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

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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.,

 

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

 

Electric Dipole Fields

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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,

 

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

 

or,

 

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

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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

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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

 

The time harmonic case

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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

 

Footnotes

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

References

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  • 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.