Tutorial example for computing the radiation pattern of axisymmetric and nonaxisymmetric linearly polarized dipoles in cylindrical coordinates

[oskooi]

Closes #2656.

Adds a new tutorial which demonstrates the computation of the radiation pattern of axisymmetric and nonaxisymmetric dipoles with linear polarization in cylindrical coordinates. The results are validated using the analytic formula for a dipole antenna in vacuum.

For some reason, I was not able to set this up using step 2 of the procedure described in Tutorial/Nonaxisymmetric Dipole Sources:

To get this working, it was necessary to replace step 2 with the full expansion of the Fourier series involving the computation of the $+m$ and $-m$ fields separately and taking their sum via $\sum_{m=1}^M E_m + E_{-m}$ rather than using $2 \sum_{m=1}^M \Re{E_m}$. Unfortunately, this roughly doubles the number of simulations required.

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[oskooi]

I determined why step 2 of the procedure to compute the far fields described in the tutorial does not work. This has to do with the fact that the $\pm m$ fields are not complex conjugates for $\phi = 0$. This property had been a key assumption which we used to simplify the Fourier-series expansion involving the azimuthal dependence.

The actual relationship between the fields for an $E_r$ point source is shown below which was determined empirically. Note that for an $E_r$ source and $m = 0$, only the $(E_r, H_\phi, E_z)$ fields are nonzero. The Fourier series expansion preserves this property.

(An $E_\phi$ point source involves flipping the sign of the right hand side for the expressions below for $E_r$, $E_\phi$, $E_z$, $H_r$, $H_\phi$, and $H_z$. For an $E_\phi$ source and $m = 0$, only the $(H_r, E_\phi, H_z)$ fields are nonzero.)

[stevengj]

We need the solution $\text{fields}(\omega, {-m})$ in addition to $\text{fields}(\omega, {+m})$. These are not complex conjugates because flipping $+m$ to $-m$ conjugates the equations but not the source term and time dependence $\sim e^{-i\omega t}$ of the Fourier fields.

However, you can still get both the $\pm$ fields from a single simulation if you do two things:

1. Make sure the source current $J$ is purely real, so that it doesn't change if you complex-conjugate everything. This can be done by adding a second source term with minus the frequency.
2. Collect the fields at both $+\omega$ and $-\omega$.

In this case you will have $\text{fields}(\omega, {-m}) = \text{fields}(-\omega, {+m})^*$: you can just take the $+m$ fields at $-\omega$ and conjugate them to get the $-m$ fields at $\omega$.