## Mathematic model analysis of Gaussian beam propagation through an arbitrary thickness random phase screen |

Optics Express, Vol. 19, Issue 19, pp. 18216-18228 (2011)

http://dx.doi.org/10.1364/OE.19.018216

Acrobat PDF (1037 KB)

### Abstract

In order to research the statistical properties of Gaussian beam propagation through an arbitrary thickness random phase screen for adaptive optics and laser communication application in the laboratory, we establish mathematic models of statistical quantities, which are based on the Rytov method and the thin phase screen model, involved in the propagation process. And the analytic results are developed for an arbitrary thickness phase screen based on the Kolmogorov power spectrum. The comparison between the arbitrary thickness phase screen and the thin phase screen shows that it is more suitable for our results to describe the generalized case, especially the scintillation index.

© 2011 OSA

## 1. Introduction

2. B. Wang, Z.-y. Wang, J.-l. Wang, J.-Y. Zhao, Y.-H. Wu, S.-X Zhang, L. Dong, and M. Wen, “Phase-diverse speckle imaging with two cameras,” Opt. Precision Eng. **19**(6), 1384–1390 (2011). [CrossRef]

3. H. G. Booker, J. A. Ferguson, and H. O. Vats, “Comparison between the extended-medium and the phase-screen scintillation theories,” J. Atmos. Terr. Phys. **47**(38), 1–399 (1985). [CrossRef]

7. L. C. Andrews and W. B. Miller, “Single- and double-pass propagation through complex paraxial optical systems,” J. Opt. Soc. Am. A **12**(1), 137–150 (1995). [CrossRef]

8. S. V. Mantravadi, T. A. Rhoadarmer, and R. S. Glas, “Simple laboratory system for generating well-controlled atmospheric-like turbulence,” Proc. SPIE **5553**, 290–300 (2004). [CrossRef]

9. X. J. Gan, J. Guo, and Y. Y. Fu, “The simulating turbulence method of laser propagation in the inner field,” J. Phys. Conf. Ser. **48**, 907–910 (2006). [CrossRef]

5. L. C. Andrews, R. L. Phillips, and A. R. Weeks, “Propagation of a Gaussian-beam wave through a random phase screen,” Waves Random Media **7**(2), 229–244 (1997). [CrossRef]

## 2. Gaussian beam parameters

_{00}wave, in the plane of the emitting aperture of the transmitter at

*z*= 0, can be expressed as [4,12,13

13. Y.-h. Gao, Z.-y. An, N.-n. Li, W.-x Zhao, and J.-s. Wang, “Optical design of Gaussian beam shaping,” Opt. Precision Eng. **19**(7), 1464–1471 (2011). [CrossRef]

*r*= (

*x*

^{2}+

*y*

^{2})

^{1/2},

*α*

_{0}= 2/(

*kW*

_{0}

^{2}) +

*i*/

*F*

_{0};

*a*

_{0}is the optical amplitude,

*F*

_{0}and

*W*

_{0}denote the phase front radius of curvature and beam radius at which the field amplitude falls to 1/

*e*of that on the beam axis, respectively, and

*k*is the optical wave number. The particular cases

*F*

_{0}= ∞,

*F*

_{0}>0, and

*F*

_{0}<0 correspond to collimated, convergent, and divergent beam formats, respectively. Based on paraxial theory, the free space optical wave in the receiver plane at

*z*=

*L*is another Gaussian beam described by [4,12]

*W*and

*F*denote the receiver plane phase front radius of curvature and beam radius, respectively, and

*p*(

*L*) = 1 +

*iα*

_{0}

*L*is the propagation parameter.

*d*=

*L*

_{2}to modulate the phase of optical field. In the paper, all the three beam case,

*F*

_{0}= ∞,

*F*

_{0}>0 and

*F*

_{0}<0,will be contained, and the beam waist appears in the middle of the optical path. Based on this figure, the nondimensional input plane beam parameters [4–7

7. L. C. Andrews and W. B. Miller, “Single- and double-pass propagation through complex paraxial optical systems,” J. Opt. Soc. Am. A **12**(1), 137–150 (1995). [CrossRef]

*W*and

*F*as

7. L. C. Andrews and W. B. Miller, “Single- and double-pass propagation through complex paraxial optical systems,” J. Opt. Soc. Am. A **12**(1), 137–150 (1995). [CrossRef]

## 3. Rytov method

*L*, the optical field can be expressed as:

*U*

_{0}(

*r*,

*L*) is the free space Gaussian beam, described in Eqs. (2), Ψ(

*r*,

*L*) is a complex phase perturbation due to turbulence, and

*ψ*

_{1}(

*r*,

*L*) and

*ψ*

_{2}(

*r*,

*L*) are the first-order and second-order complex phase perturbations, respectively. It is known that the first two order complex phase perturbations can nearly determinate all the long time statistical property, so we just consider

*ψ*

_{1}(

*r*,

*L*) and

*ψ*

_{2}(

*r*,

*L*) in this paper. For the case of homogeneous and isotropic extended turbulence, it has been shown by Andrews and Miller [6

6. L. C. Andrews, W. B. Miller, and J. C. Ricklin, “Propagation through complex optical system: a phase screen analysis,” SPIE **2312**, 122–129 (1994). [CrossRef]

**12**(1), 137–150 (1995). [CrossRef]

**r**= (

**r**

_{1}+

**r**

_{2})/2,

**p**=

**r**

_{1}-

**r**

_{2},

*r*= |

**r**|,

*ρ*= |

**p**|,

*κ*is the space wave number,

*J*

_{0}(

*x*) is the Bessel function of the first kind, the brackets < > denote an ensemble average, and the asterisk * denotes the complex conjugate field.

### 3.1 Arbitrary thickness phase screen

*d*=

*L*

_{2}, thus we can assume that random media exist only in [

*L*

_{1},

*L*

_{1}+

*L*

_{2}] and change the integral variable:

*C*

_{n}

^{2}is assumed constant, because the random media only exit over

*L*

_{1}≤

*z*≤

*L*

_{1}+

*L*

_{2}, so we can set Φ

*(*

_{n}*κ*) as:

### 3.2 Rytov Variance of Phase Screen

15. D. L. Fried and J. B. Seidman, “Laser-beam scintillation in the atmosphere,” J. Opt. Soc. Am. **57**(2), 181–185 (1967). [CrossRef]

*κ*

_{0}→0

^{+},

*κ*→∞,and

_{m}*κ*

_{0}/

*κ*→0

_{m}^{+}. Based on von Kármán spectrum and using the integral definition of confluent hypergeometric function and asymptotic formula [4],

5. L. C. Andrews, R. L. Phillips, and A. R. Weeks, “Propagation of a Gaussian-beam wave through a random phase screen,” Waves Random Media **7**(2), 229–244 (1997). [CrossRef]

*L*is the propagation path in atmosphere, and

_{A}*L*is that in the phase screen model. The left side of Eq. (26) represents the turbulence effects, which are turbulence strength and propagation path, on optical field, and the right side of Eq. (26) represents simulation effects of the phase screen.

## 4. Mutual coherent function

_{2}

^{0}(

**r**

_{1},

**r**

_{2},

*L*) is the MCF in free space

*E*

_{1}(0,0) +

*E*

_{2}(

**r**

_{1}+

**r**

_{2}) directly, but make some simplification

*d*=

*L*

_{2}, they are defined as

*I*

_{0}(

*x*) =

*J*

_{0}(

*ix*) is the modified Bessel function of the first kind.

### 4.1 Mean irradiance

**r**

_{1}=

**r**

_{2}=

**r**, the MCF determines the mean irradiance

*I*

_{0}(

*x*), and the property of Gamma function, Γ(x),

5. L. C. Andrews, R. L. Phillips, and A. R. Weeks, “Propagation of a Gaussian-beam wave through a random phase screen,” Waves Random Media **7**(2), 229–244 (1997). [CrossRef]

*d*

_{3}=

*L*

_{3}/

*L*.

*L*= 1m, and move the phase screen between transmitter and receiver to change the receive plane optical field. Figure 2 gives the comparison of the mean irradiance for two models, as a function of (

*kρ*

^{2}/

*L*)

^{1/2}. We can find out that there are few differences between Eqs. (37) and (38), even for thick phases (

*d*/

*L*= 0.1,

*d*= 100

*mm*), so the thin phase screen model for the mean irradiance can approximate to the arbitrary model well.

### 4.2 Mutual coherent function

*ρ*, when they are central symmetry about optical axis,

**r**

_{1}= -

**r**

_{2},

**7**(2), 229–244 (1997). [CrossRef]

*kρ*

^{2}/

*L*)

^{1/2}, for arbitrary thickness model and thin phase screen model. From the comparison in Fig. 3, the difference of MCF for two models, Eqs. (45) and (46), is not apparent when the phase screen relative thickness

*d*/

*L*is less than 0.01 (

*d*= 10

*mm*), but it increase as the thickness increasing. The MCF for thin phase screen approximation is more accurate when the phase screen is near to the receiver plane than it is in other position. Therefore, it is better to use Eq. (45), MCF for arbitrary thickness model, to describe the normalized MCF when the relative thickness

*d*/

*L*is greater than 0.01. As a whole tendency, the MCF decreases as the separation distance

*ρ*of two observation points increasing, which means that the further the two observation points separate, the worse the relativity of the two observation points is. And the weakening of the relativity of the two points is more severe when the phase screen is close to input plane.

### 4.3 Modulus of the complex degree of coherent

**r**

_{1},

**r**

_{2},

*L*) = Re[∆(

**r**

_{1},

**r**

_{2},

*L*)] is the wave structure function (WSF). Written as a function of separation distance

*ρ*, DOC(

*ρ*,

*r*,

*L*) and D(

*ρ*,

*r*,

*L*), the spatial coherence radius

*ρ*

_{0}is defined by DOC(

*ρ*

_{0},

*r*,

*L*) = 1/

*e*or D(

*ρ*

_{0},

*r*,

*L*) = 2.

**r**

_{1}= -

**r**

_{2}, the WSF for Gaussian beam propagation through an arbitrary thickness phase screen can be expressed as

**7**(2), 229–244 (1997). [CrossRef]

*L*

_{1}/

*kW*

_{0}

^{2}, for different thickness phase screen. The DOC, deduced by thin phase screen approximation, can only approximate the spatial coherent loss when the phase screen is located near the beam waist, and the further away from beam waist the phase screen is, the greater the differences between the two models, Eqs. (48) and (49), are. For the arbitrary thickness model, the DOC is near unit when the phase screen is close to beam waist, and it gets down as the phase screen is far away from beam waist. So the DOC [5

**7**(2), 229–244 (1997). [CrossRef]

## 5. Scintillation index

*σ*

_{I}^{2}(

**r**,

*L*), one of the most important components of fourth order field moment, describes the irradiance fluctuation on receiver plane. When the log-amplitude variance is sufficiently small

*σ*

_{χ}^{2}= 1, which is usually valid, the scintillation index [4] is defined as

*σ*

_{I,r}^{2}(

**,**

*r**L*) and

*σ*

_{I,l}^{2}(

*L*), in the form [4]

15. D. L. Fried and J. B. Seidman, “Laser-beam scintillation in the atmosphere,” J. Opt. Soc. Am. **57**(2), 181–185 (1967). [CrossRef]

**7**(2), 229–244 (1997). [CrossRef]

*d*/

*L*are 10

^{−4}and 0.01, as a function of the phase screen position 2

*L*

_{1}/

*kW*

_{0}

^{2}. From Fig. 4, it is clear that there are apparent differences between the two models, Eqs. (56) and (57), even for the thin phase screen,

*d*/

*L*= 10

^{−4}. For the relationship described by Eq. (56), the scintillation index is get down firstly and then increasing, as the phase screen moving from input plane to output plane, and it obtain the minimum value when the phase screen is located in beam waist. The scintillation index is in direct proportion to the thickness of phase screens when they are located in the same position. The results predicted by scintillation index model, Eq. (56), are according with those shown by DOC model, Eq. (47), because the weakening of coherence results in the fluctuation of irradiance as shown in Fig. 4 and Fig. 5 .

## 6. Summary

**12**(1), 137–150 (1995). [CrossRef]

*d*/

*L*< 0.01), the two models have few differences, so both of them can be used to describe these quantities accurately. But for MCF of thick phase screen, DOC, and scintillation index, we’d better to utilize the arbitrary thickness phase screen model to describe them, because the thin phase screen model has intrinsic error due to the approximation. The paper offers a theory fundamental, for AO and laser communication application research in laboratory.

## Acknowledgments

## References and links

1. | C. Chao, L. Hu, and Q. Mu, “Bandwidth requires of adaptive optical system for horizontal turbulence correction,” Opt. Precision Eng. |

2. | B. Wang, Z.-y. Wang, J.-l. Wang, J.-Y. Zhao, Y.-H. Wu, S.-X Zhang, L. Dong, and M. Wen, “Phase-diverse speckle imaging with two cameras,” Opt. Precision Eng. |

3. | H. G. Booker, J. A. Ferguson, and H. O. Vats, “Comparison between the extended-medium and the phase-screen scintillation theories,” J. Atmos. Terr. Phys. |

4. | L. C. Andrews and R. L. Phillips, |

5. | L. C. Andrews, R. L. Phillips, and A. R. Weeks, “Propagation of a Gaussian-beam wave through a random phase screen,” Waves Random Media |

6. | L. C. Andrews, W. B. Miller, and J. C. Ricklin, “Propagation through complex optical system: a phase screen analysis,” SPIE |

7. | L. C. Andrews and W. B. Miller, “Single- and double-pass propagation through complex paraxial optical systems,” J. Opt. Soc. Am. A |

8. | S. V. Mantravadi, T. A. Rhoadarmer, and R. S. Glas, “Simple laboratory system for generating well-controlled atmospheric-like turbulence,” Proc. SPIE |

9. | X. J. Gan, J. Guo, and Y. Y. Fu, “The simulating turbulence method of laser propagation in the inner field,” J. Phys. Conf. Ser. |

10. | B. D. Zhang, S. Qin, and X. S. Wang, “Accurate and fast simulation of Kolmogorov phase screen by combining spectral method with Zernike polynomials method,” Chin. Opt. Lett. |

11. | M. X. Qian, W. Y. Zhu, and R. Rao, “Phase screen distribution for simulating laser propagation along an inhomogeneous atmospheric path,” Acta Phys. Sinica |

12. | M. Gao and Z. Wu, “Experiments of effect of beam spreading of far-field on aiming deviation,” Opt. Precision Eng. |

13. | Y.-h. Gao, Z.-y. An, N.-n. Li, W.-x Zhao, and J.-s. Wang, “Optical design of Gaussian beam shaping,” Opt. Precision Eng. |

14. | V. I. Tatarskii, |

15. | D. L. Fried and J. B. Seidman, “Laser-beam scintillation in the atmosphere,” J. Opt. Soc. Am. |

**OCIS Codes**

(010.1080) Atmospheric and oceanic optics : Active or adaptive optics

(010.1290) Atmospheric and oceanic optics : Atmospheric optics

(010.1300) Atmospheric and oceanic optics : Atmospheric propagation

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(060.4510) Fiber optics and optical communications : Optical communications

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: July 7, 2011

Revised Manuscript: August 15, 2011

Manuscript Accepted: August 22, 2011

Published: September 1, 2011

**Citation**

Yuzhen Tian, Jin Guo, Rui Wang, and Tingfeng Wang, "Mathematic model analysis of Gaussian beam propagation through an arbitrary thickness random phase screen," Opt. Express **19**, 18216-18228 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-19-18216

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### References

- C. Chao, L. Hu, and Q. Mu, “Bandwidth requires of adaptive optical system for horizontal turbulence correction,” Opt. Precision Eng.18(10), 2137–3142 (2010).
- B. Wang, Z.-y. Wang, J.-l. Wang, J.-Y. Zhao, Y.-H. Wu, S.-X Zhang, L. Dong, and M. Wen, “Phase-diverse speckle imaging with two cameras,” Opt. Precision Eng.19(6), 1384–1390 (2011). [CrossRef]
- H. G. Booker, J. A. Ferguson, and H. O. Vats, “Comparison between the extended-medium and the phase-screen scintillation theories,” J. Atmos. Terr. Phys.47(38), 1–399 (1985). [CrossRef]
- L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, 2nd ed. (SPIE, 2005).
- L. C. Andrews, R. L. Phillips, and A. R. Weeks, “Propagation of a Gaussian-beam wave through a random phase screen,” Waves Random Media7(2), 229–244 (1997). [CrossRef]
- L. C. Andrews, W. B. Miller, and J. C. Ricklin, “Propagation through complex optical system: a phase screen analysis,” SPIE2312, 122–129 (1994). [CrossRef]
- L. C. Andrews and W. B. Miller, “Single- and double-pass propagation through complex paraxial optical systems,” J. Opt. Soc. Am. A12(1), 137–150 (1995). [CrossRef]
- S. V. Mantravadi, T. A. Rhoadarmer, and R. S. Glas, “Simple laboratory system for generating well-controlled atmospheric-like turbulence,” Proc. SPIE5553, 290–300 (2004). [CrossRef]
- X. J. Gan, J. Guo, and Y. Y. Fu, “The simulating turbulence method of laser propagation in the inner field,” J. Phys. Conf. Ser.48, 907–910 (2006). [CrossRef]
- B. D. Zhang, S. Qin, and X. S. Wang, “Accurate and fast simulation of Kolmogorov phase screen by combining spectral method with Zernike polynomials method,” Chin. Opt. Lett.8(10), 969–971 (2010).
- M. X. Qian, W. Y. Zhu, and R. Rao, “Phase screen distribution for simulating laser propagation along an inhomogeneous atmospheric path,” Acta Phys. Sinica58(9), 6633–6639 (2009).
- M. Gao and Z. Wu, “Experiments of effect of beam spreading of far-field on aiming deviation,” Opt. Precision Eng.18(3), 602–608 (2010).
- Y.-h. Gao, Z.-y. An, N.-n. Li, W.-x Zhao, and J.-s. Wang, “Optical design of Gaussian beam shaping,” Opt. Precision Eng.19(7), 1464–1471 (2011). [CrossRef]
- V. I. Tatarskii, Wave Propagation in a Turbulent Medium (New York: McGraw-Hill, 1961).
- D. L. Fried and J. B. Seidman, “Laser-beam scintillation in the atmosphere,” J. Opt. Soc. Am.57(2), 181–185 (1967). [CrossRef]

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