Intensity variation of straight rays I'm trying to understand what the author (Robert S. Elliott, Antenna theory and design, IEEE Press Series on Electromagnetic Wave Theory, pages 488-489) has in mind with that picture. It is substantially demonstrating that even in a geometric perspective the power density decreases with the square of the distance.

When the medium is homogeneous, so that the rays are straight lines,
the intensity law can be expressed in a different form. With reference
to Figure 10.3, consider a tube of rays with transverse cross sections
that are rectangular.

Projections back toward the source will locate apparent ray centers P
and P' for pairs of sides of the tube. It is evident that:
$dS_1=RR'd\theta d\phi$
and
$dS_2=(R+l)(R'+l)d\theta d\phi$
and hence
$\frac{\mathcal{P}_2}{\mathcal{P}_1}=\frac{S_1}{S_2}=\frac{RR'}{(R+l)(R'+l)}$

in which $\mathcal{P}_i$ are the power densities.
What are the points P and P' for him? Two different point sources? If so, why didn't he draw two different flow tubes?
Edit:
Seeing the picture "from above", do we have this situation?

 A: He must not be talking about a point source for the EM waves. For example, this may be a rectangular wave guide with a horn. Microwaves propagate inside such a wave guide. Wave fronts are planar. As they leave the horn guides them into an expanding pattern.
In reality, light is a wave. But it is common to talk about rays. Rays are lines perpendicular to the wavefront. That show the direction of motion of the wavefront. Using rays to show what wavefronts are doing is called geometric optics. A common situation is a spherically expanding wave. In this case rays are straight.
Waves diffract when they pass through a slit or opening. When they do, rays curve. They quickly become nearly straight. If you are interested in what happens far from an opening, straight rays is often a good approximation.
In this case, the waves would diffract to a pattern that spreads. Near the horn, rays curve. But far from the horn, you can talk about straight rays. The divergence angle is bigger for the smaller side of the rectangle.
If you follow the straight rays backward, they don't necessarily arrive at a point source because there was no point source.
In this case, it looks like there would be an apparent "arc" source. The arc would be centered on P and pass through P'. An expanding wedge of rays issues from the arc.

Given such an arc source and no diffraction, what would be the power density far from the source?
There really isn't an arc. But the wavefronts far away are the same as if there was. So this is a useful fiction. It helps you calculate.
