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

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

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

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

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

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

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

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

References edit

  • 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.