Synchronization in an optomechanical cavity

Keren Shlomi, D. Yuvaraj, Ilya Baskin, Oren Suchoi, Roni Winik, and Eyal Buks

Phys. Rev. E 91, 032910 – Published 13 March 2015

Abstract

We study self-excited oscillations (SEO) in an on-fiber optomechanical cavity. Synchronization is observed when the optical power that is injected into the cavity is periodically modulated. A theoretical analysis based on the Fokker-Planck equation evaluates the expected phase space distribution (PSD) of the self-oscillating mechanical resonator. A tomography technique is employed for extracting PSD from the measured reflected optical power. Time-resolved state tomography measurements are performed to study phase diffusion and phase locking of the SEO. The detuning region inside which synchronization occurs is experimentally determined and the results are compared with the theoretical prediction.

Received 5 October 2014

DOI:https://doi.org/10.1103/PhysRevE.91.032910

Authors & Affiliations

Keren Shlomi, D. Yuvaraj, Ilya Baskin, Oren Suchoi, Roni Winik, and Eyal Buks

Department of Electrical Engineering, Technion, Haifa 32000, Israel

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Vol. 91, Iss. 3 — March 2015

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Images

Figure 1
Experimental setup. (a) A schematic drawing of the sample and the experimental setup. An on-fiber optomechanical cavity is excited by a tunable laser with modulated power. The reflected light intensity is measured and analyzed. (b) Electron micrograph of a suspended micromechanical mirror (false color code: blue-silica fiber, yellow—gold mirror, gray—zirconia ferrule), the view is tilted by $52^{\circ}$ . (c) Spectral decomposition of the reflected light power $P_{R}$ vs. excitation wavelengths $λ_{L}$ . The SEO, visible as sharp peaks (black regions on color map) in the reflected power spectrum, are obtained at optical excitation wavelengths corresponding to positive slopes of the sample's reflectivity (shown by a dotted curve). The cavity resonance used in the synchronization experiments is denoted by a rectangle.
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Figure 2
The PSD as a function of modulation amplitude at resonance $ω_{p} = ω_{0}$ . The panels on the left exhibit the measured PSD whereas the panels on the right exhibit the calculated PSD obtained from Eq. (14). The modulation amplitude in (a), (b), (c), and (d) is $P_{1} / P_{0} = 0.3 \times 10^{- 3}$ , $0.8 \times 10^{- 3}$ , $3.3 \times 10^{- 3}$ , and $6.7 \times 10^{- 3}$ , respectively. The following device parameters have been employed in order to calculate the PSD according to Eq. (14): $m = 1.1 \times 10^{- 12} k g$ , $ω_{0} = 2 π \times 225 k H z$ and $(I_{0} / I_{0}^{'} δ_{m}) {(1 + κ^{2} / ω_{p}^{2})}^{- 1 / 2} = 25$ . The measured (calculated) normalized standard deviation $σ_{ϕ} / σ_{u}$ for the distributions presented in (a)–(d) are 0.98 (0.98), 0.76 (0.60), 0.59 (0.25), and 0.08 (0.17), respectively.
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Figure 3
The PSD as a function of modulation amplitude at angular frequency $ω_{p} = 2 ω_{0}$ . The panels on the left exhibit the measured PSD whereas the panels on the right exhibit the calculated PSD obtained from Eq. (14). The modulation amplitude in (a), (b), and (c) is $P_{1} / P_{0} = 0.67 \times 10^{- 2}$ , $2 \times 10^{- 2}$ , and $3.3 \times 10^{- 2}$ , respectively. The parameter $λ_{L} β ω_{0} / θ = 4800$ together with the other parameters that are listed in the caption of Fig. 2 have been employed in order to calculate the PSD according to Eq. (14). The measured (calculated) normalized standard deviation $σ_{ϕ} / σ_{u}$ for the distributions presented in (a), (b), and (c) are 1.0 (0.99), 0.90 (0.92), and 0.88 (0.89), respectively.
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Figure 4
Dephasing of SEO. The panels on the left exhibit the measured PSD whereas the panels on the right exhibit the calculated PSD obtained from numerically integrating the Fokker-Planck equation (13). The normalized dwell time $γ_{0} t_{d}$ in (a), (b), and (c) is $γ_{0} t_{d} = 0.15$ , 2.5, and 75, respectively. The device parameters are the same as those given in the caption of Fig. 2. The measured (calculated) normalized standard deviation $σ_{ϕ} / σ_{u}$ for the distributions presented in (a), (b), and (c) are 0.04 (0.11), 0.48 (0.60), and 0.99 (0.83), respectively.
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Figure 5
Rephasing of SEO. The relative amplitude of the modulation, which is turned on at time $t_{0} = 1 s$ after the first time window, is $P_{1} / P_{0} = 0.01$ . The normalized dwell time $γ_{0} t_{d}$ in (a), (b), and (c) is $γ_{0} t_{d} = 0.05$ , $0.95$ , and $1.7$ , respectively. The device parameters are the same as those given in the caption of Fig. 2. The panels on the left exhibit the measured PSD whereas the panels on the right exhibit the calculated PSD obtained from numerically integrating the Fokker-Planck equation (13). The measured (calculated) normalized standard deviation $σ_{ϕ} / σ_{u}$ for the distributions presented in (a), (b), and (c) are 0.99 (1.0), 0.71 (0.80), and 0.39 (0.69), respectively.
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Figure 6
The normalized standard deviation $σ_{ϕ} / σ_{u}$ vs. $D$ and $F$ . The solid line is the steady state bifurcation line $F_{-} (D)$ [see Eq. (25)]. The device parameters are the same as those given in the caption of Fig. 2.
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