In recent years major advances have been made in the theory of wave propagation and scattering in random media. Examples are optical beam propagation through the atmospheric turbulence, ocean acoustic and optical fluctuations, wave scintillations in the planetary atmospheres, ultrasound scattering in biological media, and wave scattering in geophysical media. This paper presents an introduction to the theory of waves in random media with some applications. The media may be divided into random continuum, randomly distributed particles, and random surfaces. General formulations are given with a unified point of view. First-order scatter model, multiple scattering model, transport theory, and diffusion model are discussed together with their range of validity. The mutual coherence function formulation and its relationship with other formulations is discussed. Both cw and pulse cases are presented, and the relationships with coherence bandwidth and coherence time are clarified. Rough surface scattering is also discussed using the perturbation theory and the Kirchhoff formulation.
The problem of determining a two-dimensional indexof refraction profile from optical scattering experiments is treated within the Born approximation. Specific attention is devoted to the case where the data is generated in a set of scattering experiments employing monochromatic incident plane waves of varying directions of propagation. It is shown that this inverse scattering problem can be viewed as a form of computed tomography where, however, diffraction and scattering of the incident wave must be taken into account in the reconstruction algorithm. An algorithm for reconstructing the object profile from the scattered field data is presented and shown to be the generalization of the filtered backprojection algorithm of conventional tomography to the case where diffraction and scattering of the wavefields is taken into account. Reconstructions obtained by use of the algorithm on computer generated scattered field data are presented.
The prevalent mathematical models for treating light scat-tering from shaped objects, atmospheric turbulence, and rough surfaces are summarized with emphasis on the more modern methods. The common features of these techniques are discussed as well as the physical problems motivating their development.
We present a number of new polar sampling theorems of use in optics. The theorems are useful in image data compression, image coding, and tomography. We illustrate their applica-tion in several instances.
This paper presents a tutorial survey of the utility of the "angular spectrum of plane waves" representation of wave fields for the theoretical analysis of optical fields. It compares the representation with those more commonly employed in optical theory and describes the types of theoretical problems that benefit most from its special features.
The formal analogy between Hamiltonian classical mechanics and quantum mechanics on the one hand and geometrical optics and physical optics on the other hand is systematically presented. When the time is replaced by the axial coordinate of a cylindrically uniform optical system and Hamilton's principle replaced by Fermat's principle, classical particle trajectories correspond to rays and quantum mechanical wave functions to physical optics fields. The operator formalism of quantum mechanics then provides elegant solutions for problems associated with propagation of beams in multimode optical fibers.
Keller's geometrical theory of diffraction (GTD) represents a major breakthrough in solving a wide variety of electromagnetic (EM) radiation and scattering problems at high frequencies. In particular, the GTD is an extension of geometrical optics to include the class of diffracted rays via a generalization of Fermat's principle. These diffracted rays are initiated at certain localized parts of an illuminated surface, e.g., edges, vertices, and shadow boundaries associated with smooth convex bodies. However, being a purely ray optical theory, the original GTD fails within the transition regions adjacent to geometric optical shadow boundaries. This and other limitations of the GTD can be overcome via a new uniform version of the GTD-or the UTD, which requires the diffracted field to make the total high frequency field continuous across the optical shadow boundaries. Attention is focussed on some UTD solutions developed recently at the O.S.U. ElectroScience Laboratory; e.g., for the problems of EM diffraction by perfectly-conducting edges, vertices and convex surfaces, and also by a thin dielectric half plane. A few selected applications of these UTD solutions to predict the EM radiation and scattering from complex structures such as aircraft, missile or spacecraft shapes are illustrated.
A unified formulation for the scattering from objects in the resonance and intermediate frequency regions is presented in terms of integral operators arising from the electric and magnetic field integral equations (EFIE and MFIE) of Maxwell's equations. The boundary conditions for a wide class of conducting and permeable scatterers are expressed in terms of coupled Fredholm equations. These equations are solved by the method of moments (MM) technique for rotationally symmetric bodies with ka 30, where a is the characteristic dimension of the body. Generalizations of these solutions for coated bodies with various anisotropies are presented. The solutions are applicable to bodies that are in part concave, convex, and with voids. The results, obtained for the backscatter and bistatic cross sections using these methods, are compared with those obtained by the extended boundary condition method, the finite element method, or measured experimentally.
In the past decade the analysis of electromagnetic scattering and antenna problems has been greatly advanced by two different theories, the method of moments (MM) and the geometrical theory of diffraction (GTD). More recently, hybrid techniques have been developed which combine the moment with the GTD and also combine the moment method or the GTD with eigenfunction solutions. It is the intent of this paper to present an overview of methods which formally combine the moment method with either eigenfunction or asymptotic techniques (i.e. GTD or optically derived currents). Each method is briefly explained and the advantages and limitations of each are given.
The extended boundary condition method (EBCM) is being used by many investigators to study a variety of electromagnetic (EM) problems such as the scattering of EM waves from non-spherical dielectric objects. This paper reviews briefly the theoretical development of EBCM involving the expansion of electromagnetic fields in terms of the vector spherical wave functions and solving the resulting matrix equations, and describes application of the method to specific problems: (1) scattering from layered objects, (2) near-zone scattering when both source and detector are near an object, and (3) fluorescent scattering resulting from active molecules embedded in an oblate spheroid.
The objective of this review is to summarize for possible users of surface scatter models the basic principles and significant characteristics of some common surface scatter models and to illustrate their applications to the interpretation of scatter measurements from man-made and sea surfaces. Other surface scatter theories which have not been applied extensively to data interpretation will only be mentioned briefly. Theoretical conditions and conditions found from computer simulation for the validity of the common surface scatter models are discussed. The common surface scatter models considered are the Kirchhoff model, the geometric optics model, the small perturbation model and a two-scale model. The less common surface scatter theories introduced are the facet method, the variational method, the diagram method, the full-wave method, and numerical methods.
The average received intensity and the intensity fluctuations in weakly turbulent media are derived for finite-sized sources of arbitrary degree of coherence (both spatial and temporal). Spatial partial coherence is modelled by random shifts and tilts of the source phase. With this simple model of source coherence, we are able to obtain the correct average intensity when the source is coherent, partially coherent and incoherent. The scintillation index as measured by a slow detector is formulated by introducing the temporal as well as spatial effects in the source coherence. Transmitter and frequency averaging effects are shown as special cases of our results.
A new approach to the problem of imaging through a time-varying randomly inhomogeneous medium is presented. This method assumes that an ensemble of short exposure images of a single fixed scene or object is available. It also assumes that knowledge of the average short-term optical transfer function is either known or estimated empirically from a point reference source. Restoration is performed by averaging a modified log gradient of the observed image spectra. This log gradient operator is an exact, mapping of discrete convolution into finite-field addi-tion, and is closely related to the complex "cepstrum," popularized by Oppenheim, It is shown to be invertible to within a constant multiplicative factor by a series of eigenvector computations. The mathematics of this gradient operator are developed, and two-dimensional simulations are presented.
Wolf's formulation for inverse scattering in the first Born approximation is reviewed and discussed from an experimental point-of-view. Some of the experimental problems associated with obtaining enough data to satisfy the requirements of this theory are described and a method for using symmetry to reduce the data requirements for a class of scatterers is explained. A successful experiment which employed this theory to reconstruct structure in an object with rectangular structure is reviewed. It appears that this theory is extremely difficult to apply to a more general scatterer with no simplifying symmetries.
A practical technique is presented for finding the shape of smooth, reflecting objects from their scattered fields. The method assumes the field on the surface of the objects satisfy Dirichlet or Neumann boundary conditions. A surface identifier combines eigenfunctions satisfying the boundary conditions for the particular object shape. The eigenfunctions are found from the T matrix. The T matrix is found from a near field recording by a holographic imaging technique which truncates the estimated T matrix guaranteeing a stable inversion. Many successive illuminations of the object are required to determine the T matrix completely.
The basis for developing projection tomographic reconstruction algorithms has been the assumption of straight-line ray-path propagation. But in the case in which propagation occurs within discretely inhomogeneous media at wavelengths of the order of the size of the scatterer, phenomena such as refraction, reflection and diffraction can no longer be neglected and a straight-line projection tomographic approach fails. This is especially evident when a large difference in refractive index occurs, such as that encountered with dm-to-mm-wave propagation in inhomogeneous atmospheric media, representing hydrometeorite distributions, the marine ocean boundary layer, the ground surface underburden, or bone and soft layers within soft tissue. An exact solution for the general vector scattering case which strictly requires a polarimetric radiative transfer approach is not available, and in this research, the assumption is made that the media are weakly diffracting so that the Born and Rytov approximations are valid. Based on this assumption, various diffraction imaging methods were developed most recently, and we are basing our studies on Devaney's back-propagation tomographic approach which was developed upon scalar wave theory. It is the main objective of this research to extend this work to the realm of electromagnetic vector wave theory for the improved diffraction-corrected imaging of radar targets embedded in clutter within the dm-to-mm-wavelength region of the electromagnetic spectrum.
Computer tomography can be used to reconstruct objects such as refractive index distributions in transparent media, if refraction is small enough that the probing optical rays are straight lines. Here we consider a ray-optical formulation of the corresponding problem, when refractive ray bending is significant. In the special case of axisymmetric objects the problem has an analytical solution, which is used in geophysical and atmospheric remote sensing. An iterative computational algorithm has been derived to effect reconstructions of strongly refracting objects and has been shown to work well in the case of relatively smooth distributions of refractive index.
The scattering phase-shifts δlf(k) (f = 0, 1, 2, ...) are generally assumed to be retrievable from scattering data on cross sections. The problem of reconstructing the object generating these data at fixed frequency (energy) was solved in the quantum mechanical framework by Newton and Sabatier, for cases with spherical symmetry. Their work applies equally well to scalar waves satisfying the wave equation with a radially distributed index of refraction. The main purpose of the paper is to show how these phase-shifts can be used to reconstruct the index of refraction numerically using a procedure that avoids the possible emergence of "transparent" distributions, yielding a unique solution.
We describe an algorithm for two-dimensional inverse scattering that is based on a finite element approximation of the scalar Helmholtz equation. The algorithm is iterative, so that nonlinearity in the direct scattering can be accounted for, but computationally, the method is slow and, at present, is limited to small problems. Nevertheless, model simulations indicate that the algorithm can produce accurate reconstructions in cases where high-resolution linear methods cannot be applied.
The parabolic approximation, applied to the propagation of time harmonic scalar waves, replaces the governing Helmholtz equation with a SchrOdinger equation. The primary computational advantage of this approximation is that it is first order in the range coordinate. The validity of the approximation limits variations in an inhomogeneous wave number field to be both small, over the total range of the experiment, and slow, as measured on a length scale determined by an averaged wavelength. Thus, the validity of the approximation can be severely strained in a number of applications; e.g., it is difficult to justify its application in the presence of regions of rapid, even discontinuous, changes near a surface that runs in the range direction. In this paper we consider the derivation of extended parabolic wave theories which retain the feature that they are first order in the range coordinate. Distinguishing the extended theories from the ordinary one, i.e. the SchrOdinger equation, is the manner in which the cross-range coordinates enter. In the ordinary theory, they appear via the two-dimensional Laplacian, a differential or local operator; in the extended theories they appear via non-local operators. Obtaining explicit forms for specific nonlocal cross-range operators requires the solution of factor-ordering problems which have their counterpart in the correspondence between quantized theories governing the evolution of dynamical systems and their related classical formulations. The operator construction can be related to the construction of both coordinate and phase space path integrals for the appropriate wave propagator and further provides for the formulation and exact solution of a multidimensional nonlinear inverse problem appropriate for ocean acoustic and seismic modelling.
Recent studies in underwater acoustics have shown how coherence theory may be used to determine the correct intensity distribution in the neighborhood of caustics. These studies may be of interest to the optics community and we review here the method developed. We begin with the basic equation governing the coherence function in the parabolic approximation. The radiation propagates in a medium with a variable mean index of refraction. A lowest order approximation to the basic equation yields the geometric optics solution. To next order we use a two-scale expansion to include the correction term to the geometric optics calculation. This gives the correct intensity distribution in the neighborhood of caustics and it is shown that in the limit of a point source the results reduce to those obtained previously. The results of a numerical example are given. Finally it is pointed out that if there is volume scattering, resulting from a stochastic variation in the index of refraction field, the method may be extended to determine the intensity corrections near a caustic due to this effect.
Various efficient methods for computing fields in focal regions are described. The methods are based on the Kirchhoff integral, the Rayleigh-Sommerfeld integrals, the boundary-diffraction-wave integral, or the angular-spectrum representation. Efficient numerical algorithms are obtained from these diffraction integrals by employing local phase and amplitude approximations, asymptotic approximations, and fast-Fourier-transform algorithms. Both ordinary lenses and holographic lenses are considered.
The roughness dependence of fully developed speckle patterns is known to manifest itself through spectral and angular correlations, i.e. correlations in the reciprocal (K-) space. In this paper we show how this dependence can be given a simple physical interpretation by use of the concept of Bragg diffraction. To that end, we briefly investigate the formal analog between the first Born scattering theory and Beckmann's theory of rough surface diffraction. Then, on the basis of earlier results, we present a K-space version of the van Cittert-Zernike theorem which enables us to use an equivalent Bragg-diffraction problem to analyse the K-space correlations. Finally, we show that earlier results for the roughness dependent spectral and angular correlations, as well as the roughness-dependence of polychromatic speckle, can be found from our analysis by simple physical considerations.
Wave scattering from a dielectric medium or a perfect conductor with a periodic rough surface is studied. Our theoretical method is based on the extended boundary condition which is derived from Green's theorem. In this formulation, the amplitudes of the diffracted Floquet waves can be related directly to the amplitudes of the incident plane waves by a transition matrix. This matrix has a simple form which is suitable for calculations, and it saves computational time over other numerical methods. The accuracy of the calculations can be checked by (1) power conservation, and (2) the reciprocity theorem. Numerical results are given, and the influence of the rough surface as compared with the flat interface case will be shown.
This paper contains a treatment of paraxial wave optics which is expressed in terms of operators acting on vectors in an abstract space. The abstract vectors represent the electromagnetic field (in scalar approximation). The effects of propagation, reflection, and refraction, etc., are implemented by operators acting on the abstract field vectors. This leads to an operator algebraic formulation of paraxial wave optics.
In this paper the principal elements of the full wave approach to problems of scattering and depolarization by rough surfaces are summarized. Both deterministic and random rough surfaces are considered in detail and the full wave solutions are compared with earlier solutions based on physical optics and perturbation theories. It is shown that since the full wave approach accounts for both specular point scattering as well as Bragg scattering in a self-consistent manner, it resolves the discrepancies between the physical optics and perturbation solutions and bridges the wide gap between them. Thus, on applying the full wave approach to scattering by composite random rough surfaces it is not necessary to restrict the analysis to surfaces that can be treated as perturbed physical optics problems. The full wave solutions are shown to satisfy duality,reciprocity and realizability relations in electromagnetic theory and the results are invariant to coordinate transformations. The full wave approach also accounts for coupling between the radiation fields, the lateral waves and the surface waves that constitute the complete expansions of the fields and it can be applied directly to problems of scattering at near grazing angles.
Coupling between guided TE modes of an asyruetric dielectric slab waveguide with a sinusoidal boundary is discussed. A general Floquet approach is developed from which analytic expressions for the dispersion characteristics near the stopband are obtained using perturbation theory.