## Propagation of a radial phased-locked Lorentz beam array in turbulent atmosphere |

Optics Express, Vol. 19, Issue 24, pp. 24699-24711 (2011)

http://dx.doi.org/10.1364/OE.19.024699

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### Abstract

A radial phased-locked (PL) Lorentz beam array provides an appropriate theoretical model to describe a coherent diode laser array, which is an efficient radiation source for high-power beaming use. The propagation of a radial PL Lorentz beam array in turbulent atmosphere is investigated. Based on the extended Huygens-Fresnel integral and some mathematical techniques, analytical formulae for the average intensity and the effective beam size of a radial PL Lorentz beam array are derived in turbulent atmosphere. The average intensity distribution and the spreading properties of a radial PL Lorentz beam array in turbulent atmosphere are numerically calculated. The influences of the beam parameters and the structure constant of the atmospheric turbulence on the propagation of a radial PL Lorentz beam array in turbulent atmosphere are discussed in detail.

© 2011 OSA

## 1. Introduction

1. A. Naqwi and F. Durst, “Focusing of diode laser beams: a simple mathematical model,” Appl. Opt. **29**(12), 1780–1785 (1990). [CrossRef] [PubMed]

1. A. Naqwi and F. Durst, “Focusing of diode laser beams: a simple mathematical model,” Appl. Opt. **29**(12), 1780–1785 (1990). [CrossRef] [PubMed]

3. W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” J. Quantum Electron. **11**(7), 400–402 (1975). [CrossRef]

4. O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. **8**(5), 409–414 (2006). [CrossRef]

5. G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. **55**(6), 993–1002 (2008). [CrossRef]

6. G. Zhou, “Propagation of vectorial Lorentz beam beyond the paraxial approximation,” J. Mod. Opt. **55**(21), 3573–3579 (2008). [CrossRef]

7. G. Zhou, “The beam propagation factors and the kurtosis parameters of a Lorentz beam,” Opt. Laser Technol. **41**(8), 953–955 (2009). [CrossRef]

8. G. Zhou, “Generalized *M*^{2} factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. **12**(1), 015701 (2010). [CrossRef]

9. C. Zhao and Y. Cai, “Paraxial propagation of Lorentz and Lorentz-Gauss beams in uniaxial crystals orthogonal to the optical axis,” J. Mod. Opt. **57**(5), 375–384 (2010). [CrossRef]

10. J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Propagation properties of Lorentz beam in uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. **43**(3), 506–514 (2011). [CrossRef]

11. O. El Gawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangular symmetric optical fields,” Opt. Commun. **269**(2), 274–284 (2007). [CrossRef]

12. A. Torre, W. A. B. Evans, O. E. Gawhary, and S. Severini, “Relativistic Hermite polynomials and Lorentz beams,” J. Opt. A, Pure Appl. Opt. **10**(11), 115007 (2008). [CrossRef]

13. H. T. Eyyuboğlu and Y. Baykal, “Hermite-sine-Gaussian and Hermite-sinh-Gaussian laser beams in turbulent atmosphere,” J. Opt. Soc. Am. A **22**(12), 2709–2718 (2005). [CrossRef] [PubMed]

*ABCD*optical system has been also examined in a turbulent atmosphere [19

19. C. Zhao 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. **58**(9), 810–818 (2011). [CrossRef]

20. J. P. Hobimer, D. R. Myers, T. M. Brennan, and B. E. Hammons, “Integrated injection-locked high-power cw diode laser arrays,” Appl. Phys. Lett. **55**(6), 531–533 (1989). [CrossRef]

24. X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. **282**(10), 1993–1997 (2009). [CrossRef]

25. J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field radiation of coherent and incoherent combined Lorentz-Gaussian laser array,” Proc. SPIE **7749**, 77491Z, 77491Z-6 (2010). [CrossRef]

25. J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field radiation of coherent and incoherent combined Lorentz-Gaussian laser array,” Proc. SPIE **7749**, 77491Z, 77491Z-6 (2010). [CrossRef]

26. P. Zhou, X. Wang, Y. Ma, and Z. Liu, “Propagation properties of a Lorentz beam array,” Appl. Opt. **49**(13), 2497–2503 (2010). [CrossRef]

27. P. Zhou, X. Wang, Y. Ma, H. Ma, and Z. Liu, “Irradiance tailoring by fractional Fourier transform of a radial Gaussian beam array,” Proc. SPIE **7822**, 78220J, 78220J-6 (2010). [CrossRef]

28. H. Tang, B. Ou, B. Luo, H. Guo, and A. Dang, “Average spreading of a radial Gaussian beam array in non-Kolmogorov turbulence,” J. Opt. Soc. Am. A **28**(6), 1016–1021 (2011). [CrossRef] [PubMed]

29. Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express **16**(22), 18437–18442 (2008). [CrossRef] [PubMed]

32. X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express **18**(7), 6922–6928 (2010). [CrossRef] [PubMed]

## 2. Propagation of a radial phase-locked Lorentz beam array in turbulent atmosphere

*z*-axis is taken to be the propagation axis. The scheme of a radial Lorentz beam array in the source plane is shown in Fig. 1 . The radial Lorentz beam array considered here is phase-locked (PL).

33. P. P. Schmidt, “A method for the convolution of lineshapes which involve the Lorentz distribution,” J. Phys. B **9**(13), 2331–2339 (1976). [CrossRef]

*M*is the term number of the expansion.

*m*

_{1}-th and 2

*m*

_{2}-th order Hermite polynomials, respectively.

33. P. P. Schmidt, “A method for the convolution of lineshapes which involve the Lorentz distribution,” J. Phys. B **9**(13), 2331–2339 (1976). [CrossRef]

*m*, however, the value of the weight coefficient

*σ*

_{2}

*dramatically decays. In the practical calculations, therefore,*

_{m}*M*takes a small integer. For an example, the original expression of the Lorentz beam in the source plane well coincides with the expansion in the Hermite-Gaussian functions with

*M*= 5.

34. S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. **69**(9), 1297–1304 (1979). [CrossRef]

36. K. Zhu, G. Zhou, X. Li, X. Zheng, and H. Tang, “Propagation of Bessel-Gaussian beams with optical vortices in turbulent atmosphere,” Opt. Express **16**(26), 21315–21320 (2008). [CrossRef] [PubMed]

**=**

*r**x*

**e**

*+*

_{x}*y*

**e**

*. (*

_{y}**,**

*r**z*) is the observation plane.

*ψ*(

*r*_{0},

**) is the solution to the Rytov method that represents the random part of the complex phase.**

*r**k*= 2

*π*/

*λ*is the wave number.

*λ*is the optical wavelength. The average intensity of the radial PL Lorentz beam array in the observation plane is given by

34. S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. **69**(9), 1297–1304 (1979). [CrossRef]

*C*

_{n}^{2}is the structure constant of the atmospheric turbulence. In Eq. (8), the quadratic structure functions are employed. Substituting Eqs. (4) and (8) into Eq. (7) and using the following mathematical formulae [37]: we can obtain the analytical average intensity of the radial PL Lorentz beam array in the observation plane:with

*j*=

*x*or

*y*in the all locations including the subscript. [(

*l*

_{2}+

*l*

_{4})/2] gives the greatest integer less than or equal to (

*l*

_{2}+

*l*

_{4})/2, and the auxiliary parameters are defined as follows:

*x*- and

*y*-directions of the observation plane is defined as [38

38. W. H. Carter, “Spot size and divergence for Hermite Gaussian beams of any order,” Appl. Opt. **19**(7), 1027–1029 (1980). [CrossRef] [PubMed]

_{1}

*F*

_{1}(⋅)is a Kummer function and Γ(⋅) is a Gamma function. The auxiliary parameters are defined as

## 3. Numerical calculations and analyses

*λ*= 0.8μm.

*w*

_{0}

*=*

_{x}*w*

_{0}

*results in the symmetry of the beam spot, which is the familiar case. Therefore, we consider*

_{y}*w*

_{0}

*being equal to*

_{x}*w*

_{0}

*.*

_{y}*C*

_{n}^{2}= 10

^{−14}m

^{-2/3},

*w*

_{0}

*=*

_{x}*w*

_{0}

*= 2mm,*

_{y}*R*= 3mm, and

*N*= 6 in Fig. 2. In the near-field such as

*z*, the central dark region disappears, and the on-axis intensity increases from zero to the maximum value. Moreover, the normalized intensity distribution in the central region tends to be uniform. In Fig. 3 ,

*C*

_{n}^{2}= 10

^{−15}m

^{-2/3}and the rest of parameters are same as those in Fig. 2. When the structure constant of the atmospheric turbulence decreases, the propagation distance where the central dark region disappears apparently increases, which can be interpreted as follow. The coherence of the radial PL Lorentz beam array will be damaged by the atmospheric turbulence. If the structure constant of the atmospheric turbulence decreases, the extent of damage of the coherence of the radial PL Lorentz beam array also decreases. In the source plane, the beam spot of the radial PL Lorentz beam array takes on a central dark distribution. The better the coherence is, the longer the axial propagation distance within which the beam spot doesn’t distort is. With decreasing the structure constant of the atmospheric turbulence, therefore, the propagation distance where the central dark region disappears increases.

*w*

_{0}

*=*

_{x}*w*

_{0}

*= 1mm in Fig. 4 , and the other parameters are same as those in Fig. 2. In the near-field such as*

_{y}*z*=

*z*, we can distinctly distinguish six petals from the whole pattern. However, the six petals are conjoint. When

_{r}*z*= 5

*z*, the outer distribution in the contour graph of normalized intensity takes on a gear shape. When

_{r}*z*= 100

*z*, the on-axis intensity is still zero and the two isolated side lobes are located at the

_{r}*y*-axis. When

*z*= 1500

*z*, the on-axis intensity is no longer zero and two side lobes are connected to the central dominating spot.

_{r}*R*= 5mm in Fig. 5 , and the rest of parameters are same as those in Fig. 2. Comparing Fig. 4 with Fig. 5, the effect of the decrease of

*w*

_{0}

*on the beam pattern is equivalent to that of the increase of*

_{x}*R*on the beam pattern, which leads to the similar beam pattern in the different observation planes. Moreover, When

*z*= 100

*z*, the on-axis intensity is nonzero. When

_{r}*z*= 200

*z*, the on-axis intensity is nearly the maximum.

_{r}*N*= 8 in Fig. 6, and the other parameters are same as those in Fig. 2. Comparing Fig. 6 with Fig. 2, we nearly cannot distinguish one from the other. When the number of the beamlet is large enough, the effect of the further increase of the beamlet is saturated. To sum up, the fill-factor in the near-field pattern is low in the case of

*w*

_{0}

*being small or*

_{x}*R*being large. If one wants to get a dark hollow uniform beam in the near-field, the parameters should be appropriate, e.g.,

*w*

_{0}

*is close to*

_{x}*R*and

*N*is large enough. As optical communications and remote sensing is involved in the far distance and a radial PL Lorentz beam array propagating in turbulent atmosphere finally turns out to be a solid beam, a radial PL Lorentz beam array is applicable to optical communications and remote sensing.

*z*in turbulent atmosphere are depicted in Figs. 7 and 8 . As the effective beam sizes in the

*x*- and

*y*-directions have the similar variational rule, only

*W*is shown in Figs. 7 and 8. The radial Lorentz beam array with the larger

_{xz}*w*

_{0}

*and*

_{x}*w*

_{0}

*spreads more rapidly, which seems to contradict the spreading of a single laser source. The combination of the different initial phase and the dark hollow distribution in the source plane maybe result in the difference. When reducing to*

_{y}*R*= 0 and

*N*= 1, the result obtained here is consistent with the behavior of a single laser source. The radial Lorentz beam array spreads more rapidly in turbulent atmosphere for a larger structure constant of the atmospheric turbulence. The structure constant of the refractive index fluctuations of the turbulence increasing denotes that the turbulence strengthens, which results in the large effective beam size. The radial PL Lorentz beam array with the larger radial radius spreads more rapidly. As to the effective beam size, the effect of the increase of

*w*

_{0}

*is similar to that of the increase of*

_{x}*R*. When the radius of the beam array is not far larger than the beam width parameter, the spreading of a radial PL Lorentz beam array in turbulent atmosphere is insensitive to the further increase of the number of beamlet under the condition of the initial number of beamlet being large. When the radius of the beam array is far larger than the beam width parameter, the spreading of a radial PL Lorentz beam array in turbulent atmosphere is first sensitive to the increase of the number of beamlet under the condition of the initial number of beamlet being not large enough. If the initial number of beamlet is large enough, the spreading of a radial PL Lorentz beam array in turbulent atmosphere is insensitive to the further increase of the number of beamlet. The variational law revealed in Figs. 7 and 8 is consistent with that denoted by Figs. 2-6.

## 4. Conclusions

*w*

_{0}

*,*

_{x}*w*

_{0}

*,*

_{y}*R*,

*N*, and

*C*

_{n}^{2}will affect the propagation distance within which the radial PL Lorentz beam array propagating in turbulent atmosphere keeps a dark hollow distribution. The radial PL Lorentz beam array in turbulent atmosphere with the larger

*w*

_{0}

*and*

_{x}*w*

_{0}

*or with the larger radial radius or for a larger structure constant of the atmospheric turbulence spreads more rapidly. When the number of beamlet is large enough, the spreading of a radial PL Lorentz beam array in turbulent atmosphere is insensitive to the further increase of the number of beamlet. This research is useful to the practical application of a coherent diode laser array in turbulent atmosphere, which is used for optical communications, remote sensing, and optical imaging. When the parameters are not appropriate, the near-field beam pattern of the radial PL Loretnz beam array is not ideal due to the low fill-factor. If one wants to achieve a higher fill-factor, the P × Q rectangular PL Lorentz beam array maybe alternative, which can be handled in the similar analytical approach as here.*

_{y}## Acknowledgments

## References and links

1. | A. Naqwi and F. Durst, “Focusing of diode laser beams: a simple mathematical model,” Appl. Opt. |

2. | J. Yang, T. Chen, G. Ding, and X. Yuan, “Focusing of diode laser beams: a partially coherent Lorentz model,” Proc. SPIE |

3. | W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” J. Quantum Electron. |

4. | O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt. |

5. | G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt. |

6. | G. Zhou, “Propagation of vectorial Lorentz beam beyond the paraxial approximation,” J. Mod. Opt. |

7. | G. Zhou, “The beam propagation factors and the kurtosis parameters of a Lorentz beam,” Opt. Laser Technol. |

8. | G. Zhou, “Generalized |

9. | C. Zhao and Y. Cai, “Paraxial propagation of Lorentz and Lorentz-Gauss beams in uniaxial crystals orthogonal to the optical axis,” J. Mod. Opt. |

10. | J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Propagation properties of Lorentz beam in uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol. |

11. | O. El Gawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangular symmetric optical fields,” Opt. Commun. |

12. | A. Torre, W. A. B. Evans, O. E. Gawhary, and S. Severini, “Relativistic Hermite polynomials and Lorentz beams,” J. Opt. A, Pure Appl. Opt. |

13. | H. T. Eyyuboğlu and Y. Baykal, “Hermite-sine-Gaussian and Hermite-sinh-Gaussian laser beams in turbulent atmosphere,” J. Opt. Soc. Am. A |

14. | H. T. Eyyuboğlu and Y. Baykal, “Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere,” Appl. Opt. |

15. | Y. Cai and S. He, “Average intensity and spreading of an elliptical Gaussian beam propagating in a turbulent atmosphere,” Opt. Lett. |

16. | Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, and Y. Baykal, “Average intensity and spreading of an elegant Hermite-Gaussian beam in turbulent atmosphere,” Opt. Express |

17. | F. Wang, Y. Cai, H. T. Eyyuboğlu, and Y. K. Baykal, “Average intensity and spreading of partially coherent standard and elegant Laguerre-Gaussian beams in turbulent atmosphere,” Prog. Electromagn. Res. |

18. | P. Zhou, X. Wang, Y. Ma, H. Ma, X. Xu, and Z. Liu, “Average intensity and spreading of Lorentz beam propagating in a turbulent atmosphere,” J. Opt. |

19. | C. Zhao and Y. Cai, “Propagation of partially coherent Lorentz and Lorentz-Gauss beams through a paraxial |

20. | J. P. Hobimer, D. R. Myers, T. M. Brennan, and B. E. Hammons, “Integrated injection-locked high-power cw diode laser arrays,” Appl. Phys. Lett. |

21. | X. Du and D. Zhao, “Propagation characteristics of stochastic electromagnetic array beams,” Appl. Phys. B |

22. | S. P. Ng and P. B. Phua, “Coherent polarization locking of a diode emitter array,” Opt. Lett. |

23. | L. Wang, L. Wang, and S. Zhu, “Formation of optical vortices using coherent laser beam array,” Opt. Commun. |

24. | X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun. |

25. | J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field radiation of coherent and incoherent combined Lorentz-Gaussian laser array,” Proc. SPIE |

26. | P. Zhou, X. Wang, Y. Ma, and Z. Liu, “Propagation properties of a Lorentz beam array,” Appl. Opt. |

27. | P. Zhou, X. Wang, Y. Ma, H. Ma, and Z. Liu, “Irradiance tailoring by fractional Fourier transform of a radial Gaussian beam array,” Proc. SPIE |

28. | H. Tang, B. Ou, B. Luo, H. Guo, and A. Dang, “Average spreading of a radial Gaussian beam array in non-Kolmogorov turbulence,” J. Opt. Soc. Am. A |

29. | Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express |

30. | L. Wang and W. Zheng, “The effect of atmospheric turbulence on the propagation properties of optical vortices formed by using coherent laser beam array,” J. Opt. A, Pure Appl. Opt. |

31. | X. Li and X. Ji, “Angular spread and directionality of the Hermite-Gaussian array beam propagating through atmospheric turbulence,” Appl. Opt. |

32. | X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express |

33. | P. P. Schmidt, “A method for the convolution of lineshapes which involve the Lorentz distribution,” J. Phys. B |

34. | S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am. |

35. | H. T. Eyyuboğlu, “Hermite-cosine-Gaussian laser beam and its propagation characteristics in turbulent atmosphere,” J. Opt. Soc. Am. A |

36. | K. Zhu, G. Zhou, X. Li, X. Zheng, and H. Tang, “Propagation of Bessel-Gaussian beams with optical vortices in turbulent atmosphere,” Opt. Express |

37. | I. S. Gradshteyn and I. M. Ryzhik, |

38. | W. H. Carter, “Spot size and divergence for Hermite Gaussian beams of any order,” Appl. Opt. |

**OCIS Codes**

(010.1300) Atmospheric and oceanic optics : Atmospheric propagation

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(140.2010) Lasers and laser optics : Diode laser arrays

(140.3290) Lasers and laser optics : Laser arrays

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: August 16, 2011

Revised Manuscript: October 17, 2011

Manuscript Accepted: November 10, 2011

Published: November 17, 2011

**Citation**

Guoquan Zhou, "Propagation of a radial phased-locked Lorentz beam array in turbulent atmosphere," Opt. Express **19**, 24699-24711 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-24-24699

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### References

- A. Naqwi and F. Durst, “Focusing of diode laser beams: a simple mathematical model,” Appl. Opt.29(12), 1780–1785 (1990). [CrossRef] [PubMed]
- J. Yang, T. Chen, G. Ding, and X. Yuan, “Focusing of diode laser beams: a partially coherent Lorentz model,” Proc. SPIE6824, 68240A, 68240A-8 (2007). [CrossRef]
- W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” J. Quantum Electron.11(7), 400–402 (1975). [CrossRef]
- O. E. Gawhary and S. Severini, “Lorentz beams and symmetry properties in paraxial optics,” J. Opt. A, Pure Appl. Opt.8(5), 409–414 (2006). [CrossRef]
- G. Zhou, J. Zheng, and Y. Xu, “Investigation in the far field characteristics of Lorentz beam from the vectorial structure,” J. Mod. Opt.55(6), 993–1002 (2008). [CrossRef]
- G. Zhou, “Propagation of vectorial Lorentz beam beyond the paraxial approximation,” J. Mod. Opt.55(21), 3573–3579 (2008). [CrossRef]
- G. Zhou, “The beam propagation factors and the kurtosis parameters of a Lorentz beam,” Opt. Laser Technol.41(8), 953–955 (2009). [CrossRef]
- G. Zhou, “Generalized M2 factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt.12(1), 015701 (2010). [CrossRef]
- C. Zhao and Y. Cai, “Paraxial propagation of Lorentz and Lorentz-Gauss beams in uniaxial crystals orthogonal to the optical axis,” J. Mod. Opt.57(5), 375–384 (2010). [CrossRef]
- J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Propagation properties of Lorentz beam in uniaxial crystals orthogonal to the optical axis,” Opt. Laser Technol.43(3), 506–514 (2011). [CrossRef]
- O. El Gawhary and S. Severini, “Lorentz beams as a basis for a new class of rectangular symmetric optical fields,” Opt. Commun.269(2), 274–284 (2007). [CrossRef]
- A. Torre, W. A. B. Evans, O. E. Gawhary, and S. Severini, “Relativistic Hermite polynomials and Lorentz beams,” J. Opt. A, Pure Appl. Opt.10(11), 115007 (2008). [CrossRef]
- H. T. Eyyuboğlu and Y. Baykal, “Hermite-sine-Gaussian and Hermite-sinh-Gaussian laser beams in turbulent atmosphere,” J. Opt. Soc. Am. A22(12), 2709–2718 (2005). [CrossRef] [PubMed]
- H. T. Eyyuboğlu and Y. Baykal, “Average intensity and spreading of cosh-Gaussian laser beams in the turbulent atmosphere,” Appl. Opt.44(6), 976–983 (2005). [CrossRef] [PubMed]
- Y. Cai and S. He, “Average intensity and spreading of an elliptical Gaussian beam propagating in a turbulent atmosphere,” Opt. Lett.31(5), 568–570 (2006). [CrossRef] [PubMed]
- Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, and Y. Baykal, “Average intensity and spreading of an elegant Hermite-Gaussian beam in turbulent atmosphere,” Opt. Express17(13), 11130–11139 (2009). [CrossRef] [PubMed]
- F. Wang, Y. Cai, H. T. Eyyuboğlu, and Y. K. Baykal, “Average intensity and spreading of partially coherent standard and elegant Laguerre-Gaussian beams in turbulent atmosphere,” Prog. Electromagn. Res.103, 33–56 (2010). [CrossRef]
- P. Zhou, X. Wang, Y. Ma, H. Ma, X. Xu, and Z. Liu, “Average intensity and spreading of Lorentz beam propagating in a turbulent atmosphere,” J. Opt.12, 01540–01549 (2010).
- C. Zhao 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.58(9), 810–818 (2011). [CrossRef]
- J. P. Hobimer, D. R. Myers, T. M. Brennan, and B. E. Hammons, “Integrated injection-locked high-power cw diode laser arrays,” Appl. Phys. Lett.55(6), 531–533 (1989). [CrossRef]
- X. Du and D. Zhao, “Propagation characteristics of stochastic electromagnetic array beams,” Appl. Phys. B93(4), 901–905 (2008). [CrossRef]
- S. P. Ng and P. B. Phua, “Coherent polarization locking of a diode emitter array,” Opt. Lett.34(13), 2042–2044 (2009). [CrossRef] [PubMed]
- L. Wang, L. Wang, and S. Zhu, “Formation of optical vortices using coherent laser beam array,” Opt. Commun.282(6), 1088–1094 (2009). [CrossRef]
- X. Du and D. Zhao, “Statistical properties of correlated radial stochastic electromagnetic array beams on propagation,” Opt. Commun.282(10), 1993–1997 (2009). [CrossRef]
- J. Li, Y. Chen, S. Xu, Y. Wang, M. Zhou, Q. Zhao, Y. Xin, and F. Chen, “Far-field radiation of coherent and incoherent combined Lorentz-Gaussian laser array,” Proc. SPIE7749, 77491Z, 77491Z-6 (2010). [CrossRef]
- P. Zhou, X. Wang, Y. Ma, and Z. Liu, “Propagation properties of a Lorentz beam array,” Appl. Opt.49(13), 2497–2503 (2010). [CrossRef]
- P. Zhou, X. Wang, Y. Ma, H. Ma, and Z. Liu, “Irradiance tailoring by fractional Fourier transform of a radial Gaussian beam array,” Proc. SPIE7822, 78220J, 78220J-6 (2010). [CrossRef]
- H. Tang, B. Ou, B. Luo, H. Guo, and A. Dang, “Average spreading of a radial Gaussian beam array in non-Kolmogorov turbulence,” J. Opt. Soc. Am. A28(6), 1016–1021 (2011). [CrossRef] [PubMed]
- Y. Zhu, D. Zhao, and X. Du, “Propagation of stochastic Gaussian-Schell model array beams in turbulent atmosphere,” Opt. Express16(22), 18437–18442 (2008). [CrossRef] [PubMed]
- L. Wang and W. Zheng, “The effect of atmospheric turbulence on the propagation properties of optical vortices formed by using coherent laser beam array,” J. Opt. A, Pure Appl. Opt.11(6), 065703 (2009). [CrossRef]
- X. Li and X. Ji, “Angular spread and directionality of the Hermite-Gaussian array beam propagating through atmospheric turbulence,” Appl. Opt.48(22), 4338–4347 (2009). [CrossRef] [PubMed]
- X. Ji, H. T. Eyyuboğlu, and Y. Baykal, “Influence of turbulence on the effective radius of curvature of radial Gaussian array beams,” Opt. Express18(7), 6922–6928 (2010). [CrossRef] [PubMed]
- P. P. Schmidt, “A method for the convolution of lineshapes which involve the Lorentz distribution,” J. Phys. B9(13), 2331–2339 (1976). [CrossRef]
- S. C. H. Wang and M. A. Plonus, “Optical beam propagation for a partially coherent source in the turbulent atmosphere,” J. Opt. Soc. Am.69(9), 1297–1304 (1979). [CrossRef]
- H. T. Eyyuboğlu, “Hermite-cosine-Gaussian laser beam and its propagation characteristics in turbulent atmosphere,” J. Opt. Soc. Am. A22(8), 1527–1535 (2005). [CrossRef] [PubMed]
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