## Abstract

The propagation of a partially coherent cosine-Gaussian beam through a paraxial *ABCD* optical system in turbulent atmosphere is investigated. Analytical expression for the average intensity in the output plane is derived. The presented formula only covers optical systems without inherent apertures, where *ABCD* are all real-valued. As a special case of the general formula, the analytical formula for the average intensity of a partially coherent cosh-Gaussian beam through an *ABCD* optical system in turbulent atmosphere is also presented, respectively. The properties of the average intensity of the partially coherent cosine-Gaussian beam are investigated with a numerical example, and the dependence of the average intensity distribution on the spatial correlation length of a partially coherent cosine-Gaussian beam is mainly discussed.

©2009 Optical Society of America

## 1. Introduction

Due to applications in optimizing the efficiency of laser amplifiers, cosine-Gaussian and cosh-Gaussian beams have received considerable interest [1,2]. The propagation properties of unapertured and apertured cosh-Gaussian beams have been investigated [3–5]. The propagation characteristics of Hermite-cosh-Gaussian beams and elegant Hermite-cosh-Gaussian beams have been examined in free space [6,7]. Coherent combination of certain cosh-Gaussian beams results in a flattened beam with an axial shadow [8]. Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned *ABCD* optical system has been studied [9]. The beam propagation factor and kurtosis parameter of the off-axial Hermite-cosh-Gaussian beams have been presented, respectively [10]. The analytical expression characterizing the propagation of non-paraxial truncated cosine-Gaussian beams has been derived in free space [11]. A group of virtual sources that generate a cosh-Gaussian beam has been proposed [12]. The structural properties of cosine-Gaussian and cosh-Gaussian beams have been separately examined in the far field [13,14]. The reciprocity of cosine-Gaussian and cosh-Gaussian beams has been analyzed in turbulent atmosphere [15]. The effects of turbulent atmosphere on cosine-Gaussian beams have been summarized [16]. Propagation characteristics of Hermite-cosine-Gaussian and Hermite-cosh-Gaussian beams in turbulent atmosphere have been revealed [17,18]. The average intensity and spreading of cosh-Gaussian beams in turbulent atmosphere have been examined [19]. The scintillation index of cosine-Gaussian and cosh-Gaussian beams in atmospheric turbulence has been demonstrated [20,21]. Properties of cosh-Gaussian and truncated cosh-Gaussian beams through a paraxial *ABCD* optical system in turbulent atmosphere have been investigated [22,23]. In fact, laser beams are almost partially coherent. Therefore, the complex degree of coherence for a partially coherent cosine-Gaussian beam has been calculated in atmospheric turbulence [24]. By incorporating atmospheric turbulence, average relative power transmittance has been evaluated for a partially coherent cosine-Gaussian beam [25]. Angular spread of partially coherent Hermite-cosh-Gaussian beams through atmospheric turbulence has been examined [26]. As to the practical applications of a laser beam in turbulent atmosphere, a series of optics systems is often used to direct or redirect a laser beam to a distant target plane. Therefore, the analysis of propagation of a laser beam through an *ABCD* optical system in turbulent atmosphere is important. In this paper, the propagation of a partially coherent cosine-Gaussian beam through an arbitrary *ABCD* optical system in turbulent atmosphere is investigated and illustrated by a numerical example.

## 2. Average intensity of a partially coherent cosine-Gaussian beam through an *ABCD* optical system in turbulent atmosphere

In the Cartesian coordinate system, the *z*-axis is taken to be the propagation axis. The cross spectral density of a partially coherent cosine-Gaussian beam in the source plane *z* = 0 takes the form as

*j*=

*x*or

*y*(hereafter).

*w*

_{0}is the waist width of the Gaussian part, and Ω is the beam parameter associated with the cosine part.

*σ*is the spatial correlation length of the laser source in the plane

*z*= 0.

Based on Huygens-Fresnel diffraction integral, the average intensity of a partially coherent cosine-Gaussian beam passing through an *ABCD* optical system in turbulent atmosphere can be obtained by

*k*= 2

*π*/

*λ*with

*λ*the incident wavelength.

*ψ*(

*x*

_{01},

*y*

_{01},

*x*,

*y*) is the solution to the Rytov method that represents the random part of the complex phase. The angle brackets denote the ensemble average over the medium statistics covering the log-amplitude and phase fluctuations due to the turbulent atmosphere. The asterisk means the complex conjugation.

*A*,

*B*,

*C*, and

*D*are matrix elements of the optical system between the source and the output planes. Moreover, there is no inherent aperture between the source and the output planes. Therefore,

*A*,

*B*,

*C*, and

*D*are all real-valued. The ensemble average term is given by

*ρ*

_{0}is the spherical-wave lateral coherence radius due to the turbulence of the entire optical system and defined as [27]

*C*

_{n}^{2}is the constant of refraction index structure and describes the turbulence level.

*b*(

*z*) corresponds to the approximate matrix element for a ray propagating backwards through the system.

*L*is the axial distance between the source and the output planes. Inserting Eqs. (1) and (4) into Eq. (3), the average intensity of the partially coherent cosine-Gaussian beam in the output plane yieldswith <

*I*(

*j*,

*z*)> given by

*B* = 0 corresponds to an image forming system [28]. In this case, Eqs. (7) and (8) are simplified to

When a partially coherent cosine-Gaussian beam passes through an image forming system in turbulent atmosphere, the average intensity is independent of the spatial correlation length *σ*. If the spatial correlation length *σ* tends to infinity, it corresponds to the fully coherent case. The corresponding Eqs. (7) and (9) reduce to

Equation (12) is just the average intensity of a coherent cosine-Gaussian beam in the transverse direction of the output plane. If Ω in Eq. (7) is replaced by iΩ, we can obtain the average intensity of the partially coherent cosh-Gaussian beam through an *ABCD* optical system in turbulent atmosphere:

*I*(

*j*,

*z*)> given by

Though the average intensities of the partially coherent cosine-Gaussian and cosh-Gaussian beams in atmospheric turbulence can be calculated by Eq. (5) in Ref [24], they are rather complicated, which is caused by the use of the general beam formulation. The advantage is that Eq. (5) in Ref [24]. is also applicable to annular and higher-order Gaussian beams. As the final purpose is to present the complex degree of coherence, no optical system is taken into account in Ref [24]. The formulae obtained here are concise and valid to an arbitrary optical system without inherent apertures.

## 3. Numerical calculations and discussions

As a numerical example, the optical system of two thin lenses is considered, which is shown in Fig. 1
. The dimension of the thin lens is assumed to be larger than the corresponding beam diameter. Therefore, the diffraction due to the lens is neglected. In this case, *A* and *B* yield [29]

*L*=

*z*

_{1}+

*z*

_{2}+

*z*

_{3}. Now, we consider the special case:

*z*

_{1}=

*z*

_{3}=

*L*/6,

*f*

_{1}=

*f*

_{2}=

*L*/3, and

*z*

_{2}=

*f*

_{1}+

*f*

_{2}. In this case,

*A*= −1 and

*B*=

*L*/3. The corresponding

*b*(

*z*) reads as

The parameters used in calculations are chosen as follows. *w*
_{0} = 0.2m, and *λ* = 1.06μm. As the *x* and *y* directions are separable in Eq. (6), only the intensity in the *x* direction is considered. Figure 2
shows the typical intensity distribution of a cosine-Gaussian beam in the source plane. Ω affects the initial intensity distribution of a cosine-Gaussian beam. When Ω is small, there is no side apex in the intensity distribution. When Ω is large enough, some side apices are detected. As the influences of the propagation distance, the turbulence level, and the beam parameters besides the spatial correlation length on the average intensity distribution have been extensively investigated elsewhere, here we mainly concentrate on the effect of the spatial correlation length. Figures 3
-5
represent the average intensity distribution in the *x* direction of a partially coherent cosine-Gaussian beam at different propagation distance in turbulent atmosphere. The solid and the dashed curves correspond to *σ* = 0.2m and 0.007m, respectively. Subfigure (a), (b), and (c) correspond to *L* = 2km, 3.5km, and 10km, respectively. In Fig. 3, Ω = 30m^{−1} and *C _{n}*

^{2}= 10

^{−14}m

^{-2/3}. In Fig. 4 , Ω = 10m

^{−1}and

*C*

_{n}^{2}=10

^{−14}m

^{2/3}; while in Fig. 5, Ω = 30m

^{−1}and

*C*

_{n}^{2}= 10

^{−16}m

^{-2/3}. The influence of the spatial correlation length is not independent and has a close relation with other parameters. When Ω is small such as 10m

^{−1}, the effect of the spatial correlation length is relatively small. With increasing the propagation distance, moreover, the only influence of the spatial correlation length on the profile of the average intensity is just that the peak of the average intensity decreases, which is shown in Figs. 4(b) and 4(c). When Ω is large such as 30m

^{−1}, the influence of the spatial correlation length is complicated. In this case, the profile of the average intensity mainly depends on the smaller of

*τ*

_{1}and

*τ*

_{2}.

*σ*= 0.007m and 0.2m correspond to

*τ*

_{1}= 0.035 and 1. In Figs. 3(a)-3(c),

*τ*

_{2}= 0.055, 0.041, and 0.021, respectively. In Figs. 5(a)-5(c),

*τ*

_{2}= 0.87, 0.62, and 0.33, respectively. Therefore, the profile of the average intensity denoted by the dashed curve in Fig. 3 is nearly same as that in Fig. 5, and the corresponding profile of the average intensity denoted by the solid curve in Fig. 3 is apparently different from that in Fig. 5.

## 4. Conclusion

The propagation of a partially coherent cosine-Gaussian beam through a paraxial *ABCD* optical system in turbulent atmosphere is investigated. An analytical expression of the average intensity is derived. The analytical average intensities of a coherent cosine-Gaussian beam and a partially coherent cosh-Gaussian beam through an *ABCD* optical system in turbulent atmosphere are also presented, respectively. As a numerical example, the properties of the average intensity of the partially coherent cosine-Gaussian beam through the optical system of two thin lenses in turbulent atmosphere are demonstrated, and we mainly concentrate on the influence of the spatial correction length on the average intensity distribution. The results show that the influence of the spatial correction length is complicated and has a close relation with other parameters. In should be noted that the presented results are valid for weak turbulence and hardly hold for strong turbulence.

## References and links

**1. **L. W. Casperson and A. A. Tovar, “Hermite-sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A **15**(4), 954–961 (1998). [CrossRef]

**2. **A. A. Tovar and L. W. Casperson, “Production and propagation of Hermite-sinusoidal-Gaussian laser beams,” J. Opt. Soc. Am. A **15**(9), 2425–2432 (1998). [CrossRef]

**3. **B. Lü, B. Zhang, and H. Ma, “Beam-propagation factor and mode-coherence coefficients of hyperbolic-cosine Gaussian beams,” Opt. Lett. **24**(10), 640–642 (1999). [CrossRef]

**4. **B. Lü, H. Ma, and B. Zhang, “Propagation properties of cosh-Gaussian beams,” Opt. Commun. **164**(4-6), 165–170 (1999). [CrossRef]

**5. **B. Lü and S. Luo, “Beam propagation factor of hard-edge diffracted cosh-Gaussian beams,” Opt. Commun. **178**(4-6), 275–281 (2000). [CrossRef]

**6. **X. Wang and B. Lü, “The *M*^{2} factor of Hermite-cosh-Gaussian beams,” J. Mod. Opt. **48**, 2097–2103 (2001).

**7. **S. Yu, H. Guo, X. Fu, and W. Hu, “Propagation perties of elegant Hermite-cosh-Gaussian laser beams,” Opt. Commun. **204**, 59–66 (2002).

**8. **K. Zhu, H. Tang, X. Wang, and T. Liu, “Flattened light beams with an axial shadow generated through superposing cosh-Gaussian beams,” Optik (Stuttg.) **113**(5), 222–226 (2002).

**9. **D. Zhao, H. Mao, W. Zhang, and S. Wang, “Propagation of off-axial Hermite-cosine-Gaussian beams through an apertured and misaligned *ABCD* optical system,” Opt. Commun. **224**(1-3), 5–12 (2003). [CrossRef]

**10. **Z. Mei, D. Zhao, D. Sun, and J. Gu, “The *M*^{2} factor and kurtosis parameter of the off-axial Hermite-cosh-Gaussian beams,” Optik (Stuttg.) **115**(2), 89–93 (2004).

**11. **X. Kang and B. Lü, “Characterization of nonparaxial truncated cosine-Gaussian beams and the beam quality in the far field,” Chin. Phys. Lett. **23**(9), 2430–2433 (2006). [CrossRef]

**12. **Y. Zhang, Y. Song, Z. Chen, J. Ji, and Z. Shi, “Virtual sources for a cosh-Gaussian beam,” Opt. Lett. **32**(3), 292–294 (2007). [CrossRef] [PubMed]

**13. **G. Zhou, R. Qü, and L. Sun, “The structural properties of cosine-Gaussian beam in the far field,” J. Mod. Opt. **55**(15), 2485–2495 (2008). [CrossRef]

**14. **G. Zhou and F. Liu, “Far field structural characteristics of cosh-Gaussian beam,” Opt. Laser Technol. **40**(2), 302–308 (2008). [CrossRef]

**15. **H. T. Eyyuboğlu and Y. Baykal, “Reciprocity of cos-Gaussian and cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Express **12**, 4659–4674 (2004). [CrossRef] [PubMed]

**16. **H. T. Eyyuboğlu and Y. Baykal, “Cosine-Gaussian laser beam intensity in turbulent atmosphere,” Proc. SPIE **5743**, 131–141 (2004). [CrossRef]

**17. **H. T. Eyyuboğlu, “Hermite-cosine-Gaussian laser beam and its propagation characteristics in turbulent atmosphere,” J. Opt. Soc. Am. A **22**(8), 1527–1535 (2005). [CrossRef]

**18. **H. T. Eyyuboğlu, “Propagation of Hermite-cosh-Gaussian laser beams in turbulent atmosphere,” Opt. Commun. **245**(1-6), 37–47 (2005). [CrossRef]

**19. **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]

**20. **H. T. Eyyuboğlu and Y. Baykal, “Scintillations of cos-Gaussian and annular beams,” J. Opt. Soc. Am. A **24**, 156–162 (2007). [CrossRef]

**21. **H. T. Eyyuboğlu and Y. Baykal, “Scintillation characteristics of cosh-Gaussian beams,” Appl. Opt. **46**(7), 1099–1106 (2007). [CrossRef] [PubMed]

**22. **X. Chu, “Propagation of a cosh-Gaussian beam through an optical system in turbulent atmosphere,” Opt. Express **15**(26), 17613–17618 (2007). [CrossRef] [PubMed]

**23. **X. Chu, Y. Ni, and G. Zhou, “Propagation of cosh-Gaussian beams diffracted by a circular aperture in turbulent atmosphere,” Appl. Phys. B **87**(3), 547–552 (2007). [CrossRef]

**24. **H. T. Eyyuboğlu, Y. Baykal, and Y. Cai, “Complex degree of coherence for partially coherent general beams in atmospheric turbulence,” J. Opt. Soc. Am. A **24**, 2891–2901 (2007). [CrossRef]

**25. **H. T. Eyyuboğlu and Y. Baykal, “Transmittance of partially coherent cosh-Gaussian, cos-Gaussian and annular beams in turbulence,” Opt. Commun. **278**(1), 17–22 (2007). [CrossRef]

**26. **A. Yang, E. Zhang, X. Ji, and B. Lü, “Angular spread of partially coherent Hermite-cosh-Gaussian beams propagating through atmospheric turbulence,” Opt. Express **16**(12), 8366–8380 (2008). [CrossRef] [PubMed]

**27. **H. T. Yura and S. G. Hanson, “Optical beam wave propagation through complex optical systems,” J. Opt. Soc. Am. A **4**(10), 1931–1948 (1987). [CrossRef]

**28. **S. Wang, and D. Zhao, *Matrix Optics* (CHEP and Springer, Beijing, 2000), p. 15.

**29. **X. Chu, Z. Lu, and Y. Wu, “The propagation of a flattened circular Gaussian beam through an optical system in turbulent atmosphere,” Appl. Phys. B **92**(1), 119–122 (2008). [CrossRef]