The content of these notes is based on the lectures by Prof. Graeme W. Milton (University of Utah) given in a course on metamaterials in Spring 2007.
Expanding a point source in plane waves[edit | edit source]
In the previous lecture we had determined that a two-dimensional point source
could be expanded into plane waves. We may think of such a point source as
a line source in three dimensions.
We can similarly try to expand true three-dimensional point sources in terms
of plane waves. To do that, let us start with a three-dimensional scalar
wave equation of the form
As before, assume that has a small positive
imaginary part (it is a slightly lossy material), i.e.,
If we express (1) in spherical coordinates and solve the
resulting differential equation, we get
where the symmetry of the equations with respect to the and
directions can be observed.
Alternatively, we can try to solve (1) using Fourier transforms.
To do that, let us assume that a Fourier transform of exists
and the inverse Fourier transform has the form
where , , and
Plugging (3) into (1) and using the observation
gives (for all not all zero)
Since the above equation holds for all values of , the Fourier components
must agree, i.e.,
Plugging (4) into (3) gives
Let us consider the integral over first. The poles are at
Now, for the integral is exponentially decreasing when . Therefore, the integral over can be split into
the sum of an integral along the real line + an integral over an arc of a
circle of radius infinity = sum of the residues at each of the poles (see
Figure 1 for a sketch of the situation).
Figure 1. Poles and integration path for integration over
Using the Residue theorem
we can show that
where is the value of at the poles, i.e.,
When , one takes the semicircular contour in the lower half plane and
picks up the residue at . The result for all can therefore be
The integral is over plane waves. The waves are evanescent, i.e., is
imaginary when .
Comparing equations (6) and (2), we get the Weyl identity Weyl19 for the solution of the wave equation in spherical coordinates
So far we have dealt with just planar wave equations. What about the full
From Maxwell's equation
Using the identity
Now, for an isotropic homogeneous medium
Plugging this into (8) we get
Plugging this into (9) gives
This equation has the form of the scalar wave equation
The only difference is that (10) consists of three scalar
wave equations and the source term is given by
Recall that, using the Green's function method, we can find the solution of
the scalar wave equation (11) (see Chew95 p.24-28
for details) as
In an analogous manner we can find the solution of (10), and
For electric dipole fields, if one has a point current source directed in the
direction, then the current density is given by
where is the current dipole moment, i.e., as and
, remains constant. If the origin is taken at
the point , we get
Plugging (13) into (12) gives
and using the identity , the magnetic field is given by
Substituting the Weyl identity (7) into these expression
gives formulae for and in terms of plane waves.
Scattering of radiation from a sphere[edit | edit source]
Recall the Airy solution for the scattering of light by a raindrop. In the
following we sketch the Mie solution which generalizes the analysis to
the scattering of electromagnetic radiation by a spherical object. The problem
remains similar, i.e., we wish to determine the scattering of a plane wave
incident on a sphere of refractive index . However, we now consider the
case where the wavelength of the incident radiation is not necessarily much
smaller than the size of the sphere.
Consider the sphere shown in Figure 2. We set up our
coordinate system such that the origin is at the center of the sphere. The
sphere has a magnetic permeability of and a permittivity .
The medium outside the sphere has a permittivity and a
permeability . The electric field is oriented parallel to the
axis and the axis points out of the plane of the paper.
Figure 2. Scattering of radiation from a sphere.
Let us now consider the situation where the material inside the sphere is
non-magnetic. Then we may write
where is the relative permittivity of the material inside the sphere.
Also, the incident plane wave is given by
where is the unit vector in the direction.
The solution of this problem was first given by Mie Mie08. A detailed
derivation is given in Kerker69. We follow the abbreviated version in
Before we can go into the details, we need to discuss vector potentials for
Vector potentials for electromagnetism[edit | edit source]
Since , there exists a vector potential such that
Also, from Maxwell's equation
In terms of the vector potential , we then have
Therefore, there exists a scalar potential such that
At this stage there is some flexibility in the choice of and .
A restriction that is useful is to require the potentials to satisfy the
Lorenz condition Lorenz67 (which is equivalent to requiring that the
charge be conserved)
Then, in the absence of free charges and currents in an isotropic
homogeneous medium, both potentials satisfy the wave equation, i.e.,
Even after these restriction the potentials are not uniquely defined and
one is free to make the gauge transformations
to obtain new potentials , provide satisfies the wave
The preceding potentials are well known. However, one can go one step
further and define superpotentials (see, for example, Bowman69).
The most widely used superpotentials are the electric and magnetic
Hertz vector potentials and (also known as
The terms of these potentials, the and can be expressed as
Comparing equations (17) with (16) and
(15) one sees that the superpotentials lead to symmetric
representations of and unlike when standard vector and scalar
potentials are used.
Of course, the superpotentials and are not uniquely
defined and one is free to make gauge transformations
where and are arbitrary scalar potential
Plugging these definitions into the Maxwell's equation lead to the
equations being satisfied if
where is an arbitrary scalar potential which is a function of position
The Lorentz condition is satisfied if
In fact, the potentials and can be expressed in terms of
For time harmonic problems, an important class of Hertz vector potentials
are those of the form (for spherical symmetry)
The vector is the radial vector from the origin in a spherical
coordinate system. The functions and are scalar potentials
(called Debye potentials) which satisfy the homogeneous wave
One important result is that every electromagnetic field defined in a
source-free region between two concentric spheres can be represented
there by two Debye potentials Wilcox57.
In spherical coordinates, the components of the fields between
two concentric spheres are given by
Recall the residue theorem which states that
and if is non-singular at , then the residue at is .
- J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi. Electromagnetic and Acoustic Scattering by Simple Shapes. North-Holland Publishing Company, Amsterdam, 1969.
- W. C. Chew. Waves and fields in inhomogeneous media. IEEE Press, New York, 1995.
- A. Ishimaru. Wave Propagation and Scattering in Random Media. Academic Press, New York, 1978.
- M. Kerker. The Scattering of Light. Academic Press, New York, 1969.
- L. Lorenz. On the identity of the vibrations of light with electrical currents. Philosphical Magazine, 34:287--301, 1867.
- G. Mie. Beitraege zur optik trueber medien speziell kolloidaler metalloesungen. Ann. Physik, 25:377--445, 1908.
- H. Weyl. Ausbreitung electromagnetischer wellen uber einem ebenen leiter. Annalen der Physik, 60:481--500, 1919.
- C. H. Wilcox. Debye potentials. J. Math. Mech., 6:167--201, 1957.