On the stored energy of planar apertures

The volume integrals in the complex Poynting theorem are usually assumed to represent stored magnetic and electric energy. But for a planar aperture radiating into a lossless homogeneous medium it is shown that the only physically observable part of the volume integrals is twice that produced in the invisible region of the pattern space factor by those field components that vanish in the plane outside of the aperture. The inductive and capacitive reactive powers corresponding to the observable parts
\langle W_{m} \rangle
and
\langle W_{e} \rangle
, respectively, of the volume integrals are found to be given by the following integrals over the invisible region
k_{y}^{2} + k_{z}^{2} > k^{2}
,
2\omega\langle W_{m} \rangle = \frac{(2\pi)^{2}}{2kZ_{0}} \int \int_{k_{y}^{2}+k_{z}^{2} >k^{2}} \frac{|k_{z}F_{y}-k_{y}F_{z}|^{2}} {\sqrt{k_{y}^{2}+k_{z}^{2}-k^{2}} dk_{y}dk_{z}
,
2\omega\langle W_{e} \rangle = \frac{(2\pi)^{2}}{2kZ_{0}} \int \int_{k_{y}^{2}+k_{z}^{2} >k^{2}} \frac{k^{2}}(|F_{y}|^{2}+|F_{z}|^{2})} {\sqrt{k_{y}^{2}+k_{z}^{2}-k^{2}} dk_{y}dk_{z}
, for an arbitrary aperture consisting of holes in a conducting plane. They are expressed in terms of the rectangular components
F_{y}
and
F_{z}
of the pattern space factor obtained from the tangential components
E_{y}
and
E_{z}
of electric field over the holes
S
,
F_{y}(k_{y},k_{z},k) = \frac{1}{(2\pi)^{2}} \int \int\_{s} E_{y}(0,y,z,k)e^{i (k_{y}y+k_{z}z)}dydz
F_{z}(k_{y},k_{z},k) = \frac{1}{(2\pi)^{2}} \int \int\_{s} E_{z}(0,y,z,k)e^{i (k_{y}y+k_{z}z)}dydz
where
Z_{0} = \sqrt{\mu/\epsilon}
, and
k_{y}
and
k_{z}
are two components of the vector propagation constant of magnitude
k=\omega\sqrt{\mu\epsilon}
. For the complementary aperture whose space factor is obtained from
Z_{0}
times the tangential components of magnetic field in the aperture, the formulas for inductive and capacitive reactive power simply interchange. The new formulas are used to make the first known test of the widely held assumption that the reciprocal relationship between
Q
and bandwidth of nonradiating systems applies also to radiating systems. By direct computation of
Q
and of bandwidth independently it is shown that the reciprocal relationship does hold for the test case treated (the planar dipole).