Time evolution of parametric instability in large-scale gravitational-wave interferometers

We present a study of three-mode parametric instability in large-scale gravitational-wave detectors. Previous work used a linearized model to study the onset of instability. This paper presents a nonlinear study of this phenomenon, which shows that the initial stage of an exponential rise of the amplitudes of a higher-order optical mode and the mechanical internal mode of the mirror is followed by a saturation phase, in which all three participating modes reach a new equilibrium state with constant oscillation amplitudes. Results suggest that stable operation of interferometers may be possible in the presence of such instabilities, thereby simplifying the task of suppression.

Figure 1

Schematic of the three-mode interaction that gives rise to parametric instability in optomechanical systems: (i) Acoustic modes of the mirror, excited by thermal fluctuations, create motional sidebands of the pump mode carrier frequency offset by acoustic frequency ${\mathrm{\Omega }}_m$; (ii) the lower-frequency Stokes sideband is enhanced by a higher-order optical cavity mode—a beat note between this and the fundamental pump mode creates a radiation pressure force at the acoustic frequency ${\mathrm{\Omega }}_m$; (iii) this leads to a growth of the acoustic oscillation amplitude, which, in turn, increases the amplitude of the Stokes sideband, thereby raising radiation pressure force amplitude and closing the instability feedback loop. The strength of the three-mode optomechanical interaction, and, therefore, the chance of instability development, depends on the following three factors: (i) The extent to which the spatial distributions of all three participating modes overlap, characterized by overlapping factor ${\mathrm{\Lambda }}_{0S}$, defined in Eq. (6); (ii) the accuracy of three-mode frequency tuning—i.e., detuning must be smaller than the larger of optical modes bandwidth, ${\mathrm{\Delta }}_m={\omega }_0-{\omega }_S-{\mathrm{\Omega }}_m\ll \max [{\kappa }_0,{\kappa }_S]$; and (iii) the energy loss rates in all three modes, which need to be lower than the rate of power transfer between the modes, which itself is characterized by the coupling strength $G_{0S}$. These three conditions yield the definition of parametric gain, ${\mathcal{R}}_0$, given in Eq. (10), and the condition for PI given by Eq. (11).

Figure 2

Temporal behavior of three interacting modes for different values of PI gain ${\mathcal{R}}_0$. Thin lines show temporal dynamics in the detuned case with frequency mismatch ${\mathrm{\Delta }}_m=0.1{\kappa }_S$. The amplitudes of optical modes are normalized by the maximal value of optical power circulating in the fundamental mode which corresponds to PI gain value of ${\mathcal{R}}_0=8$. The acoustic mode amplitude is scaled by a displacement amplitude $b_0$ necessary to shift the Stokes mode frequency by one-half linewidth, as per definition in Eq. (7). The greyed-out area and thin vertical line (indicating the maximum of the Stokes mode curve for ${\mathcal{R}}_0=8$) marks the time interval where inverse scattering process, $\hslash {\omega }_S+\hslash {\mathrm{\Omega }}_m\to \hslash {\omega }_0$, intensifies in the system with PI gain larger than that defined by condition (20).

Figure 3

Upper panel: Dependence of power circulating in the fundamental mode (red solid line) and in the higher-order Stokes mode (blue dashed line) on input power, calculated from Eqs. (27)–(29), using parameters from Table 1. Inset plot compares the behavior of circulating power with (thick red line) and without (thin red line) PI. Lower panel: Dependence of acoustic mode amplitude on input laser power for parameters listed in Table 1.

Figure 4

Dependence of reflected light power at the carrier frequency ${\omega }_0$ (red dash-dotted line) and at the HOM frequency (blue line) as a function of parametric gain ${\mathcal{R}}_0$ and, therefore, of input power. Note the nonlinear and nonmonotonic relation between the input and reflected powers. The critical PI gain value of ${\mathcal{R}}_0=4$ corresponds to the case of critical coupling, i.e., when the nonlinear loss of carrier photons to the Stokes mode and the acoustic mode reaches the level of bare loss of the interferometer defined by the reflectivities of the mirrors.

Figure 5

Characteristic time for nonlinearity in parametric instability to take over. Thin dashed line shows the effect of nonzero frequency mismatch (${\mathrm{\Delta }}_m=\pm 0.1{\kappa }_S$ for this plot).

Figure 6

Scaling law for dual-recycled Advanced LIGO interferometers: Common and differential modes of a balanced interferometer, representing the sum and difference of the optical fields in the arm cavities, can be modeled as two independent effective Fabry-Pérot cavities coupled to common and differential acoustic modes of the arm cavity mirrors.