Stationary Phase Points Research Articles

Scattering by randomly rough surfaces. III. Phase approximations

In the limit as the roughness vanishes, the solution for the pressure scattered by a rough surface of infinite extent should reduce to the image solution. Approximate image solutions for an infinite, pressure-release plane surface are studied for an omnidirectional source using the 2nd, 3rd, and 4th order phase approximations. The results are compared to the exact image solution to examine the effects of the phase approximations. The result based on the 2nd order (Fresnel phase) approximation reproduces the image solution for all geometries. Surprisingly, the results for the 3rd and 4th order phase approximations are never better than the Fresnel result, and are substantially worse for most geometries. This anomalous behavior is investigated and the cause is found to be the multiple stationary phase points produced by the 3rd and 4th order phase approximations.

Discriminating physical and non-physical diffracted energy in source–receiver interferometry

SUMMARY We show how physical and non-physical diffracted waves can be discriminated in source– receiver interferometry (SRI), a wavefield interferometric method that constructs Green’s functions between any source–receiver pair. These can be estimated by convolution and crosscorrelation of wavefields propagating to or from other sources or receivers. SRI has been observed to work surprisingly well in practical applications when theoretical requirements (e.g. complete enclosing boundaries of other sources and receivers) are contravened as is standard in practical applications. As a partial explanation to this, it has been demonstrated previouslythatSRIforsinglediffractorproblemsproducescorrectGreen’sfunctionsestimates without requiring wavefields to be generated around specific stationary-phase points in space (as is required by other forms of interferometry). In this paper, we show that also for multiplediffractorproblems,anyboundarysource–receiverpairproducesstationary,correctkinematics for singly and multiply diffracted waves. However, when multiple diffractors are present, non-physical artefacts are produced in Green’s functions estimated by SRI with incomplete boundaries. We introduce a purely data-driven algorithm that allows traveltimes of any order ofphysical, diffractedwaves tobepredicted. Thisreduces ambiguity ininterpreting wavefields generated using SRI with only partial boundaries: this also permits spurious or non-physical energy in the constructed Green’s functions to be identified and either interpreted or ignored.

Measurement of higher order chromatic dispersion in a photonic bandgap fiber: comparative study of spectral interferometric methods

Chromatic dispersion of a 37 cm long, solid-core photonic bandgap (PBG) fiber was studied in the wavelength range of 740-840 nm with spectral interferometry employing a Mach-Zehnder interferometer and a high resolution spectrometer. The interferometer was illuminated by a Ti:sapphire laser providing 20 fs pulses. A comparative study has been carried out to find the most accurate spectral phase retrieval method that is suitable for measuring higher order chromatic dispersion. The stationary phase point, the minima-maxima, the cosine function fit, the Fourier transform, and the windowed Fourier transform methods were tested. It was shown that out of these five techniques, the Fourier-transform method provided the dispersion coefficients with the highest accuracy, and it could also detect rapid phase changes in the vicinity of leaking mode frequencies within the transmission band of the PBG fiber.

An Alternative Treatment of Saddle Stationary Phase Points in Physical Optics for Smooth Surfaces

In a recent paper on computing the physical optics (PO) integral with a saddle stationary phase point (SPP) by numerical steepest descend method (NSDM) (F. Vico-Bondia, “A new fast physical optics for smooth surfaces by means of a numerical theory of diffraction,” IEEE Trans. Antennas Propag., vol. 58, no. 3, pp. 773-789, Mar. 2010), a one dimensional integral was obtained, on the path of which there existed higher-order (up to third) poles. To avoid tackling the singular integral, the Abel's summation technique was used to solve the problem ingeniously. However, it has been shown in some literatures that the integral would be divergent if there were even-order poles on the path of integral. Superficially, the integral in F. Vico-Bondia, 's paper, does not obey this law and the main aim of the communication is to clear this contradiction. We show rigorously that higher-order poles have no contributions to the final result. To acquire a general law, we further consider arbitrary polynomials of higher degrees for the PO integral, in which the orders of poles can be arbitrary number. In such a general case, we still show that all higher-order poles have no contributions and thus solve the contradiction satisfactorily. Numerical examples are presented to validate the new derivations and illustrate the accuracy and efficiency of NSDM.

The Stationary Phase Approximation, Time-Frequency Decomposition and Auditory Processing

The principle of stationary phase (PSP) is re-examined in the context of linear time-frequency (TF) decomposition using Gaussian, gammatone and gammachirp filters at uniform, logarithmic and cochlear spacings in frequency. This necessitates consideration of the use the PSP on non-asymptotic integrals and leads to the introduction of a test for phase rate dominance. Regions of the TF plane that pass the test and don't contain stationary phase points contribute little or nothing to the final output. Analysis values that lie in these regions can thus be set to zero, i.e. sparsity. In regions of the TF plane that fail the test or are in the vicinity of stationary phase points, synthesis is performed in the usual way. A new interpretation of the location parameters associated with the synthesis filters leads to: (i) a new method for locating stationary phase points in the TF plane; (ii) a test for phase rate dominance in that plane. Together this is a TF stationary phase approximation (TFSFA) for both analysis and synthesis. The stationary phase regions of several elementary signals are identified theoretically and examples of reconstruction given. An analysis of the TF phase rate characteristics for the case of two simultaneous tones predicts and quantifies a form of simultaneous masking similar to that which characterizes the auditory system.

Two-Dimensional Spectrum for BiSAR Derivation Based on Lagrange Inversion Theorem

A 2-D spectrum for bistatic synthetic aperture radar is derived in this letter. The derivation is based on the commonly used mathematic principles such as the method of stationary phase and the Fourier transform and the Lagrange inversion theorem in order to find the point of stationary phase in the method of stationary phase. Using the Lagrange inversion theorem allows minimizing the initial assumptions or the initial approximations. The derived 2-D spectrum is compared with the commonly used 2-D spectrum to verify it and illustrate its accuracy.

On high-frequency noise scattering by aerofoils in flow

AbstractA theoretical model is developed for the sound scattered when a sound wave is incident on a cambered aerofoil at non-zero angle of attack. The model is based on the linearization of the Euler equations about a steady subsonic flow, and is an adaptation of previous work which considered incident vortical disturbances. Only high-frequency sound waves are considered. The aerofoil thickness, camber and angle of attack are restricted such that the steady flow past the aerofoil is a small perturbation to a uniform flow. The singular perturbation analysis identifies asymptotic regions around the aerofoil; local ‘inner’ regions, which scale on the incident wavelength, at the leading and trailing edges of the aerofoil; Fresnel regions emanating from the leading and trailing edges of the aerofoil due to the coalescence of singularities and points of stationary phase; a wake transition region downstream of the aerofoil leading and trailing edge; and an outer region far from the aerofoil and wake. An acoustic boundary layer on the aerofoil surface and within the transition region accounts for the effects of curvature. The final result is a uniformly-valid solution for the far-field sound; the effects of angle of attack, camber and thickness are investigated.

Stationary phase method of calculating Čerenkov radiation spectrum of a charged particle moving in curved path

Based on the general expression for spectral-angular distribution of the radiation emitted by a charged particle moving in exterior medium, the essential role of stationary phase points in Čerenkov effect is investigated. And the stationary phase method of calculating spectral-angular distribution of Čerenkov radiation from a charged particle moving in curved path is proposed. By the stationary phase method, the asymptotic form of spectral-angular distribution of synchrotron-Čerenkov radiation is calculated, and the result indicates that the spectrum of synchrotron-Čerenkov radiation near critical angle c is quite different from that near the plane of particle's orbit.

Computing highly oscillatory physical optics integral on the polygonal domain by an efficient numerical steepest descent path method

In this work, the computation of physical optics (PO) type integral with the integrand of quadratic phase and amplitude is studied. First, we apply the numerical steepest descent path (NSDP) method to calculate the highly oscillatory PO integral on the triangular patch. Then, we rigorously extend the proposed NSDP method to analyze the PO integral on polygonal domains. Furthermore, the contributions of critical points on polygonal domains, including the stationary phase point, resonance and vertex points, are comprehensively studied in terms of the NSDP method. Compared to the traditional high frequency asymptotic (HFA) method, when the wave frequency is not very high but in the high frequency regime, the NSDP method has improved the PO integral accuracy by one to two digits. Meanwhile, the computational cost by using the proposed NSDP method is independent of the wave frequency.

Phase retrieval from spectral interferograms including a stationary-phase point

We report on a simple processing procedure to retrieve the phase from spectral interferograms including a stationary-phase point. First, the numerical simulations are performed to demonstrate high precision of the phase retrieval from the spectral interferogram. Second, the feasibility of the procedure is confirmed in processing the experimental data from a dispersive Michelson interferometer comprising a cube beam splitter and a plate made of BK7 optical glass. From the retrieved spectral phase, the effective thickness of the BK7 optical glass is determined precisely.

Difference of scattering geometrical optics components and line integrals of currents in modified edge representation

Equivalent edge currents (EECs) are widely used to asymptotically extract and express the diffraction from the periphery of the scatterers, as in the concept suggested by Young. Authors proposed novel EECs based on the unique concept named as Modified Edge Representation (MER), for surface to line integral reduction of Physical Optics (PO) radiation integral. MER is based upon not only asymptotic approximation but also Stokes' theorem, and it has remarkable accuracy even for small scatterers and is uniformly applicable at the geometrical boundaries. Moreover MER‐EECs are clearly defined not only at the edges but also everywhere on the scatterer surface, with the singularity at the stationary phase point (SPP) if any. It comes that the radiation integral in general, consisting of both the diffraction and the Geometrical Optics (GO) components, could be reduced into two sets of line integration of MER‐EECs along the periphery and the indentation at SPPs. In this paper, the GO reflection is numerically compared with the MER indentation integral around the reflected point for the dipole scattering from an ellipsoid; the error is empirically derived in analytical form.

Efficient Evaluation of the Physical-Optics Integrals for Conducting Surfaces Using the Uniform Stationary Phase Method

The computational time to evaluate the physical optics (PO) expression by numerical integration increases rapidly with the increase of electrical size of scattering surfaces. However, the computational time of PO integrals for electrically large object can be greatly reduced by using the stationary phase method, which is independent of the wavenumber. For this method, the theory and numerical implementations for isolated critical points have been well developed. However, for cases of nearby critical points, there are still a few issues to be considered, especially, in numerical implementations. In this paper, we mainly study the numerical implementations for several most common cases of nearby critical points. In particular, the cases of two nearby inner stationary phase points and complex inner stationary phase points are discussed in more details. Such cases occur frequently when the scattering surface includes convex-concave parts, but the numerical implementations to such cases have not been reported to our knowledge. The difficulty lies in how to identify whether two inner stationary points and complex inner stationary points on the surfaces with arbitrary shapes are close to each other or not. A strategy is designed to solve this difficulty. By validation in some typical examples, we find that the stationary phase method is robust enough to evaluate the PO integrals accurately. Finally, some interesting phenomena observed in numerical validations are interpreted.

Calculation of Physical Optics Integrals Over NURBS Surface Using a Delaminating Quadrature Method

Physical optics (PO) is a widely used radar cross section (RCS) estimation approach for electrically large-scaled objects, and it is finally reduced to the calculation of highly oscillatory integrals. In this paper the PO integrals over non-uniform rational B-spline (NURBS) surfaces is proposed to be calculated with a delaminating quadrature method. The method has the merits of being very stable, accurate, and fast, and its computation cost does not increase with frequency. Moreover, in order to keep these good properties for PO integrals involving critical points (stationary phase point, resonant point), a scheme of integral-subdividing is proposed, and new approaches for locating the critical points are also presented. Numerical examples of flat surface, cylindrical surface, and spherical surface well show the advantages of the new method.

Elastic-wavefield interferometry using P-wave source VSPs at the Wamsutter field, Wyoming

In this study, elastic-wavefield interferometry was used to recover P- and S-waves from the 3D P-wave vibrator VSP data at Wamsutter field in Wyoming. S-wave velocity and birefringence is of particular interest for the geophysical objectives of lithology discrimination and fracture characterization in naturally fractured tight gas sand reservoirs. Because we rely on deconvolution interferometry for retrieving interreceiver P- and S-waves in the subsurface, the output fields are suitable for high-resolution, local reservoir characterization. In 1D media where the borehole is nearly vertical, data at the stationary-phase point is not conducive to conventional interferometry. Strong tube-wave noise generated by physical sources near the borehole interfere with S-wave splitting analyses. Also, converted P- to S-wave (PS-wave) polarity reversals occur at zero offset and cancel their recovery. We developed methods to eliminate tube-wave noise by removing physical sources at the stationary-phase point and perturbing the integration path in the integrand based on P-wave NMO velocity of the direct-arrival. This results in using nonphysical energy outside a Fresnel radius that could not have propagated between receivers. To limit the response near the stationary-phase point, we also applied a weighting condition to suppress energy from large offsets. For PS-waves, a derivative-like operator was applied to the physical sources at zero offset in the form of a polarity reversal. These methods resulted in effectively recovering P-wave dipole and PS-wave quadrupole pseudosource VSPs. The retrieved wavefields kinematically correspond to a vertical incidence representation of reflectivity/transmissivity and can be used for conventional P- and S-wave velocity analyses. Four-component PS-wave VSPs retrieve S-wave splitting in transmitted converted waves that provide calibration for PS-wave and P-wave azimuthal anisotropy measurements from surface-seismic data.

Crosscorrelation of Earthquake Data Using Stationary Phase Evaluation: Insight into Reflection Structures of Oceanic Crust Surface in the Nankai Trough

Seismic interferometry (SI) has been recently employed to retrieve the reflection response from natural earthquakes. We perform experimental study to apply SI to Ocean Bottom Seismogram (OBS) records in the Nankai Trough, southwest Japan in order to reveal the relatively shallow geological boundaries including surface of oceanic crust. Although the local earthquakes with short raypath we use to retrieve reflection response are expected to contain the higher-frequency components to detect fine-scale structures by SI, they cannot be assumed as plane waves and are inhomogeneously distributed. Since the condition of inhomogeneous source distribution violates the assumption of SI, the conventional processing yields to the deteriorated subsurface images. Here we adopt the raypath calculation for stationary phase evaluation of SI in order to overcome this problem. To find stationary phase, we estimate the raypaths of two reflections: (1) sea-surfaceP-wave reflection and (2) sea-surface multipleP-wave reflection. From the estimated raypath, we choose the crosscorrelation traces which are expected to produce objective reflections considering the stationary phase points. We use the numerical-modeling data and field data with 6 localized earthquakes and show that choosing the crosscorrelation traces by stationary phase evaluation improves the quality of the reflections of the oceanic crust surface.

The point spread function analysis in a wavefront coding system based on stationary phase method

Wavefront coding system with a cubic phase mask is one of important methods to extend depth of the field. This paper analyzes the characteristics of the system based on stationary phase method in space domain. By analyzing the point spread function of an arbitrary strip of the cubic phase mask, this paper points out that two stationary phase points lead to oscillations in point spread function while one stationary phase point causes a smooth curve in point spread function. Theoretical analysis illustrates that the oscillations exist in point spread function if and only if there are symmetrical components about the optical symmetrical axis. Furthermore, different areas of the point spread function correspond to different positions of symmetrical components of the mask, which is useful in the manufacture and test of the cubic phase mask. In addition, the optical symmetrical axis does not coincide with the geometrical symmetrical axis in the defocused system, causing smooth curve in defocused point spread function. As a contrast, smooth curve can not be observed in focused point spread function because of the coincidence of the symmetrical axis and the geometrical symmetrical axis.

AN EFFICIENT METHOD FOR COMPUTING HIGHLY OSCILLATORY PHYSICAL OPTICS INTEGRAL

In this work, we use the numerical steepest descent path (numerical SDP) method in complex analysis theory to calculate the highly oscillatory physical optics (PO) integral with quadratic phase and amplitude variations on the triangular patch. The Stokes' phenomenon will occur due to various asymptotic behaviors on difierent domains. The stationary phase point contributions are carefully studied by the numerical SDP method and complex analysis using contour deformation. Its result agrees very well with the leading terms of the traditional asymptotic expansion. Furthermore, the resonance points and vertex points contributions from the PO integral are also extracted. Compared with traditional approximate asymptotic expansion approach, our method has signiflcantly improved the PO integral accuracy by one to two digits (10 i1 to 10 i2 ) for evaluating the PO integral. Moreover, the computation efiort for the highly oscillatory integral is frequency independent. Numerical results for PO integral on the triangular patch are given to verify the proposed numerical SDP theory.

Complex stationary-phase points in dynamical-tunneling emissions

In a recent paper [Phys. Rev. A 83, 023827 (2011)] it has been suggested, based on numerical simulation, that tunneling emission in dielectric microcavities can be explained through a two-step process: first dynamical tunneling to reach the critical line and then interference among tangential wave components refractively escaped from the critical line. Here we investigate again the tunneling-emission mechanism by using a stationary-phase method in a circular microcavity. It is shown that the stationary-phase point becomes complex for a tunneling emission and gives a tunneling-ray picture. The imaginary part of the point determines the distance {delta} between the tunneling ray and the cavity surface. In addition, we demonstrate numerically that the two-step tunneling process with a proper window function works very well, even in deformed microcavities.

Fresnel Zone Criterion to Implement Locality in the Method of Moments and PO-MoM Hybrid Method for the Reduction of Unknowns

Locality of high frequency electromagnetic scattering phenomena is embodied and imported to the Method of Moments (MoM) to reduce computational load. The proposed method solves currents on small areas only around inner and edge stationary phase points (SPPs) on the scatterer surfaces. The range of MoM area is explicitly specified in terms of Fresnel zone number as a function of frequency, source and observer positions. Based upon this criterion, scatterer of arbitrary size and shape can be solved with almost frequency independent number of unknowns. In some special cases like focusing systems, locality disappears and the method reduces to the standard MoM. The hybrid method called PO-MoM is complementarily introduced to cope with these cases, where Fresnel zone number with analogous but different definition is used. The selective use of Local-MoM and PO-MoM provides frequency insensitive number of unknowns for general combination of source and observation points. Numerical examples of RCS calculation for two dimensional flat and curved surfaces are presented to demonstrate the accuracy and reduction of unknowns of this method. The Fresnel zone, introduced in the scattering analysis for the first time, is a useful indicator of the locality or the boundary for MoM areas.

Analysis Method of Ground Wave Propagation over Land-to-Sea Mixed-Path by Using Helmholtz-Kirchhoff Integral Theorem

In this paper, we have derived a novel integral representation for the ground wave propagation over land-to-sea mixed-paths by applying the Helmholtz-Kirchhoff integral theorem. By using the method of stationary phase applicable uniformly as the stationary phase point approaches the endpoint of the integral, we have derived the asymptotic solution for the scattered fields consisting of the first-order and the second-order diffraction terms. We show that the asymptotic solution thus derived agrees with the asymptotic solution derived by applying the aperture field method (AFM) and the method of stationary phase. We have confirmed the validity and the utility of the novel integral representation and its asymptotic solution by comparing with the widely used mixed-path theorem and the experimental measurement performed in Kanto area and Tokyo bay.