Question

Background: The gravitational wave detector LIGO is basically a Michelson interferometer, and has 4 km long arms. The two arm lengths are equal, but when a gravitational wave passes through LIGO the arm lengths change, shifting the interference fringes on the detector. However, gravitational waves are so week that we need long arm lengths to detect them. A space based gravitational wave detector should have no problem in having a much longer arm length than LIGO.] If we are interested in detecting a gravitational wave producing a relative change in the length of one arm, Δ?/? = 10-20, how long should the arms be for the gravitational wave to shift the interference pattern at a given point on the detector by 1/ 1000 fringes? Assume that the wavelength of the laser used is 1064 nm, the arm lengths of the interferometer when there is no gravitational wave are equal (?1 = ?2 = ?), and the interferometer is oriented such that the gravitational wave changes the length of one arm only

Answer 1

According to the standard theory of the Michelson - Morley experiment, the change in the number of fringe is related to the change in the length of the arm of the interferometer as
   
  
where, \Delta m is the change in the number of fringes.
And so, from the required accuracy of the experiment
   
  
   
Now for the given values
   
And
  
And so, we get the required length of the arm of the interferometer is
   
  
  
This is huge by the way. To get a feel of how big this is, this is almost 10000 times bigger than the earth radius.