# Three-mode opto-acoustic parametric interactions with coupled cavity

## Abstract

We theoretically analyze three-mode opto-acoustic parametric interactions in a coupled Fabry-Perot cavity, where one acoustic mode interacts with two optical modes. We show explicitly that extra degrees of freedom in a coupled cavity allow explorations of both parametric instability and cooling regimes with high parametric gain in a single table top experiment. This work can motivate experimental realizations of the three-mode parametric instability which might be an issue in next-generation gravitational-wave detectors with high optical-power cavities, helping in the development of better models, and in developing techniques for controls. In addition, we show that the same scheme can be implemented in the resolved-sideband acoustic-mode cooling.

###### pacs:

## I Introduction

Parametric interactions have wide applications and arouse great interests in various fields of physics, from high-sensitivity transducers to low-noise amplifiers and optical parametric oscillators. Two-mode parametric interactions have been used to cool acoustic modes of mechanical oscillators. In Ref. (1), the authors cooled the normal mode of a 1.5 tonne Nb bar down to 5mK by using a microwave parametric transducer. With the same principle, a high-frequency acoustic mode of a nano-mechanical oscillator is cooled near its quantum ground state through coupling to a superconducting single-electron transistor (2). More recently, by coupling mechanical resonators to optical cavities, various table top experiments have demonstrated significant cooling of acoustic modes (3); (4); (5); (6); (7); (8); (9); (10); (11); (12); (13). These experiments show great potential of achieving the quantum ground state of macroscopic mechanical oscillators, which would be a breakthrough in both theoretical and experimental physics.

Three-mode opto-acoustic parametric interactions were first investigated by Braginsky et al. (14); (15) in the context of high optical-power Fabry-Perot cavities for interferometric gravitational-wave detectors. It was shown that three-mode interactions led to a risk of parametric instability (PI) in which the amplitude of acoustic modes of test masses could grow exponentially, thus undermining sensitivities of the detectors. The cause of parametric instability is the radiation pressure mediated coupling between optical modes and acoustic modes. It can occur if the shape of high-order cavity modes, hereafter denoted by TEM, have a substantial overlap with that of acoustic modes and simultaneously if the mode gap between TEM and TEM is equal to acoustic-mode frequency up to an error of the linewidth of cavity modes. Further theoretical studies have followed up this pioneering work. Zhao et al. (16) extended their analysis and took into account 3D structures of the optical and acoustic modes. Ju et al. (17) further considered the overall contributions from multiple optical modes, showing that multiple interactions increase the risk of instability. Gurkovsky et al. (18) analyzed PI in a signal-recycled (SR) interferometer and shows that the chance of PI was reduced due to the narrow linewidth of SR interferometer. Additionally, many ideas have been proposed to prevent PI, e.g., changing the radius of curvature (RoC) of the mirror (16), using an optical spring tranquilizer (19) or a ring damper (20). In the light of the above predictions, it is important to develop experimental techniques for their investigations. In recent progress, the University of Western Australia group demonstrated three-mode interactions in an 80m-long Fabry-Perot cavity (21). They used a compensation plate as a thermal lens to tune the effective RoC of the mirror, which changes the Gouy phase or equivalently the mode gap, such that the resonant condition mentioned is satisfied. By capacitively exciting an acoustic mode to interact with the mode, they observed resonance of a high-order optical mode at the predicted frequency. This experiment confirmed the principle of three-mode interactions, but due to low optical power and small overlap, it had not yet achieved self-sustained parametric instability or cooling. In this paper, we propose an alternative method of tuning the Gouy phase. It was first introduced by Mueller (22) as a means of designing a stable recycling cavity for next-generation gravitational-wave detectors. By adding a mirror and a lens onto a single cavity, which forms a coupled cavity, one can obtain any desired Gouy phase shift simply by adjusting relative positions of these optical components. In following sections, we will show explicitly that a table top experiment can be set up, allowing observations of three-mode interactions with high parametric gain. This can help us better understand possible parametric instability in interferometric gravitational-wave detectors, and can also be applied, in general, as a design concept for opto-acoustic parametric amplifiers, specifically to the resolved-sideband cooling (23); (24) of acoustic modes.

This paper is organized as follows: In Sec. II, we will review the theory of three-mode opto-acoustic interactions. In Sec. III, we will analyze three-mode interactions in a coupled cavity. Finally, we will summarize our results in Sec IV.

## Ii Review of three-mode opto-acoustic parametric interaction theory

For the coherence of this article, we will briefly review the theory of three-mode opto-acoustic parametric interactions. They can be understood qualitatively in both quantum and classical pictures. In the former picture, we can treat the Fabry-Perot cavity as a multi-level quantum system occupied by photons, while the mechanical oscillator (end mirror) consists of phonon (acoustic) modes in different energy eigenstates. For simplicity, we only consider one phonon mode, which represents the situation in this proposed scheme.

Initially, the photons stay in one eigenstate (the TEM mode). Resonant amplification occurs when the photons are scattered by the phonons into another eignstate (the TEM mode). The phonons can absorb energy from the photons (conventionally known as a Stokes process in Raman scattering), causing instability. Alternatively, they can release energy into the optical field (Anti-Stokes process), which is also called cooling in the literature. We show both processes schematically in Fig. 1. In the classical picture (14), mechanical oscillations modulate the carrier frequency into two sidebands. The lower sideband corresponds to the Stokes mode with frequency and the upper sideband is the Anti-Stokes mode with . If they match the frequency of TEM , higher-order optical mode will be excited. In turn, these resonant optical modes TEM and TEM will exert a radiation pressure on the mechanical oscillator at the beating frequency . Depending on the phase, this force will either amplify or damp the motion of the oscillator. To achieve good coupling, a substantial spatial overlap between the acoustic mode and the TEM mode is required.

To quantify the interactions, we follow the formalism in Ref. (14). In the paper, they introduced parametric gain to quantify the strength of three-mode interactions, which is defined as,

(1) |

Here subscript denotes Stokes mode and for Anti-Stokes mode; is the total length of the cavity; is the intra-cavity power; is the mass and is the mechanical quality factor. The detunings and is decay rates of optical modes. Overlapping factor is given by,

(2) |

where is optical-mode shapes; is the component of normal to the mirror surface; The integrals and correspond to integration over the mirror surface and volume respectively. For simplicity, we assume that , which can be achieved in real experiments. In the tuned case that , can be simply written as

(3) |

with for the Stokes process and for the Anti-Stokes process. The resulting decay rate of the acoustic mode can be written as

(4) |

Since usually , then

(5) |

When , we obtain the trivial case . For the Stokes process with (positive gain), the decay rate of the acoustic mode decreases and the mechanical oscillation will be amplified. PI happens if , where we have . For the Anti-Stokes process with (negative gain), the acoustic mode will be cooled as the effective decay rate increases.

## Iii Three-mode interactions with a coupled cavity

In this section, we will discuss how to explore three-mode interactions using a coupled cavity. To make our analysis close to realistic experiments, we consider a torsional acoustic mode with frequency MHz interacting with the optical TEM and TEM modes. The configuration is chosen because MHz can be easily achieved in a mm-scale structure and the torsional mode has a large spatial overlap with the TEM mode.

To begin with, let us consider a single Fabry-Perot cavity to see why a coupled cavity is necessary. The free spectral range of a single cavity with length cm is approximately 1GHz. Therefore, as shown in Fig. 2, one has to build either a near-planar or near-concentric cavity to obtain a desired mode gap around 1MHz between TEM and TEM. For both cases, the cavity is marginally stable and susceptible to misalignment. It is also difficult to get accesses to both instability and cooling regimes in a single setup. Creating a coupled Fabry-Perot cavity solves these problems. As we will show later, the resulting scheme is stable. Additionally, we can easily tune between instability and cooling regimes. The coupled cavity is showed schematically in Fig. 3. It is similar to the configuration of power or signal recycling interferometers (25); (26) when one considers either the common mode or the differential mode. The field dynamics can be easily obtained as shown in Ref. (27) by treating the sub-cavity as an effective mirror with frequency and mode-dependent transmissivity and reflectivity. Specifically, the effective transmissivity is

(6) |

and the effective reflectivity is given by

(7) |

The corresponding inside the main cavity can be written as

(8) |

The resonance occurs when the phase factor in Eq. (8) is equal to , which critically depends upon the phase angle of the effective reflectivity, namely . Specifically, when the TEM mode resonates inside the main cavity, which requires that , the phase shift of the TEM mode is

(9) |

where is the mode gap between TEM and TEM; is the Gouy phase and are integers. In order to satisfy the resonant condition for three-mode interactions, we need to adjust , which changes , such that . To achieve this, one obvious way is to change the length of sub-cavity , but this turns out to be impractical due to a small tuning range. An alternative and more practical approach, as shown in Fig. 4, is to add another lens or concave mirror inside the sub-cavity to tune the Gouy phase . The resulting scheme is similar to the proposed stable recycling cavity for next-generation gravitational-wave detectors (22). With the additional lens, the Gaussian beam gets focused inside the sub-cavity. Since the Gouy phase changes almost from to within one Rayleigh range around the waist, one can easily obtain a desired simply by adjusting the position of M near the waist. This might lead to problems with power density due to the small waist size, but for the table top experiment we consider here, the power density is quite low.

The corresponding with an additional lens can be derived straightforwardly by using the ray transfer relation for a Gaussian beam, which is given by

(10) |

Here is the focal length of ; ; is the displacement relative to the waist; is the Rayleigh range and superscript denotes quantities after the lens. This dictates

(11) | |||||

(12) |

The resulting Gouy phase at any point is given by

(13) |

where is the position of relative to the waist. Gouy phase is the difference between wavefront at and , namely

(14) |

where and are the positions of and relative to the waist respectively. Therefore, by adjusting the positions of and as shown in Fig. 4, we can continuously tune such that .

Equations (6) to (14) provide the design tools of the coupled cavity for three-mode interactions. To realize the experiment, we first need to design the main cavity and specify , , the RoCs of and the focal length of . From Eq. (8) and Eq. (9), we can find out the required which gives the right mode gap between TEM and . This will gives us one constraint. Combining with the requirement of mode matching to , we can fix two degrees of freedom of the system, namely the positions of and . To demonstrate this principle explicitly, we present a solution that is close to a realistic experimental setup. We assume the following,

Here are RoCs; denotes the amplitude reflectivity; is the amplitude transmissivity and is the optical loss which satisfy .

The results of mode matching for both positive (instability) and negative gain (cooling) configurations are shown in Fig. 5. In Fig. 6, we show the mode gap between TEM and TEM as a function of the position of M relative to L and the position of L relative to M. In this particular case, the dependence is almost linear with a slope for both panels. This indicates that to tune within a cavity linewidth MHz, it requires the mirror position to be adjusted within several , which can be achieved easily. Therefore, we can continuously tune between instability and cooling regimes. Fig. 7 shows the resulting resonance curves for both cases with the corresponding mode matching shown in Fig. 5. The corresponding mode gap between the TEM and TEM modes is equal to . More importantly, there is no symmetric mode on the opposite side of the TEM mode, whose presence could contribute a parametric gain with the opposite sign, thereby suppressing the overall effects. The absolute value of parametric gain could be larger than 1, if we further assume that the intra-cavity power is 100 mW, , the mass of the oscillator mg and the wavelength of light is 1064 nm. Since the cavity is in the resolved-sideband regime where the cavity linewidth is much smaller than the mechanical frequency (23), this configuration can also be applied in the resolved-sideband cooling of acoustic modes, which is less susceptible to quantum noise. Although the quantum noise analysis in Refs. (23); (24); (28) only discusses a two-mode system, their results can be extended to the three-mode system by viewing the mode as the carrier light. This can be justified by writing down the Hamiltonian of this system, which is

(15) |

Here the first two terms represent the energy of the acoustic mode, where is displacement and is momentum which satisfy the commutation relation ; and are the annihilation operators of the and modes respectively; the fifth term quantifies the interactions between the optical modes and the acoustic mode with the coupling constant ; the last term corresponds to external driving. Since only the mode is pumped externally, the interaction term can be linearized as

(16) |

which is the same interaction considered in Ref. (23); (28) after linearizing. Therefore, this three-mode Hamiltonian can be mapped into an effective two-mode Hamiltonian by replacing with its classical amplitude . All the conclusions reached in the two-mode case are also valid here. The only difference is that the carrier light TEM is also on resonance rather than far detuned in the two-mode case (23); (28). Therefore, we can achieve the resolved-sideband limit without compromising intra–cavity optical power. What’s more, the resonant condition here is equivalent to the frequency detuning equal to in the two-mode case. This optimizes the energy transfer from the acoustic mode to the optical fields in the resolved-sideband limit (23); (28).

## Iv Conclusion

We have shown that a table top experiment based on a coupled cavity can be set up to realize self-sustained three-mode opto-acoustic parametric interactions. Small adjustments of the positions of the optical components enable us to continuously tune between instability and cooling regimes. This relative simple scheme can be applied to design opto-acoustic amplifiers and experimentally investigate the three-mode parametric instability, a possible issue of next-generation gravitational-wave detectors with high optical-power cavities. Besides, it can also be used to realize the resolved-sideband cooling of acoustic modes with three-mode interactions.

ACKNOWLEDGEMENTS

We thank all the participants attending the Parametric Instability Workshop held at the Australian International Gravitational Observatory (AIGO) site, especially Prof. S. P. Vyatchanin and Dr. S. E. Strigin for stimulating discussions and giving precious advices on the improvement of the draft. We thank D. Price for pointing out several errors in the manuscripts. Miao thanks Prof. Y. Chen for the invitation to visit the Albert-Einstein-Institut. The visit was supported by Alexander von Humboldt Foundation’s Sofja Kovalevskaja Programme. This research has been supported by the Australian Research Council and the Department of Education, Science and Training and by the U.S. National Science Foundation. We thank the LIGO Scientific Collaboration International Advisory Committee of the Gingin High Optical Power Facility for their supports.

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