# Waveguide horn antenna & inverse square law

With a lab waveguide / horn antenna setup similar to image (without bends), we made the following measurements. Now we thought this could be used to demonstrate the observation of the inverse square law, but can't see how(?)

SWR meter (dB setting), measured at various distances (d) :

 0.0dB @86cm (reference)
-1.1dB @136cm
-1.3dB @186cm
-1.5dB @236cm


Converting dB to power ratio, we get -1.1dB = 0.78 comparing first two values. Then thought a power ratio based on distance only (P2/P1 = d1^2/d2^2) could confirm that measurement. But this seems not straight forward at least by the measuring points used. So what is the problem / how should this be solved? • you seem to have been in the near field of the horns, to verify the $\frac1{d^2}$ power dependence you have to be in the far field. You should also make sure that there are no spurious reflections around. – hyportnex May 17 '17 at 13:59
• You won't see the inverse square law with a setup like that unless you measure for distances that are much greater than the Rayleigh Distance for the horn. Roughly your horn antennas need to look like point sources / receivers. At the separation in the photograph, the received power will be almost independent of $d$ over quite a range, which is what your data are telling you: this should be intuitively clear. – Selene Routley May 17 '17 at 14:00
• BTW there is no need to convert from DB to power to check for inverse-square behavior. Just recognize that you are plotting in log-log space. – dmckee --- ex-moderator kitten May 18 '17 at 0:18

You won't see the inverse square law with a setup like that unless you measure for distances that are much greater than the Rayleigh Distance for the horn. Roughly your horn antennas need to look like point sources / receivers and Fraunhofer (farfield) diffraction needs to prevail. At the separation in the photograph, the received power will be almost independent of $d$ over quite a range, which is what your data are telling you: this should be intuitively clear. I'm not sure that measuring the VSWR from the reflected wave will be accurate enough at the distances you will need.

Also note that, with a setup like yours where you are measuring the power reflected by a distant object, the farfield dependence is not $1/r^2$ but $1/r^4$ - your setup is somewhat analogous to a radar. In the farfield, the scattered power is proportional to $1/r^2$. But the fraction of the scattered power received is also proportional to $1/r^2$.

• I see.. Is there a way to estimate the near field path loss, that would illustrate what the measurements show? – bretddog May 17 '17 at 15:29
• @bretddog I should think your setup is a good one to measure that. It depends in detail on the shape of the aperture and the apodization. You could use something like Mathematica to do the freespace diffraction of a uniform scalar field on the aperture - that would give you a good estimate. – Selene Routley May 17 '17 at 23:44

Given that the wavelength of the microwaves are about 3 cm and the aperture size of the horn is probably about 5 cm, the Rayleigh distance is about 4 cm.

So there is a chance that the experiment can be used to illustrate the inverse square.
It might be better to use a diode probe without a horn as the receiver which will reduce the chances of a standing wave pattern being set up between the transmitter and the receiver?
To this end also try and make sure there are no objects which might reflect microwaves around the apparatus including those doing the experiment.

As with similar experiments eg gamma radiation, ultrasound etc where the exact position of the "point" source is not known the standard method is say that the measured distance $d$ between two fiducial marks on the apparatus has a constant error $e$.
The correct source receiver distance is $d+e$.

If $I \propto \frac {1}{r^2}$ then $I = \frac{k}{r^2}$ where $k$ is a constant.

$\Rightarrow I = \frac{k}{(d+e)^2} \Rightarrow d = \frac{k'}{\sqrt I}-e$ where $k'$ is another constant.

A plot of $d$ against $\frac{1}{\sqrt I}$ should give a straight line if the inverse square law is being followed and the intercept will give a value for $e$.

As has been pointed out by others you will probably need to improve the accuracy of your detector reading and take many more readings.
The importance of this is shown by the analysis of the data which you have presented leading to an excellent straight line if the first point is ignored!!! giving a very strange value of intercept $e$!!!

If the inverse square law is not followed then other relationships could be investigated.