Optical Review

, Volume 24, Issue 3, pp 253–259 | Cite as

Effects of inner and outer scale on the modulation transfer function for a Gaussian wave propagating through anisotropic non-Kolmogorov turbulence

Regular Paper


Both experimental results and empirical research have shown that the atmospheric turbulence can present the anisotropic property not only at a few meters above the ground but also at high altitudes of up to several kilometers. This paper investigates the modulation transfer function of a Gaussian beam propagating along a horizontal path in weak anisotropic non-Kolmogorov turbulence. Mathematical expressions are obtained based on the generalized exponential spectrum for anisotropic turbulence, which includes the spectral power law value, the finite inner and outer scales of turbulence, the anisotropic factor, and other essential optical parameters of the Gaussian beam. The numerical results indicate that the atmospheric turbulence would produce less negative effects on the wireless optical communication system with an increase in the anisotropic factor.


Anisotropy Gaussian beam Modulation transfer function Non-Kolmogorov turbulence Inner and outer scales 


  1. 1.
    Andrews, L. C., Phillips, R. L.: Laser-beam propagation through random media, 2nd ed. SPIE Optical Engineering Press (2005)Google Scholar
  2. 2.
    Ishimaru, A.: Wave propagation and scattering in random media. Wiley-IEEE Press (1999)Google Scholar
  3. 3.
    Henniger, H., Wilfert, O.: An introduction to free-space optical communications. Radio Eng. 19(2), 203–212 (2010)Google Scholar
  4. 4.
    Luo, Q., Luo, X., Li, X.: Optical axis jitter rejection for double overlapped adaptive optics systems. Opt. Rev. 23(2), 273–283 (2016)CrossRefGoogle Scholar
  5. 5.
    Xiao, X., Voelz, D. G., Toselli, I., et al.: Gaussian beam propagation in anisotropic turbulence along horizontal links: theory, simulation, and laboratory implementation. Appl. Optics. 55(15), 4079–4084 (2016)ADSCrossRefGoogle Scholar
  6. 6.
    Mansour, A., Mesleh, R., Abaza, M.: New challenges in wireless and free space optical communications. Optics Laser Eng. 89, 95–108 (2017)Google Scholar
  7. 7.
    Li, Y., Gao, C., Liang, H., Miao, M., Li, X.: Performance of an adaptive phase estimator for coherent free-space optical communications over gamma–gamma turbulence. Optics Commun. 388, 47–52 (2017)Google Scholar
  8. 8.
    Cui, L., Xue, B.: Influence of anisotropic turbulence on the long-range imaging system by the MTF model. Infrared Phys. Technol. 72(1), 229–238 (2015)ADSCrossRefGoogle Scholar
  9. 9.
    Cui, L., Xue, B., Cao, X., et al.: Generalized atmospheric turbulence MTF for wave propagating through non-Kolmogorov turbulence. Opt. Express. 18(20), 21269–21283 (2010)CrossRefGoogle Scholar
  10. 10.
    Gao, C., Li, X.: Modulation transfer function of a Gaussian beam based on the generalized modified atmospheric spectrum. Int. J. Optics 2016, 1–8 (2016)Google Scholar
  11. 11.
    Kolmogorov, A. N.: The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Proc. R. Soc. A. 434(1892), 9–13 (1991)ADSMathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Toselli, I., Andrews, L. C., Phillips, R. L., et al.: Free space optical system performance for a Gaussian beam propagating through non-Kolmogorov weak turbulence. IEEE Trans. Antennas Propag. 57(6), 1783–1788 (2009)ADSCrossRefGoogle Scholar
  13. 13.
    Consortini, A. A., Ronchi, L., Stefanutti, L.: Investigation of atmospheric turbulence by narrow laser beams. Appl. Optics. 9(11), 2543–2547 (1970)ADSCrossRefGoogle Scholar
  14. 14.
    Dalaudier, F., Sidi, C.: Direct evidence of “sheets” in the atmospheric temperature field. J. Atmos. Sci. 51(2), 237–248 (1994)ADSCrossRefGoogle Scholar
  15. 15.
    Toselli, I., Agrawal, B., Restaino, S.: Light propagation through anisotropic turbulence. J. Opt. Soc. Am. A. 28(3), 483–488 (2011)ADSCrossRefGoogle Scholar
  16. 16.
    Toselli, I.: Introducing the concept of anisotropy at different scales for modeling optical turbulence. J. Opt. Soc. Am. A. 31(8), 1868–1875 (2014)ADSCrossRefGoogle Scholar
  17. 17.
    Toselli, I., Korotkova, O.: General scale-dependent anisotropic turbulence and its impact on free space optical communication system performance. J. Opt. Soc. Am. A. 32(6), 1017–1025 (2015)ADSCrossRefGoogle Scholar
  18. 18.
    Gao, C., Li, Y., Li, Y., et al.: Irradiance scintillation index for a Gaussian beam based on the generalized modified atmospheric spectrum with aperture averaged. Int. J. Opt. (1), 8730609 (2016)Google Scholar
  19. 19.
    Cui, L., Xue, B., Zhou, F.: Generalized anisotropic turbulence spectra and applications in the optical waves’ propagation through anisotropic turbulence. Opt. Express. 23(23), 30088–30103 (2015)ADSCrossRefGoogle Scholar
  20. 20.
    Young, C. Y., Masino, A. J., Thomas, F. E., Subich, C. J.: The wave structure function in weak to strong fluctuations: an analytic model based on heuristic theory. Waves Random Media 14(1), 75–96 (2004)Google Scholar
  21. 21.
    Olver, F. W. J., Lozier, D. W., Boisvert, R. F., et al.: NIST Handbook of mathematical functions. Cambridge University Press (2010)Google Scholar
  22. 22.
    Andrews, L. C.: Special functions of mathematics for engineers, 2nd ed. SPIE Optical Engineering (1998)Google Scholar
  23. 23.
    Gradshteyn, I. S., Ryzhik, I. M.: Table of integrals, series, and products, 7th edn. Academic Press (2007)Google Scholar
  24. 24.
    Erdelyi A., Magnus W., Oberhettinger, F.: Tables of integral transforms. McGraw-Hill (1954)Google Scholar

Copyright information

© The Optical Society of Japan 2017

Authors and Affiliations

  1. 1.School of Astronautics and AeronauticUniversity of Electronic Science and Technology of ChinaChengduChina

Personalised recommendations