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Plane wave propagating through infinite PEC slab

Dear reader,

For my research I am looking to simulate the RCS of rough surfaces at mm-wave frequencies. Because of the short wavelength, the model sizes that can be evaluated are small. In order to still get an impression of how a larger surface scatters the incident radiation in various directions, we would like to use periodic boundary conditions for our simulations.

To check whether the simulation was set-up correctly, instead of using a rough surface, I simulated a block of PEC. Here, periodic boundary conditions were defined to coincide with the side faces of the block and as a source I used a plane wave incident from the z-axis. However, when plotting the far field/RCS in POSTFEKO (as shown in the picture), there seems to be as much of the wave propagating through the PEC block as it is scattered back.

Since, as far as I understand, a PEC block in combination with periodic boundary conditions should effectively represent an infinite PEC slab, the incident wave should be completely reflected back and no radiation should pass through the slab in contrary to the embedded element pattern the simulation predicts.

I was wondering if anybody has an idea of what’s causing this, has suggestions on which settings I need to chance, or has come across similar issues before. Or is this expected behaviour?

I am looking forward to your response(s).

Best regards,

Wietse Bouwmeester

test.cfx

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Feko

v2021.1

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Sören Poulsen

I would assume that the plot shows the scattered field (E_s) and that you need to add the incident field (E_i) to get the total field (E_tot = E_s + E_i). The total field should vanish in the forward direction, as you say.

Wietse Bouwmeester

Thanks for the suggestion, this could be the case. I looked around a bit in CADFEKO and found that the Far Field request has an option "Calculate only the scattered part of the field" which is enabled by default. I disabled this option and reran the simulation. Unfortunately, results are the same as before.

Is this what you had in mind? Or are there some other things that need to be done in order to include the incident field?

Sören Poulsen

Thanks for the suggestion, this could be the case. I looked around a bit in CADFEKO and found that the Far Field request has an option "Calculate only the scattered part of the field" which is enabled by default. I disabled this option and reran the simulation. Unfortunately, results are the same as before.

Is this what you had in mind? Or are there some other things that need to be done in order to include the incident field?

For some reason, I couldn't open your file, but I tried some own experiments with periodic boundaries and a PEC rectangle, illuminated by a plane wave and calculating the far field, as in your setup. I get the impression that Feko calculates the scattering contribution from one cell only, see the advanced tab under the far field definition dialog, "Periodic boundary condition options".

Also, it hits me regarding my first comment above, that it doesn't make sense to add the incident field in this scattering problem, since the far field of the plane wave is a delta function. Hence, I still think it is scattered field which is shown, you can experiment with oblique angle incidence to check the directions of the backscattered and forward scattered lobes.

Wietse Bouwmeester

For some reason, I couldn't open your file, but I tried some own experiments with periodic boundaries and a PEC rectangle, illuminated by a plane wave and calculating the far field, as in your setup. I get the impression that Feko calculates the scattering contribution from one cell only, see the advanced tab under the far field definition dialog, "Periodic boundary condition options".

Also, it hits me regarding my first comment above, that it doesn't make sense to add the incident field in this scattering problem, since the far field of the plane wave is a delta function. Hence, I still think it is scattered field which is shown, you can experiment with oblique angle incidence to check the directions of the backscattered and forward scattered lobes.

I have tried using the periodic boundary condition options for the far field some time before this forum post, with 100 elements along both directions. I found that this forward scattered lobe still was present.I'm not sure how the periodic boundary condition options for the far-field computation are implemented but what I believe is that it is a multiplication of something like an embedded element pattern with an array factor. I am not sure about this though, as using this 100x100 grid seemed to increase solver time, which I would not have expected as I think multiplying the embedded element pattern with an array factor shouldn't be too expensive computationally speaking.

To check if the far-field request shows the contribution of only one isolated element, I removed the periodic boundary conditions and ran a simulation with that, the result is shown below.

This result differs from the one with the periodic boundaries conditions shown in the original post, so I figure that the Feko indeed calculates something akin to an embedded element pattern.

I also ran a simulation with a plane wave angle of incidence of 45 degrees. This gives a far field like this:

Again, the field is mirrored along the xy-plane. But as you said, I would also expect the far-field of a plane wave to be a delta function, so I'm not sure what to make of this result.

Mel

Feko calculates the far field based on the currents of the finite plate only, but these currents are calculated as if this plate forms part of an infinite array. The infinite array of course has no edge effects.
You will get the same response if you use the PO solver on the plate. PO does not model edge effects.

The reason for the "mirror image" of the far field is due to the source. It is a plane wave and causes the scattered wave below the plate to also be a plane wave, so the flat plate behaves like a mirror.

For far fields, Feko by default ignores the incident wave, so checking/unchecking the "calculate only the scattered part of the field" will not make a difference. It will, however, for near field requests.

If you want to see how much the PEC block "shields" or blocks an incident wave, how much reflects back, it would be better to request Reflection/Transmission coefficients.

Wietse Bouwmeester

Feko calculates the far field based on the currents of the finite plate only, but these currents are calculated as if this plate forms part of an infinite array. The infinite array of course has no edge effects.
You will get the same response if you use the PO solver on the plate. PO does not model edge effects.

The reason for the "mirror image" of the far field is due to the source. It is a plane wave and causes the scattered wave below the plate to also be a plane wave, so the flat plate behaves like a mirror.

For far fields, Feko by default ignores the incident wave, so checking/unchecking the "calculate only the scattered part of the field" will not make a difference. It will, however, for near field requests.

If you want to see how much the PEC block "shields" or blocks an incident wave, how much reflects back, it would be better to request Reflection/Transmission coefficients.

Thanks for your reply. I've been thinking about it, however I am not sure if I understand it completely. Firstly, good to hear that you confirm my expectations that Feko calculates the currents as if the plate were part of an infinite array, this is what I am looking for.

However, I'm confused about your point on the scattered field below the plate. How can currents be induced on the bottom part of the plate if there physically should be no incident field present below the plate and thus no scattering should occur? How would this work then if instead of a PEC plate, the plate would have been some dielectric material? Wouldn't the results below the dielectric plate be physically incorrect as well as the incident field is not taken in account?

Would it be possible for you to provide some more details on how the plane wave source and far field calculations are implemented? My guess from the responses to this post would be that the incident field is defined analytically at every point in the empty simulation domain according to the plane wave source settings. Subsequently, the mesh is loaded in and based on the previously computed incident field at some point p, the current on the corresponding mesh cell at point p is computed. Then finally, using the free space (I guess) green's function, the fields are computed. Is this correct or does it work in a different way? If it turns out that this process is indeed more or less how it's implemented in Feko, then I am wondering about the results one would obtain within a PEC enclosed hollow volume, for example a hollow PEC sphere. Namely, in this case an incident field would be present within the sphere as well, which in turn induces currents resulting in a non-zero field within the sphere which physically should not be there. Could you elaborate on this?

Furthermore, if Feko ignores the incident wave and checking/unchecking the "calculate only scattered field" box makes no difference, it makes me wonder why this box is even there. Could you explain what the function of this box is? Maybe this box is present because of legacy reasons?

Lastly, I am aware of the reflection/transmission coefficients request, however this is not what I am trying to calculate. I would like to compute how a rough surface scatters an incident wave from some direction at least in all upper hemispherical directions (0 <= theta 0 <= 90) and if possible the lower hemisphere as well, i.e. I would like to know how much of the incident wave is scattered in the forward and other directions besides only backscattering. The reason I came up with this infinite PEC plate simulation first is to see if the results would make physical sense to verify if the simulation is set-up correctly. Hence, to gain trust in the far-field results from the rough surface, it is important for me to understand in detail how this mirror image is caused exactly so I can understand how this impacts results from rough surface and how I can deal with this or even neglect it. I hope this explanation was clear and that you understand where I am coming from. Lastly, I would like to stress that I really appreciate your help in resolving this.

Mel

The plane wave illuminating the plate is not the same as shining a finite flash light on the plate.
A plane wave has infinite energy and Feko cannot add this infinite contribution to the far field, which, is calculated at infinity (1/R dependence removed), to get the total field.
For the plane wave, only the scattered far field can be computed.

If the source is rather a Hertzian dipole, it is has finite energy and here you can choose for the far fields to calculate the total or scattered field using the check box "Calculate only the scattered..."

The currents on the plate radiate into free space and this is why the scattered field will be symmetrical on both sides of the plate. The "mirror image" below the plate is that field that is required to "cancel out" the incident field.

As for a PEC hollow sphere, you can calculate the near field inside the sphere. Leaving the check box unchecked this will be the total field which is the sum of the incident plus the scattered field. The accuracy will be limited by the method and the mesh. For example for a default Method of Moments solution, using a fine mesh the "dynamic range" is roughly 60 dB.

You can do the same with a flat plate - i.e. calculate the near field below the plate which by default will be the total field (shielding then from the source above)