{1] Imagine that the LIED experiment used a normal Hichelson interferometer that had no FabryPerot cavities in it. The two arms of the interferometer begin with the same length L, but. then something happens [gravitational wave passes byl} to cause one of them to change by an amount ELL- How much does this affect the relative phase difference of the two beams 5 when they exit the beam splitter on their way out of the interferometer if the light has wavelength A? What is your answer numerically for values appropriate for LIED, where A = l' nm, L = :1 km and h = Ly'L ~ Ill21'? The strain h describes the fractional change of the arm's length due to the passing gravitational wave- Take a minute to appreciate how small L is, by comparing it to the size of a proton. fill Repeat the previous problem, but now for the actual LIGU experiment where each arm of the interferometer contains a FabryPerot cavity that causes the laser beams to elfectively traverses the arms 450 times before exiting {that is 225 round trip-s inside each FabryPerot, and assume the FabryF'erot spans the whole length of the arm]. How will this affect. the phase dilference 45? By adding the FabryPerot cavities1 the effective length of each arm is greatly enhanced which leads to a huge increase in sensitivity, while keeping the size of the experiment still relatively compact. Side note: the number of times that each Fabry-Perot cavity is traversed is called the finesse of the cavity. {3] In the absence of passing gravitational waves, the two laser beams from the two arms of the LIED interferometer are set up to destructively interfere when they exit the interferometer, so that no light will reach the photo~detector {zero photon count]. The passing gravitational wave causes a tiny phase shift so that the two beams no longer perfectly destructively interfere, and the resulting light intensity is detected by the photo-detector- Assume the detector is able to detect. the presence of a single photon. Using the result in [2}, if we are to detect the signal from the passing gravitational wave, how many photons do we need in each arm of the interferometer? How much energyr in Joules is that per arm, for J. = HIE-1 nm? Recall that the energy of a single photon is E = hid- In LIED, the high laser power required to reach its designed sensitivity is achieved by using multiple FabryPerot cavities- 'We saw in Problem 2 {c} above that the intracavity intensity of a high nesse FabryPerot is greatly enhanced on resonance