Scattering by Weakly Nonlinear Objects

Abstract

An attempt is made to construct a consistent scattering theory for systems involving nonlinear objects. Weak nonlinearity is assumed, such that harmonic generation is present, but shock wave formation is excluded. Mathematically this is described by constitutive relations in the form of Volterra series. For periodic (as opposed to monochromatic) waves, this procedure facilitates algebraic constitutive relations and dispersion equations in the transform space. Weak nonlinearity, as defined here, implies phase matching, i.e., all harmonics of the tundamental wave possess the same phase velocity (this is the reason that shock waves cannot be formed). Due to this stipulation superposition is allowed in a restricted sense, facilitating the construction of arbitrary wave solutions, e.g., cylindrical and spherical waves, by using sums (integrals) of plane waves.Using wave solutions from the linear theory, various boundary value problems can be discussed. Plane interfaces are considered, displaying well known properties relevant to new technological applications, such as bistability, and nonlinearly induced transparency. Scattering from cylinders and spheres is discussed, pointing out the new phenomena that should be observed, due to the presence of nonlinearity.Deterministic multiple scattering theory is extended to include the present class of problems. Propagation of waves in random media of distinct nonlinear objects is discussed in a cursory manner, essentially to demonstrate that the Foldy-Twersky method of medium synthesis can be extended to the present case. This suggests a new area of interest for future research. Introductory remarks. The status of scattering theory in the presence of objects involving nonlinear media is radically different from what we have in the linear case. Although there are many open problems in the linear theory, it has been around for many decades and is in a quite advanced stage. References to early results are well documented, e.g., see Stratton [1]. The mathematical methods and the properties of the relevant special functions are given, e.g., Morse and Feshbach [2]. This maturity reached for the class of linear problems is not matched in the case of scattering in the presence of nonlinear media, probably because of the complexity of the latter subject. Many aspects of nonlinear wave propagation have been discussed, but a thorough review of the literature is out of the scope of the present study. Some linkage to the existing (mostly early) literature is provided by the books cited here [3]-[12].It appears that a scattering theory as we know it for the linear case is practically nonexistent, except for specialized discussions, mainly concerning plane waves and plane interfaces (and their logical extension in the context of geometrical optics).An attempt is made here to construct a scattering theory for nonlinear systems, along the lines of the classical linear theory. In order to achieve this goal, the principle of superposition, valid for linear systems, must be salvaged. This is feas...

Assessment of the Effectiveness of Iterative Techniques for the Evaluation of the Electromagnetic Scattering by Weakly Nonlinear Dielectric Cylinders

“…The main objective of this example is to evaluate the capabilities of the iterative approaches versus the statistical cooling solution. However, it is worth noting that, only for circular cylinders, results obtained by the distorted-wave Born approximation have already been compared in [9] with those obtained by Hasan and Uslenghi [13] by using a per-turbation method and with those obtained by Censor [14] by using an iterative method. Figure 1 gives the values for k = 1 , k = 2 , and k = k * (this value of k turned out to be sufficient for the assumed convergence, i.e., at step k * the residual error {k * } turned out to be less than a fixed threshold value, th ; in all the simulations, we assumed th = 10 −4 ).…”

"Waves," "Objects" and Special Relativity

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Electrodynamics, Topsy-Turvy Special Relativity, and Generalized Minkowski Constitutive Relations for Linear and Nonlinear Systems

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