How Does LIGO Detect Gravitational Waves Using Laser Interferometry?

[EGN123]

Homework Statement

In-phase light from a laser with an effective power of 2x10⁵J and a wavelength of 1064nm is sent down perpendicular 4km arms of the LIGO detector.
(i) Determine the number of photons traveling in the interferometer arms.
(ii) Assuming the detector is sensitive enough to detect single photons at the initial position, estimate the precision with which a change in the length of one of the arms can be detected.

Homework Equations

c=ƒλ
E=hƒ
ρ=E/c
ΔpΔx ≥ ħ/2

The Attempt at a Solution

(i) Photon frequency = c/λ = (3x10⁸)/(1064x10^-9) = 2.82x10¹⁴Hz
Photon Energy = hƒ = (6.626x10^-34)(2.82x10¹⁴) = 1.87x10^-19J

Time for light to travel the length of the arms twice (i.e. return to initial position)= 2 x (4000/c) = 2.66x10^-5s

Laser releases 2x10⁵J per second, which is 1.07x10²⁴ photons per second

1.07x10²⁴ photons per second x 2.66x10^-5s = 2.85 x 10¹⁹ photons

(ii) I was very unsure how to do this part. My initial thought was the uncertainty principle so I tried that but I don't think it is correct.
Sensitive to single photons, ΔE = hf = 1.87x10^-19J
Δρ = ΔE/c = 6.23x10^-28Ns

ΔpΔx ≥ ħ/2
Δx = ħ/2Δp = 8.46 x 10^-8m

I don't think my answers are correct, especially (ii). Am I using the wrong methods? Or correct methods but have made a mistake?

 

[scottdave]

A question: your problem statement says that Power is 2 x 10^5 Joules. (Joule is energy, Watt is power). Then you state that it emits 2x10^5 Joules per second, in order to solve part i. If your assumption is correct then those calculations look fine. (it might not be far off, since I looked up about the actual LIGO on Wikipedia, which states LIGO to have 100 kW power (10^5 J/s).
I also found some information, indicating that the sensitivity of the LIGO is on the order of 10^-19 m which is a Lot less than your answer. I also found this ebook link via ligo.org which you may find interesting. http://www.gwoptics.org/ebook/index.php

 

[EGN123]

A question: your problem statement says that Power is 2 x 10^5 Joules. (Joule is energy, Watt is power). Then you state that it emits 2x10^5 Joules per second, in order to solve part i. If your assumption is correct then those calculations look fine. (it might not be far off, since I looked up about the actual LIGO on Wikipedia, which states LIGO to have 100 kW power (10^5 J/s).
I also found some information, indicating that the sensitivity of the LIGO is on the order of 10^-19 m which is a Lot less than your answer. I also found this ebook link via ligo.org which you may find interesting. http://www.gwoptics.org/ebook/index.php

Yeah that was my mistake, the power in the question is given as J/s not just J. It was more the method I was using to calculate the number of photons that I was questioning. It was the way that seemed logical to me however I have no solution to compare with. In regards to part (ii) I felt the sensitivity had to be much lower which is why I was so sure my answer was wrong, but I'm not sure what's wrong with what I've done.
I'll have a look at that ebook.

 

[scottdave]

The way I understand it. The inferometer is set up to have destructive interference between the perpendicular tunnels under normal conditions. So if everything is perfect, then no light hits the sensor. If one of the tunnels is not the same length as the other one (possibly due to a gravitational wave), then some light (1 or more photons) will strike the sensor.
I am thinking, the way perhaps to solve this problem is this: take the amount of energy in 1 photon, then find out how much length change must occur for the interference to change, such that the energy increases by more than the energy of 1 photon.

 

[EGN123]

The way I understand it. The inferometer is set up to have destructive interference between the perpendicular tunnels under normal conditions. So if everything is perfect, then no light hits the sensor. If one of the tunnels is not the same length as the other one (possibly due to a gravitational wave), then some light (1 or more photons) will strike the sensor.
I am thinking, the way perhaps to solve this problem is this: take the amount of energy in 1 photon, then find out how much length change must occur for the interference to change, such that the energy increases by more than the energy of 1 photon.

That actually makes more sense to me than how I interpreted it. I'll give that a try, thanks!