Theory of Scattering from Periodic Random Surfaces and Random Media

1998-03-25

1998-03-25

This thesis studies a probabilistic theory of wave scattering from periodic random surfaces and random media. New concepts for analysis and several numerical results are described. We assume a model such that a periodic random surface is generated by a periodic　translation of a local surface boss; the height of each boss is modulated by a stationary random sequence. Mathematically, such a periodic random surface is a periodic stationary process that is a function of position and a sample point in the sample space. Its average and correlation function　become periodic functions of position. A periodic stationary　process is known to have a harmonic series representation,　similar to the Fourier series, where “Fourier coefficients” are mutually correlated stationary processes rather than constants. Moreover, it is invariant under the two-dimensional translation which translates a sample function　with a distance of the period and a sample　point into another sample point. By the group theoretic translation of such a translation invariance,　we first find that the scattered wave for an incident plane wave has a stochastic Floquet form, that is a product of a periodic stationary process and an exponential phase factor. This means that the coherent wave is diffracted into only discrete directions, because　the average of a periodic stationary process is a periodic function. Furthermore, the scattering problem is mathematically reduced to finding out such a periodic stationary process, which is written by the harmonic series representation with stationary processes as the “Fourier coefficients”. We represent such stationary processes in terms of functional expansions with unknown kernel functions, a set of equations for which is obtained from the boundary condition. Once kernel functions are　determined, we reversely obtain the scattered wave field as a random function, from which any statistical properties of the scattering can be calculated. First, we consider the case　where the periodic random surface is generated by a Gaussian random sequence. For a TE plane incident wave, we approximately determine kernel functions. Then, we find that the incoherent scattering has ripples in angular distributions, because incoherent waves generated by coherently diffracted waves with different orders interfere. ln the properties of the scattering and diffraction, we find several anomalies for a TM incident wave, which are caused by the guided surface waves propagating along the periodic random surface. When a coherently diffracted wave couples with the guided surface waves, Wood's anomaly appears as a rapid variation of the diffracted　power against the angle of incidence. Since the guided surface waves are excited by the surface randomness and diffracted again, strong anomalous peaks, which we call incoherent Wood's anomaly, appear in the angular distribution of the incoherent scattering. It is shown that the incoherent Wood’s anomaly is independent of the angle of incidence but is determined by only period and wavelength. Further, because the guided surface waves are scattered and diffracted again, we find the enhanced backscattering and diffracted backscattering enhancement. Second, we discuss the case where the height of the periodic surfaces is randomly deformed by a binary stationary sequence. Though the stochastic functional theory of the binary stationary sequence has been discussed, no mathematical formulas have given in explicit form. However, we first obtain several new explicit formulas. Then, we apply these formulas to diffraction and scattering problems, where periodic random surface is deformed by binary stationary sequence. We find that diffraction efficiencies are related with both the height of average periodic surface and the amplitude of random deformation, but the total diffraction power depends on only the amplitude of random deformation. Finally, we discuss the scattering from a two-dimensional thin film with randomly fluctuating permittivity. For a Gaussian random fluctuation, the first order solution is derived explicitly by a probabilistic method. Using the solution, we calculate several statistical properties of the scattering. Then, we demonstrate that ripples appear in the angular distribution of the incoherent scattering. Moreover, the incoherent scattering can be enhanced in the directions of backscattering and specular reflection. We clarify that these phenomena occur from a combination of ‘single-scattering’ by the random thin film and multiple reflection between the top and bottom surfaces of the thin film.

博士（学術）

京都工芸繊維大学

14303甲第168号

http://hdl.handle.net/10212/2237

en

ETD

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