Babinet's Principle: Effective Aperture Dipoles (Matlab)

In summary, the incoming wave has the electric field in the z-direction, the magnetic B / Auxillary H-field in the x-direction, and the Poynting Vector in the y-direction (normal to the plane of the aperture). The fields can be calculated from the magnetic vector potential, and the poynting vector can be found by adding back the plane waves.
  • #1
PhDeezNutz
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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 long-wavelength 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.
Image 6-9-20 at 8.28 PM.jpg


For clarification on "normal incidence" without drawing a picture.

I'm going to assume the incoming wave has the electric field in the z-direction, the magnetic B / Auxillary H-field in the x-direction, and the Poynting Vector in the y-direction (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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8A95B4A4-DCC8-40AA-A216-899585D47551.png


I think I’m going to try and solve the analogous scattering problem of a circular disk with normal incidence and see what I can learn from it. Hopefully, I actually have enough knowledge of the relevant PDE solutions and basis functions. I’m not the most knowledgeable when it comes to Bessel functions.
 
  • #3
badbabinetsprinciple2.jpg


Several things

1) My flux pattern is upside down

2) It looks like it could be cylindrically symmetric but it's not dipping to zero and coming back up (or down rather) even when I extend my views. There shouldn't be shelves of sorts.

3) For my plane wave in the y-direction I arbitrarily made it ##E_z = E_0 \frac{e^{iky}}{r}## (polarized in z-direction). That inverse r power should not be there, it's extremely contrived but it has worked better than anything I've done thus far.
 

Related to Babinet's Principle: Effective Aperture Dipoles (Matlab)

1. What is Babinet's Principle?

Babinet's Principle is a fundamental concept in physics that states that the diffraction pattern produced by an opaque object is the same as the diffraction pattern produced by the complementary aperture (opening) of the same size and shape. In other words, the diffraction patterns of an object and its complementary aperture are identical.

2. How does Babinet's Principle apply to effective aperture dipoles?

In the context of effective aperture dipoles, Babinet's Principle states that the radiation pattern of a dipole antenna can be obtained by subtracting the radiation pattern of its complementary dipole from the radiation pattern of a full aperture dipole. This allows for easier and more efficient calculations of the radiation pattern of a dipole antenna.

3. What is the significance of Babinet's Principle in antenna design?

Babinet's Principle is important in antenna design because it allows for the simplification of calculations for the radiation pattern of dipole antennas. By using the principle, engineers can design antennas with more complex shapes and configurations without having to perform complicated calculations for each individual element.

4. How is Babinet's Principle implemented in Matlab?

In Matlab, Babinet's Principle can be implemented by using the "baperture" function. This function takes in the radiation pattern of a full aperture dipole and the radiation pattern of its complementary dipole as inputs, and then calculates the radiation pattern of the effective aperture dipole.

5. Can Babinet's Principle be applied to other types of antennas?

Yes, Babinet's Principle can be applied to other types of antennas, such as slot antennas and horn antennas. As long as the antennas have complementary apertures, the principle can be used to simplify calculations and design processes.

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