Light interferometry as described in Thorne's black holes and time warps ?

[matteo210]

light interferometry as described in Thorne's "black holes and time warps"?

hello,

maybe someone who read the book can help me with this, but i'll give all the information so that even someone who hasn't read it can understand.
At page 383 and following, and at fig. 10.6 and box 10.3, Thorne describes the basic functioning of a laser interferometric gravitational wave detector. My question is not relative to the "gravitational" part but only to the laser interferometry. In the box and in the figure (link below), Thorne writes that the "waves" reaching the photodetector interfere destructively while the "waves" reaching back the laser/emitter/source interfere constructively. Shouldn't it be the opposite? i was thinking that the "waves" reaching the photodetector, traveling the same distance in space at the same time, (both being reflected twice and passing thru the mirror once), would interfere constructively. Why Thorne writes the opposite instead?
For anyone who has not access to the book, i found this site which has a picture similar to the one in the book
http://plus.maths.org/issue18/features/thorne/
it's the drawing rapresenting laser interferometry. In the book Thorne writes that there should be no light in the photodetector due to destructive interference. Why it is not the opposite (all the light in the photodetector due to costructive interference)?
thanks

 

[PeterDonis]

Moderator's note: Thread moved to relativity forum.

 

[PeterDonis]

i was thinking that the "waves" reaching the photodetector, traveling the same distance in space at the same time, (both being reflected twice and passing thru the mirror once), would interfere constructively.

You're leaving out a key component of the system: the beam splitter. The laser light comes from the source, goes through the beam splitter, travels out the two arms, gets reflected, travels back, and reaches the beam splitter again. If the arms are exactly the same length, then what happens is that the beam "splitter" just functions as a beam "recombiner"--it just combines the two split beams back into one beam that is traveling in the direction it originally came from--i.e., towards the laser source. That's because the beams coming back are just the exact reverse of the beams going out, so going back through the beam splitter just reverses the splitting and sends all the light back to the source (as long as the arm lengths have stayed the same).

Thorne phrases it in terms of constructive and destructive interference, but he's really saying the same thing as the above. If you want to break it down in more detail, you need to consider each step in the process and take into account what the beam splitter does. So to start, we have a beam coming from the laser source and being split at the beam splitter into two beams: one beam goes straight through and is unchanged, and the other gets reflected by 90 degrees and has its phase reversed (i.e., shifted by 180 degrees). Call the beam that goes straight through from the laser source beam A, and the beam that gets reflected by 90 degrees beam B.

Now the beams each travel out their respective arms, get reflected, and come back along the two arms and pass through the beam splitter again. Beam A now gets split into a part that goes straight back to the laser detector with its phase unchanged, and a part that gets reflected by 90 degrees and goes towards the photodetector; but the reflected part also has its phase unchanged, because reflection only changes the phase on the front side of the beam splitter (the side facing the laser source), not the back side (the side facing the arm beam A travels along). Beam B also gets split into a part that goes straight through to the photodetector, and another part that gets reflected by 90 degrees; and since this reflection is off the front side of the beam splitter, it shifts the phase by 180 degrees.

So on the laser source side, we have the part of beam A that went straight through both times with phase unchanged, and the part of beam B that got reflected twice with a phase shift of 180 degrees each time. So the two are in phase and interfere constructively.

But on the photodetector side, we have the part of beam A that got reflected once off the back side of the splitter, so its phase is unchanged, and the part of beam B that got reflected once off the front side of the beam splitter, so its phase got shifted by 180 degrees. So the two are 180 degrees out of phase and interfere destructively.

 

[Ibix]

A few neat demos of waves moving from media with one refractive index to another can be found here: https://www.acs.psu.edu/drussell/Demos/reflect/reflect.html