## Propagation of a partially coherent Lorentz-Gauss beam through a paraxial *ABCD* optical system

Optics Express, Vol. 18, Issue 5, pp. 4637-4643 (2010)

http://dx.doi.org/10.1364/OE.18.004637

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### Abstract

Based on the generalized Huygens-Fresnel integral and the Hermite-Gaussian expansion of a Lorentz distribution, analytical expressions for the mutual coherence function, the effective beam size, and the spatial complex degree of coherence of a partially coherent Lorentz-Gauss beam through a paraxial and real *ABCD* optical system are derived, respectively. As a numerical example, the focusing of a partially coherent Lorentz-Gauss beam is considered. The normalized intensity distribution, the effective beam size, and the spatial complex degree of coherence for the focused partially coherent Lorentz-Gauss beam are numerically demonstrated in the focal plane. The influence of the spatial coherence length on the normalized intensity distribution, the effective beam size, and the spatial complex degree of coherence is mainly discussed.

© 2010 OSA

## 1. Introduction

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

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

3. J. Yang, T. Chen, G. Ding, and X. Yuan, “Focusing of diode laser beams: a partially coherent Lorentz model,” Proc. SPIE **6824**, 68240A (2008). [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]

14. G. Zhou, “Beam propagation factors of a Lorentz-Gauss beam,” Appl. Phys. B **96**(1), 149–153 (2009). [CrossRef]

*ABCD*matrix, which is very simple and convenient to the practical applications. To properly design an optical system that includes a single mode diode laser, the analysis of propagation of a Lorentz-Gauss beam passing through an

*ABCD*optical system is prerequisite. In the remainder of this paper, therefore, the propagation of a partially coherent Lorentz-Gauss beam through a paraxial and real

*ABCD*optical system is investigated. Moreover, analytical expressions for the mutual coherence function, the effective beam size, and the spatial complex degree of coherence are derived by means of the mathematical techniques. A numerical example is also demonstrated.

## 2. Propagation of a partially coherent Lorentz-Gauss beam through a paraxial and real *ABCD* optical system

*x*- and

*y*-directions of the output plane is defined as [18

18. 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**(13), 11130–11139 (2009). [CrossRef] [PubMed]

_{1}

*and Ω*

_{j}_{2}

*given by*

_{j}*x*

_{1},

*y*

_{1},

*z*) and (

*x*

_{2},

*y*

_{2},

*z*) turns out to [19

19. A. T. Friberg and J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A **5**(5), 713–720 (1988). [CrossRef]

## 3. Numerical calculations and analyses

*f*is placed in front of the single mode diode laser, so that the partially coherent Lorentz-Gauss beam is transformed into a converging beam. In the case of diode-fiber coupling, a fiber end is placed in the focal region. The matrix elements of this optical arrangement are

*A*=0,

*B*=

*f*,

*C*=−1/

*f*, and

*D*=1. The normalized light intensity, the effective beam size, and the spatial complex degree of coherence for the partially coherent Lorentz-Gauss beam in the focal plane are calculated by using the formulae derived above. As the

*x*- and

*y*-directions are separable, only the

*x*-direction is considered in the following calculations. Moreover, we mainly concentrate on the effect of the spatial coherence length. Figure 1 represents the normalized intensity distribution in the focal plane. The parameters used are chosen as follow:

*λ*=0.8μm,

*f*=1m and

*w*

_{0}

*=1mm.*

_{x}*w*

_{0}=2mm in Fig. 1(a) and

*w*

_{0}=∞ in Fig. 1(b). The normalized intensity distribution of the partially coherent Lorentz-Gauss beam spreads with decreasing the spatial coherence length

*σ*. The effective beam size of partially coherent Lorentz-Gauss beams in the focal plane versus the coherence length

_{x}*σ*is depicted in Fig. 2 . The inset shows the detail change of

_{x}*W*within the range of 0.2mm≤

_{xz}*σ*≤1mm. With increasing the spatial coherence length

_{x}*σ*, the effective beam size first quickly decreases and then tends to a minimum value. The better coherence the partially coherent Lorentz-Gauss beam has, the smaller effective beam size it has. If the spatial coherence length

_{x}*σ*keeps invariant, the partially coherent Lorentz-Gauss beam with smaller

_{x}*w*

_{0}and

*w*

_{0}

*has the larger effective beam size. Figure 3 shows the spatial complex degree of coherence in the focal plane.*

_{x}*w*

_{0}=2mm and

*w*

_{0}

*=1mm.*

_{x}*σ*=1mm, 2mm, and

_{x}*x*

_{1},

*f*) and (

*x*

_{2},

*f*), their spatial complex degree of coherence increases with increasing the spatial coherence length

*σ*. To quantitatively evaluate the influence of the spatial coherence length on the spatial complex degree of coherence, we calculate the spatial complex degree of coherence at two points (0.1mm,

_{x}*f*)and (0.4mm,

*f*) by altering the spatial coherence length

*σ*, which is shown in Fig. 4 . With increasing the spatial coherence length

_{x}*σ*, the spatial complex degree of coherence first quickly increases and then tends to the saturated value 1. With a given

_{x}*σ*, the partially coherent Lorentz-Gauss beam with the smaller

_{x}*w*

_{0}and

*w*

_{0}

*has the larger spatial complex degree of coherence.*

_{x}21. G. Zhou, “Generalized *M*^{2} factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. **12**(1), 015701 (2010). [CrossRef]

## 4. Conclusions

*ABCD*optical system is derived. Moreover, the analytical formulae for the effective beam size and the spatial complex degree of coherence are also presented. As a numerical example, the normalized light intensity, the effective beam size, and the spatial complex degree of coherence for the partially coherent Lorentz-Gauss beam focused by a thin lens are calculated in the focal plane. The effect of the spatial coherence length is mainly discussed. With increasing the spatial coherence length, the partially coherent Lorentz-Gauss beam has the smaller effective beam size and the higher spatial complex degree of coherence. As apertures usually exist in the practical optical system, the propagation of a partially coherent Lorentz-Gauss beam through a paraxial and complex

*ABCD*optical system also deserves to be investigated. This research is useful to the optical designs that are involved in the single mode diode laser.

## Acknowledgements

## References and links

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

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

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

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

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

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

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

8. | G. Zhou, “Analytical vectorial structure of a Lorentz-Gauss beam in the far field,” Appl. Phys. B |

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

10. | G. Zhou, “Focal shift of focused truncated Lorentz-Gauss beam,” J. Opt. Soc. Am. A |

11. | G. Zhou, “Nonparaxial propagation of a Lorentz-Gauss beam,” J. Opt. Soc. Am. B |

12. | G. Zhou, “Fractional Fourier transform of Lorentz-Gauss beams,” J. Opt. Soc. Am. A |

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

14. | G. Zhou, “Beam propagation factors of a Lorentz-Gauss beam,” Appl. Phys. B |

15. | L. Mandel, and E. Wolf, |

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

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

18. | 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 |

19. | A. T. Friberg and J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A |

20. | M. Born, and E. Wolf, |

21. | G. Zhou, “Generalized |

**OCIS Codes**

(030.1640) Coherence and statistical optics : Coherence

(350.5500) Other areas of optics : Propagation

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: November 20, 2009

Revised Manuscript: January 14, 2010

Manuscript Accepted: February 5, 2010

Published: February 22, 2010

**Citation**

Guoquan Zhou, "Propagation of a partially coherent Lorentz-Gauss beam through a paraxial ABCD optical system," Opt. Express **18**, 4637-4643 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-5-4637

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### References

- W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. 11(7), 400–402 (1975). [CrossRef]
- A. Naqwi and F. Durst, “Focus 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. SPIE 6824, 68240A (2008). [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]
- O. Elgawhary 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]
- 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, “Analytical vectorial structure of a Lorentz-Gauss beam in the far field,” Appl. Phys. B 93(4), 891–899 (2008). [CrossRef]
- G. Zhou, “Propagation of vectorial Lorentz beam beyond the paraxial approximation,” J. Mod. Opt. 55(21), 3573–3577 (2008). [CrossRef]
- G. Zhou, “Focal shift of focused truncated Lorentz-Gauss beam,” J. Opt. Soc. Am. A 25(10), 2594–2599 (2008). [CrossRef]
- G. Zhou, “Nonparaxial propagation of a Lorentz-Gauss beam,” J. Opt. Soc. Am. B 26(1), 141–147 (2009). [CrossRef]
- G. Zhou, “Fractional Fourier transform of Lorentz-Gauss beams,” J. Opt. Soc. Am. A 26(2), 350–355 (2009). [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, “Beam propagation factors of a Lorentz-Gauss beam,” Appl. Phys. B 96(1), 149–153 (2009). [CrossRef]
- L. Mandel, and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, UK, 1995).
- P. P. Schmidt, “A method for the convolution of lineshapes which involve the Lorentz distribution,” J. Phys. B 9(13), 2331–2339 (1976). [CrossRef]
- I. S. Gradshteyn, and I. M. Ryzhik, Table of integrals, series, and products (Academic Press, New York, 1980).
- 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(13), 11130–11139 (2009). [CrossRef] [PubMed]
- A. T. Friberg and J. Turunen, “Imaging of Gaussian Schell-model sources,” J. Opt. Soc. Am. A 5(5), 713–720 (1988). [CrossRef]
- M. Born, and E. Wolf, Principles of Optics 7th ed. (Cambridge University Press, Cambridge, UK, 1999).
- G. Zhou, “Generalized M2 factors of truncated partially coherent Lorentz and Lorentz-Gauss beams,” J. Opt. 12(1), 015701 (2010). [CrossRef]

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