Scalar Wave Field Research Articles

Noninterferometric characterization of partially coherent scalar wave fields and application to scattered light

We report on the application of a simple propagation-based phase-space tomographic technique to the determination of characteristic projections through the mutual optical intensity and the generalized radiance of a scalar, quasi-monochromatic partially coherent wave field. This method is applied to the reconstruction of the coherence functions of an initially spatially coherent optical wave field that has propagated through a suspension of polystyrene microspheres. As anticipated, we see that the field separates into a ballistic, or unscattered, component and a scattered component with a much shorter coherence length. Good agreement is obtained between experimental results and the results of a model based on a wave-transport equation.

High-frequency diffraction corrections to backscattering cross sections from a rough three-dimensional surface

Diffraction corrections to scalar wave fields at perfectly free and rigid rough surfaces were derived by two iterations of the corresponding integral equations. These diffraction corrections to the pressure or normal velocity (which, in the geometrical optics limit, are doubled at perfectly rigid and free surfaces, respectively) were obtained with an accuracy of approximately 1k(2), where k is the wave number of incidence radiation. Based on these corrections to the surface fields, the backscattering cross sections at normal incidence from the statistically rough Gaussian surfaces were derived. It was found that for the gentle roughness, diffraction results in effective "smoothing" of roughness for rigid and free surfaces and increasing of the backscattering cross sections, but for a rigid surface with steep roughness, the "fictitious" surface can be more rough than the real one, and the diffraction corrections become negative.

The local density of states in finite size photonic structures, small particles approach

The local density of states is studied with the help of the local perturbation method for vector and scalar wave fields propagating in finite size photonic structures. The local density of states is numerically calculated for one-, two-, and three-dimensional finite size photonic structures considering them as made of small particles.

Modeling the spatial coherence of seismic surface waves at short propagation distances

Scattering from small, random inhomogeneities in the ground (such as rocks and voids) diminishes the spatial coherence of seismic waves, much as turbulence in the atmosphere diminishes spatial coherence of propagating sound waves. In this paper, modeling of seismic spatial coherence for surface waves propagating over short distances (roughly 50 m to 1 km) and at relatively high frequencies (greater than approximately 1 Hz) is considered. The random inhomogeneities are distributed broadly in size and are described with a quasi‐wavelet (QW) model. For simplicity, we consider propagation of a 2‐D scalar wavefield. Scattering from an individual QW is first calculated using a single‐scattering approximation. Then, the contributions from all randomly placed QWs, including all size classes, are cross‐correlated to find the spatial coherence of the total wavefield. As an example, we consider modeling of seismic coherence at a site in the Mojave Desert with random basaltic inhomogeneities surrounded by sand. The impact of coherence loss on direction‐finding with geophone arrays is discussed.

A non-iterative reconstruction method for direct and unambiguous coherent diffractive imaging

We develop a deterministic algorithm for coherent diffractive imaging (CDI) that employs a modified Fourier transform of a Fraunhofer diffraction pattern to quantitatively reconstruct the complex scalar wavefield at the exit surface of a sample of interest. The sample is placed in a uniformly-illuminated rectangular hole with dimensions at least two times larger than the sample. For this particular scenario, and in the far-field diffraction case, our non-iterative reconstruction algorithm is rapid, exact and gives a unique analytical solution to the inverse problem. The efficacy and stability of the algorithm, which may achieve resolutions in the nanoscale range, is demonstrated using simulated X-ray data.

A systematic study of theoretical relations between spatial correlation and Green's function in one-, two- and three-dimensional random scalar wavefields

SUMMARY It has been demonstrated theoretically and experimentally that Green’s function between two receivers can be retrieved from the spatial correlation of an ambient noise field. While studies on the spatial correlation of ambient noises have been mainly conducted in the frequency domain, the retrieval of Green’s function has been done in the time domain. Recently, it was pointed out that both studies could be connected via the inverse Fourier transform. In this paper, a systematic study of theoretical relations is conducted between the spatial correlation and the Green’s function of wave equation in 1-, 2- and 3-D random scalar fields under an assumption that mutually uncorrelated plane waves are incident on two receivers from various directions. Different relation is obtained for a different dimensional case: Integration, Hilbert transform and differentiation have to be operated on the normalized spatial correlation to retrieve Green’s function for 1-, 2- and 3-D, respectively. It is reconfirmed that the Green’s function can be retrieved from the spatial correlation between two receivers only for isotropic random fields. The effect of the anisotropy of incident waves on the retrieval of Green’s function is evaluated analytically. Even for the anisotropic incident waves, it is possible to retrieve Green’s function from the spatial correlation by taking an azimuthal average in 2-D and a spherical average in 3-D.

Coherence Current, Coherence Vortex, and the Conservation Law of Coherence

Introducing scalar and vector densities for a mutual coherence function, we present a new conservation law for optical coherence of scalar wave fields in the form of a continuity equation. This coherence conservation law provides new insights into topological phenomena for the complex coherence function. Some properties related to the newly introduced coherence vector density, such as a circulating coherence current associated with a coherence vortex, are investigated both theoretically and experimentally for the first time.

Nonparaxial fields with maximum joint spatial-directional localization I Scalar case

In paraxial optics, the spatial and angular localization of a beam are usually characterized through second moments in intensity. For these measures, Gaussian beams have the property of achieving a minimum angular spread for a given spatial spread (or beam waist). For wide-angle fields, however, the standard measures of spatial and angular localization become inappropriate, and new definitions must be used. Previously proposed definitions [J. Opt. Soc. Am. A 17, 2391 (2000)] are adopted, and the scalar monochromatic wave fields that achieve a minimum angular spread for a given spatial spread are found.

Attenuation, transport and diffusion of scalar waves in textured random media

Most theoretical investigations of seismic wave scattering rely on the assumption that the underlying medium is statistically isotropic. However, deep seismic soundings of the crust as well as geological observations often reveal the existence of elongated or preferentially oriented scattering structures. In this paper, we develop mean field and radiative transfer theories to describe the attenuation and multiple scattering of a scalar wavefield in an anisotropic random medium. The scattering mean free path is found to depend strongly on the propagation direction. We derive a radiative transfer equation for statistically anisotropic random media from the Bethe–Salpeter formalism and propose a Monte Carlo method to solve this equation numerically. At longer times, the energy density is shown to obey a tensorial diffusion equation. The components of the diffusion tensor are obtained in closed form and excellent agreement is found between Monte Carlo simulations and analytical solutions of the diffusion equation. The theory has important potential implications for lithospheric models where scatterers are for example flat structures preferentially aligned along the surface. In this simple geometry, analytical expressions of the Coda Q parameter can be given explicitly in the diffusive regime. Our results suggest that pulse broadening and coda decay are controlled by different parameters, related to the eigenvalues of the diffusion tensor. These eigenvalues can differ by more than one order of magnitude. This theory could be applied to probe the anisotropy of length scales in the lithosphere.

Off-axis holographic tomography for diffracting scalar wavefields

Inverse scattering theories are available that permit tomographic reconstruction of the complex-valued refractive index distribution of weakly scattering objects from knowledge of intensity measurements. These imaging methods are valuable in applications that involve high-frequency optical wavefields, in which direct wavefield phase measurements can be difficult experimentally. The so-called in-line holographic imaging geometry has been well-studied, in which two measurements of the forward scattered wavefield intensity are acquired on distinct parallel detector planes that are perpendicular to the direction of the illuminating wavefields. In this work, based on the principles of intensity diffraction tomography, a reconstruction theory for off-axis holographic tomography with diffracting scalar wavefields is developed and investigated. A distinct feature of the method is that, at each tomographic view angle, the object is illuminated by use of two plane waves that propagate in different directions. The intensities of the forward scattered wavefields are measured on a single detector behind the object. This permits the direction of the probing wavefield to be varied for acquisition of the necessary measurement data, rather than the detector placement as required by conventional in-line holographic methods. The developed image reconstruction method is validated and investigated by use of computer-simulation studies.

Effect of absorption on the spatial coherence in scalar fields generated by statistically homogeneous and isotropic sources

We investigate the behavior of the spectral degree of coherence of fluctuating scalar wave fields generated by statistically homogeneous and isotropic sources within absorbing media. In the limit of negligible losses, the degree of coherence is shown to be of a known universal form (sinc function), whereas in the limit of infinite losses it is shown to asymptotically approach the degree of coherence of the source. The intermediate cases are studied for fields produced by sources with Gaussian and sinc-type coherence functions. These studies show that the concept of negligible losses is meaningful only when defined with respect to specific source distribution statistics, and that even for arbitrarily small nonzero losses, the coherence length of the field can be arbitrarily short.

Kinetics of scalar wave fields in random media

This paper concerns the derivation of kinetic models for high frequency scalar wave fields propagating in random media. The kinetic equations model the propagation in the phase space of the energy density of a wave field or the correlation function of two wave fields propagating in two possibly different media. Dispersive effects due to, e.g. spatial and temporal discretizations, which are modeled as non-local pseudo-differential operators, are taken into account. The derivation of the models is based on a multiple-scale asymptotic expansion of the spatio-temporal Wigner transform of two scalar wave fields.

Reflection and transmission in multilayered media in terms of surface integrals

We derive a rigorous analytic description of multiple reflection/transmission phenomena of 3-D scalar harmonic wave fields in a multilayered inhomogeneous medium with smooth curvilinear infinite interfaces of arbitrary shape. Each individual event is given in terms of a surface integral over the corresponding interface. The derivation is based on a generalisation of the technique that was used for description of corresponding phenomena at a single interface in our previous paper [K. Klem-Musatov, A. Aizenberg, H.B. Helle, J. Pajchel, Reflection and transmission at curvilinear interface in terms of surface integrals, Wave Motion (39)(1) (2004) 77–92].

Physical Interpretation of the Motion of a Laser Pulse within the Framework of the Four-Oscillator Model

A physical multidimensional local model of the propagation process of a laser pulse in air having no analogy to famous models in laser physics is constructed. It is based on the representations of classical wave mechanics characteristic of the theory of scalar wave fields.

X-ray omni microscopy.

The science of wave-field phase retrieval and phase measurement is sufficiently mature to permit the routine reconstruction, over a given plane, of the complex wave-function associated with certain coherent forward-propagating scalar wave-fields. This reconstruction gives total knowledge of the information that has been encoded in the complex wave-field by passage through a sample of interest. Such total knowledge is powerful, because it permits the emulation in software of the subsequent action of an infinite variety of coherent imaging systems. Such 'virtual optics', in which software forms a natural extension of the 'hardware optics' in an imaging system, may be useful in contexts such as quantitative atom and X-ray imaging, in which optical elements such as beam-splitters and lenses can be realized in software rather than optical hardware. Here, we develop the requisite theory to describe such hybrid virtual-physical imaging systems, which we term 'omni optics' because of their infinite flexibility. We then give an experimental demonstration of these ideas by showing that a lensless X-ray point projection microscope can, when equipped with the appropriate software, emulate an infinite variety of optical imaging systems including those which yield interferograms, Zernike phase contrast, Schlieren imaging and diffraction-enhanced imaging.

Wave-function reconstruction of complex fields obeying nonlinear parabolic equations.

We present a generalized Gerchberg-Saxton (GS) algorithm for reconstructing a (2+1)-dimensional complex scalar wave field which obeys a known nonlinear nondissipative parabolic differential equation, given knowledge of the wave-field modulus at three or more values of an evolution parameter such as time. This algorithm is used to recover the complex wave function of a (2+1)-dimensional Bose-Einstein condensate (BEC) from simulated modulus data. The Gross-Pitaveskii equation is used to model the dynamics of the BEC, with the modulus information being provided by a temporal sequence of simulated absorption images of the condensate. The efficacy of the generalized GS algorithm is examined for a wide range of simulation conditions, including strong nonlinearities, vortex states and Poisson noise. The general form of this algorithm, which allows one to reconstruct a time-dependent wave function, will be useful for studying the phase dynamics of topological defects in coherent quantum systems.

Reflection and transmission at curvilinear interface in terms of surface integrals

We derive a rigorous analytic description of multiple reflection/transmission phenomena of 3D scalar harmonic wave fields in two inhomogeneous half spaces separated by a smooth curvilinear interface of arbitrary shape. The resulting wave field is expressed in terms of a branching sequence of events connected by recurrence relationships. An individual event is a result of the perturbation of boundary conditions by the previous event of the sequence that, in turn, again perturbs the boundary conditions. Such an event is described by the surface Helmholtz integral of any fundamental solution satisfying conditions at infinity. Functions of the current point of the interface and their normal derivatives ( Cauchy data) are described by the surface integrals with the fundamental solutions for an absolutely absorbing interface. The Cauchy data of the latter integrals are obtained from the subsequent transformations. The perturbation of the interface and its normal derivatives are mapped into the spatial frequency domain by means of the Fourier transform where they are multiplied by the corresponding reflection/transmission coefficients and then is mapped back to an individual point on the interface by means of the inverse Fourier transform. Parameters of these transformations depend on the point where the result is computed. Therefore the computation at a new point of the interface requires the repetition of these transformations.

Scattering theory for a particle coupled to a scalar field

We establish soliton-like asymptotics for finite energy solutions to classical particle coupled to a scalar wave field. Any solution that goes to infinity as $t\to\infty$ converges to a sum of traveling wave and of outgoing free wave. The convergence holds in global energy norm. The proof uses a non-autonomous integral inequality method.

Modeling of seismic guided waves at the Dead Sea Transform

On several recordings of linear seismometer arrays crossing the Arava Fault (AF) in the Middle East, we see prominent wave trains emerging from in‐fault explosions which we interpret as waves being guided by a fault zone related low‐velocity layer. The AF is located in the Arava Valley and is considered the principal active fault of the mainly N‐S striking Dead Sea Transform System in this section. Observations of these wave trains are confined to certain segments of the receiver lines and occur only for particular shot locations. They exhibit large amplitudes and are almost monochromatic. We model them by a two‐dimensional (2‐D) analytical solution for the scalar wave field in models with a vertical waveguide embedded in two quarter spaces. A hybrid search scheme combining genetic algorithm and a local random search is employed to explore the multimodal parameter space. Resolution is investigated by synthetic tests. The observations are adequately fit by models with a narrow, only 3–12 m wide waveguide with S wave velocity reduced by 10–60% of the surrounding rock. We relate this vertical low‐velocity layer with the damage zone of the AF since the location of receivers observing and of shots generating the guided waves, respectively, match with the surface trace of the fault. The thickness of the damage zone of the AF, at least at shallow depths, seems to be much smaller than in other major fault zones. This could be due to less total slip on this fault.

Radiative transfer in strongly lossy inhomogeneous thin films

Originally, the radiative transfer theory has been developed and has evolved in the framework of the real geometrical optics. However, such a context is inappropriate to deal with radiative transfers in strongly absorbing thin films because of the complex nature of rays. Here, we present an extension of the old theory to remove this limitation and exhibit the effective geometry that is probed by a scalar wave field within such media.