Abstract

It is a standard assumption in the theory of optical propagation through the turbulent atmosphere that the refractive-index fluctuations $n_1(\mathbf{x})$ are statistically isotropic. It is well known, however, that $n_1(\mathbf{x})$ in the free atmosphere and in the nocturnal boundary layer is often strongly anisotropic, even at very small scales. Here we present and discuss a model atmosphere characterized by randomly undulating, non-turbulent and non-overturning, quasi-horizontal refractive-index interfaces, or “sheets.” We assume $n_1(\mathbf{x})=v[z-h(x,y)]$, where $v(z)$ is a random function that has a 1D spectrum $V({\kappa }_z)$, and where $h(x,y)$ is a vertical displacement that varies randomly as a function of the horizontal coordinates $x$ and $y$. We derive a closed-form expression for the 3D spectrum $\mathrm{\Phi }(\mathit{\kappa })$ and show that the horizontal 1D spectra have the same power law as $V({\kappa }_z)$ if the structure function of $h(x,y)$ is quadratic. Moreover, we evaluate the scintillation index ${\sigma }_I^2$ for a plane wave propagating horizontally through the undulating sheets, and we compare ${\sigma }_I^2$ predicted for undulating sheets with Tatarskii’s classical predictions of ${\sigma }_I^2$ for fully developed, isotropic turbulence. For Phillips-type sheets, where $V({\kappa }_z)\propto {\kappa }_z^{-2}$, in the diffraction limit we find ${\sigma }_I^2\propto k$ (where $k=2\pi /\lambda$ is the optical wavenumber), which is only slightly different from Tatarskii’s famous $k^{7/6}$ law for propagation through fully developed, Obukhov–Corrsin-type, isotropic turbulence where $\mathrm{\Phi }(\kappa )\propto {\kappa }^{-11/3}$. Our model predicts that ${\sigma }_I^2$ is inversely proportional to the sheet tilt angle standard deviation $\sqrt{〈{\theta }_x^2〉}$, regardless of whether or not diffraction plays a role and regardless of the value of the power-law exponent of $V({\kappa }_z)$.

© 2016 Optical Society of America

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