Degrees of freedom

The arm cavities of the Sagnac speed meter, like those of a Fabry-Perot Michelson interfer- ometer, must be held resonant in order to maintain the light power required for the design sensitivity, and so these cavities represent a degree of freedom that must be controlled with active feedback. Meanwhile, the error signal is insensitive to the motion of the inter-cavity mode matching mirror, M₉, since this is situated at half the total round trip distance and is sensed by the counter-propagating modes at almost the same time. Other mirrors are potentially significant: the beam splitter M₆and steering mirror M₇, as shown in figure 4.9. As these mirrors are situated near the start of one and the end of the other modes’ round trips, a velocity dependent signal is created at the balanced homodyne detector (BHD, see section 4.2.1). We neglect all other auxiliary optics.

To assess the importance of the optics to the interferometer’s sensitivity to differential arm cavity length 𝐿(−), transfer functions from individual mirrors to the BHD port, where the 𝐿₍₋₎signal should by design couple most strongly, were calculated using Optickle (see ap- pendix C.1.2). The results in figure 5.1 show that the cavity mirrors are the most important positions to control, with the arm cavity finesse enhancing the sensitivity of the BHD to the arm cavity mirrors such that they dominate the signals from M₆and M₇. These results have been confirmed both with Finesse and analytically [141].

The common mode motion of the arm cavities, 𝐿(+), will also need to be controlled by means of a photodetector placed at the input port to sense the light returning from the interferometer back towards the laser. This motion will be suppressed by applying strain and heat to the laser’s crystal to change its geometry and therefore lasing frequency. This solution involves the creation of a wide bandwidth controller able to provide large correc- tions within the audio band. While the control of 𝐿(+)is crucial to maintain the light power within the arm cavities, we focus on 𝐿(−) given that it represents the main signal appear- ing at the output port and the one which will primarily contribute to the sensitivity of the interferometer in the context of gravitational wave detectors.

While the motion of M₉ can be suppressed at the main readout given suitable mirror po- sitioning in order to cancel the signal from each of the counter-propagating modes, the effect of M₆ and M₇ is less clear cut. To assess the impact the motion from these mirrors has on 𝐿(−) sensitivity, a calculation of the effect of seismic noise from M7to the BHD can be made. M₆need not be considered separately here: the transfer function is almost iden- tical to that of M₇ and so we need only calculate one, and the suspension design—a work in progress at the time of writing—is intended to have better isolation than that of M₇’s auxiliary suspension.

104 106 108 1010 1012 Resp onse ( W m ) 𝐿₍₋₎ 𝑀₆ 𝑀₇ 101 102 103 104 105 Frequency (Hz) −180 −135 −90 −45 0 45 90 135 180 Phase (° )

Figure 5.1: Transfer functions from important mirrors or combinations of mirrors in the Sagnac speed meter experiment to the balanced homodyne detector. The 𝐿(−) degree of freedom has the

strongest response by design. The main beam splitter, M₆, and the steering mirror for cavity A, M₇, have response a factor of 10−3_{that of 𝐿}

(−). Other mirrors have significantly lower coupling. Measurements of the seismic motion present upon the ground outside the vacuum system can be propagated through a model of the passive seismic isolation within the vacuum sys- tem to obtain the effective seismic-induced motion of the tables upon which the suspensions sit. The seismic motion of M₇can then be calculated by multiplying this spectrum with the transfer function of the auxiliary suspension from the table to the test mass, taken from a state-space model. This seismic noise can be projected into an effective differential arm cavity motion displacement spectral density by multiplying it by the ratio of the transfer functions of M₇ and 𝐿(−) to the BHD port1, taken from figure 5.1. This can be compared with the requirement for sensitivity of the BHD to 𝐿(−). Figure 5.2 shows that M7’s motion, projected into 𝐿(−), will meet the requirement above 100 Hz, and the result is similar for M₆.

The results in figures 5.1 and 5.2 show that control of 𝐿(−) will be required to meet the sensitivity requirement at the BHD port above 100 Hz, where the measurement of reduced

1_{This is the same as multiplying the motion of M}

7by its transfer function to the BHD port to yield a signal

in W√Hz−1_{, and dividing by the transfer function from 𝐿}

(−)to the BHD to yield an effective motion in terms

101 102 103 Frequency (Hz) 10−22 10−21 10−20 10−19 10−18 10−17 10−16 10−15 10−14 10−13 10−12 Effe ctiv e 𝐿(−) displacement noise ( m √ Hz )

𝑀₇effective 𝐿₍₋₎noise from seismic Sagnac speed meter requirement

Figure 5.2: Effective 𝐿(−)seismic noise contribution from M7. This is calculated by first propagating

a seismic noise spectral density for the laboratory near the vacuum system through damping and suspension models to obtain the motion of the M₇ test mass. With this figure, the response at the BHD can be calculated from the transfer function shown in figure 5.1, and this in turn can be expressed in units of differential arm cavity motion by dividing it by the response of 𝐿(−) to the

BHD port. The requirement is given only for frequencies above 100 Hz where the measurement of reduced radiation pressure noise will be made, and this figure shows that seismic motion of M₇ will not represent a significant problem to the sensitivity of the experiment in the desired band. This conclusion applies also to the main beam splitter, M₆, which is expected to have even greater isolation from seismic noise.

radiation pressure noise will be made. It should be noted, however, that the desired BHD homodyne angle depends on the relative path lengths of M₁₁ to M₁₆ and M₆ to M₁₆. This length will be controlled by an auxiliary loop not considered part of the longitudinal control strategy, and will be the subject of future work alongside a strategy for the control of 𝐿(+).