1.2.1 The basic idea of interferometer
The gravitational-wave detector is a type of the Michelson interferometer, which has a Fabry–Perot cavity in each arm and other functional components. In principle, observing gravitational waves is done by measuring the change in distance between test masses at the ends of each arm and the beam splitter.
In this case, to isolate seismic noise and reduce the thermal noise, the test masses are isolated and freely suspended which can be regarded as a pendulum. A monochro-matic frequency-stabilized laser incident beam is divided by a beam splitter (BS) that allows half of the light to transmit and the other to reflect. The light after splitting travels into two arms, with distance LX and LY, and then are entirely reflected to BS by the end mirror that is referred to as an end test mass. We represent ETMX and ETMY as the end test mass at x and y arm, respectively. Then, the light in each arm is recombined at the BS and interfere with each other. Finally, the light is transmitted to a photodetector and the port where is set the laser source (Fig. 1.2).
Note that we assume that all optics are lossless, the BS is a perfect 50/50 split, and the end mirrors reflect all light.
Assuming that the length of two arms of the interferometer is L, the angular frequency of the light is ω0, and the round trip length of the light traveling back and forth is 2L, the resulting phase shift is given by
ϕ(t) = ω0tr = 2ωoL
c. (1.4)
The phase shift is a constant, and its value is proportional to L.
Figure 1.2: Basic idea of gravitational-wave detector.
Initially, the arm length of the interferometer is adjusted so that the two beams demonstrate completely destructive interference. There is no light coming into the photodetector. Because the two arms are equal in length, the phase shift produced by a single round trip in one arm is equivalent to the phase shift caused by the other arm. It means that while the lengths of the two arms are the same, the port where the laser source is set has constructive interference. Moreover, the port where the photodetector is placed has destructive interference. To simplify, the ports which are set laser source and photodetector are usually referred to as bright port and dark port, respectively. In sum, the symmetric laser interferometer gravitational-wave detector output signal is zero.
When the gravitational wave arrives, one arm initially stretches, and the other arm shortens due to its polarization characteristics. Then, half a cycle later, the stretched and shortened arms are switched. Thus, the two coherent lights have different paths, which eliminates the initial destructive interference. Therefore, the amount of the photons propagate to the photon detector; thereby, it outputs a signal. The change
in the relative arm length ∆L is used as the detector output signal to measure the gravitational-wave strain h.
∆L = ∆Lx− ∆Ly = (L + δL)− (L − δL) = 2δL = hL, (1.5)
h = ∆L
L = 2δL
L , (1.6)
where L is the detector arm length, and ∆L is the change in the relative arm length.
Note that ∆L/c is also a relative phase difference according to Eq. (1.4). From Eq.
(1.6), we can obtain the gravitational-wave strain by measuring the phase difference shown in Fig. 1.3. For typical gravitational-strain amplitude (h≈ 10−21), the typical laser-wave vector k0 = ω0/c ≈ 3 × 106 m−1 with a 4-km interferometer arm. Thus the phase shift is ∆ϕ ≈ 10−11, which is too small to measure until we improve the sensitivity of the measuring device.
Figure 1.3: Basic idea of gravitational-wave detector. The gravitational wave passes through the interferometer, which makes the test mass shift and produces a slight difference between the two arms.
1.2.2 Fabry–Perot arm cavities
Figure 1.4: A Michelson interferometer with additional ITMX and ITMY (input test mass) to form arm cavities.
From Eq. (1.6), we know that a gravitational-wave signal is related to the ratio between δL and L, where L is the distance between the BS and the end mirror.
Therefore, we can increase the sensitivity of the detector directly by lengthening the arm. However, constructing extremely long arms is expensive. Thus, rather than increasing the physical distance between the BS and the end mirror, we increase the length of the optical path. Here, the concept is that additional mirrors can be inserted into each arm between the BS and the end mirror., which is the so-called Fabry–Perot cavity.
Fig. 1.4 shows the configuration of a Michelson interferometer with Fabry–Perot cavity. We label the additional mirrors as input test masses (ITMX and ITMY) and the original mirrors as end test masses (ETMX and ETMY). The distances between
the input and end test masses are expressed as LX and LY, respectively. As a result, we increase the distances LX and LY by introducing Fabry–Perot cavities.
1.2.3 Sideband
The input monochromatic laser field Ein with frequency ω0 is reflected on the end mirror with displacement motion x(t), which caused by the effect of external force G (i.e., gravitational waves). The phase of the reflected optical field is modulated by the mechanical motion of the mirror, allowing the output field’s spectrum of Eout to contain two sideband fields with frequencies of ω0 ± Ω, where ω0 is the carrier frequency of the input laser and Ω is the frequency of the gravitational waves as shown in Fig. 1.5.
Figure 1.5: The spectrum of the optical field reflects from the moving mirror. The center peak represents the carrier light with frequency ω0, and the two smaller peaks represent the signal sideband with frequency ω0± Ω, defined by the mirror motion spectrum x(Ω)