Figure 1

The idea of 3-mode interactions. The left diagram represents the scheme by which three modes interact with each other. The carrier mode is pumped into a cavity, where it experiences scattering on the mirror surface profiled by a particular acoustic mode. If the mirror surface motion spatially overlaps with the electric field distribution of a transverse optical mode, there is a possibility for the carrier mode to be scattered into this mode. The right frequency combination of all three modes can result in amplification of acoustic modes via the transverse optical mode, as represented in the diagram by a closed loop. The frequency structure is shown in the right-hand plot. The asymmetry of the modal structure of optical cavities means that only one sideband participates in the interaction. This could be either red or blue shifted. The graph shows an example of the excitation process due to a red-shifted sideband. Note that, in practice, there is no special requirement for the pump energy to be in the form of a TEM 00 mode: this is just the conventional choice for the pump mode.Reuse & Permissions

Figure 2

Acoustic modes of the sapphire test mass model. Mode frequencies $f_m$ are represented in terms of the FSR of the optical cavity. The 800 modes plotted here cover the frequency range up to the 3rd FSR. This covers all the modes that can potentially contribute to parametric instability.Reuse & Permissions

Figure 3

Mirror shapes for the ideal concentric mirror (solid thick line), concentric Gaussian mirror CG (dashed line), concentric Mesa mirror CM (dash-dotted line), and flat Mesa mirror FM (solid thin line).Reuse & Permissions

Figure 4

Near-concentric cavity structures for Gaussian and Mesa cavities. Each dashed vertical line corresponds to a high-order mode frequency lying in the 1st FSR. The number above each vertical line refers to the mode type listed in Table . There is a distinguishable difference between these two cavity structures. The more uniform spread of high-order modes for the Mesa cavity increases the chance of parametric instability.Reuse & Permissions

Figure 5

Diffraction losses for concentric cavities. The $x$ axis corresponds to the mode number shown in Table  (Appendix A).Reuse & Permissions

Figure 6

$Q$ factors of the optical modes for CM and CG cavities. The dashed line corresponds to the coupling limit of the cavities. The $x$ axis corresponds to the mode number shown in Table  (Appendix A).Reuse & Permissions

Figure 7

Overlapping parameters $\Lambda$ as a function of opto-acoustic mode rotation. The solid/dashed line corresponds to the overlapping of CG/CM modes, respectively. These plots were produced such that ${\Lambda }^{\max }$ was at $\varphi =0$. Note that for any optical mode of a $(p,q)$ type with $q=0$, $\Lambda$ remains constant for an arbitrary $\varphi$, as expected.Reuse & Permissions

Figure 8

Ratio of Mesa beam to Gaussian beam overlapping parameters $\Lambda$, for a given acoustic mode.Reuse & Permissions

Figure 9

Histogram of the optical mode distribution in terms of the overlapping parameter $\Lambda$ for the (a) Gaussian cavity, and (b) Mesa cavity. Each bin represents the $\Lambda$ range normalized by ${\Lambda }_{\mathrm{th}}={\omega }_m^2/\vartheta Q_m$, as described in Sec. 4.Reuse & Permissions

Figure 10

Parametric gain for concentric (a) Gaussian, and (b) Mesa cavities. Dots represent gains for the resonance condition $\Delta \omega$ estimated from acoustic and optical analysis, whereas the vertical red lines represent gains in the $\Delta \omega \pm 2\pi ·2000\mathrm{Hz}$ range.Reuse & Permissions

Figure 11

An example of gain $R$ variation for the 85.134 kHz acoustic mode in the $\Delta \omega /2\pi \pm 2000\mathrm{Hz}$ range. Note, variation in $R$ corresponds to the vertical line in Fig. 10.Reuse & Permissions

Figure 12

Histogram of the acoustic modes distribution per test mass in (a) the Gaussian cavity, and (b) the Mesa cavity, respectively. Only modes with parametric gain $R\ge 0.2$ are shown.Reuse & Permissions