## Abstract

An elongated bottle microresonator with nanoscale parabolic effective radius variation can possess a series of dense equally spaced optical eigenfrequencies whose separation can match the eigenfrequency of its axially symmetric acoustic mode. It is shown that this acoustic mode can parametrically excite optical modes and give rise to a highly equidistant and moderately broadband optical frequency comb with the teeth spacing independent of the input laser power and the amplitude of mechanical vibrations.

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Among a broad class of optical microresonators including Fabry–Perot, spherical, toroidal, bottle, and photonic crystal resonators, of special interest are those that can serve as high $Q$-factor acoustic and optical resonators simultaneously. These microresonators exhibit fascinating phenomena of opto-acoustic coupling, including the resonant Brillouin and Raman scattering, radiation pressure interactions, and optomechanical cooling. An investigation of these phenomena emerged as a research direction in physics—cavity optomechanics (see [1–9] and references therein).

Microresonators of special interest are those having optical modes which can resonantly interact with acoustic modes. In such resonators, the separation of optical eigenfrequencies is close to their natural acoustic frequency or Brillouin frequency. The matching of two optical whispering gallery mode (WGMs) eigenfrequencies with an acoustic eigenfrequency $\sim 130\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{MHz}$ has been reported for a spherical microresonator with a 162 μm radius [4]. In [5], the splitting of optical WGM eigenfrequencies of two evanescently coupled 32 μm radius toroidal microresonators has been adjusted to match the 23 MHz acoustic eigenfrequency of one of them. Resonant opto-acoustic interaction has been demonstrated for much larger calcium fluoride and silica resonators, whose radii exceeded 2.5 mm and free spectral ranges matched the Brillouin frequencies equal to $\sim 18$ and 11 GHz, respectively [6,7]. Generally, for microresonators with small characteristic dimensions $\sim 100\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{\mu m}$, the resonant matching condition of optical WGMs frequencies and natural acoustic frequencies $\sim 100\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{MHz}$ is difficult to achieve because of their small dimensions and insufficient fabrication precision.

Here we note that the ultraprecise fabrication of microresonators, which are designed to enable efficient resonant interaction of optical and acoustic modes, can be realized using the surface nanoscale axial photonics platform [10,11]. For example, two coupled identical bottle microresonators having the effective radius variation (ERV) of $\sim 5\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{nm}$ were fabricated in [12] at the 38 μm diameter optical fiber with a precision better than 0.2 angstrom. The splitting of optical eigenfrequencies of these resonators was $\sim 0.5\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{GHz}$ and could be reduced further by post-processing. In another example, a bottle resonator with a 2.8 nm ERV demonstrated in [11] had the free spectral range $\sim 180\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{MHz}$. The separation of optical eigenfrequencies in both examples was comparable to the natural breathe frequencies of these resonators (see [13] and calculations below).

Recently, it has been experimentally demonstrated that the axially symmetric breathe acoustic modes of a bottle microresonator can be excited in both passive and lasing regimes by optical WGMs of this resonator [14]. The separation of optical WGM eigenfrequencies considered in [14] did not match acoustic eigenfrequencies, i.e., were not resonant. In this Letter, we introduce and study a bottle resonator with a nanoscale parabolic ERV (Fig. 1) which exhibits resonant interaction of optical and acoustic modes. We show that an acoustic mode localized in this resonator, which resonantly interacts with an optical WGM, can parametrically generate a series of optical modes with a highly equidistant spectrum forming a moderately broadband optical frequency comb (OFC).

We consider slow optical WGMs numerated by azimuthal, radial, and axial quantum numbers, $m$, $n$, and $q$, respectively. The slowness of modes is determined by the proximity of their eigenfrequency $\nu $ to a cutoff frequency ${\nu}_{\text{cut}}$(see, for example, [13]). In contrast to spherical and toroidal microresonators [1–7,15,16], the separation of eigenfrequencies of a bottle resonator along the quantum number $q$ can be made very small, without increasing the resonator cross-sectional radius ${r}_{0}$ (see, for example, [13]). This makes the bottle resonator very attractive for the investigation of interactions between its optical WGMs and acoustic modes. The acoustic modes of our interest are the axially symmetric breathe modes with zero azimuthal quantum number, ${m}_{\text{ac}}=0$, illustrated in Fig. 1(a). These modes can be excited by the radiation pressure of an optical WGM whose magnitude is modulated with the frequency of the chosen acoustic mode. In turn, an axially symmetric acoustic mode with eigenfrequency ${\nu}^{(\text{ac})}$ causes resonant coupling of optical modes with the same $m$ if the difference of their eigenfrequencies $\mathrm{\Delta}{\nu}^{(\text{op})}$ along the quantum number $q$ matches ${\nu}^{(\text{ac})}$. Crucially, the clamping losses of the considered acoustic modes can be eliminated, since the optical fiber hosting a bottle resonator can be clamped in the regions completely separated from the acoustic mode location.

Provided that the power of optical WGMs is relatively small so that the back-action effects [1] can be ignored, variation of a slow optical WGM along the bottle axis $z$ is described by the Schrödinger equation [17]

The excitation of the optical modes of a bottle resonator by its natural mechanical vibration with frequency ${\nu}^{(\text{ac})}$ is modeled by Eq. (1) with $\mathrm{\Delta}r(z,t)=\mathrm{\Delta}r(z)+\eta (z)\mathrm{sin}(2\pi {\nu}^{(\text{ac})}t)$, where $\mathrm{\Delta}r(z)$ is the unperturbed nanoscale profile of the resonator and $\eta (z)\ll \mathrm{\Delta}r(z)$ is the axial distribution of the amplitude of the excited acoustic mode. In the presence of acoustic oscillation with frequency ${\nu}^{(\text{ac})}$, the solutions of Eq. (1) can be expressed through Floquet quasi-states [19]:

The eigenfrequencies of a high $Q$-factor optical microresonator are usually detected by measuring the resonant transmission amplitude of light evanescently coupled to the resonator though a waveguide [Fig. 1(a)]. For weak coupling between the input–output waveguide and the resonator (strong under-coupled regime [16]), the inelastic output amplitude [corresponding to $|{p}_{2}|+|{p}_{1}|\ne 0$ in Eq. (5)] is found from Eqs. (2)–(4) as

Consider the important case when the input optical frequency is equal to the bottle resonator eigenfrequency, ${\nu}_{1}=\mathrm{Re}({\zeta}_{q})$. Then, the equally spaced OFC teeth are determined from Eq. (5) as

Equation (6) shows that, in contrast to the frequency of mechanical vibrations generated by optomechanical back-action [1] and the repetition rate of the OFC generated due to Kerr nonlinearity [21], the spacing between the teeth maxima in our case is fully determined by the natural acoustic frequency $\mathrm{Re}\text{\hspace{0.17em}}{\nu}^{(\text{ac})}$. Provided that the heating effects are small, this frequency does not depend on the input optical power and mechanical vibration amplitude. The power of the generated OFC teeth is determined from Eq. (5) as

From this equation, the power of the teeth is proportional to the squared product of the optical $Q$-factor, ${Q}^{(\text{opt})}=\mathrm{Re}\text{\hspace{0.17em}}{\zeta}_{q}/\mathrm{Im}\text{\hspace{0.17em}}{\zeta}_{q}$, and acoustic $Q$-factor, ${Q}^{(\text{ac})}=\mathrm{Re}\text{\hspace{0.17em}}{\nu}^{(\text{ac})}/\mathrm{Im}\text{\hspace{0.17em}}{\nu}^{(\text{ac})}$.

To illustrate the general results described by Eqs. (5)–(7), we consider the parametric excitation of optical modes in a parabolic bottle resonator perturbed by the axially symmetric acoustic mode with ${m}_{\text{ac}}=0$, ${n}_{\text{ac}}=1$, and ${q}_{\text{ac}}=0$. We simplify the problem by approximating the Gaussian axial distribution of the amplitude of this mode by a parabola. Then,

In addition, we assume that the input of light is situated in the middle of the resonator [${z}_{0}=0$ in Eq. (2) and Fig. 1(a)]. In this case, the excited optical and acoustic modes have the same reflection symmetry ($\mathrm{\Psi}(z,t)=\mathrm{\Psi}(-z,t)$), and mechanically generated transitions will only exist between the WGMs with axial quantum numbers $q$ of even parity. Parametric excitation of these modes takes place for acoustic oscillations with a frequency close to $2\mathrm{\Delta}{\nu}^{(\text{op})}=2c{(2\pi {n}_{r})}^{-1}{(-{r}_{0}{R}_{0})}^{-1/2}$:

Equation (1) with the ERV defined by Eq. (8) can be solved analytically [22]. Under the conditions of Eqs. (8) and (9), i.e., $\u03f5\ll 1$, $\delta \ll 1$, we find that the parametrically generated frequency combs are defined by Eqs. (5)–(7), where ${z}_{0}=0$, ${U}_{2q+1,p}(0)=0$, and

From this equation, the inelastic transmission amplitude, Eq. (5), and comb teeth power, Eq. (7), depend only on the ratio of relative amplitude of vibrations and relative deviation of acoustic eigenfrequency from the optical eigenfrequency spacing, $\u03f5/\delta $, which formally can be arbitrarily small. Figure 3 shows the power of the frequency comb teeth for $\u03f5/\delta =0.1$, 0.3, 0.5, and 0.9 and constant $\mathrm{Im}\text{\hspace{0.17em}}{\zeta}_{q}$. It is seen that the bandwidth of combs grows with $\u03f5/\delta $ and can be significant when this ratio approaches unity [19]. At the same time, it follows from Eq. (10) that the power of the teeth slowly vanishes as $\u03f5/\delta \to 1$. The plots in Fig. 3 also show that the behavior of comb tooth heights ${\mathrm{\Omega}}_{2q,p}$ is irregular as a function of quantum numbers $q$ and $p$. This is analogous to the behavior of transition probabilities in a quantum mechanical time-dependent harmonic oscillator described by the same Schrödinger equation [18]. The increase of asymmetry of the graphs with growing $\u03f5/\delta $ is explained by approaching the ground states at $2q+p=0$.

Note that the acoustic eigenfrequency ${\nu}^{(\text{ac})}$ of the bottle resonator is surrounded by adjacent eigenfrequencies of acoustic modes with non-zero axial quantum numbers ${q}_{a}$ [9]. The separation of these eigenfrequencies, $\mathrm{\Delta}{\nu}^{(\text{ac})}$, is usually much smaller than the width $\mathrm{Im}\text{\hspace{0.17em}\hspace{0.17em}}{\zeta}_{q}$ of optical eigenfrequencies. At room temperature, this separation is also smaller than the width of acoustic eigenfrequencies, $\mathrm{\Delta}{\nu}^{(\text{ac})}<\mathrm{Im}{\nu}^{(\text{ac})}$. The opposite condition $\mathrm{\Delta}{\nu}^{(\text{ac})}\gg \mathrm{Im}\text{\hspace{0.17em}}{\nu}^{(\text{ac})}$, which corresponds to very high mechanical $Q$-factors ${Q}_{\text{mech}}>{10}^{5}$, can be achieved at very low temperatures below $T\sim 0.1\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{K}$ [24,25]. The elimination of acoustic modes with large ${q}_{\text{ac}}$ can be performed by clamping the bottle resonator in the region where these modes are situated [Fig. 2(c)].

In summary, it is shown that an OFC can be generated in an optical bottle microresonator with an equidistant spectrum by its natural mechanical vibrations. In practice, small deviations from the spectral equidistance are introduced by fabrication errors. However, these deviations affect the OFC teeth power, rather than their equidistance, which is defined by the natural frequency of vibrations ${\nu}^{(\text{ac})}$. Generally, the power of OFCs generated mechanically is inverse proportional to the squared product of their optical and mechanical $Q$-factors. Provided that these $Q$-factors are large enough, the power required for the generation of these combs can be remarkably small and only limited by the sensitivity of the optical detectors.

## Funding

Royal Society (WM130110); Horizon 2020 Framework Programme (H2020) (H2020-EU.1.3.3, 691011); Engineering and Physical Sciences Research Council (EPSRC) (EP/P006183/1).

## Acknowledgment

The author acknowledges the Royal Society Wolfson Research Merit Award.

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