Waves in composites and metamaterials/Point sources and EM vector potentials

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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 that

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 [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

Electric Dipole Fields[edit | edit source]

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


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 we get

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


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


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

The time harmonic case[edit | edit source]

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


Footnotes[edit | edit source]

  1. Recall the residue theorem which states that
    and if is non-singular at , then the residue at is .

References[edit | edit source]

  • 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.