- #1
PhDeezNutz
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- Homework Statement
- I want to get a diffraction pattern from a circular hole in the Franhofer limit. In another thread I have been studying radiation patterns of various configurations and now I seek to study diffraction. I want to see if it's possible to place electric and magnetic dipoles in the vicinity of the hole and produce a respectable diffraction pattern on the far wall. The hole is in the XZ-plane and I want to graph the flux through a far away y = constant plane. I'm assuming normally incident plane wave through the hole.
- Relevant Equations
- From Jackson 9.16 we have the following equations for the vector potential of a radiating electric dipole and radiating magnetic dipole.
$$\vec{A} \left(\vec{r}\right) = - \frac{i \mu_0 \omega}{4 \pi} \left( \frac{e^{ikR}}{R}\right) \vec{p}$$
$$\vec{A} \left(\vec{r}\right) = \frac{\mu}{4 \pi} \left( \frac{e^{ikR}}{R^2} \right) \left( -ikR + 1\right)\vec{m} \times \vec{R}$$
(I don't think the last formula is in Jackson but I have used it to create a qualitatively accurate animation of a radiating magnetic dipole)
Naturally the ##\vec{B}## and ##\vec{E}## can be calculated by taking the curl and multiplying by the appropriate constant in the latter case.
$$\vec{B} = \nabla \times \vec{A}$$
$$\vec{E} = \frac{i}{k \sqrt{\mu \epsilon}} \nabla \times \vec{B}$$
The Poynting Vector field can then be found by taking the real parts of ##\vec{B}## and ##\vec{E}## and doing a cross product.
$$\vec{S} = \mathfrak{R} \left( \vec{E} \right) \times \mathfrak{R} \left( \vec{B}\right)$$
I oriented a magnetic dipole perpendicular to the hole (parallel to the ŷ ŷ y^ŷ direction) with one end at it's origin and I get the following pattern
I was really looking for something like this
As you can see I'm getting almost the exact opposite of what I want since I'm going for Fraunhofer Diffraction. Conventional wisdom would suggest "if you want the exact opposite do the exact opposite". I have tried making the magnetic dipole parallel to the hole and I lose rotational symmetry in the flux symmetry on the back wall which is something I would not have expected.
I am open to suggestions and although I have tried many different combinations only to start back at square 1.
I would think I need to add more sources and separate them so as to have path length differences. Should they be electric dipoles or other magnetic dipoles.
I read Han's Bethe's paper "On the Theory of Diffraction by Small Holes" and the values he derived were
|p⃗ |=13π(a3)E0|p→|=13π(a3)E0
and
|m⃗ |=23π(a3)H0|m→|=23π(a3)H0I believe he oriented the electric field vector in the z-direction and the magnetic dipole in the y-direction. When I do such a configuration I do not remotely get a valid interference pattern on the back wall. I could have misinterpreted his paper. It was hard to read.
For the record I don't even know if my approach is valid.
As you can see I'm getting almost the exact opposite of what I want since I'm going for Fraunhofer Diffraction. Conventional wisdom would suggest "if you want the exact opposite do the exact opposite". I have tried making the magnetic dipole parallel to the hole and I lose rotational symmetry in the flux symmetry on the back wall which is something I would not have expected.
I am open to suggestions and although I have tried many different combinations only to start back at square 1.
I would think I need to add more sources and separate them so as to have path length differences. Should they be electric dipoles or other magnetic dipoles.
I read Han's Bethe's paper "On the Theory of Diffraction by Small Holes" and the values he derived were
|p⃗ |=13π(a3)E0|p→|=13π(a3)E0
and
|m⃗ |=23π(a3)H0|m→|=23π(a3)H0
For the record I don't even know if my approach is valid.