Three-dimensional Gaussian Beam Research Articles

Gaussian beam tracing for three-dimensional sound propagation problems in subsonic moving medium

The study of sound propagation in moving medium is important in various fields, such as atmospheric sound and underwater acoustics. To address this problem, a three-dimensional Gaussian beam tracing model is developed for subsonic moving medium, based on the Helmholtz equation of velocity potential for high-frequency sound wave in a moving medium with arbitrary Mach numbers. The dynamic ray equations in the moving medium are derived by using the beam tracing method, and further the partial differential equation is transformed into ordinary differential equations, so as to be able to more efficiently and accurately calculate the three-dimensional sound field in the moving medium. The Gaussian beam tracing method reveals that the expansion of the beam in a moving medium is more complex than in a static medium, and the energy in the ray tube is not necessarily conserved. The model is applied to several problems, including point source sound propagation in a semi-infinite homogeneous medium, three-dimensional long-range sound propagation in horizontally layered atmospheres, and three-dimensional sound propagation in the Gulf Stream. The results of the point source sound propagation problem in the semi-infinite homogeneous medium verify the effectiveness and accuracy of the model. The results of the atmospheric sound propagation problem indicate that compared with the commonly used <i>N</i> × 2D method, the three-dimensional Gaussian beam tracing in a moving medium fully considers the effect of medium motion, especially the effect of crosswind, and can calculate the sound pressure field more accurately. Although the Mach number of the ocean current is very small, its effect cannot be ignored. The ocean current can quantitatively change the sound propagation mode and affect the convergence zone position. In some areas, the difference between calculation results with and without considering the ocean current is more than 5 dB. Moreover, the deviation of rays caused by lateral flow is much smaller, and even in the areas with complex terrain, the deviation becomes more obvious only after being reflected by the interface. Moreover, the influence of lateral flow on sound propagation is much smaller than that caused by flow velocity parallel to the propagation direction. In conclusion, the developed Gaussian beam tracing method provides an accurate and efficient approach for solving the sound propagation problem in subsonic moving media.

3-D Gaussian beam scattering from a gyromagnetic perforated layer: Quasi-static approach

This paper is devoted to the study of the scattering of a three-dimensional (3-D) Gaussian beam with the circular cross section from a double periodic perforated gyromagnetic layer with polarization independent unit-cell, in the quasi-static approximation. We used the plane-wave spectrum representation for Gaussian beam field representation and reduced it to a single integral representation. The phenomena of the lateral beam shift influenced by Faraday rotation and the nearly total beam transmission when passing through gyromagnetic slab biased with an external static magnetic field in the Faraday configuration were considered.

Nonspecular reflection of Gaussian wave beams on a two-dimensional periodic array with shorted waveguides of rectangular cross-section

This paper investigates nonspecular reflection of three-dimensional Gaussian wave beams from a two-dimensional periodic grating of short-circuited rectangular waveguides. ‘Orthogonal’ beam reflection from a periodic structure has been examined. The correlation of reflection rates in terms of power and the length of short-circuited waveguides has been estimated for specular and nonspecular beams. The relation of the reflected beams’ directional pattern in the far field to the array and incident beam parameters has been investigated. The splitting effect of the reflected beams’ directional pattern has been established in the case of p-polarized beam incidence. The possibility of applying a grid of rectangular waveguide sections in an electromagnetic field impulse compressor has been demonstrated. Various operational modes of the reflection grating used as a coupling element between the open-structure resonator and the transmission line have been described. The effects of directional pattern reduction and reflected beam directional pattern deformation have been established. These effects have been physically grounded. The results are presented in the form of three-dimensional directional patterns of the diffuse beam field in the far field. The calculation algorithm is based on the available solutions of the key vector problems of plane linearly polarized electromagnetic wave diffraction on two-dimensionally periodic structures.

Autocollimation of a Gaussian beam reflected by a beveled array of short-circuited rectangular waveguides

This paper investigates the dispersion of a three-dimensional Gaussian wave beam on a two-dimensional periodic array of short-circuited waveguides with a beveled (nonrectangular) grid in the autocollimation mode. The correlation of reflection rates in terms of the power and the length of the short-circuited waveguides has been estimated for specular and non-specular beams. The relationship between the directional pattern of the reflected beam in the far field and the array and incident beam parameters has been investigated. Three-dimensional directional patterns have been designed in terms of the power of specular and non-specular beams in the far field. Energy characteristics and three-dimensional directional patterns have been obtained for the non-specular beam in the far field. A new algorithm for analyzing three-dimensional problems of scattering of linearly polarized wave beams on two-dimensional periodic structures is proposed. This algorithm allows the investigation of wave beams with arbitrary field distributions over the cross-section. The case of oblique incidence of a linearly polarized Gaussian wave beam on a two-dimensional periodic structure is considered. A reflective array with short rectangular waveguides is chosen as the structure. The beam pattern transformation of the scattered beam in the far field is investigated. The effects of narrowing of the beam pattern and distortion of the reflected beam pattern are detected. A physical explanation of these effects is offered. The results are presented in the form of two-dimensional and three-dimensional patterns of the scattered field of the beam in the far field.

Spatial modifications of three-dimensional elliptic Gaussian beam scattered by two-dimensional periodic array.

The diffraction problem of a three-dimensional elliptic p- polarized Gaussian beam on an aperture array of rectangular holes is solved. The new algorithm for the solution of three-dimensional scattering problems of linearly polarized wave beams on two-dimensional periodic structures is offered. The given algorithm allows exploring of wave beams with any allocation of a field on cross section. The case of oblique incidence of linearly polarized elliptic Gaussian wave beam on two-dimensional periodic structure is viewed. As structure the rectangular waveguides phased antenna array is chosen. The elliptic shape of a beam cross section gives the chance to proportion energy of an incident field in a plane of an antenna array in the chosen direction. The frequency dependence of the reflection coefficient intensity for the Gaussian beam is calculated. For the analysis of patterns of the reflected and transmitted beams in a far zone the frequencies on which the strongest interaction between next waveguides channels is observed have been chosen. Dynamics of patterns transformation of the reflected and transmitted beams depending on the form of cross-section and a polarization direction of an incident beam on different frequencies is investigated. It is determined that shape of the pattern of reflected and transmitted beams (symmetry, asymmetry, bifurcation, amplitude, width) depends on chosen spatial orientation of the ellipse axes of the cross section in the incident beam. Frequency ranges, in which the form of the reflected and transmitted beams is not Gaussian, are defined. The nature of transformation of the patterns of scattered beams was examined. Narrowing effect of the pattern of transmitted beam and deformation of the pattern of reflected beam is detected. A physical explanation of these effects is given. The results are presented in the form of two- and three-dimensional patterns of the scattered field of beams in the far field.

Double refraction of a Gaussian beam into a uniaxial crystal

For a linearly polarized three-dimensional Gaussian beam in air that is normally incident upon a plane interface with a uniaxial crystal with optic axis in an arbitrary direction, we present integral representations for the transmitted field suitable for asymptotic analysis and efficient numerical evaluation and derive analytical expressions for transmitted nontruncated Gaussian beams for the cases in which the incident beam is polarized parallel to the plane containing the optic axis and the interface normal and transverse to it. The general solution for an arbitrary polarization state of an incident Gaussian beam follows by superposition of these two solutions.

THE ELLIPTIC GAUSSIAN BEAM SCATTERING ON PHASED ANTENNA ARRAY WITH RECTANGULAR WAVEGUIDES

The difiraction problem of a three-dimensional elliptic Gaussian beam on a aperture array of rectangular holes is solved. The both normal and oblique incidences of the beam are considered and the results are presented in the form of the three-dimensional patterns. The pattern lobe distortion and conditions at which the side lobes appear are studied. The conditions under which the shift of the re∞ected and transmitted fleld patterns appears are studied. The existence of higher spatial Floquet harmonics in the case of oblique beam incidence is observed.

GAUSSIAN BEAM SCATTERING ON TWO-DIMENSIONAL PERIODIC APERTURE ARRAY

The difiraction problem of three-dimensional Gaussian beam on the aperture array of rectangular holes is solved. A new algorithm for calculating scattered flelds of the beam is proposed. The conditions under which the distortion of the re∞ected fleld pattern and the narrowing of the transmitted fleld pattern appear are studied.

2.5D Quantitative Millimeter Wave Imaging of a Hidden Object on a Simplified Human Body Model Using Value Picking Regularization

In this contribution, the authors provide a proof of principle for quantitative imaging of concealed objects on the human body using millimeter waves. A two-and-a-half-dimensional (2.5D) quantitative millimeter wave imaging algorithm is applied to reconstruct a hidden dielectric object on a clothed simplified human body model. At millimeter wave frequencies, the incident field is typically a fully three-dimensional (3D) Gaussian beam, illuminating only a limited spot on the body. Due to the large dimensions of the human body in terms of wavelengths, a 3D discretization is hardly feasible. Therefore, it is assumed that the electromagnetic properties of the body do not significantly change within the illuminated spot, along the longitudinal direction of a person. Hence, only the cross-section of a human body model is discretized. This 2.5D assumption however is still not sufficient to reduce the forward problem to a feasible size. Therefore, a priori knowledge on the illumination and on the scattering properties of the clothed human body is used to deduce a simplified model to describe the cross-section of the clothed human abdomen. The complex permittivity profile of a small dielectric object, hidden underneath clothing and representing some type of explosive, is reconstructed. The complex permittivity profiles of all other scatterers are assumed to be known. The presented quantitative inverse scattering algorithm is based on a Newton-type optimization, combined with an approximate line search and regularized by applying Stepwise Relaxed Value Picking regularization. The input data of the quantitative inverse scattering problem are synthetic scattering data since the authors are not aware of any amplitude and phase measurement data for concealed weapon detection yet made available to the inversion community at these high frequencies.

THREE-DIMENSIONAL GAUSSIAN BEAM SCATTERING FROM A PERIODIC SEQUENCE OF BI-ISOTROPIC AND MATERIAL LAYERS

The three-dimensional Gaussian beam scattering from the bounded periodic sequence of one-to-one composed isotropic magnetodielectric and bi-isotropic layers are investigated. The beam field is represented by an angular continuous spectrum of plane wave. The problem of the partial plane wave diffraction on the structure is solved using the circuit theory and the transfer matrix methods. It is found that after reflection from the structure,the circular Gaussian beam becomes,in general,an elliptical Gaussian beam,in addition to a displacement of the beam axis from the position predicted by ray optics.

Angular sensitivities of volume gratings for substrate-mode optical interconnects

The angular sensitivities of slanted volume gratings (VGs) illuminated by three-dimensional (3-D) converging-diverging spherical Gaussian beams for substrate-mode optical interconnects in microelectronics are analyzed by application of 3-D finite-beam rigorous coupled-wave analysis. Angular misalignments about the z, y, and x axes that correspond to yaw, pitch, and roll misalignments resulting from manufacturing tolerances of chips are investigated. Two cases of linear polarization of the central beam of the Gaussian are considered: E perpendicular K and H perpendicular K, where K is the grating vector. From worst-case manufacturing tolerances, the ranges of yaw, pitch, and roll misalignment angles are alpha = +/-1.17 degrees, beta= +/-3.04 degrees, and gamma = +/-3.04 degrees, respectively. Based on these ranges of misalignment angles, the decreases of diffraction efficiencies for slanted VGs that are due to both the yaw and the roll misalignments are relatively small. However, the efficiency of substrate-mode optical interconnects achieved by slanted VGs could be reduced by 61.04% for E perpendicular K polarization and by 58.63% for H perpendicular K polarization because of the pitch misalignment. Thus the performance of a VG optical interconnect is most sensitive to pitch misalignment.

Three-dimensional converging–diverging Gaussian beam diffraction by a volume grating

The diffraction characteristics of a volume grating (VG) illuminated by a three-dimensional (3-D) converging-diverging Gaussian beam at conical incidence are investigated by applying 3-D finite-beam (FB) rigorous coupled-wave analysis (RCWA) based on the conventional 3-D RCWA in conjunction with two-dimensional plane-wave decomposition. The Gaussian beam is assumed to have an arbitrary incidence angle, an arbitrary azimuthal angle, and any linear polarization. The two cases with linear polarizations of the central beam of the Gaussian (E perpendicular K and H perpendicular K) are investigated. The diffraction efficiencies and the diffracted beam profiles for both unslanted VGs and slanted VGs (designed for substrate-mode optical interconnects) are presented. In general, the diffraction efficiencies of a converging-diverging spherical Gaussian beam diffracted by both unslanted VGs and slanted VGs increase and approach the central-beam results as the refractive-index modulation increases.

Generalized [formula omitted]-factor of hollow Gaussian beams through a hard-edge circular aperture

Based on the generalized truncated second-order moments, the generalized M 2 -factor ( M G 2 -factor) of three-dimensional hollow Gaussian beams (HGBs) through a hard-edge circular aperture is studied in cylindrical coordinate system analytically and numerically. The closed-form expression for the M G 2 -factor of the truncated HGBs, which is dependent on the truncation parameter β and the beam order n, can be simplified to that of the truncated, the untruncated Gaussian beams and the untruncated HGBs. Also, the power fraction is demonstrated analytically and numerically, which shows that the area of the dark region across the HGBs increases as n increasing.

Scatterer size estimation in pulse-echo ultrasound using focused sources: theoretical approximations and simulation analysis.

The speckle in ultrasound images has long been thought to contain information related to the tissue microstructure. Many different investigators have analyzed the frequency characteristics of the backscattered signals to estimate the scatterer acoustic concentration and size. Previous work has been mostly restricted to unfocused or weakly focused ultrasound sources, thus limiting its implementation with diagnostically relevant fields. Herein, we derive equations capable of estimating the size of a scatterer for any reasonably focused source provided that the velocity potential field in the focal region can be approximated as a three-dimensional Gaussian beam, scatterers are a sufficient distance from the source, and the field is approximately constant across the scatterer. The calculations show that, when estimating the scatterer size, correcting for focusing requires a generalized attenuation-compensation function that includes both attenuation and focusing along the beam axis. The Gaussian approximation is validated by comparing the ideal velocity potential field for three spherically focused sources with f-numbers of 1, 2, and 4 to the Gaussian approximation for frequencies from 2 to 14 MHz. The theoretical derivations are evaluated by simulating the backscatter by using spherically focused sources (f-numbers of 1, 2, and 4) adjacent to attenuating media (0.05 to 1 dB/cm/MHz) that contain scatterers with Gaussian impedance distributions. The generalized attenuation-compensation function yielded results accurate to 7.2% while the traditional attenuation-compensation functions that neglected focusing had errors as high as 103%.

Propagation properties of three-dimensional flattened Gaussian beams in uniaxially anisotropic crystals

Starting from the paraxial vectorial theory of beams propagating in uniaxially anisotropic media, the analytical propagation equations of three-dimensional flattened Gaussian beams (FGBs) in uniaxial crystals are derived and illustrated numerically. It is shown that, due to the anisotropy of crystals, the ordinary and extraordinary beams originated by incident FGBs propagate with different Rayleigh lengths, thus the linear polarization state and axial symmetry of incident FGBs do not remain during propagation in crystals.

Three-dimensional electromagnetic diffraction of a Gaussian beam by a perfectly conducting half-plane.

Formulas are derived for the diffraction of a three-dimensional electromagnetic Gaussian beam by a perfectly conducting half-plane. The beam can be incident from any direction, and the main component of the electric field can point in any direction on the plane of the beam waist. The center of the beam waist is on the edge of the half-plane. The incident beam is constructed as a superposition of plane waves, and the total diffracted field is obtained from a superposition of the diffracted fields that are due to each plane wave. Physical constraints that limit the size and direction of the beam relative to the half-plane are described and incorporated into the theory. The scattered field in the far zone is obtained by asymptotic evaluation of the general formulas. Graphical results for the near-field as well as far-field patterns are presented and discussed.

Determination of longitudinal and transverse shifts of three-dimensional Gaussian beams reflected at a grating near an anomaly

Spatial displacements of three-dimensional Gaussian beams diffracted at a reflection grating are studied theoretically and numerically. The complex diffraction coefficients (amplitudes and phases) of the grating diffracted plane waves calculated by a rigorous method for conical diffraction are the basis for this investigation. The classical analytical formula for the longitudinal shift depending on the gradient of the reflection phase is generalized to the simultaneous analytical treatment of the longitudinal Goos—Hänchen like shift as well as a transverse shift. A second method uses the full integration on the whole spectrum of plane waves of the diffracted beam.

Analytical approach to the transmission of a Gaussian beam into a biaxial crystal

We study the transmission of a linearly polarised three-dimensional Gaussian beam through a plane interface between an isotropic medium and a biaxially anisotropic crystal for the case in which the interface normal is along one of the principal axes of the crystal. To that end, we use recently developed exact solutions for the transmitted field inside a biaxial crystal to derive an analytical expression for the transmitted field in the paraxial approximation for the case in which the incident wave is a linearly polarised non-truncated Gaussian beam with its main propagation direction along the interface normal. We express the non-truncated transmitted beam inside the biaxial crystal in a form similar to that for a scalar Gaussian beam that propagates in a homogeneous, isotropic medium. For a truncated incident Gaussian beam we derive a simplified integral representation for the transmitted field that is valid also in the limit as the biaxial crystal reduces to a uniaxial or an isotropic medium.

Using shear wave amplitude patterns to detect metastable olivine in subducted slabs

Olivine within the lowest temperature regions of subducting oceanic lithosphere may persist metastably to depths well below the equilibrium phase boundary for transformation to β phase. This may produce a tapering low seismic velocity wedge within the slab in the transition zone; transformational faulting of such metastable olivine has been proposed as a possible explanation of deep earthquakes. Detecting seismic velocity structure associated with metastable olivine within a slab using seismic wave arrival time information alone has proven problematic. Motivated by previous two‐dimensional numerical calculations for upgoing and downgoing seismic waves, a three‐dimensional Gaussian beam method is used to assess whether the azimuthally varying teleseismic amplitude signature produced by a slab with an internal wedge of metastable olivine is distinguishable from that for a slab with equilibrium transitions. Five slab velocity structures are considered: a thermal model with horizontal phase transitions at the same depths as in ambient mantle, two models incorporating equilibrium transitions within the slab, and two models with tongues of metastable olivine. A depth‐ and temperature‐dependent thermal conductivity term and the effect of the latent heat of transformation of olivine to β phase are included in the thermal modeling. Three‐dimensional dynamic ray tracing and Gaussian beam summation are performed for events located at different depths and lateral positions for each of the slab models. For sources near 400 km depth the presence of low‐velocity metastable olivine produces relatively complex azimuthal shear wave amplitude patterns compared to thermal and equilibrium models. Comparison with observed amplitudes for four events in the Kurile slab indicates that allowing for internal slab velocity complexity in the transition zone provides better agreement with observations than simple thermal and equilibrium slab models. This indicates that careful modeling of teleseismic shear wave amplitude patterns may have the resolution to detect and characterize the presence of metastable olivine in deep slabs.

Numerical study of the displacement of a three-dimensional Gaussian beam transmitted at total internal reflection. Near-field applications

Longitudinal and transverse shifts of a light beam at total internal reflection was experimentally studied by far-field measurements on the reflected field. We propose to use a scanning tunneling optical microscope (STOM) to study these shifts in transmission, and we present a theoretical model of this proposed experiment to obtain a numerical estimation of these shifts. We study the reflection and the transmission of a three-dimensional polarized incident beam. We verify the validity of our formalism by studying the Goos-Hanchen shift in reflection and by comparing our results with published ones. Then we calculate STOM images of the transmitted field distribution. On the images the well-known Goos-Hanchen shift is easily observed. But we also encounter a smaller shift, perpendicular to the plane of incidence. This transverse shift was also observed in reflection by Imbert and Levy [Nouv. Rev. Opt. 6, 285 (1975)]. We study the variations of the two shifts versus various parameters such as the angle of incidence, the optical index, and the incident polarization. Then we discuss the feasibility of the near-field observation of these shifts.