Spatial Fourier Transform: Bessel x Sinusoidal

[tworitdash]

TL;DR Summary

I am trying to formulate an analytical expression for the spectrum of the electric fields on a circular aperture (cylindrical waveguide). The field expressions are a multiplication of Bessel function and sinusoidal function. I am attaching only one kind of integration that I have.

$$I(k_x, k_y) = \int_{0}^{R} \int_{0}^{2\pi} J_{m-1}(\alpha \rho) \sin((m + 1) \phi) e^{j\rho(k_x \cos\phi + k_y \sin\phi)} \rho d\rho d\phi$$Is there any way to do it? J is the Bessel function of the first kind. I thought of partially doing only the phi integral as $\int_{0}^{2\pi} \sin((m + 1) \phi) e^{j\rho(k_x \cos\phi + k_y \sin\phi)} d\phi$ but then again I am not able to find any solution.

 

[jasonRF]

You can definitely do the $\phi$ integral - I have been doing a lot of similar integrals recently. What you want to do is define $k_x = k \cos\psi$ and $k_y = k \sin\psi$, and split up the $\sin$ into complex exponentials. Then you get integrals of the form,
$$ \begin{eqnarray*}
\mathcal{I}^{\pm} & = & \int_0^{2\pi} \, e^{\pm j (m+1) \phi} \, e^{j k \cos(\phi-\psi)} \, d\phi \\
& = & e^{\pm j (m+1) \psi} \int_0^{2\pi} \, e^{\pm j (m+1) (\phi-\psi)} \, e^{j k \cos(\phi-\psi)} \, d\phi \\
& = & e^{\pm j (m+1) \psi} \int_{-\psi}^{2\pi-\psi} \, e^{\pm j (m+1) \xi} \, e^{j k \cos\xi} \, d\xi
\end{eqnarray*}
$$

To proceed you need the integral,
$$ J_\ell(\beta) = \frac{1}{2\pi j^\ell} \int_0^{2\pi} \, e^{j \ell \phi} \, e^{j \beta \cos\phi} \, d\phi
$$
which commonly occurs in these kinds of problems. Note that this is the integral of a periodic function over one period, so the result does not depend on which period you integrate over.

The integral over $\rho$ is harder, because you have an integrand that is basically $\rho \, J_{m-1}(a\rho) \, J_{\pm(m+1)}(b\rho)$. Similar integrals can be found at DLMF (https://dlmf.nist.gov/10.22), but the exact integral is not there. Perhaps you can use recurrence relations for the $J_n$ to get this into a form where you can do this closed-form? If not, then you may be stuck doing numerical integration, which should be pretty easy as the integrand is well behaved.

Jason

 

[jasonRF]

By the way, the integral representation of the Bessel function above follows from the expression
$$
e^{- j z \cos\phi} = \sum_{n=-\infty}^\infty j^{-n} e^{j n \phi}\, J_n(z).
$$
You could have also just used this expansion in your original integral and of course ended up with the same result.

 

[tworitdash]

Thank you so much for the inputs @jasonRF . I will try. Previously I was successful with the integration of the product of Bessel functions. So, probably for the $\rho$ variations, I can formulate something with Lommel's integral. I hope I can see some light at the end! :)