRef]

D. Liu and Z. Zhou, “Propagation of partially polarized, partially coherent beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 54, 95–101 (2009).
[CrossRef]

Lu, X.

Naqwi, A.

Palma, C.

Severini, S.

O. E. Gawhary and S. Severini, “Lorentz beam and symmetry properties in paraxial optics,” J. Opt. A 8, 409–414 (2006).
[CrossRef]

Wang, X.

Y. Zhou, X. Wang, C. Dai, X. Chu, and G. Zhou, “Nonparaxial analysis in the propagation of a cylindrical vector Laguerre–Gaussian beam in a uniaxial crystal orthogonal to the optical axis,” Opt. Commun. 305, 113–125 (2013).
[CrossRef]

Xu, S.

J. Li, Y. Chen, and S. Xu, “Propagation properties of Lorentz beam in uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 43, 506–514 (2011).
[CrossRef]

Zhang, L.

Zhao, C.

W. Du and C. Zhao, “Propagation of Lorentz and Lorentz–Gauss beams through an apertured fractional Fourier transform optical system,” Opt. Laser Eng. 49, 25–31 (2011).
[CrossRef]

C. Zhao and Y. Cai, “Paraxial propagation of Lorentz and Lorentz–Gauss beam in uniaxial crystals orthogonal to the optical axis,” J. Mod. Opt. 57, 375–384 (2010).
[CrossRef]

Zhao, D.

X. Du and D. Zhao, “Propagation of uniformly polarized stochastic electromagnetic beams in uniaxial crystals,” J. Electromagn. Waves Appl. 24, 971–981 (2010).
[CrossRef]

Zhao, X.

D. Deng, C. Chen, X. Zhao, and H. Li, “Propagation of an Airy vortex beam in uniaxial crystals,” Appl. Phys. B 110, 433–436 (2013).
[CrossRef]

Zhou, G.

Y. Zhou, X. Wang, C. Dai, X. Chu, and G. Zhou, “Nonparaxial analysis in the propagation of a cylindrical vector Laguerre–Gaussian beam in a uniaxial crystal orthogonal to the optical axis,” Opt. Commun. 305, 113–125 (2013).
[CrossRef]

G. Zhou and R. Chen, “Wigner distribution function of Lorentz and Lorentz–Gauss beams through a paraxial ABCD optical system,” Appl. Phys. B 107, 183–193 (2012).
[CrossRef]

G. Zhou, “Super Lorentz–Gauss modes and their paraxial propagation properties,” J. Opt. Soc. Am. A 27, 563–571 (2010).
[CrossRef]

G. Zhou, “Generalized M2 factors of truncated partially coherent Lorentz and Lorentz–Gauss beams,” J. Opt. 12, 1–6 (2010).

G. Zhou, “Average intensity and spreading of a Lorentz–Gauss beam in turbulent atmosphere,” Opt. Express 18, 726–731 (2010).
[CrossRef]

G. Zhou, “Propagation of a partially coherent Lorentz–Gauss beam through a paraxial ABCD optical system,” Opt. Express 18, 4637–4643 (2010).
[CrossRef]

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

G. Zhou, “Beam propagation factors of a Lorentz–Gauss beam,” Appl. Phys. B 96, 149–153 (2009).
[CrossRef]

G. Zhou, “The beam propagation factors and the kurtosis parameters of a Lorentz–Gauss beam,” Opt. Laser Technol. 41, 953–955 (2009).
[CrossRef]

G. Zhou, “Analytical vectorial structure of a Lorentz–Gauss beam in the far field,” Appl. Phys. B 93, 891–899 (2008).
[CrossRef]

G. Zhou, “Nonparaxial propagation of a Lorentz–Gauss beam,” J. Opt. Soc. Am. B 25, 2594–2599 (2008).
[CrossRef]

G. Zhou, “Focal shift of focused truncated Lorentz–Gauss beam,” J. Opt. Soc. Am. A 25, 2594–2599 (2008).
[CrossRef]

Zhou, Y.

Y. Zhou, X. Wang, C. Dai, X. Chu, and G. Zhou, “Nonparaxial analysis in the propagation of a cylindrical vector Laguerre–Gaussian beam in a uniaxial crystal orthogonal to the optical axis,” Opt. Commun. 305, 113–125 (2013).
[CrossRef]

Zhou, Z.

D. Liu and Z. Zhou, “Propagation of partially coherent flat-topped beams in uniaxial crystals orthogonal to the optical axis,” J. Opt. Soc. Am. A 26, 924–930 (2009).
[CrossRef]

D. Liu and Z. Zhou, “Propagation of partially polarized, partially coherent beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 54, 95–101 (2009).
[CrossRef]

Appl. Opt.

Appl. Phys. B

G. Zhou and R. Chen, “Wigner distribution function of Lorentz and Lorentz–Gauss beams through a paraxial ABCD optical system,” Appl. Phys. B 107, 183–193 (2012).
[CrossRef]

D. Deng, C. Chen, X. Zhao, and H. Li, “Propagation of an Airy vortex beam in uniaxial crystals,” Appl. Phys. B 110, 433–436 (2013).
[CrossRef]

G. Zhou, “Beam propagation factors of a Lorentz–Gauss beam,” Appl. Phys. B 96, 149–153 (2009).
[CrossRef]

G. Zhou, “Analytical vectorial structure of a Lorentz–Gauss beam in the far field,” Appl. Phys. B 93, 891–899 (2008).
[CrossRef]

Eur. Phys. J. D

D. Liu and Z. Zhou, “Propagation of partially polarized, partially coherent beams in uniaxial crystals orthogonal to the optical axis,” Eur. Phys. J. D 54, 95–101 (2009).
[CrossRef]

IEEE J. Quantum Electron.

W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. 11, 400–402 (1975).
[CrossRef]

J. Electromagn. Waves Appl.

X. Du and D. Zhao, “Propagation of uniformly polarized stochastic electromagnetic beams in uniaxial crystals,” J. Electromagn. Waves Appl. 24, 971–981 (2010).
[CrossRef]

J. Mod. Opt.

C. Zhao and Y. Cai, “Paraxial propagation of Lorentz and Lorentz–Gauss beam in uniaxial crystals orthogonal to the optical axis,” J. Mod. Opt. 57, 375–384 (2010).
[CrossRef]

J. Opt.

G. Zhou, “Generalized M2 factors of truncated partially coherent Lorentz and Lorentz–Gauss beams,” J. Opt. 12, 1–6 (2010).

J. Opt. A

O. E. Gawhary and S. Severini, “Lorentz beam and symmetry properties in paraxial optics,” J. Opt. A 8, 409–414 (2006).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

G. Zhou, “Nonparaxial propagation of a Lorentz–Gauss beam,” J. Opt. Soc. Am. B 25, 2594–2599 (2008).
[CrossRef]

Opt. Commun.

Y. Zhou, X. Wang, C. Dai, X. Chu, and G. Zhou, “Nonparaxial analysis in the propagation of a cylindrical vector Laguerre–Gaussian beam in a uniaxial crystal orthogonal to the optical axis,” Opt. Commun. 305, 113–125 (2013).
[CrossRef]

Opt. Express

Opt. Laser Eng.

W. Du and C. Zhao, “Propagation of Lorentz and Lorentz–Gauss beams through an apertured fractional Fourier transform optical system,” Opt. Laser Eng. 49, 25–31 (2011).
[CrossRef]

Opt. Laser Technol.

J. Li, Y. Chen, and S. Xu, “Propagation properties of Lorentz beam in uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 43, 506–514 (2011).
[CrossRef]

G. Zhou, “The beam propagation factors and the kurtosis parameters of a Lorentz–Gauss beam,” Opt. Laser Technol. 41, 953–955 (2009).
[CrossRef]

J. Li and Y. Chen, “Propagation of confluent hypergeometric beam through uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. 44, 1603–1610 (2012).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Geometry of the propagation of a laser beam in a uniaxial crystal orthogonal to the optical axis.

Fig. 2.
Fig. 2.

Contour graph of the intensity of the x component of a Lorentz–Gauss beam in several observation planes in the uniaxial crystal: e=1.5 (a) z=0.1zr, (b) z=0.5zr, (c) z=zr, and (d) z=3zr.

Fig. 3.
Fig. 3.

Contour graph of the intensity of the y component of a Lorentz–Gauss beam in several observation planes in the uniaxial crystal: e=1.5 (a) z=0.1zr, (b) z=0.5zr, (c) z=zr, and (d) z=3zr.

Fig. 4.
Fig. 4.

Contour graph of the intensity of the longitudinal component of a Lorentz–Gauss beam in several observation planes in the uniaxial crystal: e=1.5 (a) z=0.1zr, (b) z=0.5zr, (c) z=zr, and (d) z=3zr.

Fig. 5.
Fig. 5.

Contour graph of the intensity of a Lorentz–Gauss beam in several observation planes in the uniaxial crystal: e=1.5 (a) z=0.1zr, (b) z=0.5zr, (c) z=zr, and (d) z=3zr.

Fig. 6.
Fig. 6.

Contour graph of the intensity of a Lorentz–Gauss beam in the observation plane z=zr of different uniaxial crystal: (a) e=0.6, (b) e=0.8, (c) e=1, and (d) e=1.5.

Fig. 7.
Fig. 7.

Contour graph of the intensity of Lorentz–Gauss beams of different waist widths in the observation plane z=zr, e=1.5: (a) w0=w0x=w0y=0.5λ, (b) w0=1λ, w0x=w0y=0.5λ, (c) w0=1λ, w0x=0.5λ, w0y=1.5λ, (d) w0=1λ, w0x=1.5λ, w0y=0.5λ, (e) w0=5λ, w0x=0.5λ, w0y=1.5λ, and (f) w0=5λ, w0x=1.5λ, w0y=0.5λ.

Equations (34)

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ε=(ne2000no2000no2),
(Ex(r0,0)Ey(r0,0))=(Ex(x0,y0,0)Ey(x0,y0,0))=(E0exp(x02+y02,w02)w0xw0y[1+(x0/w0x)2][1+(y0/w0y)2]0),
E(x,y,z)=d2kexp(ikr)exp(ikezz)(E˜x(k)kxkyk02no2kx2E˜x(k)kezkxk02no2kx2E˜x(k))+d2kexp(ikr)exp(ikozz)(0[kxkyk02no2kx2E˜x(k)+E˜y(k)]kykoz[kxkyk02no2kx2E˜x(k)+E˜y(k)]),
E˜α(k)=1(2π)2d2r0exp(ik·r)Eα(r0,0)(α=x,y),
koz=(k02no2kx2)1/2,kez=[k02no2(ne2/no2)kx2ky2]1/2.
E(x,y,z)=exp(ik0nez)d2kexp(ik·r)exp(ine2kx2+no2ky22k0neno2z)(E˜x(k)kxkyk02no2E˜x(k)nekxk0no2E˜x(k))+exp(ik0noz)d2kexp(ik·r)exp(ikx2+ky22k0noz)(0[kxkyk02no2E˜x(k)+E˜y(k)]kykonoE˜y(k)).
Ex(x,y,z)=k0no2πizEx(x0,y0,0)Πe(r,r0,0)dx0dy0,
Ey(x,y,z)=ik0no2πz3(xx0)(yy0)Ex(x0,y0,0)×[Πe(r,ro)Πo(r,ro)]dx0dy0+k0no2πizEy(x0,y0,0)Πo(r,r0)dx0dy0,
Ez(x,y,z)=ik0no2πz2[(xx0)Ex(x0,y0,0)×Πe(r,ro)+(yy0)Ey(x0,y0,0)×Πo(r,r0)]dx0dy0,
Πe(r,r0)=exp(ik0nez)exp{k02izne[no2(xx0)2+ne2(yy0)2]},
Πo(r,r0)=exp(ik0noz)exp{k0no2iz[(xx0)2+(yy0)2]}.
Ex(x,y,z)=k0noE02πizexp(ik0nez)Γx(x,z)Γx(y,z),
Γx(x,z)=w0x(w0x2+x02)exp(x02w02)×exp[k0no22izne(xx0)2]dx0,
Γx(y,z)=w0y(w0y2+y02)exp(y02w02)×exp[k0no22izne(yy0)2]dy0.
Γx(x,z)=exp(k0no22iznex2)exp(k0no22iznex2A)×w0x(w0x2+x02)exp[k0no22izneA(x0xA)2]dx0,
Γx(y,z)=exp(k0ne2izy2)exp(k0ne2izy2B)×w0y(w0y2+y02)exp[k0ne2izB(y0yB)2]dy0,
A=1+2iznek0no2w02,B=1+2izk0new02.
f1(τ)*f2(τ)=f1(η)f2(τη)dη.
Γx(x,z)=w0xexp(k0no22iznex2)exp(k0no22iznex2A)[f1(xA)*f2(xA)],
Γx(y,z)=w0yexp(k0ne2izy2)exp(k0ne2izy2B)[f1(yB)*f2(yB)],
f1(τ)=1(w0x2+τ2),f2(τ)=exp(k0no2A2izneτ2),
f1(τ)=1(w0y2+τ2),f2(τ)=exp(k0neB2izτ2).
Ex(x,y,z)=π24k0noE02πizexp(ik0nez)exp[k02izne(no2x2+ne2y2)]×exp[k02izne(no2Ax2+ne2By2)]×[Vx++Vx][Vy++Vy],
Vx±=exp[k0no22izneA(w0x±ixA)2][1erf[k0no2A2izne(w0x±ixA)]],
Vy±=exp[k0ne2izB(w0y±iyB)2][1erf[k0neB2iz(w0y±iyB)]],
erf(x)=2π0xexp(s2)ds.
Ey(x,y,z)=ik0no2πz3(xx0)(yy0)Ex(x0,y0,0)×[Πe(r,ro)Πo(r,ro)]dx0dy0=ik0no2πz3exp(ik0nez)(xx0)(yy0)Γx(x,z)Γx(y,z)ik0no2πz3exp(ik0noz)(xx0)(yy0)Γy(x,z)Γy(y,z),
Ez(x,y,z)=ik0no2πz2(xx0)Ex(x0,y0,0)×Πe(r,ro)dx0dy0=ik0no2πz2exp(ik0nez)(xx0)Γx(x,z)Γx(y,z).
Γy(α,z)=w0α(w0α2+α02)exp(α02w02)×exp[k0no2iz(αα0)2]dα0=exp(k0no2izα2)exp(k0no2izα2C)×w0α(w0α2+α02)exp[k0noC2iz(α0αC)2]dα0=w0αexp(k0no2izα2)exp(k0no2izα2C)[f1(αC)*f2(αC)],
α=x,y,C=1+2izk0now02.
Ey(x,y,z)=π24ik0no2πz3{exp(ik0nez)exp[k02izne(no2x2+ne2y2)]×exp[k02izne(no2Ax2+ne2By2)][(xiw0x)Vx++(x+iw0x)Vx]×[(yiw0y)Vy++(y+iw0y)Vy]exp(ik0noz)exp[k0no2iz(x2+y2)]exp[k0no2izC(x2+y2)]×[(xiw0x)Ux++(x+iw0x)Ux][(yiw0y)Uy++(y+iw0y)Uy]},
Ez(x,y,z)=π24ik0noE02πz2exp(ik0nez)exp[k02izne(no2x2+ne2y2)]×exp[k02izne(no2Ax2+ne2By2)][(xiw0x)Vx++(x+iw0x)Vx]×(Vy++Vy),
Ux±=exp[k0no2izC(w0x±ixC)2][1erf[k0noC2iz(w0x±ixC)]],
Uy±=exp[k0no2izC(w0y±iyC)2][1erf[k0noC2iz(w0y±iyC)]].

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