Spectral domain double integral with singularities

[tworitdash]

TL;DR Summary

I am trying to perform an integration for mutual admittance of two modes in a rectangular waveguide. The integral is in the spectral domain ([itex](k_x, k_y) [/itex]). However, it is very difficult to make it as a form of a single integral. Therefore, I tried doing the first one integral with respect to ([itex]k_y[/itex]) for each [itex]k_x[/itex]. However, I see a null value at the points where [itex]k_x = k_0[/itex]. I choose an integration path that doesn't contain the poles on the re axis.

The integral looks like

[tex] Y_{mut, mn} = -j^{m+n}nm \int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \frac{2 ab (k^2 - k_x^2) \sin^2(\frac{k_yb}{2}) \cos^2(\frac{k_xa}{2})}{\omega \mu k_z (\frac{k_yb}{2})^2 [(n\pi)^2 - (k_xa)^2][(m\pi)^2 - (k_xa)^2]} dk_x dk_y [/tex]Here,

[tex] k_z = -1j \sqrt{(-(k_0^2 - k_x^2 - k_y^2))}[/tex]​

. Therefore, this is in the bottom Riemann sheet. This root has been taken to make sure that [tex]k_z[/itex] remains purely imaginary for [itex] k_0^2 <(k_x^2+k_y^2) , Imag(k_z)<0 [/itex] and purely real when [itex] k_0^2<(k_x^2+k_y^2) , Real(k_z) >0 [/itex] . I tried only pulling the integral involving [itex]k_y[/itex].

[tex] D(k_x) = \int_{-\infty}^{\infty} \frac{ (k^2 - k_x^2) \sin^2(\frac{k_yb}{2})} {(-1j \sqrt{(-(k_0^2 - k_x^2 - k_y^2))}) (\frac{k_yb}{2})^2} [/tex]​

I couldn't get a closed-form integral for this one. If someone happens to know, please let me know. Coming to numerical integral, I performed a numerical integral for each [itex]k_x[/itex] point in the range of [−5k0,5k0] and performed the integral with respect to[itex]k_y[/itex]when [itex]k_y[/itex] varies as [−5k0,5k0]. And I have chosen the integration path as shown in below figure (Just imagine [itex]k_y[/itex] instead of [itex]k_x[/itex] ) here.

for i = 1:length(kx)

kxi = kx(i);

del = 0.01 .* k0;Y = @(ky) (k0^2 - kxi^2)./(-1j .* sqrt(-(k0.^2 - kxi.^2 - ky.^2))) .* (sinc(ky .* b./2/pi)).^2;

y(i) = integral(Y, -50.*k0-1j*del,50.*k0+1j*del);% 'Waypoints', [(-1-1j).*del, (1+1j).*del]);

end

However, when in my problem, when [itex] (k_x,k_y)=(k0,0)[/itex] the function has a 0/0 form and this is on the integration path. Therefore, I get a discontinuity there. Basically, it can't avoid the pole when
[itex] k_x = k0[/itex]. I need to have all points continuos for this integration so that I can perform the other integral over [itex]k_x[/itex]. Am I doing something wrong?

 

[PeroK]

It's difficult to read your post.

 

[tworitdash]

It's difficult to read your post.

Yes, I am working on it.

 

[tworitdash]

It's difficult to read your post.

I just edited the post. Please have a look now.

 

[Paul Colby]

Could you describe the actual EM problem you're addressing in a little more detail?

 

[tworitdash]

Could you describe the actual EM problem you're addressing in a little more detail?

@Paul Colby

This is the mutual admittance between rectangular waveguide modes of the kind [itex]TE_{m, 0}[/itex] and [itex]TE_{n, 0}[/itex] where [itex]m, n[/itex] are odd numbers [itex]m, n = 1, 3, 5, 7, ...[/itex]. I need to find the aperture admittance of an open-ended waveguide (Open to Free Space). These integrals basically help find the excitation coefficients of higher-order modes at the aperture when the waveguide is only excited with the fundamental mode ([itex]TE_{10} [/itex]).

 

[Paul Colby]

Thanks. Okay, so the radiation from the end of the guide would be effected by any structure on the exterior to the guide, such as a flange or ground plane. So one has to specify into what the guide is radiating. I assume it's a free half space with a conductive plane as boundary. One could treat this as the coupling between the guide and a much much larger one. In this problem the admittance matrix is a simple integration of trigonometric functions. Why do the problem in k-space?

 

[tworitdash]

Thanks. Okay, so the radiation from the end of the guide would be effected by any structure on the exterior to the guide, such as a flange or ground plane. So one has to specify into what the guide is radiating. I assume it's a free half space with a conductive plane as boundary. One could treat this as the coupling between the guide and a much much larger one. In this problem the admittance matrix is a simple integration of trigonometric functions. Why do the problem in k-space?

Yes, this is a problem with a metal flange connected to the aperture. I did it in k space to consider the Green's function representation of free space. They are known in the k space. Then, I take the power of the EM fields outside the guide (free space) and equated it to the power of fields inside the guide to formulate the aperture admittance. Using boundary conditions at the aperture, I used the fields in k space (Fourier transforms of the fields inside the waveguide) into the equation of the power for the free space fields. I could use free space as a big guide but how can I represent it for free space? Even if I do, I would have to define the phase center of the system? To have a common reference coordinate system for both the guide and the large guide? (because free space is a combination of spherical modes)On the other hand, I also have expressions on how to compute the integral in the spatial domain by converting the above-mentioned functions into their IFFTs. However, when [itex] m == n [/itex], the function explodes at the origin and therefore I also don't know how to make it converge.

[tex] Y_{mn} = \frac{j8a}{b \mu \omega} \int_{0}^{a}\int_{0}^{b} (b - y) f^{m, n}(x) \frac{e^{-j\sqrt{x^2 + y^2}}}{\sqrt{x^2 + y^2}} dx dy [/tex]

When m == n,[tex] f^{m,n}(x) = \frac{1}{4\pi^2 an} \Big(\frac{1}{\pi} \Big[ k_0^2 + (\frac{n \pi}{a})^2 \Big] \sin(\frac{n \pi |x|}{a}) + \frac{n}{a}(a - |x|) \Big[ k_0^2 - (\frac{n \pi}{a})^2 \Big] \cos(\frac{n \pi |x|}{a}) \Big)[/tex]

If I change it to a one-variable integral of [itex] \rho [/itex], it looks like,

[tex] Y_{m, n} = \frac{j4}{\mu \omega \pi^2 n} \int_{0}^{\sqrt{a^2 + b^2}} \Big[\Big[ k_0^2 + (\frac{n \pi}{a})^2 \Big] H_0(\frac{n \pi \rho}{a}) + n \pi \Big[ k_0^2 - (\frac{n \pi}{a})^2 \Big] J_0(\frac{n \pi \rho}{a})\Big] e^{-j\rho} d\rho [/tex]

Here [itex] H_0 [/itex] is the StruveH function of order 0 and [itex] J_0 [/itex] is the Bessel function of the first kind and order 0.

where a and b are the dimensions of the waveguide. This function explodes at the origin (x = 0, y = 0). That's why I don't know how to perform the integral. I am attaching a picture of the function of the integrated for different sets of m and n. This is also another reason why I didn't go for the spatial domain integrals.

 

[Paul Colby]

Interesting approach but it sounds non-standard. I still don't see the charm of k-space representations for the guide fields nor understand the details your approach. The number of ways to fail at this kind of analysis are enormous. I know because I've found most of them. Not only are divergent integrals one need be concerned with but also one ultimately must solve of a linear problem of the form, ##A x = b##, which becomes an issue if your ##A## is singular or poorly conditioned. This problem arises a huge number of ways but usually is associated with an incomplete (or redundant) statement of the problem. This is what gives me concern for your power balance equations having tried similar sounding things in the past.

On the other hand, radiation from an open ended guide is a solved problem. One (potential) way is the mode matching approach I suggested. I would expect the aperture fields of the smaller guide coupled to one 10 times as large would be a reasonable approximation of an infinitely larger guide[1]. This would be a testable assertion BTW. I also know the mode matching equations lead to a well behaved system of equations. Or, one might see what solutions people have published in the past.

[1] Once an aperture field is known, computing the far field radiation pattern is a simple plane wave integral over the aperture.

 

[tworitdash]

Interesting approach but it sounds non-standard. I still don't see the charm of k-space representations for the guide fields nor understand the details your approach. The number of ways to fail at this kind of analysis are enormous. I know because I've found most of them. Not only are divergent integrals one need be concerned with but also one ultimately must solve of a linear problem of the form, ##A x = b##, which becomes an issue if your ##A## is singular or poorly conditioned. This problem arises a huge number of ways but usually is associated with an incomplete (or redundant) statement of the problem. This is what gives me concern for your power balance equations having tried similar sounding things in the past.

On the other hand, radiation from an open ended guide is a solved problem. One (potential) way is the mode matching approach I suggested. I would expect the aperture fields of the smaller guide coupled to one 10 times as large would be a reasonable approximation of an infinitely larger guide[1]. This would be a testable assertion BTW. I also know the mode matching equations lead to a well behaved system of equations. Or, one might see what solutions people have published in the past.

[1] Once an aperture field is known, computing the far field radiation pattern is a simple plane wave integral over the aperture.

Thank you for your reply.

I use mode matching when I do the transition between 2 waveguides. There it is simple as I have tried with similar kind of waveguides. Their coordinate systems are the same and therefore their fields can be represented from the same coordinate system.

However, when it comes to free space as a second waveguide, with the mode matching technique I don't know how to represent the fields in the second waveguide. The mode matching technique I found is called boundary condition mode matching for aperture problems that deal with the spherical modes of the free space. Then, it becomes very difficult for me to match it to the waveguide because I have to choose the same origin for both and that only can happen when I use the phase center (I might be wrong too). That is too involved and that is why I left it.

 

[Paul Colby]

However, when it comes to free space as a second waveguide, with the mode matching technique I don't know how to represent the fields in the second waveguide. The mode matching technique I found is called boundary condition mode matching for aperture problems that deal with the spherical modes of the free space. Then, it becomes very difficult for me to match it to the waveguide because I have to choose the same origin for both and that only can happen when I use the phase center (I might be wrong too). That is too involved and that is why I left it.

If you should have a mode matching code that allows you to analyze two rectangular guides of different dimensions coupled through an aperture you could test my suggestion to use a larger yet finite guide as an approximation of free space. Run the code twice, once with the second (exit) guide 5 times larger than the entrance guide. Repeat for an exit guide 10 times the entrance one. Compare the aperture fields you obtain for the smaller guide in the two cases. If they are close, I'd say you're done.

I don't follow your hangup on phase centers. Is this a program complexity or a conceptual one for you?

For grins I'm looking at what it might take to write a mode patching code. So far it doesn't look too onerous.

 

[tworitdash]

If you should have a mode matching code that allows you to analyze two rectangular guides of different dimensions coupled through an aperture you could test my suggestion to use a larger yet finite guide as an approximation of free space. Run the code twice, once with the second (exit) guide 5 times larger than the entrance guide. Repeat for an exit guide 10 times the entrance one. Compare the aperture fields you obtain for the smaller guide in the two cases. If they are close, I'd say you're done.

I don't follow your hangup on phase centers. Is this a program complexity or a conceptual one for you?

For grins I'm looking at what it might take to write a mode patching code. So far it doesn't look too onerous.

Ah! I understood what you wanted to say. And, yes implementing it is quite simple. I've done it for circular waveguides and it is really accurate and analytical. No doubt about that.

All I was confused about how to imagine the free space as a waveguide because the papers I found use a Boundary condition mode matching with spherical modes of the free space (Not exactly a very big waveguide). Therefore, one needs the phase center and conceptually that was difficult for me because one has to find the surface on which the phase remains constant for the waveguide.

So, basically what you are saying is this:

1. First, find the number of modes that can be excited at a very big waveguide as a result of the fundamental mode excitation on the smaller one.
2. Do mode matching and find the reflection coefficients for all the modes and draw the field pattern.

Am I right? That way it is very easy to implement. I will try that. Thank you!

I could have imagined this way. I don't know why I read the papers first and got me confused with 2 different coordinate systems :P . Then, I jumped to spectral-domain technique.

 

[Paul Colby]

Great, glad to hear it hope it works. Remember to check convergence. Increasing the second guides dimensions and resolving will tell you if the approximation is valid as far as the aperture fields are concerned. It would be interesting to plot the convergence as a function of relative guide size.

 

[tworitdash]

Great, glad to hear it hope it works. Remember to check convergence. Increasing the second guides dimensions and resolving will tell you if the approximation is valid as far as the aperture fields are concerned. It would be interesting to plot the convergence as a function of relative guide size.

Thanks! I already have a relation for the mode convergence which depends on the geometry. So, checking the convergence with the size of the guide can be simple.