Parabolic Equation Extended to the Analysis of Wide-Angle Scattering in Random Propagation Media

For the last 36 years, the analysis of electromagnetic wave propagation through random media, such as atmospheric turbulence, has been aided by the parabolic equation method. The resulting theory has been applied to everything from laser beam propagation to image propagation within the Earth’s atmosphere and ocean as well as in other planetary atmospheres. Here, the general stochastic scalar Helmholtz wave equation is reduced to a corresponding stochastic parabolic differential equation for the random electric field. From this equation, various statistical moments of the propagating wave field can be derived.

This approximate treatment is applicable so long as the wavelength λ of the electromagnetic wave is small in comparison to the characteristic size I₀ of the random inhomogeneity of the permittivity field of the medium. In this case, the scattered radiation from the inhomogeneities is concentrated in a narrow cone with a vertex angle θ~λ/I₀ << 1. Hence, the scattered waves propagate in essentially the same direction as the primary wave. This defines the small scattering angle scenario of wave propagation to which the parabolic equation applies. The most important statistical quantity derived from the parabolic equation is the mutual coherence function (MCF), which is defined by the ensemble average of the product of the electric field with its complex conjugate in a plane transverse to the direction of propagation. This is an important quantity for describing the degradation of a laser beam or an image propagating through a random medium.

The use of the parabolic equation comes into question, however, when is on the order of I₀ and the scattering angle becomes large. This situation is encountered in the propagation of millimeter waves through the Earth’s atmosphere or through sandstorms on the surface of Mars. Consequently, at the NASA Glenn Research Center, a complete mathematical analysis of the basis of the parabolic equation was made to extend its applicability to wide-scattering-angle situations in which θ~λ/I₀ ~ 1. An operator form of the extended parabolic equation was written on which operator techniques were employed in the solution for a corresponding MCF. This theoretical exercise not only extended the parabolic equation to wide scattering angles, but also revealed the multiple scattering structure of the Green functions that enter into the theory. This resulted in an extended parabolic equation that can describe wide-angle scattering and that, unlike the traditional treatment, can consider longitudinal variations of the MCF of a wave field along the direction of propagation. The latter capability is due to the incorporation of higher order scattering processes into the theory.

The figures show the behavior of the on-axis longitudinal MCF, Γ₁₁(x_c, x_d, 0), as a function of longitudinal separation, x_d, for a typical Earth atmosphere propagation scenario for a wave source of wavelength λ = 0.63 μm (red light from, e.g., a helium-neon laser), placed x_c = 5 km from the measurement point, for two values of inner scale size (i.e., the smallest inhomogeneity of the turbulent fluctuations): I₀ = 1.0 mm and I₀ = 1.0 cm, respectively. In both figures, the structure constant of refractive index fluctuations is C_n² = 1×10^-12 m^-2/3. Both the real and imaginary values of the MCF are displayed. The oscillatory behavior of the longitudinal MCF is due simply to the movement of the difference coordinate through the Fresnel zones intersecting the longitudinal axis. The oscillations decay because of the loss of longitudinal coherence as the difference coordinate becomes larger.

Γ₁₁ = (x_c, x_d, 0) versus x_d (in meters) for l = 0.63 μm, I₀ = 1.0 mm, x_c = 5 km, and C_n² = 10^-12 m^-2/3.

Γ₁₁ = (x_c , x_d , 0) versus x_d (in meters) for l = 0.63 μm, I₀ = 1.0 cm, x_c = 5 km, and C_n2 = 10^-12 m^-2/3.

The characteristic behavior of the longitudinal MCF suggests a potential remote-sensing technique for atmospheric turbulence whereby the inner scales of turbulence can be ascertained on the basis of the spatial frequency of the longitudinal variations. Other results from this study are (1) a complete expression for the transverse and longitudinal MCF for any random medium for which the condition λ ≤ I₀ prevails and (2) a verification of the fact that the classical parabolic equation is applicable for use in a medium characterized by the Kolmogorov spectrum of fluctuations even though λ ~ I₀. This latter fact is due to the Kolmogorov spectral density level near the inner scale of turbulence being much smaller than it is at the larger scale sizes at which most of the narrow-angle scattering occurs.

Bibliography

Manning, Robert M.: Application of an Extended Parabolic Equation to the Calculation of the Mean Field and the Transverse and Longitudinal Mutual Coherence Functions Within Atmospheric Turbulence. NASA/TM--2005-213841, 2005. http://gltrs.grc.nasa.gov/cgi-bin/GLTRS/browse.pl?2005/TM-2005-213841.html

Manning, Robert M.: An Extended Parabolic Equation and Its Application to the Calculation of Transverse and Longitudinal Mutual Coherence Functions in Atmospheric Turbulence. Waves in Random and Complex Media, vol. 15, no. 3, 2005, pp. 405-416.

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