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Homework Statement:

A monochromatic plane wave with fields ##\vec{E}_0## and ##\vec{B}_0## scatters from a thin conducting disk of radius ##a##. In the longwavelength limit, the scattered field is described by electric and magnetic dipole radiation fields with moments
##\vec{p}_d =  \frac{16}{3} a^3 \epsilon_0 \hat{n} \times \left( \hat{n} \times \vec{E}_0 \right)##
and
##\vec{m}_d =  \frac{8}{3 \mu_0} a^3 \left( \hat{n} \cdot \vec{B}_0 \right) \hat{n}##
The unit vector ##\hat{n}## points in the direction of the incident wave propagation vector when the latter is normal to the plane of the disk. Use Babinet's principle to deduce the effective dipole moments which characterize the diffracted field when a circular hole of radius ##a## in a flat conducting plane is illuminated by a plane wave with aperture fields ##\vec{E}_a## and ##\vec{B}_a##.
I want to program a diffraction flux pattern from a circular aperture in an infinite plane perfect conductor. I think the result I'm trying to get is a Bessel function type profile with a central maxima. I think this is valid when assuming normal incidence which is what I'll assume from here on out. Maybe I misunderstand Babinet's principle and could use some help/clarification.
Relevant Equations:
 See solution below.
For clarification on "normal incidence" without drawing a picture.
I'm going to assume the incoming wave has the electric field in the zdirection, the magnetic B / Auxillary Hfield in the xdirection, and the Poynting Vector in the ydirection (i.e. normal to the plane of the aperture).
That being the case,
##\vec{m}_a = \frac{16}{3} a^3 \epsilon_0 \vec{E}_0 \left( \hat{n} \cdot \hat{n} \right) = \frac{16}{3} z^3 \epsilon_0 E_0 \hat{y} ##
and
##\vec{p}_a = 0##
So now we have to calculate the fields from effective aperture magnetic dipole, we can do this directly from the magnetic vector potential which according to Jackson 9.33 is
##\vec{A} \left( \vec{r} \right) = \frac{i k \mu_0}{4 \pi} \left( \hat{r} \times \vec{m} \right) \frac{e^{ikr}}{r} \left( 1  \frac{1}{ikr} \right)##
for a time harmonic oscillating source
Of course
##\vec{H} = \frac{1}{\mu} \nabla \times \vec{A}##
I think
##\vec{E} = \frac{i}{k}\sqrt{\frac{\mu}{\epsilon}} \nabla \times \vec{H}##
Both of which can be numerically computed
When I add back the plane waves per Babinet's principle I get the following
which is nothing like the typical Bessel function like diffraction profile.
Can someone walk me through what I should be doing, and how I should be interpreting it?
It should be worth noting that I found the total fields E and H before finding the total poynting vector so I did not neglect cross terms
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