3.5.1
Is There a Gouy Phase ‘Sweet Spot’ ?
As shown in sections3.4.2and 3.4.3, the number of unstable modes, and the parametric gain of these, depends on the phase accumulated in the Signal-Recycling Cavity. While there is no ideal region with no PIs, there are regions where both of these qualifiers are relatively low. Adjusting the tuning (i.e. propagation phase) of the SRC in order to minimise PI is ill-advised, since this will change the operational mode of the interferometer and therefore the overall sensitivity curve of the detector. However, the Gouy phase accumulated in the recycling cavities could be adjusted without changing this operating point, by changing the radius of curvature of the mirrors. It should be noted that this will change the geometric stability of the cavity, which will have consequences for the control the detector, as explored in [56,57].
s
Figure 3.13: Total number of unstable modes and summed parametric gain for different choices of one-way Gouy phase, ψ in the Power- and Signal-Recycling Cavities.
plot uses the same technique as that used to create figure3.11, but now explores the 2-dimensional parameter space formed by the Gouy phase accumulated in both the SRC, ψS and PRC, ψP. Each
horizontal slice of the 2D plots forms another version of figure3.11(b)with a different value of PRC 1-way Gouy phase, counting the number of modes that are unstable at their Comsol-computed mechanical resonant frequency, and summing the total gain of these unstable modes. As usual, we include calculation of all couplings to higher order Hermite-Gauss optical modes up to 10th order. The ψS and ψP labels on the axes correspond to the values of Gouy phase applied directly
to the space between the S(P)RM and S(P)R2 (see figure3.2). This is a single, symmetric value applied to Gouy phase in both the x- and y-axes, meaning that the astigmatic effects created in the recycling cavities are removed in this simulation.
Rather than a unique ‘sweet spot’, figure3.13shows that there are many small regions with few, or zero, PIs. The Gouy phase of the recycling cavities for the Advanced LIGO design was determined based on the geometric stability of the cavities and the need for good alignment signals from these cavities. Compromising between these two led to chosen values of ψS ∼ 19◦,ψP ∼ 25◦ [2], as
indicated in figure3.11(b). Figure3.13shows that the effect of recycling cavity Gouy phase on PIs should also be used when the same decision is being made for future detectors. For the case of our Advanced LIGO model, which is dependent on the exact set of optical parameters used and set of mechanical modes tried, this plot indicates that values of ψS ∼ 50◦ and ψP ∼ 40◦may experience
fewer PIs than the current design.
3.5.2
ITM vs. ETM
As noted in section3.2.3, the simulations above only explore the behaviour of a single mechanical mode of the End Test Mass in the X-arm of the Advanced LIGO interferometer model. All four test masses are suspended in the same manner, have the same general geometry, and interact directly with the same optical field in the arm cavities. Therefore in the Advanced LIGO model we expect them to produce PIs for the same surface motion maps. In reality, the four test masses are not completely identical, but are extremely similar, meaning that the same type of surface motion has a slightly different resonant frequency for each test mass (typically a few Hz, see figure 3.1). Based on this, we can expect that the estimated number of stable modes is four times the numbers predicted by plots such as figure3.11.
applied to an ITM is not the same as that for an ETM. In the figure, we plot the parametric gain of mode 37 as a function of mechanical mode frequency as usual. This time, however, we apply this same map to all four arm cavity mirrors in turn.
Figure 3.14: Parametric gain of mode 37 when the surface motion occurs at each of the Input- and End Test Masses in the X- and Y-arms.
The dominant force driving the motion of the optic is the optical field in the arm cavity, since this is enhanced by the cavity finesse and therefore the power is a factor of several hundred higher than the field in the PRC. However, while an ETM only experiences radiation pressure from the arm cavity directly, ITMs interact with light from both inside and outside the cavity, and the combined radiation pressure forces from both sides will therefore affect the total parametric gain. The change in optical response for input versus end mirrors comes from the location of the mirrors and the optical fields that are therefore incident on them.
In reality this interaction would be more complicated still, since the test masses are bulky, composite optics, and so the optical field can interact with multiple surfaces of the mirror. In our Finesse model, only the highly reflective surface of the optic has a defined surface motion which interacts with the optical field on both sides. The bulk material and anti-reflective surfaces act only as a simple low-reflectivity plane and a change in the optical path length for existing fields, without introducing additional radiation-pressure driven interactions.
Further investigation is needed to verify this explanation for the difference between ITM and ETM behaviour, for example by building a minimal Finesse model of a power-recycled arm cavity using thin optics and exploring how the parametric gain of a mode applied to the ITM or ETM varies with power recycling gain for a given arm power.
3.5.3
Modelling PIs using parameters from the Livingston detector
The study above explores the influence of recycling on parametric instabilities in an optical model of the Advanced LIGO design. The optical configurations of the two sites differ slightly from this design (and each other) in small but important ways—for example, the radii of curvatures of the test masses are not all identical, meaning that the arm cavities do not have identical beam parameters and there is a small amount of mode mismatch at the beamsplitter.
Studies [48,20] have shown that arm mismatch will influence parametric gain–since, as illustrated by figure3.6, changing the radii of curvature of the test masses will change the optical response– but that this effect is not expected to significantly change the number of PIs that the detector experiences.
Figure 3.15: Parametric gain of mode 37 in a model of the LIGO Livingston detector (LLO) compared to the Advanced LIGO (aLIGO) design, shown for the case of a single arm cavity (‘1 arm’) and the full DRFPMi configuration (DR).
Figure3.15agrees with this statement: here, we plot the parametric gain of mode 37 as a function of mechanical mode frequency, comparing the results for the Advanced LIGO design file (as seen in figure3.5) to a model of the LIGO Livingston detector, ‘LLO’. This model [58] uses measured values for the optics [59] and setup.
The traces for the Advanced LIGO and LLO are not significantly different in form. The response of the single arm cavity is shifted up by ∼ 0.2 kHz. In the dual-recycled (DR) configuration, this means that the common mode peak is further away from the arm cavity envelope and becomes smaller, while the differential peak is amplified. However, while this shows that mismatched arms will change the absolute behaviour of parametric instabilities in the interferometer, it does not
fundamentally change the results presented above.