## Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence |

Optics Express, Vol. 18, Issue 20, pp. 21269-21283 (2010)

http://dx.doi.org/10.1364/OE.18.021269

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### Abstract

A generalized exponential spectrum model is derived, which considers finite turbulence inner and outer scales and has a general spectral power law value between the range 3 to 5 instead of standard power law value 11/3. Based on this generalized spectrum model, a new generalized long exposure turbulence modulation transfer function (MTF) is obtained for optical plane and spherical wave propagating through horizontal path in weak fluctuation turbulence. When the inner scale and outer scale are set to zero and infinite, respectively, the new generalized MTF is reduced to the classical generalized MTF derived from the non-Kolmogorov spectrum.

© 2010 OSA

## 1. Introduction

3. D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE **2120**, 43–55 (1994). [CrossRef]

6. A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov trubulence,” Atmos. Res. **88**(1), 66–77 (2008). [CrossRef]

8. N. S. Kopeika, A. Zilberman, and E. Golbraikh, “Generalized atmospheric turbulence: implications regarding imaging and communications,” Proc. SPIE **7588**, 758808 (2010). [CrossRef]

## 2. Generalized exponential spectrum

### 2.1 Exponential spectrum

*κ*is the spatial wave number,

8. N. S. Kopeika, A. Zilberman, and E. Golbraikh, “Generalized atmospheric turbulence: implications regarding imaging and communications,” Proc. SPIE **7588**, 758808 (2010). [CrossRef]

9. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of Arrival Fluctuations for Free Space Laser Beam Propagation through non Kolmogorov turbulence,” Proc. SPIE **6551**, 65510E (2007). [CrossRef]

8. N. S. Kopeika, A. Zilberman, and E. Golbraikh, “Generalized atmospheric turbulence: implications regarding imaging and communications,” Proc. SPIE **7588**, 758808 (2010). [CrossRef]

9. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of Arrival Fluctuations for Free Space Laser Beam Propagation through non Kolmogorov turbulence,” Proc. SPIE **6551**, 65510E (2007). [CrossRef]

### 2.2 Refractive-index structure function for generalized exponential spectrum model

*R*. For Kolmogorov turbulence, the relationship between

### 2.2.1 Expression derivation of A ^ ( α )

**7588**, 758808 (2010). [CrossRef]

### 2.2.2 Expression derivation of c ( α )

### 2.3 Generalized exponential spectrum model

*α*are plotted and shown in Fig. 1(a) and Fig. 1(b). When

## 3. Generalized atmospheric turbulence MTF based on the generalized exponential spectrum

11. R. E. Hufnagel and N. R. Stanley, “Modulation Transfer Function associated with Image Transmission through Turbulent Media,” J. Opt. Soc. Am. **54**(1), 52–61 (1964). [CrossRef]

12. D. L. Fried, “Optical Resolution Through a Randomly Inhomogeneous Medium for Very Long and very Short Exposures,” J. Opt. Soc. Am. **56**(10), 1372–1379 (1966). [CrossRef]

12. D. L. Fried, “Optical Resolution Through a Randomly Inhomogeneous Medium for Very Long and very Short Exposures,” J. Opt. Soc. Am. **56**(10), 1372–1379 (1966). [CrossRef]

*ρ*is the separation between points in the image plane transverse to the direction,

*ν*is the spatial frequency measured in cycles per unit length,

*F*is focal length.

*L*is the optical path.

*α*is further restricted to the range 3 to 4 just as [7,8

**7588**, 758808 (2010). [CrossRef]

13. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical Propagation in non-Kolmogorov Atmospheric Turbulence,” Proc. SPIE **2471**, 181–196 (1995). [CrossRef]

### 3.1 Plane wave structure function for non-Kolmogorov turbulence

*z*, as a result, Eq. (27) is expressed as

13. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical Propagation in non-Kolmogorov Atmospheric Turbulence,” Proc. SPIE **2471**, 181–196 (1995). [CrossRef]

### 3.2 Spherical wave structure function for non-Kolmogorov turbulence

*z*, Eq. (36) becomesWhere,

13. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical Propagation in non-Kolmogorov Atmospheric Turbulence,” Proc. SPIE **2471**, 181–196 (1995). [CrossRef]

### 3.3 Generalized atmospheric turbulence MTF for plane wave and spherical wave

*D*is the receiver aperture diameter.

## 4. Numerical results

*α*,

### 4.1 Effect of outer scale’s variation on the MTF

*α*and

*α*is set to a constant value of 11/3. Then outer scale

14. A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. **47**(34), 6385–6391 (2008). [CrossRef] [PubMed]

15. W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Log-amplitude variance and wave structure function: a new perspective for Gaussian beams,” J. Opt. Soc. Am. A **10**(4), 661–672 (1993). [CrossRef]

9. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of Arrival Fluctuations for Free Space Laser Beam Propagation through non Kolmogorov turbulence,” Proc. SPIE **6551**, 65510E (2007). [CrossRef]

### 4.2 Effect of inner scale’s variation on MTF

*α*and

*α*is chosen to be 11/3 and

14. A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. **47**(34), 6385–6391 (2008). [CrossRef] [PubMed]

## 4.3 Effect of *α*’s variation on MTF

*α*’s influence on MTF,

*α*’s changes on MTF. In this simulations,

*α*is set to 10/3, 11/3 and 3.9 respectively, and the corresponding MTFs as a function of normalized spatial frequency are plotted in Fig. 4 , from which we can infer that various

*α*brings different impact on the form of MTF just as in [7,8

**7588**, 758808 (2010). [CrossRef]

## 5. Conclusions

## Acknowledgments

## References and links

1. | V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, (trans.for NOAA by Israel Program for Scientific Translations, Jerusalem, 1971). |

2. | L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media. (SPIE Optical Engineering Press, Bellingham, 2005). |

3. | D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE |

4. | M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE |

5. | M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE |

6. | A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov trubulence,” Atmos. Res. |

7. | A. Zilberman, and N. S. Kopeika, “Slant-path generalized atmospheric MTF,” in Proceedings of IEEE 25th Convention of Electrical and Electronics Engineers (Institute of Electrical and Electronics Engineers, Israel, 2008), pp. 217–221. |

8. | N. S. Kopeika, A. Zilberman, and E. Golbraikh, “Generalized atmospheric turbulence: implications regarding imaging and communications,” Proc. SPIE |

9. | I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of Arrival Fluctuations for Free Space Laser Beam Propagation through non Kolmogorov turbulence,” Proc. SPIE |

10. | L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE Optical Engineering Press, Bellingham, Wash., 1998). |

11. | R. E. Hufnagel and N. R. Stanley, “Modulation Transfer Function associated with Image Transmission through Turbulent Media,” J. Opt. Soc. Am. |

12. | D. L. Fried, “Optical Resolution Through a Randomly Inhomogeneous Medium for Very Long and very Short Exposures,” J. Opt. Soc. Am. |

13. | B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical Propagation in non-Kolmogorov Atmospheric Turbulence,” Proc. SPIE |

14. | A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. |

15. | W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Log-amplitude variance and wave structure function: a new perspective for Gaussian beams,” J. Opt. Soc. Am. A |

**OCIS Codes**

(010.1290) Atmospheric and oceanic optics : Atmospheric optics

(010.1330) Atmospheric and oceanic optics : Atmospheric turbulence

(110.4100) Imaging systems : Modulation transfer function

(110.0115) Imaging systems : Imaging through turbulent media

**ToC Category:**

Atmospheric and Oceanic Optics

**History**

Original Manuscript: August 9, 2010

Revised Manuscript: September 1, 2010

Manuscript Accepted: September 5, 2010

Published: September 22, 2010

**Citation**

Cui Lin-yan, Xue Bin-dang, Cao Xiao-guang, Dong Jian-kang, and Wang Jie-ning, "Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence," Opt. Express **18**, 21269-21283 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-20-21269

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### References

- V. I. Tatarskii, The Effects of the Turbulent Atmosphere on Wave Propagation, (trans.for NOAA by Israel Program for Scientific Translations, Jerusalem, 1971).
- L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media. (SPIE Optical Engineering Press, Bellingham, 2005).
- D. T. Kyrazis, J. Wissler, D. B. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in the upper troposphere and lower stratosphere,” Proc. SPIE 2120, 43–55 (1994). [CrossRef]
- M. S. Belen’kii, S. J. Karis, J. M. Brown, and R. Q. Fugate, “Experimental study of the effect of non-Kolmogorov stratospheric turbulence on star image motion,” Proc. SPIE 3126, 113–123 (1997). [CrossRef]
- M. S. Belen’kii, E. Cuellar, K. A. Hughes, and V. A. Rye, “Experimental study of spatial structure of turbulence at Maui Space Surveillance Site (MSSS),” Proc. SPIE 6304, 63040U (2006). [CrossRef]
- A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosol turbulence characteristics in the troposphere: Kolmogorov and non-Kolmogorov trubulence,” Atmos. Res. 88(1), 66–77 (2008). [CrossRef]
- A. Zilberman, and N. S. Kopeika, “Slant-path generalized atmospheric MTF,” in Proceedings of IEEE 25th Convention of Electrical and Electronics Engineers (Institute of Electrical and Electronics Engineers, Israel, 2008), pp. 217–221.
- N. S. Kopeika, A. Zilberman, and E. Golbraikh, “Generalized atmospheric turbulence: implications regarding imaging and communications,” Proc. SPIE 7588, 758808 (2010). [CrossRef]
- I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of Arrival Fluctuations for Free Space Laser Beam Propagation through non Kolmogorov turbulence,” Proc. SPIE 6551, 65510E (2007). [CrossRef]
- L. C. Andrews, Special Functions of Mathematics for Engineers, 2nd ed. (SPIE Optical Engineering Press, Bellingham, Wash., 1998).
- R. E. Hufnagel and N. R. Stanley, “Modulation Transfer Function associated with Image Transmission through Turbulent Media,” J. Opt. Soc. Am. 54(1), 52–61 (1964). [CrossRef]
- D. L. Fried, “Optical Resolution Through a Randomly Inhomogeneous Medium for Very Long and very Short Exposures,” J. Opt. Soc. Am. 56(10), 1372–1379 (1966). [CrossRef]
- B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical Propagation in non-Kolmogorov Atmospheric Turbulence,” Proc. SPIE 2471, 181–196 (1995). [CrossRef]
- A. Zilberman, E. Golbraikh, and N. S. Kopeika, “Propagation of electromagnetic waves in Kolmogorov and non-Kolmogorov atmospheric turbulence: three-layer altitude model,” Appl. Opt. 47(34), 6385–6391 (2008). [CrossRef] [PubMed]
- W. B. Miller, J. C. Ricklin, and L. C. Andrews, “Log-amplitude variance and wave structure function: a new perspective for Gaussian beams,” J. Opt. Soc. Am. A 10(4), 661–672 (1993). [CrossRef]

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