Kirchhoff Integral Research Articles

Vector solution of the diffraction task using the Hertz vector

A large class of diffraction problems can be solved on the basis of the Huygens principle. However, methods of solving diffraction problems based on this principle exhibit narrow boundaries of applicability. The goal of the present work is to offer a relatively simple physically based and mathematically strict "dipole wave" vector theory of non-paraxial diffraction of electromagnetic radiation which allows analytical solutions of typical diffraction problems. The suggested theory logically retains the wave approach used in the Kirchhoff method and does not exhibit strict limitations to applicability inherent in the Kirchhoff integral. The diffraction problem is solved by using the Hertz vector in the Kirchhoff integral instead of the field vector. The method efficiency is illustrated in several examples. Analytical solutions of diffraction base problems have been obtained for linearly polarized radiation on an infinite slit and on various-shaped holes at an arbitrary angle of incidence and polarization. It was shown the possibility of vector addition particular solutions to obtain diffraction patterns from several holes. The diffraction of radiation with azimuthal and radial directions of polarization on a ring slit is also considered. The main qualitative feature of the obtained solutions is the presence of "poles" one or two points of zero field in the diffraction pattern which are superimposed on the common system of light and dark fringes. The poles are located along electrical field vector directions. The vector analytical formulas describing the propagation of some laser beams in the free space have been obtained too. The solutions of the diffractive problems satisfy the Maxwell equations and the reciprocity principle.

Matter diffraction at oblique incidence: Higher resolution and theHe34Efimov state

We study the diffraction of atoms and weakly bound three-atomic molecules from a transmission grating at non-normal incidence. Due to the thickness of the grating bars, the slits are partially shadowed. Therefore, the projected slit width decreases more strongly with the angle of incidence than the projected period, increasing, in principle, the experimental resolution. The shadowing, however, requires a revision of the theory of atom diffraction. We derive an expression in the style of the Kirchhoff integral of optics and show that the diffraction pattern exhibits a characteristic asymmetry which must be accounted for when comparing with experimental data. We then analyze the diffraction of weakly bound trimers and show that their finite size manifests itself in a further reduction of the slit width by $(3∕4)⟨r⟩$ where $⟨r⟩$ is the average bond length. The improved resolution at non-normal incidence may in particular allow us to discern, by means of their bond lengths, between the small ground state of the helium trimer [$⟨r⟩\ensuremath{\approx}1\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$, Barletta and Kievsky, Phys. Rev. A 64, 042514 (2001)] and its predicted Efimov-type excited state ($⟨r⟩\ensuremath{\approx}8\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$, ibid.), and in this way to experimentally prove the existence of this long-sought Efimov state.

Parameter inversion and angle migration in anisotropic elastic media

Seismic parameter inversion using the inverse generalized Radon transform (GRT) was originally formulated as an integral over source and receiver surface coordinates, involving a Jacobian from angle to surface coordinates. By formulating the inversion directly in terms of angle coordinates at the subsurface image point, I obtain simplified inversion formulas for point scatterers and subsurface reflectors. Amplitude‐preserving angle migration based on the inverse GRT gives an estimate of the scattering coefficient in the Born integral as a function of scattering angle and azimuth. The scattering coefficient is directly related to the linearized plane‐wave reflection coefficient between two anisotropic media, which limits the result to small angles and small contrasts in the elastic parameters. I use the Kirchhoff modeling integral for anisotropic media to formulate a Kirchhoff angle‐migration integral that is valid for large angles and large parameter contrasts. It gives an estimate of the plane‐wave reflection coefficient normalized with respect to amplitude. The Kirchhoff integrals, however, are non reciprocal, so I propose to use new reciprocal integrals for modeling and angle migration. The new migration integral, which is much simpler than the Kirchhoff integral, gives an estimate of the plane‐wave reflection coefficient normalized with respect to the energy flux normal to the reflector. A restricted inverse GRT has been proposed for migration‐velocity analysis. By simplifying this, I derive an expression for amplitude‐compensated migration that is the estimated reflection coefficient divided by the norm of the radiation pattern vector. This partly removes the effects of amplitude variations with scattering angle and azimuth.

Polarization of transmission scattering simulated by using a multiple-facets model.

A Mueller matrix for scattering by a rough plane surface of a glass hemisphere was simulated by using a micro-facet model. The algorithms are formulated in vector representation in terms of the input and output directions. The single-facet scattering simulation used the results of the Kirchhoff integral for medium rough surfaces with exponential height distribution. Scatterings by two or more facets were also simulated. For a fixed angle between the incident and the detection directions, the transmission scattering and its polarization properties were symmetric when plotted against the off-specular incident angle. The single-facet model generated no depolarization or polarization change. When double-facet scattering was included, polarizations were changed appreciably while depolarization was still very small. Depolarization increased appreciably when scattering by higher orders was included. The simulated results that include all orders of scattering fit excellently to the measured scattering transmittance and its polarization and depolarization.

Analytic Element Modeling of Cylindrical Drains and Cylindrical Inhomogeneities in Steady Two‐Dimensional Unsaturated Flow

The analytic element method, first developed for modeling flow in saturated porous media, is adapted to the solution of the governing equation for steady flow in unsaturated porous media. The governing equation for steady‐state unsaturated flow is made amenable to analytic element solution through transformation with the Kirchhoff integral, representation of the hydraulic conductivity by an exponential function of the pressure head, and use of a coordinate transformation. A number of analytic elements are available for the resulting modified Helmholtz equation; analytic element equations are presented for uniform flow, cylindrical drains, and cylindrical inhomogeneities. The analytic element solution allows for the analytic evaluation of the pressure head, saturation, and Darcy flux at any point in the vadose zone. The applicability of the analytic element method to simulate unsaturated flow is demonstrated by solving for several cases of steady flow in a region containing arbitrarily located cylindrical inhomogeneities, and for flow in a region containing one cylindrical inhomogeneity and two cylindrical drains.

Analytic Element Modeling of Cylindrical Drains and Cylindrical Inhomogeneities in Steady Two-Dimensional Unsaturated Flow

The analytic element method, first developed for modeling flow in saturated porous media, is adapted to the solution of the governing equation for steady flow in unsaturated porous media. The governing equation for steady-state unsaturated flow is made amenable to analytic element solution through transformation with the Kirchhoff integral, representation of the hydraulic conductivity by an exponential function of the pressure head, and use of a coordinate transformation. A number of analytic elements are available for the resulting modified Helmholtz equation; analytic element equations are presented for uniform flow, cylindrical drains, and cylindrical inhomogeneities. The analytic element solution allows for the analytic evaluation of the pressure head, saturation, and Darcy flux at any point in the vadose zone. The applicability of the analytic element method to simulate unsaturated flow is demonstrated by solving for several cases of steady flow in a region containing arbitrarily located cylindrical inhomogeneities, and for flow in a region containing one cylindrical inhomogeneity and two cylindrical drains.

Seismic reflectors beneath the central cones of Aso Volcano, Kyushu, Japan

The subsurface structure of Aso Volcano, central Kyushu, Japan, has been imaged down to 10 km below sea level and notable features of subsurface structure in the central cones were revealed by a 3-D seismic reflection analysis. The 3-D reflection analysis included reflection enhancements and 3-D migration. The reflection enhancements and the NMO correction were applied on waveform data from the seismic experiment ASO98. The 3-D migration with a Kirchhoff integral was processed for 1397 NMO corrected traces to calculate the reflectivity at about 60 000 image points in an approximately 7 km by 7.5 km square region. The results of the 3-D migration show that the subsurface structure beneath the central cones can be divided into three parts, the western part, the eastern part, and the northwestern part. Bright reflectors appear in most of them and a significant discontinuity is inferred. The reflector horizon at 2 km below sea level spreads beneath the western and the eastern part and is probably the top surface of the Pre-Aso volcanic rocks or the basement rocks. Reflection voids appear in the western part and just beneath the active crater. The reflector void in the western part implies a hot region and is most probably the known seismic anomalies, such as low velocity region or attenuation region. Another reflector void beneath the active crater can also be associated with a hot region, apparently surrounding the conduit system of the current volcanic craters. The major discontinuity is located beneath the northwest flank of the central cones. This discontinuity divides the northwest region with weak reflectors from other parts. The discontinuity may be the trace of the Oita–Kumamoto Tectonic Line, which is the major tectonic line in central Kyushu, crossing the Aso caldera.

Diffraction theory and focusing of light by a slab of left-handed material

A diffraction theory in a system consisting of left- and right-handed materials is proposed. The theory is based upon the Huygens's principle and the Kirchhoff's integral and it is valid if the wavelength is smaller than any relevant length of the system. The theory is applied to the calculation of the smearing of the foci of the Veselago lens due to the finite wavelength. We show that the Veselago lens is a unique optical instrument for the 3D imaging, but it is not a “superlens” as it has been claimed recently.

Correlation dimension of self-similar surfaces and application to Kirchhoff integrals

For surfaces generated by a class of asymptotically self-similar processes we define a probability measure, supported by the surface. We show that the correlation dimension of that surface measure is linked to the self-similarity exponent almost surely. This result is applied to the Kirchhoff integral well known in scattering from rough surfaces. We show that a certain average of the scattered intensity exhibits almost surely a scaling that allows us to recover the self-similarity index of the surface in an experiment involving only one sample of the surface.

Optical Diffraction in Close Proximity to Plane Apertures. II. Comparison of Half-Plane Diffraction Theories.

The accuracy and physical significance of the classical Rayleigh-Sommerfeld and Kirchhoff diffraction integrals are assessed in the context of Sommerfeld’s rigorous theory of half-plane diffraction and Maxwell’s equations. It is shown that the Rayleigh-Sommerfeld integrals are in satisfactory agreement with Sommerfeld’s theory in most of the positive near zone, except at sub-wavelength distances from the screen. On account of the bidirectional nature of diffraction by metallic screens the Rayleigh-Sommerfeld integrals themselves cannot be used for irradiance calculations, but must first be resolved into their forward and reverse components and it is found that Kirchhoff’s integral is the appropriate measure of the forward irradiance. Because of the inadequate boundary conditions assumed in their derivation the Rayleigh-Sommerfeld and Kirchhoff integrals do not correctly describe the flow of energy through the aperture.

Some remarks on intensity calculations near the caustic

Summary Using various formulae describing the field near the caustic, the amplitude distribution along the direction perpendicular to the smooth branch of the caustic of A2 type was calculated and then compared with the results of numerical calculations based on the Kirchhoff integral.

Earth imaging using seismic migration

Most subsurface imaging of the Earth is accomplished using seismic migration, which uses the entire wave forms recorded at the surface. The traditional method for performing migration was to use the Kirchhoff integral using a ray-based Green function to backproject the recorded data to positions of reflectors within the subsurface. Recent developments in Kirchhoff migration include the use of multivalued traveltime tables calculated with methods like wave-front construction ray tracing. Additional advances include the use of ray amplitudes calculated using dynamic ray tracing. Recently, more attention has been given to the development of wave-equation-based methods in which the recorded wave field is backprojected using some approximation to the wave equation. Wave-equation-based methods have grown in acceptance because better implementation of the backprojection operators have been developed and computing capability has allowed the generally slower methods to become more feasible.

Adiabatic energy levels and electric dipole moments of Rydberg states ofRb2andCs2dimers

We calculate potential energy curves for heavy alkali-metal dimers, ${\mathrm{Rb}}_{2}$ and ${\mathrm{Cs}}_{2}$ in which one of the atoms is in a highly excited Rydberg states. The method combines numerical integration of coupled equations, describing interaction of electron with the ground-state atom in the field of the Coulomb core of the Rydberg atom, with subsequent matching of the obtained wave function with the Coulomb Green's function in the form of the Kirchhoff integral. The spin-orbit interaction for the Rydberg electron is also included. The results show the existence of several groups of states. Most interesting of them are dominated either by the ${}^{3}S$ symmetry near the ground-state atom or by the ${}^{3}{P}_{J}$ symmetry, $J=0,1,2.$ All states, except the ${}^{3}{P}_{1}$ state, exhibit oscillatory dependence of energy on the internuclear distance that can support long-range molecular Rydberg states [e.g., ``trilobite states'' for the ${}^{3}S$ symmetry, Greene et al., Phys. Rev. Lett. 85, 2458 (2000)]. These states possess large diagonal and transition dipole moments which are expressed analytically in terms of the Coulomb wave functions and calculated in a broad range of internuclear separations.

Dipole-wave theory of electromagnetic diffraction

A theory of diffraction is presented which systematically employs the wave approach used in the Kirchhoff method and, unlike the Kirchhoff integral, does not have severe applicability limitations. The diffraction problem is solved by using the Hertz vector instead of the field vector used in the Kirchhoff integral. The basic diffraction problems for linearly, radially, and azimuthally polarized radiation are solved analytically. The key qualitative feature of the solutions is the presence of 'poles' — zero field points within the diffraction pattern of light and dark fringes. The poles lie along the direction of the electric field vector. The solutions obtained satisfy the Maxwell equations and the reciprocity principle.

Wave front reconstruction of Fresnel off-axis holograms with compensation of aberrations by means of phase-shifting digital holography

Off-axis holograms recorded with a CCD camera are numerically reconstructed in amplitude by calculating through the Fresnel–Kirchhoff integral. A phase-shifting Mach–Zehnder interferometer is used for recording four-quadrature phase-shifted off-axis holograms. The basic principle of this technique and its experimental verification are described. We show that the application of this algorithm allows for the suppression of the zero order of diffraction and of the twin image and that the contrast of the reconstructed images can be further enhanced by digital compensation of the aberrations introduced by the holographic recording system

Refractive lenses for coherent x-ray sources

Incoherent x rays in the wavelength interval from approximately 0.5-2 A have been focused with refractive lenses. A single lens would have a long focal length because the refractive index of any material is close to unity; but with a stack of N lens elements the focal length is reduced by the factor N, and such a lens is termed a compound refractive lens (CRL). Misalignment of the parabolic lens elements does not alter the focusing properties and results in only a small reduction in transmission. Based on the principle of spontaneous emission amplification in a FEL wiggler, coherent x-ray sources are being developed with wavelengths of 1-1.5 A and source diameters of 50-80 mum; and the CRL can be used to provide a small, intense image. Chromatic aberration increases the image size by an amount comparable with the diffraction-limited size, and so chromatic correction is important. Pulse broadening through the lens that is due to material dispersion is negligible. The performance of a CRL used in conjunction with a coherent source is analyzed by means of the Kirchhoff integral. For typical parameters, intensity gain is 10(5)-10(6), where gain is defined as the intensity ratio in an image plane with and without the lens in place. (There may be some confusion concerning the usage of the word intensity. As employed in this manuscript, intensity, also called irradiance, refers to power per unit area. This is a commonly accepted usage for intensity, although there are places in the literature where the term radiant incidence is reserved for this definition and intensity refers to power per unit solid angle.) The image intensity is maximized when the CRL is placed 100-200 m from the source, and the diameter of the diffraction-limited spot is approximately 0.12 mum.

Bistatic scattering coefficient from one- and two-dimensional random surfaces using the stationary phase and scalar approximation with shadowing effect: comparisons with experiments and application to the sea surface

In this paper, the bistatic scattering coefficient from one- and two-dimensional random surfaces using the stationary phase method and scalar approximation with shadowing effect is investigated. Both of these approaches use the Kirchhoff integral. With the stationary phase, the bistatic cross section is formulated in terms of the surface height joint characteristic function where the shadowing effect is investigated. In the case of the scalar approximation, the scattering function is computed from the previous characteristic function and in terms of expected values for the integrations over the slopes, where the shadowing effect is analysed analytically. Both of these formulations are compared with experimental data obtained from a Gaussian one-dimensional randomly rough perfectly-conducting surface. With the stationary-phase method, the results are applied to a two-dimensional sea surface.

Theoretical study of the Kirchhoff integral from a two-dimensional randomly rough surface with shadowing effect: application to the backscattering coefficient for a perfectly-conducting surface

In this paper, the backscattering coefficient of a two-dimensional randomly rough perfectly-conducting surface is investigated using the Kirchhoff approach with a shadowing function. The rough surface height/slope correlations assumed to be Gaussian are accounted for in this analysis. The scattering coefficient is then formulated in terms of a characteristic function for the integrations over the surface heights, in terms of expected values for the integrations over the surface slopes. Numerical comparisons of Kirchhoff's approach (KA) with the stationary-phase (SP) approximation are made with respect to the choice of the one-dimensional surface height autocorrelation function and the shadowing effect. For an isotropic surface the results show that SP underestimated the incoherent backscattering coefficient compared with KA. Moreover, when the correlation between the slopes and the heights is neglected, the shadowing effect may be ignored.

Narrow-waisted Gaussian beam discretization for short-pulse radiation from one-dimensional large apertures

We develop a Gabor-based Gaussian beam (GB) algorithm for representing two-dimensional (2-D) radiation from finite aperture distributions with short-pulse excitation in the time domain (TD). The work extends previous results using 2-D frequency-domain (FD) narrow-waisted Gaussian beams. The FD algorithm evolves from the rigorous Kirchhoff integration over the aperture distribution, which is then parameterized via the discrete Gabor basis and evaluated asymptotically for high frequencies to furnish the GB propagators to the observer. The TD results are obtained by Fourier inversion from the FD and yield pulsed beams (PB). We describe the resulting TD algorithm for several aperture distributions, ranging from simple linearly phased (linear delay) to arbitrary time delay profiles; the latter accommodate the important case of focusing TD aperture fields. For modulated pulses with Gaussian envelopes, we compute accurate closed form analytic solutions, which have been calibrated against numerical reference data. Our results confirm that the previously established utility of the Gabor-based narrow-waisted FD-GB algorithm for radiation from distributed apertures remains intact in the TD.

A discrete wavelet transform (DWT)‐based compression technique for the computation of Kirchhoff integrals in MR/FDTD

The multiple-region FDTD (MR/FDTD) has recently been introduced as an extension of the classical FDTD to multiple subregions within a problem domain. In MR/FDTD, the problem domain is broken down into several volumes interacting with each other by means of a near-field to near-field transformation. One of the available methods is the Kirchhoff integrals, a method that has been successfully applied to FDTD, but that suffers from a high demand for computation resources. In this paper, a new discrete wavelet transform (DWT)-based compression technique for the computation of Kirchhoff integrals is presented that leads to important savings both in memory size and computation time, while combining excellent accuracy and ease of implementation. © 2000 John Wiley & Sons, Inc. Microwave Opt Technol Lett 27: 312–316, 2000.