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
![{\displaystyle {\text{(1)}}\qquad \left[{\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}+k_{0}^{2}\right]~\varphi (x,y,z)=-\delta (x)~\delta (y)~\delta (z)~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe5e70dca155b908abdd68e858377453cd00996a)
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
that

gives (for all
not all zero)
![{\displaystyle {\cfrac {1}{8\pi ^{3}}}~\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\left[-k_{x}^{2}-k_{y}^{2}-k_{z}^{2}+k_{0}^{2}\right]~{\widehat {\varphi }}(\mathbf {k} )~e^{i~\mathbf {k} \cdot \mathbf {x} }~{\text{d}}\mathbf {k} =-{\cfrac {1}{8\pi ^{3}}}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }e^{i\mathbf {k} \cdot \mathbf {x} }~{\text{d}}\mathbf {k} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/857e9b4c297ae91e1a91b4a5fec9fb8ed361f231)
Since the above equation holds for all values of
, the Fourier components
must agree, i.e.,
![{\displaystyle ~\left[-k_{x}^{2}-k_{y}^{2}-k_{z}^{2}+k_{0}^{2}\right]~{\widehat {\varphi }}(\mathbf {k} )=-1}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3f2f263c390225e85e03e7c6c92a55bd710e931a)
Therefore,

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
[1]
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
written as

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
Maxwell's equations?
From Maxwell's equation

Using the identity

we get

Now, for an isotropic homogeneous medium

Plugging this into (8) we get

Recall that

Plugging this into (9) gives

or,
![{\displaystyle {\text{(10)}}\qquad {\left[\nabla ^{2}+k^{2}\right]~\mathbf {E} (\mathbf {r} )=-i\omega \mu ~\left[{\cfrac {1}{k^{2}}}~{\boldsymbol {\nabla }}({\boldsymbol {\nabla }}\cdot \mathbf {J} (\mathbf {r} ))+\mathbf {J} (\mathbf {r} )\right]~.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97dd3f682d3360839f1090eeaed25396b941f91b)
This equation has the form of the scalar wave equation
![{\displaystyle {\text{(11)}}\qquad \left[\nabla ^{2}+k^{2}\right]~\varphi (\mathbf {r} )=s(\mathbf {r} )~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7cce35758e4ee2245f6629712cefb4ca81095a1c)
The only difference is that (10) consists of three scalar
wave equations and the source term is given by
![{\displaystyle \mathbf {s} (\mathbf {r} ):=-i\omega \mu ~\left[{\cfrac {1}{k^{2}}}~{\boldsymbol {\nabla }}({\boldsymbol {\nabla }}\cdot \mathbf {J} (\mathbf {r} ))+\mathbf {J} (\mathbf {r} )\right]~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b23246ca95051d387082cfeb9cd1c04f8eccf6ea)
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
we get
![{\displaystyle {\text{(12)}}\qquad {\mathbf {E} (\mathbf {r} )={\cfrac {i\omega \mu }{4\pi }}\int {\cfrac {e^{ik|\mathbf {r} -\mathbf {r} '|}}{|\mathbf {r} -\mathbf {r} '|}}~\left[{\cfrac {1}{k^{2}}}~{\boldsymbol {\nabla }}({\boldsymbol {\nabla }}\cdot \mathbf {J} (\mathbf {r} '))+\mathbf {J} (\mathbf {r} ')\right]~{\text{d}}\mathbf {r} '~.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/951cfc9699eaed0ea996ac22d9ba1c2448e87102)
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
![{\displaystyle \mathbf {E} (\mathbf {r} )={\cfrac {i\omega \mu }{4\pi }}\int {\cfrac {e^{ikr}}{r}}~\left[{I~l}~\left\{{\cfrac {1}{k^{2}}}{\boldsymbol {\nabla }}({\boldsymbol {\nabla }}\cdot {\widehat {\boldsymbol {\alpha }}}~\delta (\mathbf {r} '))+{\widehat {\boldsymbol {\alpha }}}~\delta (\mathbf {r} ')\right\}\right]~{\text{d}}\mathbf {r} '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1de3d933ed238c51c548064e56889e4885135c25)
or,
![{\displaystyle {\text{(14)}}\qquad {\mathbf {E} (\mathbf {r} )=i\omega \mu ~I~l~\left[{\cfrac {1}{k^{2}}}{\boldsymbol {\nabla }}({\boldsymbol {\nabla }}\cdot {\widehat {\boldsymbol {\alpha }}})+{\widehat {\boldsymbol {\alpha }}}\right]{\cfrac {e^{ikr}}{4\pi r}}~.}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/16436c09d8456fae9536a988f371e3cad1d48584)
Also, from

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
Ishimaru78.
Before we can go into the details, we need to discuss vector potentials for
electromagnetism.
Vector potentials for electromagnetism[edit | edit source]
Since
, there exists a vector potential
such that
. Hence,

Also, from Maxwell's equation

In terms of the vector potential
, we then have

Therefore, there exists a scalar potential
such that

i.e.,

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
equation

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
polarization potentials).
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
functions.
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
and time.
The Lorentz condition is satisfied if

In fact, the potentials
and
can be expressed in terms of
and
as

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
equations

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

and

- ↑
Recall the residue theorem which states that

If

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.