volume | PIER Journals

Abstract

Analytical expressions of the elements of a cross spectral density matrix are derived to describe the partially coherent Lorentz-Gauss beam propagating in uniaxial crystals orthogonal to the x-axis. The intensity and degree of polarization for the partially coherent Lorentz-Gauss beam propagating in uniaxial crystals orthogonal to the x-axis are also presented. The evolution properties of the partially coherent Lorentz-Gauss beam are numerically demonstrated. The influences of the uniaxial crystal and coherence length on the propagation properties of the partially coherent Lorentz-Gauss beam in uniaxial crystals orthogonal to the x-axis are examined. The uniaxial crystal considered here has the property of the extraordinary refractive index being larger than the ordinary refractive index. The partially coherent Lorentz-Gauss beam in the direction along the x-axis spreads more rapidly than that in the direction along the y-axis. With increasing the ratio of the extraordinary refractive index to the ordinary refractive index, the spreading of the partially coherent Lorentz-Gauss beam increases in the direction along the x-axis, but decreases in the direction along the y-axis. Meanwhile, the degree of polarization in the edges of the long and short axes of the beam spot increases. With increasing the coherence length, the beam spot of the partially coherent Lorentz-Gauss beam uniformly becomes less, and the maximum degree of polarization in the edge of the beam spot decreases.

1. Naqwi, A. and F. Durst, "Focus of diode laser beams: A simple mathematical model," Appl. Opt., Vol. 29, 1780-1785, 1990.
doi:10.1364/AO.29.001780

2. Yang, J., T. Chen, G. Ding, and X. Yuan, "Focusing of diode laser beams: A partially coherent Lorentz model," Proc. SPIE, Vol. 6824, 68240A, 2008.
doi:10.1117/12.757962

3. Gawhary, O. E. and S. Severini, "Lorentz beams and symmetry properties in paraxial optics," J. Opt. A: Pure Appl. Opt., Vol. 8, 409-414, 2006.
doi:10.1088/1464-4258/8/5/007

4. Zhou, G., "Focal shift of focused truncated Lorentz-Gauss beam," J. Opt. Soc. Am. A, Vol. 25, 2594-2599, 2008.
doi:10.1364/JOSAA.25.002594

5. Zhou, G., "Beam propagation factors of a Lorentz-Gauss beam," Appl. Phys. B, Vol. 96, 149-153, 2009.
doi:10.1007/s00340-009-3460-9

6. Torre, A., "Wigner distribution function of Lorentz-Gauss beams: A note," Appl. Phys. B, Vol. 109, 671-681, 2012.
doi:10.1007/s00340-012-5236-x

7. Zhou, G. and R. Chen, "Wigner distribution function of Lorentz and Lorentz-Gauss beams through a paraxial ABCD optical system," Appl. Phys. B, Vol. 107, 183-193, 2012.
doi:10.1007/s00340-012-4889-9

8. Zhao, C. and Y. Cai, "Paraxial propagation of Lorentz and Lorentz-Gauss beams in uniaxial crystals orthogonal to the optical axis," J. Mod. Opt., Vol. 57, 375-384, 2010.
doi:10.1080/09500341003640079

9. Wang, X., Z. Liu, and D. Zhao, "Nonparaxial propagation of Lorentz-Gauss beams in uniaxial crystal orthogonal to the optical axis," J. Opt. Soc. Am. A, Vol. 31, 872-878, 2014.
doi:10.1364/JOSAA.31.000872

10. Zhou, G., "Fractional Fourier transform of Lorentz-Gauss beams," J. Opt. Soc. Am. A, Vol. 26, 350-355, 2009.
doi:10.1364/JOSAA.26.000350

11. Du, W., C. Zhao, and Y. Cai, "Propagation of Lorentz and Lorentz-Gauss beams through an apertured fractional Fourier transform optical system," Opt. Lasers in Eng., Vol. 49, 25-31, 2011.
doi:10.1016/j.optlaseng.2010.09.004

12. Zhou, G. and X. Chu, "Average intensity and spreading of a Lorentz-Gauss beam in turbulent atmosphere," Opt. Express, Vol. 18, 726-731, 2010.
doi:10.1364/OE.18.000726

13. Zheng, H., R. Chen, and C. H. R. Ooi, "Self-focusing dynamics of Lorentz-Gaussian beams in Kerr media," Lasers Eng., Vol. 24, 345-354, 2013.

14. Keshavarz, A. and G. Honarasa, "Propagation of Lorentz-Gaussian beams in strongly nonlocal nonlinear media," Commun. Theor. Phys., Vol. 61, 241-245, 2014.
doi:10.1088/0253-6102/61/2/16

15. Saraswathi, R. C., K. Prabakaran, K. B. Rajesh, and Z. Jaroszewicz, "Tight focusing properties of radially polarized Lorentz-Gaussian beam," Optik, Vol. 125, 5339-5342, 2014.
doi:10.1016/j.ijleo.2014.06.058

16. Sun, Q., A. Li, K. Zhou, Z. Liu, G. Fang, and S. Liu, "Virtual source for rotational symmetric Lorentz-Gaussian beam," Chin. Opt. Lett., Vol. 10, 062601, 2012.
doi:10.3788/COL201210.062601

17. Jiang, Y., K. Huang, and X. Lu, "Radiation force of highly focused Lorentz-Gauss beams on a Rayleigh particle," Opt. Express, Vol. 19, 9708-9713, 2011.
doi:10.1364/OE.19.009708

18. Mandel, L. and E. Wolf, Optical Coherence and Quantum Optics, Cambridge University Press, Cambridge, 1995.
doi:10.1017/CBO9781139644105

19. Zhou, G., "Propagation of a partially coherent Lorentz-Gauss beam through a paraxial ABCD optical system," Opt. Express, Vol. 18, 4637-4643, 2010.
doi:10.1364/OE.18.004637

20. Zhao, C. and Y. Cai, "Propagation of partially coherent Lorentz and Lorentz-Gauss beams through a paraxial ABCD optical system in a turbulent atmosphere," J. Mod. Opt., Vol. 58, 810-818, 2011.
doi:10.1080/09500340.2011.573591

21. Eyyuboğlu, H. T., "Partially coherent Lorentz-Gaussian beam and its scintillations," Appl. Phys. B, Vol. 103, 755-762, 2011.
doi:10.1007/s00340-011-4414-6

22. Zhou, G. and X. Chu, "M²-factor of a partially coherent Lorentz-Gauss beam in a turbulent atmosphere," Appl. Phys. B, Vol. 100, 909-915, 2010.
doi:10.1007/s00340-010-4046-2

23. Xu, Y. and G. Zhou, "Kurtosis parameter of partially coherent Lorentz-Gauss beam in turbulent atmosphere," High Power Laser and Particle Beams, Vol. 22, 1187-1191, 2010.
doi:10.3788/HPLPB20102206.1187

24. Zhou, G., "Generalized M²-factors of truncated partially coherent Lorentz and Lorentz-Gauss beams," J. Opt., Vol. 12, 015701, 2010.
doi:10.1088/2040-8978/12/1/015701

25. Ciattoni, A., G. Cincotti, and C. Palma, "Propagation of cylindrically symmetric fields in uniaxial crystals," J. Opt. Soc. Am. A, Vol. 19, 792-796, 2002.
doi:10.1364/JOSAA.19.000792

26. Lü, B. and S. Luo, "Propagation properties of three-dimensional flatted Gaussian beams in uniaxially anisotropic crystals," Opt. Laser Technol., Vol. 36, 51-56, 2004.
doi:10.1016/S0030-3992(03)00132-4

27. Liu, D. and Z. Zhou, "Various dark hollow beams propagating in uniaxial crystals orthogonal to the optical axis," J. Opt. A: Pure Appl. Opt., Vol. 10, 095005, 2008.
doi:10.1088/1464-4258/10/9/095005

28. Tang, B., "Hermite-cosine-Gaussian beams propagating in uniaxial crystals orthogonal to the optical axis," J. Opt. Soc. Am. A, Vol. 26, 2480-2487, 2009.
doi:10.1364/JOSAA.26.002480

29. Li, J., Y. Chen, Y. Xin, and S. Xu, "Propagation of higher-order cosh-Gaussian beams in uniaxial crystals orthogonal to the optical axis," Eur. Phys. J. D, Vol. 57, 419-425, 2010.
doi:10.1140/epjd/e2010-00068-9

30. Zhou, G., R. Chen, and X. Chu, "Propagation of Airy beams in uniaxial crystals orthogonal to the optical axis," Opt. Express, Vol. 20, 2196-2205, 2012.
doi:10.1364/OE.20.002196

31. Khonina, S. N., A. A. Morozov, and S. V. Karpeev, "Effective transformation of a zero-order Bessel beam into a second-order vortex beam using a uniaxial crystal," Laser Phys., Vol. 24, 056101, 2014.
doi:10.1088/1054-660X/24/5/056101

32. Yumashev, K. V. and P. A. Loiko, "Depolarization in c-cut tetragonal laser crystals," Laser Phys., Vol. 26, 015002, 2016.
doi:10.1088/1054-660X/26/1/015002

33. Ghaffar, A. and Q. A. Naqvi, "Focusing of electromagnetic plane wave into uniaxial crystal by a three dimensional plano convex lens," Progress In Electromagnetics Research, Vol. 83, 25-42, 2008.
doi:10.2528/PIER08041404

34. Hennani, S., L. Ez-Zariy, and A. Belafhal, "Transformation of finite Olver-Gaussian beams by an uniaxial crystal orthogonal to the optical axis," Progress In Electromagnetics Research M, Vol. 45, 153-161, 2016.
doi:10.2528/PIERM15102402

35. Wolf, E., "Unified theory of coherence and polarization of random electromagnetic beams," Phys. Lett. A, Vol. 312, 263-267, 2003.
doi:10.1016/S0375-9601(03)00684-4

36. Schmidt, P. P., "A method for the convolution of lineshapes which involve the Lorentz distribution," J. Phys. B: At. Mol. Opt. Phys., Vol. 9, 2331-2339, 1976.
doi:10.1088/0022-3700/9/13/018

37. Ciattoni, A. and C. Palma, "Optical propagation in uniaxial crystals orthogonal to the optical axis: Paraxial theory and beyond," J. Opt. Soc. Am. A, Vol. 20, 2163-2171, 2003.
doi:10.1364/JOSAA.20.002163

38. Born, M. and E. Wolf, Principles of Optics, 7th Ed., Pergamon, Oxford, 1999.
doi:10.1017/CBO9781139644181

39. Gradshteyn, I. S. and I.M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, New York, 1980.

Dr. Guoquan Zhou

Dr. Zhiyue Ji

Dr. Guoyun Ru