High resolution extended image near field
optics
3. Exact radiating solutions to Maxwell’s
equations in a vacuum
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Copyright (c) Malcolm
Kemp 2010
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The explanation of the unusual properties of the idealised
optical layout described in Section 2 lies in the
behaviour of certain types of exact solutions of Maxwell’s equations in the
presence of idealised plane mirrors.
Before exploring these further, let us first the nature of
radiating solutions to Maxwell’s equations in a vacuum. These can be
written as superpositions of (potentially infinitely many) outwardly and
inwardly radiating electric and magnetic dipoles.
Born & Wolf (1980) describe the behaviour of a single outwardly
radiating electric dipole, characterised by source location ,
a unit vector, ,
describing the direction in which the dipole is pointing, and an electric
polarization vector whose
value at point and
at time is
given by ,
where is the Dirac function and is a function
of time. The full (i.e. exact) solution to Maxwell’s equations (in a vacuum)
for such a dipole then has the following form, where ,
,
and
are
the electric field, electric displacement, magnetic and magnetic induction
vectors respectively:
Here and
is
the speed of light. Square brackets denote retarded values, i.e. .
The form of this solution is slightly easier to visualise in
spherical polar coordinates taking
the origin as the source location, ,
as
the angle between and
and
as
the angle that the projection of onto
the plane perpendicular to makes
with a constant vector perpendicular to .
If ,
and
are
unit vectors in the direction of increasing ,
and respectively
then the outwardly radiating electric dipole has the form and
where:
The form of the inwardly radiating electric
dipole, i.e. the time reversed solution, can be found by replacing by and by
(since
)
and by placing a negative sign in front of the corresponding expressions for and (since ).
The corresponding outwardly and inwardly radiating magnetic
dipoles have replaced
by and
replaced
by ,
given the symmetric nature of Maxwell’s equations in a vacuum. For reasons that
will become obvious later on, we will concentrate on these latter types of
dipoles in the remainder of this analysis.
We can further decompose each of these dipoles into
superpositions of sinusoidally time-varying dipoles all with the same origin,
using Fourier analysis. These will be the types of dipoles that we will
concentrate on in the remainder of this analysis. For magnetic dipoles
with ,
and
constant,
these have the following form (where is
the real part of the complex number and is the square
root of ):
Outwardly radiating (magnetic) dipoles
Inwardly radiating (magnetic) dipoles
where
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