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optics
4. Exact radiating solutions to Maxwell’s
equations in the presence of idealised plane mirrors
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Copyright (c) Malcolm
Kemp 2010
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Consider now the behaviour of inwardly and outwardly
radiating (magnetic) dipoles in the presence of an idealised plane mirror, i.e.
the solution, say, in the half space arising from a dipole whose
origin in Cartesian coordinates is
given by ()
and whose direction is given by if
there is:
(a) A vacuum in the
region ; and
(b) A perfectly conducting
plane mirror at .
As Born & Wolf (1980) explain, the exact boundary
condition satisfied on the plane is that the component of tangential
to is zero.
Now let and
.
The reason we focus on magnetic rather than electric dipoles
using the terminology in Section
3 is that the superposition of two such equal magnitude and in-phase
dipoles, one emanating at pointing in direction and the other
emanating at and pointing in the direction then exactly
satisfies the required boundary condition at . Suppose we
write this superposition as:
It exactly satisfies the boundary condition because at we have and
,
if in
Cartesian coordinates. So the and components of
the electric field at are both zero and is
thus exactly perpendicular to the mirror.
Consider further the special case of the above where and
.
We then have and
,
the dipole is emanating from the plane mirror itself and the solutions take the
form:
Surfaces of constant phase for this special case are
hemispheres centred about .
The direction and amplitude of the real physical values of on
each such hemisphere then have the form ,
i.e. is
perpendicular to both the direction of the corresponding radius vector and the
direction of the dipole and has a maximum amplitude proportional to the sine of
the angle between these two vectors.
Consider also the situation where we have the special case
solution form as above and we place a perfectly conducting metallic
hemispherical mirror placed at (in
the region ) for some constant . As is
exactly tangential to each such hemisphere, any exact outwardly
radiating (magnetic) dipole from will strike the hemisphere, be
reflected with a 180 degree phase transition and create exactly the right
boundary conditions to create an exact inwardly radiating (magnetic dipole).
If the hemisphere was centred at ,
some point on the plane mirror not far from ,
then outwardly radiating dipoles from would
not have the right characteristics to generate the exact boundary conditions
needed for an equivalent inwardly radiating dipole, at least not one that
radiates back to .
However, any dipole emanating from that
bounced a second time off the plane mirror and then of the hemispherical mirror
would then have the right characteristics, to first order, to create the
required boundary conditions. So, if is sufficiently large compared
to then
the layout would again create an arbitrarily accurate inwardly radiating
(magnetic) dipole with destination .
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