Electromagnetic field of a dipole on a two‐layer earth

Geophysics ◽  
1981 ◽  
Vol 46 (3) ◽  
pp. 309-315 ◽  
Author(s):  
W. C. Chew ◽  
J. A. Kong
Keyword(s):  
Electromagnetic Field ◽  
Saddle Point ◽  
Branch Point ◽  
Asymptotic Approximation ◽  
Spherical Wave ◽  
Geometrical Optics ◽  
Parabolic Cylinder ◽  
Point Singularity ◽  
Image Source ◽  
Free Part

The electromagnetic field due to a horizontal electric dipole placed on top of a two‐layer earth is represented in terms of fields due to a dipole over a half‐space earth and its image source fields. Integral representations of image source fields are evaluated with uniform asymptotic approximations. Leading order ordinary saddle‐point approximation, giving rise to the geometrical optics approximation (GOA), is shown to be inaccurate. This is especially true when the angle of observation is close to the critical angle, which corresponds to the presence of a branch‐point singularity near the saddle point. In the uniform asymptotic approximation, the integrand of the image source integral is split into a branch‐point free part and another part containing the branch‐point singularity. The branch‐point free part can be approximated with a spherical wave function, while the part containing the branch point can be approximated with parabolic cylinder functions. Vertical magnetic field components and the horizontal electric field component near the surface are illustrated and compared with the geometrical optics approximation, giving the direct numerical result as well as experimental measurement. It is shown that the uniform asymptotic approximation yields excellent agreement with numerical and experimental results compared to the geometrical optics approximation.

scholarly journals A Green’s Function for Acoustic Problems in Pekeris Waveguide Using a Rigorous Image Source Method

2021 ◽  
Vol 11 (6) ◽  
pp. 2722
Author(s):  
Zhiwen Qian ◽  
Dejiang Shang ◽  
Yuan Hu ◽  
Xinyang Xu ◽  
Haihan Zhao ◽  
...  
Keyword(s):  
Green’S Function ◽  
Shallow Water ◽  
Green's Function ◽  
Plane Waves ◽  
Spherical Wave ◽  
Sound Field ◽  
Efficient Computation ◽  
Image Source ◽  
Source Method ◽  
Pekeris Waveguide

The Green’s function (GF) directly eases the efficient computation for acoustic radiation problems in shallow water with the use of the Helmholtz integral equation. The difficulty in solving the GF in shallow water lies in the need to consider the boundary effects. In this paper, a rigorous theoretical model of interactions between the spherical wave and the liquid boundary is established by Fourier transform. The accurate and adaptive GF for the acoustic problems in the Pekeris waveguide with lossy seabed is derived, which is based on the image source method (ISM) and wave acoustics. First, the spherical wave is decomposed into plane waves in different incident angles. Second, each plane wave is multiplied by the corresponding reflection coefficient to obtain the reflected sound field, and the field is superposed to obtain the reflected sound field of the spherical wave. Then, the sound field of all image sources and the physical source are summed to obtain the GF in the Pekeris waveguide. The results computed by this method are compared with the standard wavenumber integration method, which verifies the accuracy of the GF for the near- and far-field acoustic problems. The influence of seabed attenuation on modal interference patterns is analyzed.

Resonant Raman scattering at the saddle-point singularity inInxGa1−xAs

1985 ◽  
Vol 32 (2) ◽  
pp. 1005-1008 ◽  
Author(s):  
K. P. Jain ◽  
R. K. Soni ◽  
S. C. Abbi ◽  
M. Balkanski
Keyword(s):  
Raman Scattering ◽  
Saddle Point ◽  
Point Singularity ◽  
Resonant Raman ◽  
Resonant Raman Scattering

scholarly journals Electromagnetic Radiation from the Geometrical Point of View

2018 ◽  
Vol 186 ◽  
pp. 01002
Author(s):  
Divakov Dmitriy ◽  
Malykh Mikhail ◽  
Tiutiunnik Anastasiia
Keyword(s):  
Electromagnetic Field ◽  
Electromagnetic Radiation ◽  
Coordinate System ◽  
Maxwell's Equations ◽  
Plane Waves ◽  
Geometrical Optics ◽  
Point Of View ◽  
Curvilinear Coordinates ◽  
Point To Point ◽  
The Relationship

The article describes the relationship between the solutions of Maxwell's equations which can be considered at least locally as plane waves and the curvilinear coordinates of geometrical optics. We introduce phase-ray coordinate system for any electromagnetic field if vectors E and H are orthogonal to each other and their directions do not change with time t, but may vary from point to point in the domain G.

Optical Electromagnetics I

2006 ◽  
Author(s):  
Michael E. Thomas
Keyword(s):  
Electromagnetic Field ◽  
Optical Spectrum ◽  
Geometrical Optics ◽  
Theoretical Description ◽  
Index Of Refraction ◽  
Wave Number ◽  
Physical Optics ◽  
Electromagnetic Propagation ◽  
Optical Propagation ◽  
Precise Knowledge

In this chapter, the optical spectrum is defined and subdivided into many sub-bands, which are traditionally determined by transparency in various media. Propagation of the electromagnetic field in vacuum, as based on Maxwell’s equations, and basic notions of geometrical and physical optics, are covered. The theoretical and conceptual foundation of the remaining chapters is established in this chapter and the next. Optical electromagnetic propagation is generally and often accurately described by classical geometrical optics or ray optics. When diffraction or wave interference is of concern, then the more complete field of physical optics is used. Geometrical optics requires precise knowledge of the spatial and spectral dependence of the index of refraction. This requires electrodynamics, which is most appropriately described by quantum optics. These topics are covered in the first five chapters. The definitions of the optical spectrum and the various models for describing propagation are introduced in the following. The optical electromagnetic field covers the range of frequencies from microwaves to the ultraviolet (UV) or wavelengths from 10 cm to 100 nm. This is a very liberal definition covering six orders of magnitude, yet the description of propagation is very similar over this entire band, and distinct from radio-wave propagation and x-ray propagation. A listing of the nomenclature for the different spectral bands within the range of optical wavelengths is given in Table 1.1. Other commonly used units of spectral measure such as wave number, frequency, and energy are also listed in the table. These various quantities are related to wavelength by the following formulas: where c is the speed of light (c = 2.99792458 × 108 m/sec), λ is wavelength, f is frequency in hertz, E is energy, h is Planck’s constant (h = 6.6260755(40) × 10−34 J sec), and ν is frequency in wave numbers (the number of wavelengths per centimeter). Although wavelength is commonly used by applied scientists and engineers, frequency is the most appropriate unit for the theoretical description of light–matter interactions. Because of the importance of spectroscopy in the discussion of optical propagation, the spectroscopic unit of wave number will be consistently used.

scholarly journals Topological phase diagram and saddle point singularity in a tunable topological crystalline insulator

2015 ◽  
Vol 92 (7) ◽  
Author(s):  
Madhab Neupane ◽  
Su-Yang Xu ◽  
R. Sankar ◽  
Q. Gibson ◽  
Y. J. Wang ◽  
...  
Keyword(s):  
Phase Diagram ◽  
Saddle Point ◽  
Topological Phase ◽  
Point Singularity ◽  
Topological Crystalline Insulator

Evolution of the polarization of electromagnetic waves in weakly anisotropic inhomogeneous media — a comparison of quasi-isotropic approximations of the geometrical optics method and the Stokes vector formalism

Open Physics ◽  
2008 ◽  
Vol 6 (3) ◽  
Author(s):  
Yury Kravtsov ◽  
Bohdan Bieg
Keyword(s):  
Electromagnetic Field ◽  
Electromagnetic Waves ◽  
Normal Modes ◽  
Geometrical Optics ◽  
Inhomogeneous Media ◽  
Stokes Vector ◽  
Coupled Equations ◽  
Isotropic Approximation ◽  
Total Phase ◽  
Electromagnetic Beams

AbstractThe main methods describing polarization of electromagnetic waves in weakly anisotropic inhomogeneous media are reviewed: the quasi-isotropic approximation (QIA) of geometrical optics method that deals with coupled equations for electromagnetic field components, and the Stokes vector formalism (SVF), dealing with Stokes vector components, which are quadratic in electromagnetic field intensity. The equation for the Stokes vector evolution is shown to be derived directly from QIA, whereas the inverse cannot be true. Derivation of SVF from QIA establishes a deep unity of these two approaches, which happen to be equivalent up to total phase. It is pointed out that in contrast to QIA, the Stokes vector cannot be applied for a polarization analysis of the superposition of coherent electromagnetic beams. Additionally, the ability of QIA to describe a normal modes conversion in inhomogeneous media is emphasized.

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