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Abstract
We demonstrate the suppression of cavity optomechanical parametric oscillations with the generation of a Turing pattern comb (Kerr comb) in an anomalous dispersion toroidal microcavity. If the pump light alone influences the optomechanical oscillation, it should be amplified because the strong pump light is blue-detuned from the cavity resonance. However, once the Kerr comb was generated, all the comb lines contributed to the cavity optomechanical behavior and the optomechanical oscillation was suppressed. A calculation taking account of the effective damping rate from all the comb lines is in good agreement with the experimental observation.
© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
A Kerr comb in a microcavity has been of great interest due to its ability to generate multi-wavelength light from a weak continuous wave (CW) laser light [1, 2]. When CW laser light is coupled to an optical microcavity with a high quality factor (Q) and a small mode volume, cascaded four-wave mixing (FWM) occurs inside the cavity. This approach allows us to obtain frequency combs with large mode spacings that have a low driving power, small size, and low fabrication cost. These features are attractive for applications such as astronomy, spectroscopy, arbitrary waveform generation, optical clocks, and telecommunications. The mechanism of Kerr comb formation in a microcavity has been studied both numerically and experimentally. Many earlier studies have employed a numerical model with a Lugiato-Lefever equation (LLE) [3–6]. The first experimental demonstration of a Kerr comb was reported in the infrared regime with equally spaced comb broadening of over 500 nm [1]. Recent progress on this topic includes comb generation with pulse shaping [7], revealing the dynamics of comb formation [8,9], soliton pulse generation [10], a coherent microwave-to-optical link [11], and mode coupling assisted comb generation in the normal dispersion regime [12].
A silica toroid microcavity has a high Q and a small mode volume, which is an excellent platform on which to demonstrate photonics applications such as biochemical sensing, optical signal processing, cavity optomechanics, and Kerr combs [13–17]. In particular, toroid microcavities exhibit a strong cavity optomechanical effect [18–23]. Cavity optomechanics is a phenomenon that involves an interaction between light and mechanical motion [24–27]. A high Q toroid microcavity pumped with a CW light leads to a strong circulating power inside a cavity. The high optical density causes radiation pressure, which excites the mechanical oscillation modes of the microcavity structure leading to cavity optomechanical parametric oscillations (OMPOs) [18,19]. In many previous studies of cavity optomechanics using a toroid microcavity, the oscillation modes were excited with a low pump power that was much smaller than the threshold power for FWM. Although the cavity optomechanical behavior in comb generation has been reported [20], it has not been well understood. Here we try to understand the cavity optomechanical behavior when a Kerr comb is generated in a toroid microcavity. The strong pump light generates a Kerr comb inside the cavity, which leads to multi-optomechanical coupling with the cavity resonances.
Figure 1 shows the amplification and damping mechanisms caused by symmetric optomechanical sidebands. This illustration includes two new photon generation processes. The first is optomechanical oscillation, which generates Doppler-shifted photons at blue and red sidebands. When pumping with blue detuning (the pump has a higher frequency than the resonance), the cavity resonance suppresses the blue sideband and enhances the red sideband. In this case, the optomechanical oscillation is enhanced. On the other hand, the optomechanical oscillation is suppressed when pumping with red detuning (the pump has a lower frequency than the resonance). The second process is FWM, which generates photons in other resonances with equal mode spacings. Pump scanning from a high to a low frequency is employed for FWM generation because the resonance frequency is shifted by thermal and Kerr effects. In this research, we generated Turing pattern combs (also known as primary combs) [28–30] whose pump line was always on the blue-detuned side [9,10]. Therefore, in theory, the microcavity structure should oscillate if we consider only the influence of the strong pump light. However, the key is that the generated comb lines also influence the optomechanical amplification and damping. While generating a Kerr comb (Turing pattern comb), the suppression of OMPO is both expected and observed as we show in the next section.
Fig. 1 Amplification and damping mechanisms in an optomechanical coupled system. Optomechanical oscillation is suppressed (amplified) when the light is coupled to the resonance with red detuning (blue detuning). When a Turing pattern comb is generated, the pump line is always at blue detuning and the generated comb lines are at red detuning. Since the pump line and all the generated comb lines contribute to the cavity optomechanical behavior, the system may exhibit amplification and suppression depending on the parameters.
2. Experiment
Figure 2(a) shows our experimental setup. Amplified CW light, whose polarization was aligned with a polarization controller, was coupled to a silica toroid microcavity using a tapered fiber. The output light was monitored using an optical spectrum analyzer and two photodetectors connected to an oscilloscope and an electrical spectrum analyzer. In addition, a weak probe laser was launched into a microcavity in the counter direction to measure the detuning values between the resonance and generated comb line frequencies while generating a Kerr comb. Figure 2(b) shows a resonance mode with a beat signal between the probe laser and the comb line. The measured detuning was calibrated using a fiber Mach-Zehnder interferometer with a free spectrum range (FSR) of 19.7 MHz. Table 1 summarizes the parameters of the microcavity structure and the mechanical properties. The minor radius is that of the cross-section of a toroidal structure. Ω/2π is the mechanical oscillation frequency. The effective mass (meff) was calculated by using a finite element simulation, whose mechanical mode is shown in Fig. 1(a) (inset). The calculated mechanical frequency is in good agreement with the measurement (Ω/2π = 40 MHz). In the experiment, a tapered fiber was brought into contact with the microcavity surface to realize a stable coupling condition that reduced the mechanical Q from 130 to 100 [31] (The calculation of mechanical modes did not take the fiber into account). The optical intrinsic Q (Qi) of the modes used for the comb generation was about 1 × 107. Figure 1(c) shows the dispersion of the mode family used for the comb generation, which was obtained from the FSR distances. The cold resonant frequencies are given with relative mode number μ (the pump mode corresponds to μ = 0) as [8]
where ω0/2π is the pump resonant frequency and D1/2π is the cavity FSR. D2 is the second-order dispersion, which is given as D2 = −cβ2D12/n, where c, β2, and n are the speed of light, group velocity dispersion, and refractive index, respectively. The dispersion of the mode family used for the comb generation is anomalous; the values are D1/2π = 560.5 GHz and D2/2π ≈ 17 MHz. The higher order terms are negligible.
Table 1. Parameters of the toroid microcavity used in this experiment
Fig. 2 (a) Experimental setup for comb generation and measurement of detuning between resonance and comb line frequencies. During Kerr comb generation, a weak probe laser was swept through the resonance in the counter direction to the pump laser. The inset shows a mechanical mode excited in this experiment and whose frequency is calculated as 40 MHz by using a finite element simulation. TLD: tunable laser diode, EDFA: erbium doped fiber amplifier, FPC: fiber polarization controller, OSA: optical spectrum analyzer, PD: photodetector, OSC: oscilloscope, ESA: electrical spectrum analyzer, MZI: fiber Mach-Zehnder interferometer, BPF: tunable band pass filter. (b) Resonance mode profile with a beat signal between the probe laser and the comb line (green), whose detuning was calibrated using a fiber Mach-Zehnder interferometer (gray). (c) Measured dispersion of the mode family used for comb generation (blue points) with respect to the mode number offset from the pump mode. The red curve is a parabolic fit using a D2/2π value of 17 MHz. The deviated points are shifted by mode coupling with other mode families.
Figure 3(a) shows the transmission power when scanning the pump laser from a short to a long wavelength. The cold resonance wavelength was 1560.85 nm, which corresponds to a 0 ms scan time. The total Q (Qt) was about 3.5×106. Figures 3(b) and 3(c) show typical measured optical spectra and radio frequency (RF) signals in states (i), (ii), and (iii). First, neither a Kerr comb nor an OMPO was observed in state (i) because the intracavity power was below the threshold. Then RF signals at multiples of 40 MHz appeared in state (ii), which is the result of the OMPO excited by the CW pump. The oscillation frequency agrees with the calculated frequency of the mechanical mode shown in the inset of Fig. 2(a). Since no comb has been generated at this condition, the transmission fluctuation in state (ii) was caused solely by the OMPO. Then a Kerr comb starts to appear in state (iii) which was a Turing pattern comb that had a smooth envelope with a mode spacing of 3-FSR (= 560.5 GHz × 3). Surprisingly, the OMPO was suppressed after the comb was generated. Since a Turing pattern comb is generated via a strong blue-detuned pump, the microcavity structure should oscillate only if we consider the influence of the pump light. Our observation of the suppression of the optomechanical oscillation indicates that the generated comb lines influenced the optomechanical behavior in the toroid microcavity, as we predicted in Fig. 1. In the next section, we compare the mechanical and optomechanical damping rates by taking generated comb lines into account.
Fig. 3 (a) Transmission when scanning pump laser from a short to a long wavelength. The pump power was 280 mW. In state (ii), OMPOs occurred without comb generation, which caused the transmission fluctuation. After a Turing pattern comb generation, the fluctuation was suppressed as shown in state (iii). (b) and (c) Typical measured optical spectra and detected RF signals in state (i), (ii) and (iii). The RF signal in state (ii) had harmonics of 40 MHz, which was in agreement with a calculated mechanical frequency.
3. Discussion
3.1. Model
The optomechanical behavior of a cavity is defined by the effective mechanical damping rate (Γeff) [21,26]. The effective damping rate obeys the following relationship between the mechanical (Γm) and optomechanical (Γopt) damping rates.
The optomechanical damping rate is given by where |aμ|2 is the number of photons circulating inside the cavity, which is coupled to the resonant mode μ (ħωμ|aμ|2 × D1/2π corresponds to the circulating power). ħ is the Planck constant divided by 2π. κ is the optical cavity decay rate (= ω0/Qt) and Δωμ/2π is the detuning between the coupled light and resonance frequencies at resonance mode μ. gom is the optomechanical single-photon coupling strength: G is the resonance angular frequency shift per displacement, xZPF is the mechanical zero-point fluctuation amplitude, and R is the cavity radius. κ and gom are regarded as independent of the resonance mode μ. Since the optomechanical damping rate can be both positive and negative, the effective damping rate increases or decreases depending on the sign. The change in the effective damping rate leads to the amplification or damping of the mechanical motion. Note that Γeff < 0 induces an OMPO, which causes a large change in the photodetector signal in the RF regime as shown in Fig. 3(c).3.2. Excitation of mechanical mode with continuous wave
We calculated the effective damping rate by using Eq. (2) but with a single resonance mode (Γeff = Γm + Γopt,0) that assumes there is no FWM generation. The number of photons |a0|2 is obtained from a coupled mode equation, such as,
where sin is the input field (ħω0|sin|2 corresponds to the input power), κc is the optical coupling rate to the waveguide (= ω0/Qc), Qc is the coupling Q, which has the relation . The calculation parameters were chosen to follow the experimental condition. The mechanical damping rate Γm/2π was 40 kHz, which obeys Γm = Ω/Qm. By taking all the parameters into account, the pump detuning range for OMPO (Γeff < 0) was calculated to have a 7 kHz to 355 MHz range. Since this analytical approach can be applied when the system is pumped with a CW light, the transition from state (i) to (ii) in Fig. 3(a) should occur when the pump approaches a detuning of 355 MHz. According to this CW pump model, the OMPO should remain even in the detuning range of state (iii), if no FWM process has occurred and only the pump light influenced the system. However, this is not the case with the experiment as shown in Fig. 3. Since the photons in the pump mode were converted to other resonance modes via FWM in the experiment, we need to take account of the excitation of the optomechanical mode with multi-optical resonances.3.3. Excitation of mechanical mode with Kerr comb
It is straightforward to analyze the optomechanical behavior of a toroid microcavity excited with a single frequency by using Eqs. (2) and (3). On the other hand, it is not easy to couple an LLE directly with Eq. (3), because we need to obtain the effective detuning and the intracavity photon number while generating a Kerr comb. It should be noted that Δωμ in Eq. (3) is the effective detuning. Here, instead of the full calculation approach, we calculated the effective damping rate by assuming that the shape of the Turing pattern comb does not change with respect to the detuning. The intracavity photon numbers of the comb lines were obtained from the LLE calculation, and the effective pump detuning was measured in the experiment. The other detuning values of the comb sidebands (generated comb lines) are obtained by assuming the comb has a mode spacing of D1/2π, which was confirmed with the measurement.
First, to obtain the numbers of photons in each resonance mode, we use an LLE model as given by [3–6]
a(ϕ, t) is the internal field including the multi resonance mode (|a|2 indicates the intracavity photon numbers in the modes), ϕ is the angular coordinate in a microcavity, t is the slow time, Δωcold,0 is the cold pump detuning, and g is the nonlinear coefficient as . n2 is the nonlinear refractive index of silica and Aeff is the effective mode area. The toroid microcavity used in this research has an effective mode area of 12 μm2. The output field is given as . Figure 4(a) shows the calculated output spectrum for a Turing pattern comb, which is in good agreement with the experiment shown in state (iii) of Fig. 3(b). The red points in Fig. 4(a) show the number of photons inside the microcavity in each resonant mode.Fig. 4 (a) Measured and calculated output optical spectra of Turing pattern combs in state (iii) (blue lines). The black dash lines show the optical powers of calculated comb lines (μ = 0, ±3, ±6), which are in good agreement with the experiment. The red points show the photon numbers inside a microcavity (log scale). (b) The green points represent measured detuning values from each resonance frequency when generating a 3-FSR Turing pattern comb. The pump detuning was 47 MHz. The pump was blue-detuned and the generated comb lines were red-detuned, which follow the microcavity dispersion (green curve) measured in Fig. 2(c). In the condition for generating a comb shown in Fig. 3(b), the calculated pump detuning for OMPO (Γeff < 0) was from 51 to 222 MHz in the blue-detuned side.
Next, we measured the effective detuning Δωμ of the pump and the generated comb lines in a hot microcavity. So, we performed a detuning measurement while generating a 3-FSR Turing pattern comb by launching a probe laser in the counter direction [9]. The detuning values of the pump laser (μ = 0) and the first sidebands (μ = ±3) were measured as shown in Fig. 4(b) (green points). Since FWM occurs with frequency spacing at multiple of D1/2π, the angular frequencies of comb lines are given by ωp + D1μ (ωp is the pump angular frequency). Hence the detuning of the generated comb lines should follow the cavity dispersion as follows
Indeed the measured detuning in Fig. 4(b) is in good agreement with the cavity dispersion (red line) obtained from the measurement shown in Fig. 2(c). Figure 4(b) shows that the pump laser was at blue detuning and the sidebands were at red detuning. These three lines have a strong influence on the optomechanical damping rate because of the relatively high power and the detuning.It should be noted that, when the Turing pattern comb appears, all the generated comb lines are always in the red detuning regime except for the pump light in an anomalous dispersion microcavity (D2 > 0). Hence the generated comb lines are always contributing to the suppression of the OMPO as illustrated in Fig. 1. To confirm the validity of our model, we calculated an effective mechanical damping rate using Eq. (2). We used the measured pump detuning (Δω0/2π = 47 MHz) in state (iii) of Fig. 3(b). The detuning values for the generated comb lines are obtained with Eq. (7). The number of photons inside the cavity is obtained from Fig. 4(a). The calculated pump detuning for OMPO (Γeff < 0) was in the 51 to 222 MHz range shown in Fig. 4(b). The measured detuning was smaller than 51 MHz (i.e. outside the OMPO range), which indicates the suppression of the OMPO and explains the experimental behavior well. The calculation result is in good agreement with the experiment observation and our simple model sufficiently explained that the OMPO could be suppressed under the condition in state (iii) of Fig. 3.
4. Conclusion
We demonstrated the suppression of OMPOs with a Turing pattern comb, which was generated by a blue-detuned pump from the resonance frequency. When we consider only the influence of the blue-detuned pump light, optomechanical oscillations are always amplified. On the other hand, we can explain the suppression of OMPO with a Turing pattern comb by considering the influence of all the comb lines, which are in the red detuning regime. The optomechanical coupling behavior in a Kerr medium has proved of interest [32,33], and this study constitutes a step toward a better understanding of the FWM process and optomechanical coupling systems. Although we focused on Turing pattern combs, soliton combs will achieve efficient damping because the pump and comb lines are both in the red-detuned regime [10]. Furthermore, dispersion engineering is of interest to this study because detuning values depend on the cavity dispersion [34].
Appendix
In this Appendix, we performed the same measurement and calculation using a different toroid microcavity to show that our simple model is valid even when a cavity with different parameters is used. Figure 5(a)–5(c) show the transmission power, optical spectra, and RF beat signals while scanning a pump laser. The mechanical properties of the microcavity are shown in Table 2. The total Q was about 2 × 106 and the cavity dispersion D2/2π was 50 MHz. In state (iv), we observed a 3-FSR Turing pattern comb (= 970.6 GHz ×3) without the RF signal induced by OMPO. The green points in Fig. 5(d) show the measured detuning values, which followed the microcavity dispersion (D2/2π = 50 MHz), when generating the comb shown in state (iv). The calculated pump detuning for Γeff > 0 (suppression of the OMPO) in the blue-detuned side was from 139 to 219 MHz, which we obtained by assuming the optical spectrum power in state (iv). The measured pump detuning (Δω0/2π = 191 MHz) stayed within the calculated detuning range. On the other hand, in our analysis of a CW case, the calculated pump detuning values for Γeff > 0 were Δω0/2π >755 MHz and Δω0/2π < 0.19 kHz. Hence the different microcavity also suppressed the OMPO assisted by a Turing pattern comb even though the transmittance signal in Fig. 5(a) behaved differently from that in Fig. 3. The difference is due to the presence of states (iii) and (v) where the OMPO is not fully suppressed even after the generation of the comb (due to the strong influence of the pump line). This behavior is mainly due to the large cavity dispersion. FWM starts to occur at a pump detuning of 225 MHz, which is determined by cavity dispersion, optical Q, and input power. It should be noted that FWM at 3-FSR starts to occur when the red curve in Fig. 5 approaches the zero detuning line at μ = ±3. So the system starts to exhibit the suppression of OMPO. However, with these cavity parameters, the damping provided by the two FWM sidebands was insufficient to completely stop the OMPO (state (iii)). Then the OMPO stops when the input laser is further scanned (state (iv)) because the detuning values of two FWM sidebands were close to the mechanical frequency. Finally, the microcavity oscillates again because the influence of the pump becomes much larger than that of the two FWM sidebands when the detuning is further decreased (state (v)). Although the behavior in Fig. 5 is different from Fig. 3, it could be explained by using the same model. The critical parameter for this behavior was not a higher mechanical Q but the dispersion and mode number for the first comb sidebands. Here a Turing pattern comb formed again after the generation of sub-comb lines, that was often observed in our experiment. Although this formation has not been well understood, we think it is due to the influence of cavity optomechanics because sub-comb lines generated with optomechanical oscillations and a Turing pattern comb formed without the oscillations.
Table 2. Parameters of the toroid microcavity that provided the experimental results in Fig. 5
Fig. 5 (a) Transmission when scanning the pump laser with a different toroid microcavity of which parameter is given in Table 2. The power fluctuation was suppressed in state (iv). The pump power was 320 mW. (b) and (c) Typical measured optical spectra and RF signals in state (i)–(v). Although a 3-FSR Turing pattern comb was generated in state (iv), small sub-comb lines were observed in state (iii) and (v) with RF signals induced by the OMPO. (d) The green points represent measured detuning values from each resonance frequency when generating a 3-FSR Turing pattern comb, which follow the microcavity dispersion (green curve). In the condition for generating the comb shown in state (iv), whose pump detuning was 191 MHz, the calculated pump detuning for Γeff > 0 was from 139 to 219 MHz in the blue-detuned side.
Funding
JSPS KAKENHI Grant Number (JP16J04286); MEXT KAKENHI (JP15H05429).
Acknowledgments
First author acknowledges the Program for Leading Graduate Schools, “Global Environmental System Leaders Program” by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) in Japan.
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