TLDR:
You’re reading that right — one-tenth of a milliwatt radio is enough to create error-free communications between the Sun and Alpha Centauri through two FOCAL antennas. Maccone’s paper assumes two 12-meter FOCAL antennas.
Actually the proposal is a bit more complicated than using a 12 meter large antenna, it is sending this antenna at 550 AU (the FOCAL in "FOCAL antenna"). Geoffrey A. Landis wrote:
"The gravitational field of the sun will focus light from a distant source to a focal point at a minimal distance of 550 Astronomical Units from the sun.
A proposed mission to this gravitational focus could use the sun as a very large lens, allowing (in principle) a large amplification of signal from the target, and a very high magnification."
Some people as Louis Friedmann says that only moving 10 meters away from the focal point (at 550 au) would reduce the gain by 10 dB.
https://www.youtube.com/watch?v=eEZ6zYZ6lDQ
What's interesting about this is that with closed loop control (using multiple antennas on the very signal they are receiving) one might well be able to achieve that 10m accuracy.
Maybe their math is correct, but the presentation here doesn't seem right. "Not so — there is no focal ‘point’ here but a focal line. As we move through and past 550 AU, we take advantage of the fact that the focal line extends to infinity."
Light is reversible, even if the space is curved. If all light from the "focal line" gathers at the focal point on the other side of the star, where does it go in reverse?
> If all light from the "focal line" gathers at the focal point on the other side of the star...
It doesn't - the light on the focal line may be from any source that falls on the same line on the opposite side of the star. But since the chances of more than one light source (distant star) falling on this exact same line are pretty minuscule, it's still a (potentially) useful lens
I don't understand what you are trying to say. There is no such thing as a focal line. More generally, it's not possible for light from several points to focalize at the same point because you can't revert those rays. I guess Randal Monroe explains it better than me here:
However, I can see one way to read what they mean. If you take the Sun's "focal point", the point where parallel lines from one side intersect after being bent by its gravity, and move a tiny away from the Sun, then you form the image of a point that resides on the line going from you through the Sun all the way to infinity on the other side. You need to move very little for the opposite to move a lot, so for measurement purposes you are still at 590 AU from the Sun. This way with very tiny moves you could survey the planets in a different solar system, and in particular you could resolve them to multi-pixels.
