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### High resolution extended image near field optics 3. Exact radiating solutions to Maxwell’s equations in a vacuum

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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 ):

where