Parametric optomechanics

Brillouin scattering results from the nonlinear electro-striction and photo-elasticity-based interaction between two optical waves and an acoustic wave. While the spontaneous scattering results from incoherent fluctuations of the medium and does not depend on the power of the signal that is scattered, the stimulated process appears because fluctuations induced by the field and its relative efficiency depend on the field power.¹ Stimulated Brillouin scattering (SBS) can be considered as a strongly nondegenerate parametric downconversion process in which a pump photon is transformed into a lower frequency Stokes photon and a phonon.² The scattering starts when the optical pump power exceeds a certain threshold value. Geometrical confinement of the optical and acoustic waves can result in significant amplification of the scattering efficiency, leading to a strong coupling regime in which the coupling coefficient exceeds the attenuation rates of the waves.^3–6 

Resonant SBS usually results in backward scattering of the Stokes light. The offset frequency between the pump and Stokes light, coinciding with the frequency of the sound wave, is defined by the properties of the material the cavity is made of.^7,8 While the bandwidth of the SBS process is relatively small, the sound waves usually do not form modes in solid state cavities at room temperature, because the phonon lifetime is much smaller than its round trip time in the cavity. A significant broadening of the Brillouin gain profile can be observed in crystalline ring microcavities as a result of variations in the acoustic phase velocity, depending on the orientation of the crystalline structure and the sound path.⁹ 

If the spectrum of the optical cavity is equidistant in frequency, it is possible to generate multiple orders of SBS, representing an optical frequency comb. In this process, the Stokes wave becomes the pump wave for another Stokes wave as well as the sound wave propagating in the opposite direction to the initially generated sound wave.¹⁰ The sound wave excites optical modes and generates an optical frequency comb with fixed frequency spacing.¹¹ 

The resonant SBS process competes with other nonlinear processes in the cavity. Those include optical four-wave mixing (FWM) and stimulated Raman scattering (SRS). It is possible to reduce the impact of those processes on the SBS by optimizing the cavity geometry. Phase matched resonant SBS has the lowest threshold among optical nonlinear processes, and these processes can coexist. For instance, one can create a narrow-linewidth Brillouin laser and then use the monochromatic Stokes light produced by this laser to generate a FWM-based Kerr frequency comb in the same cavity.¹² 

The standard SBS has a power threshold since it corresponds to the parametric generation of the sound wave from the background fluctuations. Thresholdless excitation is possible with utilization of polychromatic light, e.g., two counter propagating optical waves separated in frequency by the SBS characteristic frequency, to pump a solid state cavity. One can also generate sub-Brillouin frequencies using opto-mechanical nonlinearity¹³ with three laser fields. Two specially configured optical pumps can also prepare the quantum states of a mechanical system.¹⁴ 

The propagation of phonons in a solid state material is nonlinear. Higher order phonon harmonics can be generated out of a monochromatic sound wave excited either optically^15–17 or electrically.^18–22 This process is not efficient at high SBS frequencies because of the significant attenuation of the hyper-sound, but it is frequently observed in low-frequency opto-mechanics, where phonons have much longer lifetimes. It was shown, for instance, that acoustic pulses can be formed in a solid state microcavity due to interaction between phonons. In this case, the cascaded opto-mechanical process is prohibited by the absence of the optical modes to support the higher order Stokes light. The light generates the fundamental sound harmonic that results in the generation of the higher frequency sound harmonics without involvement of the light, and the sound harmonics can form a sound pulse.^15–17 

In this paper, we discuss the process of generating coherent multiple phonons from a single scattering event involving two photons. Such a process is well established at lower optical frequencies and is efficient but incoherent. It results in multiphonon absorption, which leads to excessive material loss.²³ We show that the nonlinearity of the strain tensor enables coherent scattering that can be observed in a high quality microcavity. Note that this is different from the single phonon SBS followed by the nonlinear frequency conversion of the photon. We show that two phonons can be generated simultaneously without their frequency necessarily coinciding with the SBS frequency. We perform an experiment with a magnesium fluoride (MgF₂) whispering gallery mode (WGM) resonator and use the developed theory to explain the experimental results.

Optical microresonators enable the realization of high concentrations of optical fields in a small geometrical volume. If two nondegenerate counter propagating optical pump waves are coupled to a microresonator (Fig. 1), their beatnote can create a slowly moving polarization pattern that potentially can excite an acoustic wave, even if the frequency separation of the pump waves does not coincide with the Brillouin frequency.

The process is not phase matched if the polarization wave created by the two optical waves propagates faster than the sound wave in the material. Therefore, the parametric excitation of the single acoustic wave is prohibited when the frequency difference of the optical pumps deviates from the SBS frequency [Fig. 2(b)]. The situation changes when interaction of two acoustic waves, supporting a beat note that propagates $(ω̃+−ω̃−)/ΩSBS$ faster than a sound wave, and light waves become feasible. This case, corresponding to phase matching of the two-phonon scattering, is illustrated by [Fig. 2(c)].

The generated phonons can mix and multiply due to phonon nonlinearity, creating sound waves at both smaller and larger frequencies. For example, quadratic phonon nonlinearity may result in the generation of phonons at doubled frequencies. Cubic phonon nonlinearity can mix the phonons propagating in opposite directions. The higher order opto-mechanical process involving two optical waves and three phonons can also be phase matched [Fig. 2(d)]. For this process to occur, the strain tensor should include cubic terms in deformation.

The parametric opto-mechanic oscillator described above generates counter propagating phonons. The phonons do not form modes because of their short lifetime. To detect those phonons, one needs to utilize opto-mechanic light scattering. Indeed, photons propagating in the resonator can create optical harmonics by scattering from the phonons. The main conditions for the scattering are (i) availability of an optical mode in the resonator to accept the scattered photons and (ii) satisfaction of the momentum conservation law. Similarly to the previous reasoning related to parametric oscillation, only phonons with frequency Ω_SBS are phase matched with optical modes in the first order process. The phonons cannot generate optical harmonics if the resonator does not have optical modes with matching frequencies to accept them. However, higher order processes can take place. Nonlinear acoustic frequency mixing among the phonons is also possible.^18–22 Therefore, if a high amplitude wave is generated at frequency Ω_SBS, it is reasonable to expect that phonons at other frequencies will also be generated. In what follows, we assume that due to nonlinear acoustical mixing, the phonons generate frequency combs (of phonons) at frequencies NΩ_SBS.

To validate the theoretical calculations, we performed an experiment using a crystalline whispering gallery mode (WGM) resonator pumped with two independent lasers. The scheme of the experiment is depicted in Fig. (1). The resonator was manufactured out of magnesium fluoride (MgF₂) by mechanical grinding (other materials also can be utilized). It was characterized by a high loaded quality factor (Q ≃ 10⁹) and a free spectral range of 37.0 GHz. In addition to the fundamental mode family, the resonator supported multiple high order mode families characterized by similar quality factors. The rim of the resonator was shaped to permit optimal evanescent coupling of the laser light to the modes with prisms.

Two semiconductor distributed feedback lasers were self-injection locked to corresponding cavity modes. By locking the counter-propagating lasers to two modes of the same family, we observed generation of optical frequency combs due to cubic nonlinearity of the material, as well as crosstalk between the lasers that did not have optical isolators in front of them (Fig. 3). Optical frequency combs were also generated when the power of at least one of the lasers exceeded ∼2 mW. However, in the case of locking the lasers to modes belonging to different mode families, the crosstalk between the lasers was suppressed. We observed generation of optical harmonics separated by frequencies proportional to the characteristic SBS frequency of the material. The spectrum was irregular, and the harmonics could not be explained by the four-wave mixing process. The offset frequencies were proportional to the SBS phonon frequency. These observations are consistent with the parametric opto-mechanical scheme introduced above (Fig. 4).

An example of the observed optical spectra for the clockwise and counterclockwise light exiting the resonator is shown in Fig. (5). In this experiment, the frequencies of counter-propagating optical pumps were separated by 37.2 GHz, which is approximately equal to 3Ω_SBS. The pumps were coupled to two different mode families, and no FWM was observed at small enough pump powers. The resonant Rayleigh scattering was suppressed for the pumped modes due to the high intrinsic quality factor of the resonator and its operation in the overloaded state, as well as due to spatial filtering of the received light.

The resonator generated Stokes lines in the clockwise direction at Ω_SBS and 5Ω_SBS offsets with respect to the counterclockwise pump. It also generates an anti-Stokes line in the counterclockwise direction offset from the clockwise pump by 2Ω_SBS.

Let us analyze the observed spectrum in detail. The counterclockwise (ccw) phonon with frequency Ω_SBS and clockwise (cw) optical harmonic localized in mode D [Fig. 4(a)] is generated via the regular resonant SBS process.

The interaction of the two pumps localized in modes B and C also parametrically generates phonons on the frequency 2Ω_SBS. Since the pumps were separated by 3Ω_SBS, the parametric optomechanical process generated phonons at 2Ω_SBS (clockwise) and Ω_SBS (counterclockwise) [Fig. 4(b)], in accordance with (11). The ccw Ω_SBS phonon can also be seeded by the regular SBS, reducing the overall threshold of the oscillation. The 2Ω_SBS generation can be accompanied by the generation of two Ω_SBS phonons because of the third order of the process.

The clockwise sideband at a lower frequency (E) was generated by mixing the counterclockwise pump C with phonons having frequency 3Ω_SBS and propagating counterclockwise as well as 2Ω_SBS and propagating clockwise [Fig. 4(c)]. If the phonons are generated in the nonlinear phonon mixing process, the scattering of the pump on these phonons does not have any specific threshold other than the threshold of the multiphonon process.

Interestingly, the anti-Stokes sideband (A) could be generated by mixing the clockwise pump (B) and phonons with frequencies of Ω_SBS/2 (ccw) and 3Ω_SBS/2 (cw). We did not observe these fractional frequency offsets in the experiment. However, we noticed that the counterclockwise harmonic has a strong interaction with a clockwise mode A′. This might happen because of the spatial overlap of the modes as well as resonant Rayleigh scattering. The mode A′ can be generated parametrically [Fig. 4(d)].

Neither of the harmonics was generated when only one pump was coupled to the corresponding resonator mode for any frequency detuning of the pump. It is important to stress here that single phonon scattering was not permitted by the phase matching conditions when the offset frequency exceeded Ω_SBS. Only the parametric interactions are reasonable for the observed harmonics. On the other hand, since we have no ability to detect the generated acoustic phonons directly, a more detailed follow-up study is needed to confirm the resonant multiphonon scattering experimentally.

By sending the light exiting the resonator to a microwave spectrum analyzer, we observed generation of spectrally pure microwave signals at the offset frequencies. One of the signals is illustrated in Fig. (6). The cleanness of the spectrum supports the idea of the coherent nature of the observed process.

The observed nonlinear process also could be attributed to a chain of optical co- and counter-propagating³⁰ FWM interactions, where the clockwise field generated both +4Ω_SBS and −4Ω_SBS GHz harmonics. However, the harmonic at +4Ω_SBS GHz is barely visible and is within the measurement accuracy corresponding to the background. In addition, the trace of this harmonic can be explained by the residual FWM interaction of the clockwise pump (B) and the clockwise harmonic (A′).

The involvement of a few different mode families was essential in our experiment. It allowed observation of the harmonics at frequencies that do not coincide with the main mode family of the resonator. Pumping the different mode families allowed us to suppress four-wave mixing. To further study the multi-phonon parametric scattering phenomenon, one must engineer the optical spectrum of the resonator. The main criterion for successful spectral engineering is creating a cavity spectrum that does not support FWM, Raman scattering, or standard Brillouin scattering. The observation can be attempted in the nonlinear waveguides; however, it is harder to observe the process because the intermodal FWM can be less suppressed therein (Ref. 31).

In conclusion, we have shown the possibility of a parametric two-phonon opto-mechanical process. Two coherent counter-propagating phonons can be generated simultaneously from two counter-propagating optical waves when the frequency separation between the waves exceeds the characteristic SBS phonon frequency. The frequencies of the generated phonons are defined by the phase matching conditions.