In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...
, the Weyl expansion, also known as the Weyl identity or angular spectrum expansion, expresses an outgoing
spherical wave as a
linear combination of
plane wave
In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space.
For any position \vec x in space and any time t, ...
s. In a
Cartesian coordinate system
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in ...
, it can be denoted as
:
,
where
,
and
are the
wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
s in their respective coordinate axes:
:
.
The expansion is named after
Hermann Weyl
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is asso ...
, who published it in 1919.
The Weyl identity is largely used to characterize the reflection and transmission of spherical waves at planar interfaces; it is often used to derive the
Green's function
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if \operatorname is the linear differenti ...
s for
Helmholtz equation
In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation
\nabla^2 f = -k^2 f,
where is the Laplace operator (or "Laplacian"), is the eigenvalu ...
in layered media. The expansion also covers
evanescent wave
In electromagnetics, an evanescent field, or evanescent wave, is an oscillating electric and/or magnetic field that does not propagate as an electromagnetic wave but whose energy is spatially concentrated in the vicinity of the source (oscillat ...
components. It is often preferred to the
Sommerfeld identity
The Sommerfeld identity is a mathematical identity, due Arnold Sommerfeld, used in the theory of propagation of waves,
:
\frac
= \int\limits_0^\infty I_0(\lambda r) e^ \frac
where
:
\mu =
\sqrt
is to be taken with positive real part, to e ...
when the field representation is needed to be in Cartesian coordinates.
The resulting Weyl integral is commonly encountered in
microwave integrated circuit analysis and electromagnetic radiation over a stratified medium; as in the case for Sommerfeld integral, it is
numerically evaluated.
As a result, it is used in calculation of Green's functions for
method of moments for such geometries. Other uses include the descriptions of
dipolar
In physics, a dipole () is an electromagnetic phenomenon which occurs in two ways:
*An electric dipole deals with the separation of the positive and negative electric charges found in any electromagnetic system. A simple example of this syste ...
emissions near surfaces in
nanophotonics
Nanophotonics or nano-optics is the study of the behavior of light on the nanometer scale, and of the interaction of nanometer-scale objects with light. It is a branch of optics, optical engineering, electrical engineering, and nanotechnolog ...
,
holographic
Holography is a technique that enables a wavefront to be recorded and later re-constructed. Holography is best known as a method of generating real three-dimensional images, but it also has a wide range of other applications. In principle, i ...
inverse scattering problems,
Green's functions in
quantum electrodynamics
In particle physics, quantum electrodynamics (QED) is the relativistic quantum field theory of electrodynamics. In essence, it describes how light and matter interact and is the first theory where full agreement between quantum mechanics and spec ...
and
acoustic or
seismic wave
A seismic wave is a wave of acoustic energy that travels through the Earth. It can result from an earthquake, volcanic eruption, magma movement, a large landslide, and a large man-made explosion that produces low-frequency acoustic ener ...
s.
See also
*
Angular spectrum method
*
Fourier optics
*
Green's function
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if \operatorname is the linear differenti ...
*
Plane wave expansion
In physics, the plane-wave expansion expresses a plane wave as a linear combination of spherical waves:
e^ = \sum_^\infty (2 \ell + 1) i^\ell j_\ell(k r) P_\ell(\hat \cdot \hat),
where
* is the imaginary unit,
* is a wave vector of length ,
* ...
*
Sommerfeld identity
The Sommerfeld identity is a mathematical identity, due Arnold Sommerfeld, used in the theory of propagation of waves,
:
\frac
= \int\limits_0^\infty I_0(\lambda r) e^ \frac
where
:
\mu =
\sqrt
is to be taken with positive real part, to e ...
References
Sources
*
*
*
*
Mathematical identities
Mathematical physics
Electrodynamics
Wave mechanics
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