Introduction

The generation of optical frequency combs (OFCs) represents a vibrant field of research in photonics1,2 with numerous applications in precision spectroscopy, optical metrology, and high-speed optical communications3,4,5,6. Currently, the main techniques to generate OFCs use mode-locked lasers7, electro-optic (EO) modulation of continuous wave lasers8, and nonlinear optical processes in microresonators9,10. The most common method to generate OFCs in microresonators is based on the nonlinear Kerr effect11,12,13,14, but alternative approaches have been proposed, such as the application of the EO effect in microresonators made of materials with second-order nonlinearity15,16,17,18,19,20. The microresonator-based comb technologies are particularly attractive because they offer compact, efficient, and high-repetition-rate frequency combs (of the order of GHz and THz). However, applications such as high-precision spectroscopy require OFCs with lower repetition rates (a few hundreds of MHz or smaller)3,21, which is more challenging to achieve with microresonators. Indeed, the repetition rate frequency of microresonator-based OFCs is usually equal to the microresonator free spectral range (FSR), \(\Delta {\nu }_{{{\mbox{FSR}}}}\), which, for the commonly used ring, toroidal, and spherical microresonators, is inversely proportional to their size. As an example, a silica toroidal or spherical microresonator with \(\Delta {\nu }_{{{\mbox{FSR}}}}=50\) MHz has a radius \({r}_{0}=c/(2{{{{{\rm{\pi }}}}}}{n}_{0}\Delta {\nu }_{{{\mbox{FSR}}}})=\) 65 cm (here \(c\) is the speed of light and \({n}_{0}=1.46\) is the refractive index of silica). Therefore, to achieve sufficiently low repetition rates, the microresonator size has to be increased to dimensions which may be unpractical.

One approach to solve the challenge of generating OFCs with low repetition rates in microresonators consists in the adiabatic modulation of microresonator parameters at frequency \({\nu }_{{{\mbox{par}}}}\) that is much smaller than its FSR, \({\nu }_{{{\mbox{par}}}}\ll \Delta {\nu }_{{{\mbox{FSR}}}}\). In this case, the repetition rate of the OFC is equal to the modulation frequency and independent of the microresonator size. In a recent paper22, the authors demonstrated a flat frequency comb spectrum with repetition rate as low as \(50\) MHz generated by optomechanical oscillations of a toroidal silica microresonator. The adiabatically slow modulation of the microresonator eigenfrequency depends on time as \({\nu }_{{{{{{\rm{e}}}}}}}(t)={\nu }_{{{{{{\rm{e}}}}}}0}+\delta {\nu }_{{{\mbox{par}}}}{{\cos }}(2{{{{{\rm{\pi }}}}}}{\nu }_{{{\mbox{par}}}}t)\), where \(\delta {\nu }_{{{\mbox{par}}}}\) is the amplitude of modulation. This modulation can generate a close to uniform OFC formed by \(N\cong \delta {\nu }_{{{\mbox{par}}}}/{\nu }_{{{\mbox{par}}}}\) spectral resonances separated by the modulation frequency \({\nu }_{{{\mbox{par}}}}\) (see consideration below). For a microresonator with radius \({r}_{0}\), the parametric modulation corresponds to an amplitude change of the effective radius variation (ERV) defined by the scaling relation \(\delta {r}_{{{\mbox{eff}}}}={r}_{0}\delta {\nu }_{{{\mbox{par}}}}/{\nu }_{{{{{{\rm{e}}}}}}0}\). For the experimental parameters in the work of Hu et al.22, \({r}_{0}\cong 30\) µm, \({\nu }_{{{{{{\rm{e}}}}}}0}\cong 200\) THz, \({\nu }_{{{\mbox{par}}}}=50\) MHz, and \(N\cong 1000\), we find \(\delta {r}_{{{\mbox{eff}}}}\cong 8\) nm. In ref. 22, this modulation was generated by the radiation pressure of the resonant high-power input light that excited the mechanical breathing mode of the microresonator.

The application of SNAP bottle microresonators (SBMs) with shallow nanoscale ERV, which are introduced at the surface of optical fibres23,24,25,26,27, constitutes an alternative and promising approach to generate OFCs with low repetition rates. Optical eigenmodes in SNAP microresonators are whispering gallery modes (WGMs) adjacent to the fibre surface and slowly bouncing between turning points along the microresonator axial length. The adiabatic generation of OFCs with low repetition rates can be realized in SBMs by a strong input light, similar to the approach of Hu et al.22. Alternatively, the EO modulation of the microresonator refractive index can also be used to generate OFCs (see15,18,19 and references therein). The FSR of a SBM is estimated as \(\Delta {\nu }_{{{\mbox{FSR}}}}=c{(2{{{{{\rm{\pi }}}}}}{n}_{0})}^{-1}{({r}_{0}R)}^{-1/2}\) where \({r}_{0}\) is the fibre radius, \(R\gg {r}_{0}\) represents the axial radius, and \({n}_{0}\) is the refractive index of the microresonator material23,24. It is worth noting that for SBMs, the value of \(R\) can be dramatically large. For example, for the semi-parabolic SBM fabricated by Sumetsky28, the axial radius was \(R\cong 0.7\) km and, potentially, can be an order of magnitude greater. A feasible FSR for a silica (\({n}_{0}=1.46\)) SBM with \(R=10\) km and \({r}_{0}=100\) µm is \(\Delta {\nu }_{{{\mbox{FSR}}}}\cong 33\) MHz. To maximise the frequency comb bandwidth, the SBM should have equally spaced axial eigenfrequencies. This feature is achievable with the SNAP technology as we can fabricate SBMs with parabolic or semi-parabolic shape28. Experimentally, OFCs generated by conventional bottle microresonators with axial radius \(R\) of the order of 100 µm have been presented by several groups29,30,31 starting with the first demonstration by Savchenkov et al.32. The generation of OFCs in parabolic SBMs by the nonlinear Kerr effect and by resonant harmonic parametric excitation has been theoretically studied33,34. However, to the best our knowledge, OFCs generated by parametric modulation of parabolic SBMs have not been experimentally demonstrated to date.

In this paper, we theoretically investigate the formation of OFCs by the parametric modulation of parabolic SBMs going beyond the lumped-element model15,18,19,20,22 and taking into account the spatial distribution of the parametric modulation (SDPM). Based on the theory of SNAP microresonators with time-dependent parameters35, we optimize the SDPM targeting to arrive at the maximum flat and broadband OFC spectrum. Significantly, the characteristic ERV \(\delta {r}_{{{{{{\rm{par}}}}}}}\) of the SDPM can be much smaller than, as well as comparable to, the dramatically small nanometre-scale ERV \(\triangle {r}_{0}\) of the SBM. Below, we demonstrate the critical importance of the SDPM in the adiabatic (\({\nu }_{{{\mbox{par}}}}\ll \Delta {\nu }_{{{\mbox{FSR}}}}\)) and resonant (\({\nu }_{{{\mbox{par}}}}=2\Delta {\nu }_{{{\mbox{FSR}}}}\)) cases as well as for relatively small (\(\delta {r}_{{{\mbox{par}}}}\ll \triangle {r}_{0}\)) and large (\(\delta {r}_{{{\mbox{par}}}} \sim \triangle {r}_{0}\)) modulation amplitudes.

Results and discussion

We consider a parabolic SBM formed by the nanoscale ERV of an optical fibre with radius \({r}_{0}\) illustrated in Fig. 1. Light is coupled into the SBM by a transverse optical waveguide placed at \(z={z}_{0}\). In our analysis, we model the input light as a monochromatic source \({A}_{{{\mbox{in}}}}\left(z,t\right)\cdot {{{{{{\rm{e}}}}}}}^{-2{{{{{\rm{\pi }}}}}}{{{{{\rm{i}}}}}}{\nu }_{{{\mbox{par}}}}t}\) where function \({A}_{{{\mbox{in}}}}\left(z,t\right)\) is localised near the point \({z}_{0}\) and has the characteristic switching time \({\alpha }^{-1}\):

$$\begin{array}{c}{A}_{{{\mbox{in}}}}\left(z,t\right)={A}_{0}\delta \left(z-{z}_{0}\right)\left\{\begin{array}{cc}\left(1-{{{{{{\rm{e}}}}}}}^{-\alpha t}\right) & t \, \ge \,0\\ 0 & t \, < \, 0\end{array}.\right.\end{array}$$
(1)
Fig. 1: Illustration of the considered system.
figure 1

A SNAP bottle microresonator (SBM) with parabolic cutoff frequency variation (CFV) centred at \({z}_{0}=0\) introduced at the surface of an optical fibre with radius \({r}_{0}\). The SBM is coupled to the transverse waveguide that pumps light at frequency \({\nu }_{{{\mbox{in}}}}\) and excites whispering gallery modes (WGMs). The SBM is parametrically modulated with frequency \({\nu }_{{{\mbox{par}}}}\) and generates an optical frequency comb at the output of the waveguide with the repetition rate \({\nu }_{{{\mbox{par}}}}\).

If frequency \({\nu }_{{{\mbox{in}}}}\) is close to one of the cutoff frequencies of the optical fibre \({\nu }_{{{{{{\rm{c}}}}}}}\left(z,t\right)\)23, the input light excites a WGM that slowly propagates along z and bounces between the SBM turning points forming eigenmodes. The slowness of the propagating WGM depends on the proximity of its frequency \({\nu }_{{{\mbox{in}}}}\) to the cutoff frequency \({\nu }_{{{{{{\rm{c}}}}}}}\left(z,t\right)\) which is assumed to have the parabolic shape perturbed by the parametric modulation:

$${\nu }_{c}\left(z,t\right)= \, {\nu }_{c0}+\delta {\nu }_{{{\mbox{par}}}}\cdot \xi \left(z/L\right)\cdot \sin \left(2\pi {\nu }_{{{\mbox{par}}}}t\right)\\ +\left\{\begin{array}{cc}\varDelta {\nu }_{0}\left[{\left(\frac{z}{L}\right)}^{2}-1\right] & \left|z\right|\le L\\ 0 & \left|z\right| > L\end{array}.\right.$$
(2)

Here, \(2L\) is the length of the SBM, \({\nu }_{{{{{{\rm{c}}}}}}0}\) is the cutoff frequency of the uniform optical fibre away from the SBM, \(\Delta {\nu }_{0}\) is the maximum of the cutoff frequency variation (CFV) that forms the SBM (Fig. 1), \(\delta {\nu }_{{{\mbox{par}}}}\) is the maximum of the modulation amplitude, and \(\xi \left(z/L\right)\) is its dimensionless spatial distribution.

For sufficiently small and slow deformation of the optical fibre, the WGM field distribution is separable in cylindrical coordinates as \({{{{{{\rm{e}}}}}}}^{{{{{{\rm{i}}}}}}l\phi }{\Theta }_{p}(r)\Psi (z,t)\cdot {{{{{{\rm{e}}}}}}}^{-{{{{{\rm{i}}}}}}2\pi {\nu }_{{{{{{\rm{in}}}}}}}t}\), where \(l\), and \(p\), are the azimuthal and radial quantum numbers. Then, under the condition that \({\nu }_{{{{{{\rm{c}}}}}}}\left(z,t\right)\) is sufficiently close to \({\nu }_{{{\mbox{in}}}}\), the function \(\Psi \left(z,t\right)\) obeys the Schrödinger equation35:

$${{{{{\rm{i}}}}}}\frac{1}{2{{{{{\rm{\pi }}}}}}{\nu }_{{{{{{\rm{c}}}}}}0}}{\partial }_{t}\Psi(z,t) =\, \Big[-\frac{1}{2{\beta }_{{{{{{\rm{c}}}}}}0}^{2}}{\partial }_{z}^{2} +\frac{{\nu }_{{{{{{\rm{c}}}}}}}\left(z,t\right)-{\nu }_{{{\mbox{in}}}}-{{{{{\rm{i}}}}}}\gamma }{{\nu }_{{{{{{\rm{c}}}}}}0}} \\ -\frac{1}{{\beta }_{{{{{{\rm{c}}}}}}0}^{2}}{D}_{0}\delta \left(z-{z}_{0}\right)\Big]\Psi(z,t) +{A}_{{{\mbox{in}}}}\left(z,t\right).$$
(3)

Here, \({\beta }_{{{{{{\rm{c}}}}}}0}=2{{{{{\rm{\pi }}}}}}{n}_{0}{\nu }_{{{{{{\rm{c}}}}}}0}/c\) is the cutoff wavenumber, \({n}_{0}\) is the refractive index of the SBM, and \(c\) is the speed of light. In Eq. (3), the absorption losses in the SBM are determined by the parameter \(\gamma\) while the complex number \({D}_{0}\) takes into account the effect of the input-output waveguide coupled to the SBM at \({z=z}_{0}\)23.

In the presence of coupling and losses, the frequencies of the SBM are complex-valued and given by

$${\widetilde{\nu }}_{q}=\, {\nu }_{q}-{{{{{\rm{i}}}}}}{\Gamma }_{q}= \left({\nu }_{{{{{{\rm{c}}}}}}0}-\Delta {\nu }_{0}+\left(q+1/2\right)\Delta {\nu }_{{{\mbox{FSR}}}}-\delta {\nu }_{{D}_{0},q}\right)\\ -{{{{{\rm{i}}}}}}{\Gamma }_{q},\quad \Delta {\nu }_{{{\mbox{FSR}}}}=c/2{{{{{\rm{\pi }}}}}}{n}_{0}L\cdot \sqrt{2\Delta {\nu }_{0}/{\nu }_{{{{{{\rm{c}}}}}}0}}$$
(4)

where \(q\) is the axial quantum number and \(\Delta {\nu }_{{{\mbox{FSR}}}}\) is the axial FSR of the unmodulated SBM. Here \(\delta {\nu }_{{D}_{0},q}\) and \({\Gamma }_{q}\) are the frequency shift and width induced by coupling and material losses23,

$$\delta {\nu }_{{D}_{0},q}= \frac{{c}^{2}}{8{{{{{{\rm{\pi }}}}}}}^{2}{n}_{0}^{2}{\nu }_{{{{{{\rm{c}}}}}}0}}{{\mbox{Re}}}\left({D}_{0}\right)|{\varphi }_{q}\left({z}_{0}\right){|}^{2},\\ {\Gamma }_{q}= \gamma +\frac{{c}^{2}}{8{{{{{{\rm{\pi }}}}}}}^{2}{n}_{0}^{2}{\nu }_{{{{{{\rm{c}}}}}}0}}{{\mbox{Im}}}\left({D}_{0}\right)|{\varphi }_{q}\left({z}_{0}\right){|}^{2}.$$
(5)

In this equation, \({\varphi }_{q}(z)\) are the eigenmodes of Eq. (3) in the absence of source, parametric modulation, coupling effects, and losses (\({A}_{0}=\delta {\nu }_{{{\mbox{par}}}}={D}_{0}=\gamma =0\)). From expression Eq. (5), the Q-factor of the eigenmode \(q\) is calculated as

$${Q}_{q}=\frac{{\nu }_{{{\mbox{in}}}}}{{\Gamma }_{q}}.$$
(6)

The OFC spectrum is determined from the solution of Eq. (3) as follows. First, we calculate the Fourier transform of this solution at the position \({z=z}_{0}\) of the source, \(F(\nu ,{z}_{0},\delta {\nu }_{{{\mbox{par}}}})=\int {{{{{\rm{d}}}}}}t{{\exp }}(-2{{{{{\rm{\pi }}}}}}{{{{{\rm{i}}}}}}\nu t)\Psi ({z}_{0},t)\). Here, for convenience, the modulation amplitude \({\delta \nu }_{{{{{{\rm{par}}}}}}}\) from Eq. (2) is included. Next, we normalise \(F(\nu ,{z}_{0},\delta {\nu }_{{{\mbox{par}}}})\) by relating it to the maximum of the output spectrum at zero modulation amplitude, \({{\max }}\left[{|F}\left(\nu ,{z}_{0},0\right)|\right]\). Then, the output OFC power spectrum is found as

$$P\left(\nu \right)={\left|\frac{F\left(\nu ,{z}_{0},\delta {\nu }_{{{\mbox{par}}}}\right)}{{{\max }}\left[|F\left(\nu ,{z}_{0},0\right)|\right]}\right|}^{2}$$
(7)

Below, we investigate the OFCs generated by the SBM determined by Eqs. (2) and (3) for the cases of adiabatically slow parametric modulation, \({\nu }_{{{\mbox{par}}}}\ll \varDelta {\nu }_{{{\mbox{FSR}}}}\), and resonant parametric modulation, \({\nu }_{{{\mbox{par}}}}=2\varDelta {\nu }_{{{\mbox{FSR}}}}\).

Adiabatic parametric modulation of the SBM

In this section, we consider the adiabatic modulation of the SBM assuming that the frequency \({\nu }_{{{\mbox{par}}}}\) is much smaller than the microresonator FSR, \({\nu }_{{{\mbox{par}}}}\ll \varDelta {\nu }_{{{\mbox{FSR}}}}\), and the OFC resonances are well defined so that their width is much smaller than their separation, i.e., \({\Gamma }_{q}\ll {\nu }_{{{\mbox{par}}}}\). We set the input frequency \({\nu }_{{{\mbox{in}}}}\) equal to one of the SBM eigenfrequencies \({\nu }_{{q}_{0}}\). The simplest parametric modulation corresponds to an axially uniform SDPM with \(\xi (z/L)=1\) and \(\varepsilon \left(z,t\right)=\delta {\nu }_{{{\mbox{par}}}}\cdot {{\sin }}(2{{{{{\rm{\pi }}}}}}{\nu }_{{{\mbox{par}}}}t).\) In this case, the transmission power spectrum \(P\left(\nu \right)\) vanishes at frequencies \(\nu \ne {\nu }_{{q}_{0}}+n{\nu }_{{{\mbox{par}}}}\), \(n=0,\pm 1,\pm 2,\ldots\), and (see Methods)

$$P\left({\nu }_{{q}_{0}}+n{\nu }_{{{\mbox{par}}}}\right)={\left|{J}_{n}\left(\frac{{{{{{\rm{\delta }}}}}}{\nu }_{{{\mbox{par}}}}}{{\nu }_{{{\mbox{par}}}}}\right){J}_{0}\left(\frac{{{{{{\rm{\delta }}}}}}{\nu }_{{{\mbox{par}}}}}{{\nu }_{{{\mbox{par}}}}}\right)\right|}^{2}.$$
(8)

From this equation, the OFC spectrum is formed by a series of equally spaced resonances with magnitudes determined by the dependence of Bessel functions \({J}_{n}(\frac{{{{{{\rm{\delta }}}}}}{\nu }_{{{\mbox{par}}}}}{{\nu }_{{{\mbox{par}}}}})\) on the comb resonance number \(n\). From the asymptotic of the Bessel function for large order and argument, \(\left|n\right|,\frac{\delta {\nu }_{{{\mbox{par}}}}}{{\nu }_{{{\mbox{par}}}}}\gg 1\), we find that the OFC spectrum determined by Eq. (8) is relatively flat within the bandwidth including \(N \sim \frac{2\delta {\nu }_{{{\mbox{par}}}}}{{\nu }_{{{\mbox{par}}}}}\) resonances.

The comparison of the OFC spectrum obtained from the numerical solution of Eq. (3) with that given by Eq. (8) and shown in Fig. 2 demonstrates their excellent agreement. We consider an SBM with maximum CFV \(\Delta {\nu }_{0}=20\) GHz, total length \(2L=284\) \(\mu {{\mbox{m}}}\) and radius \({r}_{0}=20\) \(\mu {{\mbox{m}}}\). The input-output waveguide is placed at the centre of the SBM, \({z}_{0}=0\). The input light frequency is set to \({\nu }_{{{\mbox{in}}}}={\nu }_{{{{{{\rm{c}}}}}}0}=200\) THz which is equal to the SBM eigenfrequency \({\nu }_{{q}_{0}}\) \({{{{{\rm{w}}}}}}\)ith axial quantum number \({q}_{0}=2\). For these parameters, the axial FSR of the SBM determined by Eq. (4) is \(\Delta {\nu }_{{{\mbox{FSR}}}}=3.25\) GHz. We analyse the OFCs generated with two different relatively small modulation amplitudes, \(\delta {\nu }_{{{\mbox{par}}}}=0.1\Delta {\nu }_{0}=2\) GHz and \(\delta {\nu }_{{{\mbox{par}}}}=0.04\Delta {\nu }_{0}=0.75\) GHz, at frequency \({\nu }_{{{\mbox{par}}}}=100\) MHz. From the scaling relation \(\delta {r}_{{{\mbox{par}}}}={r}_{0}{\delta \nu }_{{{\mbox{par}}}}/{\nu }_{{{{{{\rm{c}}}}}}0}\), these amplitudes correspond to the ERV \({\delta r}_{{{\mbox{par}}}}=200\) pm and \({\delta r}_{{{\mbox{par}}}}=75\) pm, respectively, while the total ERV of the SBM considered is \({\Delta r}_{0}={r}_{0}{\Delta \nu }_{0}/{\nu }_{{{{{{\rm{c}}}}}}0}=2\) nm. The material attenuation and coupling coefficient are set to \(\gamma =2{{{{{\rm{\pi }}}}}}\) MHz and \({D}_{0}=0.001\left(1+{{{{{\rm{i}}}}}}\right)\) \(\mu {{{\mbox{m}}}}^{-1}\), which corresponds to \({Q}_{{q}_{0}}={1.2\times 10}^{7}\) calculated from Eqs. (5) and (6).

Fig. 2: Optical frequency combs (OFCs) generated by the spatially uniform adiabatic modulation of a parabolic SNAP bottle microresonator.
figure 2

The blue and red lines correspond to the OFC spectrum obtained from numerical solution of Eq. (3) for constant modulation amplitudes \(\delta {\nu }_{{{\mbox{par}}}}=2\) GHz (blue line) and \(\delta {\nu }_{{{\mbox{par}}}}=0.75\) GHz (orange line), respectively. In both cases, the Q-factor is \({Q}_{{q}_{0}}=1.2\times {10}^{7}\). The blue squares and the orange triangles correspond to the spectrum found from Eq. (8) for \(\delta {\nu }_{{{\mbox{par}}}}=2\) GHz and \(\delta {\nu }_{{{\mbox{par}}}}=0.75\) GHz, respectively.

We now investigate the effect of the SDPM, \(\xi \left(z/L\right)\), on the generation of OFCs. We assume the Gaussian SDPM, \(\xi \left(z/L\right)={{{{{{\rm{e}}}}}}}^{-{\left(z/\sigma L\right)}^{2}}\), where the dimensionless parameter \(\sigma\) determines the ratio of characteristic SDPM width and SBM length \(2L\). Excitation of a broader OFC spectrum at the minimum required power depends on the techniques to generate the parametric modulations. In particular, the required input power may be determined by the maximum amplitude of modulation, its integrated intensity, or by a more complex functional dependence of the modulation parameters on the input power. Here, we suggest that the required power is determined by the maximum amplitude of the SDPM and set the same \({\delta {{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}\) for all the cases considered.

Figure 3 shows the results of our numerical simulations for the parabolic SBM (red line) with the parameters indicated above using Gaussian SDPMs (green line) with different values of \(\sigma\). We present the OFC spectra of the signal obtained from Eq. (3) for two modulation amplitude maxima \({\delta {{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}=0.1\Delta {\nu }_{0}=2\) GHz and \({\delta {{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}=0.04\Delta {\nu }_{0}=0.75\) GHz, both much smaller that the maximum CFV of the SBM \(\Delta {\nu }_{0}=20\) GHz. In the case \(\sigma =0.02\), the Gaussian SDPM is localized near the centre of the SBM as shown in Fig. 3a and the OFC spectrum exhibits only a few comb resonances. The spectrum of the frequency combs becomes wider and flater with growing \(\sigma\) (Fig. 3b–e). In the case \(\sigma =10\) shown in Fig. 3f, the parametric modulation has a close to the uniform SDPM over the whole length of the SBM and the spectrum coincides with that obtained from Eq. (8) and illustrated in Fig. 2.

Fig. 3: Optical frequency combs (OFCs) generated by the adiabatic modulation of the parabolic SNAP bottle microresonator (SBM) with Gaussian spatial distribution of the modulations (SDPMs).
figure 3

The SDPM (green line) is defined by \(\xi \left(z/L\right)={e}^{-{\left(z/\sigma \cdot L\right)}^{2}}\), where \(\sigma\) is the dimensionless SDPM width and \(L\) is half the length of the SBM (red line). Rows show the OFC generated with different values of \(\sigma\) for two modulation amplitude maxima: \({\delta \nu }_{{{\mbox{par}}}}=0.1\varDelta {\nu }_{0}=2\) GHz and \({\delta \nu }_{{{\mbox{par}}}}=0.04\varDelta {\nu }_{0}=0.75\) GHz. a A limited number of comb resonances are formed at \(\sigma =0.02\) for both values of \({\delta \nu }_{{{\mbox{par}}}}\), when the SDPM is strongly localized near the centre of the SBM. be The OFC spectrum becomes wider and flatter with growing \(\sigma\). For \({\delta \nu }_{{{\mbox{par}}}}=0.75\) GHz, the OFC has a smaller bandwidth, though the power level is ~ 10 dB higher than in the case of \({\delta \nu }_{{{\mbox{par}}}}=2\) GHz. f The generated OFC achieves its optimal shape for a close to uniform SDPM, when \(\sigma =10\).

So far, we considered the modulation of the SBM with maximum amplitudes \({\delta \nu }_{{{\mbox{par}}}}\) much smaller than the full CFV \(\triangle {\nu }_{0}=20\) GHz of the resonator. However, the dramatically small CFV and corresponding ERV of the SBM (\({\varDelta r}_{0}=2\) nm in the cases considered) makes the modulations comparable or larger than the resonator CFV realistic as well. As an example, we modulate the same SBM with a uniform SDPM and maximum amplitude \({\delta \nu }_{{{\mbox{par}}}}=2\triangle {\nu }_{0}=40\) GHz (Fig. 4a). The corresponding ERV modulation amplitude is 4 nm (compare it with the 8 nm ERV oscillation amplitude experimentally achieved by Hu et al.22 and discussed in the introduction). It is seen from Fig. 4b that, as expected, the generated OFC spectrum expands over the bandwidth \(5\triangle {\nu }_{0}\) (from \({\nu }_{{q}_{0}}-2.5\triangle {\nu }_{0}\) to \({\nu }_{{q}_{0}}+2.5\triangle {\nu }_{0}\)). The number of OFC resonances within the central most uniform part of this spectrum, from −40 to 40 GHz, is well estimated by \(N \sim 2{\delta \nu }_{{{\mbox{par}}}}/{\nu }_{{{\mbox{par}}}}={4\triangle \nu }_{0}/{\nu }_{{{\mbox{par}}}}=\) 800, followed from the asymptotic of Eq. (8) discussed above, while the full bandwidth includes close to \({5\triangle \nu }_{0}/{\nu }_{{{\mbox{par}}}}=\)1000 OFC resonances.

Fig. 4: Optical frequency comb (OFC) generated by a uniform adiabatic modulation with amplitude exciding the bandwidth of the SNAP bottle microresonator (SBM).
figure 4

a Adiabatic modulation of the parabolic SBM (red line) with a uniform spatial distribution \(\xi \left(z/L\right)\) (green line). The maximum amplitude of the modulation is \({\delta \nu }_{{{\mbox{par}}}}=2\triangle {\nu }_{0}=40\) GHz. b The generated OFC has a bandwidth five times larger than that of the SBM, \({\triangle \nu }_{0}=20\) GHz. The number of comb lines generated are ~1000.

Resonant parametric modulation of the SBM

As in the previous section, we place the input light source in the middle of the SBM (\({z}_{0}=0\)) so that the excited eigenmodes have even parity. Then, due to the symmetry of the system, transitions between eigenmodes of different parity are suppressed and the parametric modulation frequency has to be \({{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}=2\Delta {\nu }_{{{\mbox{FSR}}}}\) to achieve an effective resonant excitation of the SBM eigenmodes. In this case, the SBM has the maximum CFV \(\Delta {\nu }_{0}=5\) GHz, the total length \(2L=9.18\) mm, and radius \({r}_{0}=80\) \(\mu {{\mbox{m}}}\). The axial FSR of this SBM is \(\Delta {\nu }_{{{\mbox{FSR}}}}=50\) MHz, so the resonant condition takes place at \({{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}=2\Delta {\nu }_{{{\mbox{FSR}}}}=100\) MHz. The input light frequency is set to \({{{{{{\rm{\nu }}}}}}}_{{{\mbox{in}}}}={\nu }_{{q}_{0}}=200\) THz, where the axial mode number is \({q}_{0}=50\). From the scaling relation \({\Delta r}_{0}={r}_{0}{\Delta \nu }_{0}/{\nu }_{{q}_{0}}\), the ERV of the SBM considered is \(\Delta {r}_{0}=2\) nm, as in the adiabatic case. The attenuation factor and coupling coefficient are set to \(\gamma =2{{{{{\rm{\pi }}}}}}\) MHz and \({D}_{0}=0.001\left(1+{{{{{\rm{i}}}}}}\right)\) \(\mu {{{\mbox{m}}}}^{-1}\). The corresponding Q-factor of mode \({q}_{0}\) calculated from Eqs. (5) and (6) is \({Q}_{{q}_{0}}=3\times {10}^{7}\).

The results of our numerical modelling of Eq. (3) are shown in Fig. 5. As in the adiabatic case, the OFCs are generated by the Gaussian SDPM determined by \(\xi \left(z/L\right)={{{{{{\rm{e}}}}}}}^{-{\left(z/\sigma L\right)}^{2}}\) with dimensionless widths \(\sigma\) varying from 0.02 to 10. We consider two modulation amplitude maxima, \(\delta {{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}=0.04\Delta {\nu }_{0}=200\) MHz and \(\delta {{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}=0.015\Delta {\nu }_{0}=75\) MHz, which are much smaller than the maximum CFV of the SBM \(\Delta {\nu }_{0}\). From the scaling relation \(\delta {r}_{{{\mbox{par}}}}={r}_{0}{\delta {{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}/{\nu }_{c0}\), these amplitudes correspond to the ERV \(\delta {r}_{{{\mbox{par}}}}=80\) pm and \(\delta {r}_{{{\mbox{par}}}}=30\) pm, respectively. Contrary to the adiabatic case, the OFC spectral bandwidth behaves nonmonotonically as a function of \(\sigma\). It is seen from Fig. 5 that, with increasing the SDPM width from very narrow at \(\sigma =0.02\) (Fig. 5a), the OFC spectral bandwidth first becomes wider (Fig. 5b, c), reaches the optimum profile at around \(\sigma =0.3\) (Fig. 5d), and then shrinks down (Fig. 5e) and vanishes for the practically uniform SDPM at \(\sigma =10\) (Fig. 5f). Vanishing of the OFC spectral bandwidth for narrow SDPM (Fig. 5a) is straightforwardly explained by the reduction of integrated modulation power. The less obvious suppression of the OFC generation for the SDPM approaching the uniform distribution (Fig. 5f) can be clarified with the help of the quantum and semiclassical perturbation theory. Indeed, the transition between SBM axial modes with different quantum numbers \(q\) caused by the spatially uniform perturbation is strongly suppressed due to the orthogonality of these modes. For example, in the first order of the quantum perturbation theory, the transition from mode \({\varphi }_{{q}_{1}}(z){{\exp }}({{{{{\rm{i}}}}}}{\nu }_{{q}_{1}}t)\) to mode \({\varphi }_{{q}_{2}}(z){{\exp }}({{{{{\rm{i}}}}}}{\nu }_{{q}_{2}}t)\) under the perturbation \(W(z){{\sin }}(2{{{{{\rm{\pi }}}}}}{{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}t)\) (see Eq. (2)) is proportional to \(\langle {\varphi }_{{q}_{1}}|W|{\varphi }_{{q}_{2}}\rangle\)36. For the constant perturbation, \(W\left(z\right)={W}_{0}\), we have \(\langle {\varphi }_{{q}_{1}}|W|{\varphi }_{{q}_{2}}\rangle ={W}_{0}\langle {\varphi }_{{q}_{1}}|{\varphi }_{{q}_{2}}\rangle =0\) due to the orthogonality of \({\varphi }_{{q}_{1}}\) and \({\varphi }_{{q}_{2}}\).

Fig. 5: Optical frequency combs (OFCs) generated with a Gaussian spatial distribution of the modulations (SDPMs) in the resonant regime.
figure 5

The parabolic SNAP bottle microresonator (SBM) (red line) is modulated with a Gaussian SDPM \(\xi \left(z/L\right)={e}^{-{\left(z/\sigma L\right)}^{2}}\) (green line) for two maximum modulation amplitudes, \(\delta {\nu }_{{{\mbox{par}}}}=200\) MHz and \(\delta {\nu }_{{{\mbox{par}}}}=75\) MHz, and for different values of \(\sigma\). a A strongly localised modulation (\(\sigma =0.02\)) leads to the formation of a narrow bandwidth OFC spectrum with a few comb resonances. b, c The bandwidth of the frequency comb spectrum increases with the width \(\sigma\) of the SDPMs. d The optimal spectrum with largest bandwidth is achieved at \(\sigma \sim 0.3\). e The bandwidth of the OFC starts to decrease when the value of \(\sigma\) is further increased. This case corresponds to a parabolic SDPM since \(\xi \left(z/L\right)={e}^{-{\left(z/\sigma L\right)}^{2}}\approx {\left(z/L\right)}^{2}\) for \(\sigma =1\). f The generation of combs is strongly suppressed in the case of a uniform SDPM.

Similar to the previous section, in addition to the cases of relatively small modulation amplitudes \(\delta {\nu }_{{{\mbox{par}}}}\ll \triangle {\nu }_{0}\), it is interesting to investigate the case when \(\delta {\nu }_{{{\mbox{par}}}}\) is comparable to the full CFV \(\triangle {\nu }_{0}.\) As an example, Fig. 6 compares the OFC spectra generated by the same SBM modulated with maximum amplitude \(\delta {\nu }_{{{\mbox{par}}}}=2\triangle {\nu }_{0}=10\) GHz and Gaussian profile \(\xi (z/L)={e}^{-{(z/\sigma \cdot L)}^{2}}\). Figure 6a shows the case with \(\sigma =0.3\) (optimal for the relatively weak modulation, see Fig. 5d) and Fig. 6b shows the case of close to uniform SDPM with \(\sigma =10\). It is seen that, contrary to the case of relatively small modulation amplitude, the uniform modulation generates an OFC which is, in general, comparable with that for \(\sigma =0.3\). However, the OFC spectrum for \(\sigma =0.3\) is closer to uniform and on average has a greater power over the larger bandwidth. We suggest that an OFC profile with better flatness and larger bandwidth can be achieved with an optimization using a more complex SDPM. The detailed solution of this problem and the investigation of effects induced by larger vibrations of SBMs, such as change of their temperature and other physical characteristics37, is beyond the scope of this paper.

Fig. 6: Optical frequency combs (OFCs) generated by parametric modulations with Gaussian spatial distributions and amplitude exciding the bandwidth of the SNAP bottle microresonator (SBM).
figure 6

a The parabolic SNAP bottle microresonator (SBM) (red line) is parametrically modulated with a Gaussian modulation profile \(\xi \left(z/L\right)={e}^{-{\left(z/\sigma L\right)}^{2}}\) (green line) with \(\sigma =0.3\). The amplitude of the modulation exceeds the bandwidth of the SBM, \(\delta {\nu }_{{{\mbox{par}}}}=2\triangle {\nu }_{0}=\)10 GHz. The OFC spectrum generated is several times larger than the SBM bandwidth \(\triangle {\nu }_{0}=5\) GHz. b A close to uniform modulation (\(\sigma =10\)) is used to parametrically modulate the parabolic SBM. The bandwidth of the resulting OFC spectrum is several times wider than the SBM bandwidth. Unlike the case of small modulation amplitudes, the frequency comb spectrum generated is comparable to the case \(\sigma =0.3\).

Conclusion

Parametric modulation of optical microresonators is a promising approach for the generation of optical frequency combs (OFCs) which has been demonstrated in spherical15, ring (racetrack)19,20, and toroidal15,18 microresonators. The repetition rate of the OFC excited by these resonators is inverse proportional to their perimeter so that the OFCs with the characteristic repetition rate smaller than a gigahertz correspond to resonators with macroscopically large dimensions (see Introduction for details). However, it was proposed in refs. 33,34,38 that a SNAP bottle microresonator (SBM) with much smaller dimensions can be used to generate OFCs with repetition rate below a gigahertz.

In this paper, we consider the generation of OFCs in a parabolic SBM by the modulation of its parameters. The main difference of an SBM compared to spherical and toroidal microresonators consists in its strong axial elongation. For this reason, the spatial distribution of the parametric modulation (SDPM) along the SBM axis becomes of major importance. Furthermore, the characteristic parametric modulation of the effective radius variation (ERV), \({{{{{\rm{\delta }}}}}}{r}_{{{\mbox{par}}}}\), can be comparable or larger than the dramatically small nanometre-scale ERV \(\triangle {r}_{0}\) of the SBMs considered. In our numerical simulations, we investigate the effect of the SDPM assuming that it has a Gaussian shape. In the adiabatic case, when the modulation frequency is much smaller than the axial free spectral range (FSR), \({{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}\ll \Delta {{{{{{\rm{\nu }}}}}}}_{{{\mbox{FSR}}}}\), we show that a uniform SDPM produces the OFC with the best flatness and largest spectral bandwidth (Fig. 3f). An adiabatic SDPM with uniform amplitude \({{{{{\rm{\delta }}}}}}{r}_{{{{{{\rm{par}}}}}}}\ge \triangle {r}_{0}\) generates a relatively flat OFC with bandwidth proportional to this amplitude (Fig. 4). For resonant modulation, \({{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}=2\Delta {{{{{{\rm{\nu }}}}}}}_{{{\mbox{FSR}}}}\), with a relatively small amplitude, \({{{{{\rm{\delta }}}}}}{r}_{{{{{{\rm{par}}}}}}}\ll \triangle {r}_{0}\), the OFC disappears both for very narrow and uniform SDPM as shown in Fig. 5a, f. However, the OFC can achieve an optimal shape with the best flatness and largest bandwidth at an intermediate SDPM width, as depicted in Fig. 5d. Our study only includes symmetric SDPMs. However, the behaviour of asymmetric SDPMs may differ significantly, particularly in the resonant case, as the spatial coupling of modes would be altered. For instance, a spatially antisymmetric modulation could couple modes with different symmetry. More complex SDPM profiles may, therefore, lead to changes in the process of OFC generation that are non-intuitive and difficult to predict. For larger modulation amplitudes, \(\delta {r}_{{par}}\ge \triangle {r}_{0}\), the resonant modulation with a nonuniform SDPM results in a flater OFC with a broader bandwidth compared to that generated by the uniform SDPM (Fig. 6). We suggest that an SDPM with a more complex dependence on the axial coordinate, which takes into account the actual axial distribution of the SBM modes, may lead to a better OFC optimization.

The optimized generation of OFCs by an SBM proposed in this paper can be realized experimentally by one of the following approaches. First, modulation of the refractive index and radiation pressure of an SBM can be induced by the input resonant pump light harmonically modulated with frequency \({{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}\). To ensure the resonant enhancement of the pump light, its spectrum should be localized near an SBM cutoff frequency \({\nu }_{{{{{{\rm{c}}}}}}p}\). While the case when the pump light induces the OFC directly (i.e., when \({\nu }_{{{{{{\rm{c}}}}}}p}={\nu }_{{{{{{\rm{c}}}}}}0}\), see Eq. (2)) is of special interest, we assumed in this paper that the cutoff frequencies \({\nu }_{{{{{{\rm{c}}}}}}p}\) and \({\nu }_{{{{{{\rm{c}}}}}}0}\) (as well as the spectrum of the pump light and the frequency \({{{{{{\rm{\nu }}}}}}}_{{{\mbox{in}}}}\) of a relatively weak input light) are separated so that the spectrum of pump light and the spectrum of the generated OFC do not overlap. The required spatial distribution of the pump light can be achieved by adjusting its spectrum. Second, the adiabatic excitation of the OFCs can be performed in a way similar to that demonstrated by Hu et al.22 by excitation of acoustic modes of an SBM with a strong pump light. Third, assuming that the ultraprecise fabrication of an SBM of material with strong second-order nonlinearities may become possible, the refractive index modulation can be performed by applying a periodic electric field to the SBM fabricated of such material18. Finally, an SBM can be fabricated at the surface of a microcapillary filled with such highly nonlinear or piezoelectric material, which can be used to mechanically modulate the SBM parameters39.

Overall, complementary to the spherical, ring, and toroidal microresonators, the elongated SBMs allows us to change the SDPM and optimise the OFC spectrum. In the case of adiabatic parametric modulation, it is possible to tune the repetition rate of the OFC without changing the SBM profile by modifying the modulation frequency \({{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}\). In the case of the resonant parametric modulation, the repetition rate frequency is determined and fixed by the FSR of the SBM, which can be made much smaller that the FSR of other microresonators with similar dimensions. In both cases, for relatively weak modulation amplitude, the comb bandwidth is limited by the bandwidth of discrete eigenfrequencies of the SBM considered.

Methods

Generation of OFCs by a spatially uniform parametric modulation

We consider a cutoff frequency that depends on time and position as \({\nu }_{{{{{{\rm{c}}}}}}}(z,t)={\nu }_{{{{{{\rm{c}}}}}}0}+\Delta \nu (z)+\delta {{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}\cdot {{\sin }}(2{{{{{\rm{\pi }}}}}}{{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}t)\), where \(|\Delta \nu (z)|\ll {\nu }_{{{{{{\rm{c}}}}}}0}\) and \(\delta {{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}={const}\). After the substitution

$$\Psi \left(z,t\right)=\psi \left(z,t\right){{{{{{\rm{e}}}}}}}^{\frac{{{{{{\rm{i}}}}}}\delta {{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}}{{{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}}\cos \left(2{{{{{\rm{\pi }}}}}}{{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}t\right)},$$
(9)

Equation (3) can be written as

$$\left[{{{{{{\rm{i}}}}}}\frac{1}{2{{{{{\rm{\pi }}}}}}}\partial }_{t}-\hat{L}\right]\psi (z,t)=S\left(z,t\right)$$
(10a)
$$\hat{L}=-\frac{1}{2}\kappa {\partial }_{z}^{2}+\left({\nu }_{{{{{{\rm{c}}}}}}0}-{{{{{{\rm{\nu }}}}}}}_{{{\mbox{in}}}}+\Delta \nu \left(z\right)-{{{{{\rm{i}}}}}}\gamma \right)-\kappa {D}_{0}\delta \left(z-{z}_{0}\right)$$
(10b)
$$S\left(z,t\right)={\nu }_{{{{{{\rm{c}}}}}}0}{A}_{{{\mbox{in}}}}\left(z,t\right){{{{{{\rm{e}}}}}}}^{-{{{{{\rm{i}}}}}}\frac{\delta {{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}}{{{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}}{{\cos }}\left(2{{{{{\rm{\pi }}}}}}{{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}t\right)}$$
(10c)

where \(\kappa ={\nu }_{{{{{{\rm{c}}}}}}0}/{\beta }_{{{{{{\rm{c}}}}}}0}^{2}\). For the parabolic CFV, \(\Delta \nu \left(z\right)=\Delta {\nu }_{0}[{(\frac{z}{L})}^{2}-1]\), we search for solution of Eq. (10a) in the form

$$\psi \left(z,t{{{{{\rm{;}}}}}}{z}_{0}\right)=\mathop{\sum }\limits_{q}{\varphi }_{q}\left(z\right){\varphi }_{q}^{* }\left({z}_{0}\right){a}_{q}\left(t\right)$$
(11)

where \({a}_{q}\left(t\right)\) are functions of time to be determined and \({\varphi }_{q}\left(z\right)\) are the normalised eigenfunctions of the operator \(\hat{L}\) with associated eigenfrequencies \({\varsigma }_{q}\). For finite material loss \(\gamma\) and coupling \({D}_{0}\) considered, the eigenfrequencies are complex-valued and equal to \({\varsigma }_{q}={\nu }_{q}-{{{{{{\rm{\nu }}}}}}}_{{{\mbox{in}}}}-{{{{{\rm{i}}}}}}{\Gamma }_{q}\) (see Eqs. (4) and (5)). Nevertheless, due to the relatively small values of \(\gamma\) and \({{{{{\rm{Im}}}}}}({D}_{0})\) assumed, the normalization of \({\varphi }_{q}\left(z\right)\) and their orthogonality can be achieved with the required accuracy.

In order to determine \({a}_{q}\left(t\right)\), we substitute Eq. (11) into Eq. (10a),

$$\mathop{\sum }\limits_{q}{\varphi }_{q}\left(z\right){\varphi }_{q}^{* }\left({z}_{0}\right)\left[{{{{{\rm{i}}}}}}\frac{1}{2{{{{{\rm{\pi }}}}}}}\frac{{{{{{\rm{d}}}}}}}{{{{{{\rm{d}}}}}}t}{a}_{q}\left(t\right)-{\varsigma }_{q}{a}_{q}\left(t\right)\right]=S\left(z,t\right),$$
(12)

multiply each side of Eq. (12) by \({\varphi }_{{q{{\hbox{'}}}}}^{* }\left(z\right)\) and integrate over \(z\). As a result, we arrive at the following equations for \({a}_{q}\left(t\right)\):

$$\frac{1}{2{{{{{\rm{\pi }}}}}}}\frac{{{{{{\rm{d}}}}}}}{{{{{{\rm{d}}}}}}t}{a}_{q}\left(t\right)+{{{{{\rm{i}}}}}}{\varsigma }_{q}{a}_{q}\left(t\right)=-{{{{{\rm{i}}}}}}{\nu }_{{{{{{\rm{c}}}}}}0}{A}_{0}\left(1-{{{{{{\rm{e}}}}}}}^{-\alpha t}\right){{{{{{\rm{e}}}}}}}^{-{{{{{\rm{i}}}}}}\frac{\delta {{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}}{{{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}}{{\cos }}\left(2{{{{{\rm{\pi }}}}}}{{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}t\right)}$$
(13)

with initial conditions \({a}_{q}\left(t=0\right)={A}_{{{\mbox{in}}}}\left(z,0\right)=0\). Solving these equations and application of the Jacobi-Anger expansion40, results in the following solution of Eq. (3) at \(z={z}_{0}\):

$$\Psi \left({z}_{0},t\right)=\, {\nu }_{{{{{{\rm{c}}}}}}0}{A}_{0}\mathop{\sum }\limits_{q}{\left|{\varphi }_{q}\left({z}_{0}\right)\right|}^{2}\mathop{\sum }\limits_{n}\mathop{\sum }\limits_{m}{\left(-1\right)}^{m}{{{{{{\rm{i}}}}}}}^{n}{J}_{n-m}\left(\frac{\delta {{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}}{{{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}}\right)\\ {J}_{m}\left(\frac{\delta {{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}}{{{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}}\right)\frac{{e}^{2{{{{{\rm{\pi }}}}}}{{{{{\rm{i}}}}}}n{{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}t}}{\left({\nu }_{q}-{{{{{{\rm{\nu }}}}}}}_{{{\mbox{in}}}}+m{{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}\right)-{{{{{\rm{i}}}}}}{\Gamma }_{q}}.$$
(14)

We assume now that the input frequency \({{{{{{\rm{\nu }}}}}}}_{{{\mbox{in}}}}\) is situated in a small vicinity of one of the SBM frequencies \({\nu }_{{q}_{0}}\) of the order of the resonance width, \(|{{{{{{\rm{\nu }}}}}}}_{{{\mbox{in}}}}-{\nu }_{{q}_{0}}| \sim {\Gamma }_{{q}_{0}}\), and the excited OFC resonances are well defined so that \({\Gamma }_{{q}_{0}}\ll {{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}\). In addition, we assume that none of the SBM frequencies \({\nu }_{q}\), except for \({\nu }_{{q}_{0}}\), are close to the parametrically modulated OFC resonances. Such situation may occur for adiabatic modulation, when \({{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}\ll \Delta {{{{{{\rm{\nu }}}}}}}_{{{\mbox{FSR}}}}\). Then, Eq. (14) is reduced to

$$\Psi \left({z}_{0},t\right)=\mathop{\sum }\limits _{n}{\Psi }_{n}{e}^{2{{{{{\rm{\pi }}}}}}{{{{{\rm{i}}}}}}n{{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}t},$$
(15)

where

$${\Psi }_{n}={{{{{{\rm{i}}}}}}}^{n+1}\frac{{\nu }_{{{{{{\rm{c}}}}}}0}}{{\Gamma }_{{q}_{0}}}{A}_{0}{\left|{\varphi }_{{q}_{0}}\left({z}_{0}\right)\right|}^{2}{J}_{n}\left(\frac{\delta {{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}}{{{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}}\right){J}_{0}\left(\frac{\delta {{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}}{{{{{{{\rm{\nu }}}}}}}_{{{\mbox{par}}}}}\right).$$
(16)

In particular, at zero modulation, we have:

$$\Psi \left({z}_{0},t\right)={\Psi }_{0}={{{{{\rm{i}}}}}}\frac{{\nu }_{{{{{{\rm{c}}}}}}0}}{{\Gamma }_{{q}_{0}}}{A}_{0}{\left|{{{{{{\rm{\varphi }}}}}}}_{{q}_{0}}\left({z}_{0}\right)\right|}^{2}.$$
(17)

Substitution of Eqs. (16), (17) into Eq. (7) yields Eq. (8).

Numerical solution of the Schrödinger equation

The Schrödinger equation is solved using the Fourier split-step method41, which can be briefly described as follows. We first rewrite Eq. (3) as ref. 42:

$${{{{{\rm{i}}}}}}{\partial }_{\tau }\hat{\Psi }=\, -\frac{1}{2}{\partial }_{x}^{2}\hat{\Psi }+\frac{1}{2}{V}_{0}\left[{x}^{2}+\delta {\hat{{{{{{\rm{\nu }}}}}}}}_{{{\mbox{par}}}}{x}_{c}^{2}\xi \left(x/{x}_{c}\right)\right]\hat{\Psi }+\Delta \hat{\nu }\hat{\Psi }-{{{{{\rm{i}}}}}}\hat{\gamma }\hat{\Psi }\\ -{\hat{D}}_{0}\delta (x-{x}_{0})\hat{\Psi }+{\hat{A}}_{{{\mbox{in}}}}(x,\tau )$$
(18)

In this equation, we have introduced the dimensionless variables \(\tau =t/{T}_{0}\), \(x=z/{L}_{0}\) and \(\hat{\Psi }=\Psi /{I}_{0}\), where \({T}_{0}\), \({L}_{0}\) and \({I}_{0}\) are the scaling factors. The dimensionless parameters in Eq. (18) are defined as \({V}_{0}=8{{{{{{\rm{\pi }}}}}}}^{2}\Delta {\nu }_{0}{\nu }_{{{{{{\rm{c}}}}}}0}{n}_{0}^{2}{L}_{0}^{4}/{c}^{2}{L}^{2}\), \(\delta {\hat{{{{{{\rm{\nu }}}}}}}}_{{{\mbox{par}}}}=\delta {\nu }_{{{\mbox{par}}}}/\Delta {\nu }_{0}\), \({x}_{c}=L/{L}_{0}\), \(\Delta \hat{\nu }=2{{{{{\rm{\pi }}}}}}({\Delta \nu }_{0}+{\nu }_{c0}-{\nu }_{{{\mbox{par}}}}){T}_{0}\), \(\hat{\gamma }=2{{{{{\rm{\pi }}}}}}\gamma {T}_{0}\), \({\hat{D}}_{0}={L}_{0}{D}_{0}\) and \({\hat{A}}_{{{\mbox{in}}}}(x,\tau )=2{{{{{\rm{\pi }}}}}}{\nu }_{{{{{{\rm{c}}}}}}0}{T}_{0}/{L}_{0}{I}_{0}\cdot {A}_{{{\mbox{in}}}}(x,\tau )\).

We can split the r.h.s. of Eq. (18) into the momentum space component

$$\hat{{{{{{\rm{D}}}}}}}=-\frac{1}{2}{\partial }_{x}^{2}$$
(19)

and the position space component

$$\hat{{{{{{\rm{N}}}}}}}=\frac{1}{2}{V}_{0}\left[{x}^{2}+\delta {\hat{\nu }}_{{{\mbox{par}}}}{x}_{{{{{{\rm{c}}}}}}}^{2}\xi \left(x/{x}_{{{{{{\rm{c}}}}}}}\right)\right]+\Delta \hat{\nu }-{{{{{\rm{i}}}}}}\hat{\gamma }-{\hat{D}}_{0}\delta (x-{x}_{0})$$
(20)

and rewrite Eq. (18) in terms of these two operators as:

$${{{{{\rm{i}}}}}}{\partial }_{\tau }\hat{\Psi }=\left[\hat{{{{{{\rm{D}}}}}}}+\hat{{{{{{\rm{N}}}}}}}\right]\hat{\Psi }+{\hat{A}}_{{{\mbox{in}}}}(x,\tau )$$
(21)

For a small time step \(\Delta \tau\), the solution of this equation can be approximated as

$$\hat{\Psi }(\tau +\Delta \tau )={{{{{{\rm{e}}}}}}}^{-{{{{{\rm{i}}}}}}\Delta \tau \hat{{{{{{\rm{D}}}}}}}}{{{{{{\rm{e}}}}}}}^{-{{{{{\rm{i}}}}}}\Delta \tau \hat{{{{{{\rm{N}}}}}}}}\hat{\Psi }(\tau )-{{{{{\rm{i}}}}}}\Delta \tau {\hat{A}}_{{{\mbox{in}}}}(x,\tau )$$
(22)

with an error of the order of \({\Delta \tau }^{2}\). To compute the term associated to the operator \(\hat{D}\), we use the spatial fast Fourier transform (FFT) of the field \(\hat{\Psi }\):

$$\hat{{{{{{\rm{D}}}}}}}\hat{\Psi }={{{{{{\rm{FFT}}}}}}}^{-1}\left\{{{{{{{\rm{e}}}}}}}^{-\frac{{{{{{\rm{i}}}}}}}{2}{k}^{2}}{{{{{\rm{FFT}}}}}}\left(\hat{\Psi }\right)\right\}$$
(23)

where \(k\) represents the coordinate in the reciprocal space. We solve Eq. (22) with a uniform space grid of \(N={2}^{n}\) points defined as \({x}_{n}=n\Delta x\), with \(n\in [-N/2,N/2-1]\). The points in the reciprocal space are defined as \({k}_{n}=2\pi n/\Delta {xN}\).