LIGO Phase Shifts

Classical Mechanics Level 3

If the laser light used in Advanced LIGO is $1064 \text{ nm}$ , an arm of the detector is $4 \text{ km}$ long, and the GW150914 system causes strains $h = 10^{-21}$ , what is the phase shift in laser light traversing a detector arm at maximum displacement versus no displacement, in radians?

See Gravitational Waves .

1.18 \times 10^{-11}

2.36 \times 10^{-11}

4.72 \times 10^{-11}

9.44 \times 10^{-11}

1 solution

Matt DeCross
Feb 12, 2016

The angular frequency of the light is $\omega = ck = \frac{2\pi c}{\lambda}$ and the wavelength is given. The phase shift is $2\omega t$ , where $t$ is the time spent by light traveling the extra distance caused by the strain. Note the factor of two because the light must travel both directions through the extra distance. The time spent in one direction is: $t = \frac{d}{c}$ and by definition of the strain, the extra distance is $4 \times 10^{-21} \text{ km}$ . Putting it all together, $\Delta \theta = 2\omega t = 2\frac{2\pi c}{\lambda} \frac{d}{c} = \frac{4\pi d}{\lambda}.$ Plugging in numbers, we find the answer.

Hi @Matt DeCross , isn't d(phi) = k•d(x)? Phi here is phase difference.

Sudhanshu Meshram - 2 years, 9 months ago

In your notation, x=2ct is the distance traveled by light in time 2t (again, factor of 2 caused by having to travel both directions). Then one gets d(phi) = 2ck dt = 2 omega dt in agreement with the above, using omega = ck.

Matt DeCross - 2 years, 7 months ago

LIGO Phase Shifts

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