Enhanced optomechanical interaction in coupled microresonators

Jiahua Fan; Lin Zhu

doi:10.1364/OE.20.020790

1. Introduction

Optomechanical systems have attracted great interest for the ability to combine nanomechanics and nanophotonics [1–6]. These systems exploit optical forces arising from the perturbation of light fields by mechanical displacements [7, 8]. Typically the optomechanical interaction is weak and strong optical power is needed to generate detectable optical forces [9]. By resonance enhancement of both optical field and mechanical motion, the optomechanical system can go beyond mesoscale [10] and allow applications in both fundamental physics [11–16] and applied engineering [17–24], including quantum ground state cooling [12], optomechanically induced transparency [3, 21], microwave oscillator [24–26], integrated pressure sensing [23], etc.

Generally, most of the reported optomechanical systems are based on the interaction between mechanical modes and a single high-Q optical resonance. When the input light is coupled to an optical resonator, the mechanical motion of the resonator modulates the intra-cavity optical field with the mechanical resonance frequency and generates two frequency sidebands. To achieve effective energy transfer between the light field and mechanical motions, the frequency of the input light must be detuned from the cavity resonance so that the resonator will favor one sideband which is closer to the optical resonance frequency. However, this also indicates that only a fraction of input power enters the resonator and contributes to the optomechanical interaction.

Compound resonator systems consist of two or more coupled resonators and generally exhibit multiple resonances. Recent progress in the fabrication of extremely high-Q microresonators and high-frequency mechanical oscillators makes it possible to couple multiple optical resonances via optomechanical interaction. The major benefit for building optomechanical system based on the compound resonator system is its controllable resonance functions. The frequency spacing between optical resonances can be continuously tuned by changing the coupling rate between the resonators [15], which provides additional freedom for designing optomechanical systems. Moreover, in contrast to the single resonator optomechanical system where mechanical modes of one mechanical resonator interact with the same optical field, mechanical modes of separate mechanical resonators can couple to the optical field inside individual resonator. This feature presents the opportunity for light-induced energy transfer and synchronization between mechanical oscillators [27–29].

In this paper, we investigate the optomechanical interaction in two coupled microresonators while utilizing the composite resonances of the system. By taking advantages of the split resonance, we can bypass the requirement of frequency detuning for the input light in a single resonator system. For the thermally driven system, we show that the optomechanical interaction can be greatly enhanced compared to the single resonator system, and a controllable energy transfer between separate mechanical oscillators is possible. For the external modulation driven system, the power efficiency to obtain the Optomechanically Induced Transparency (OMIT) phenomenon can be greatly increased.

2. Conceptual model

The schematics of optomechanical systems based on a single resonator and two coupled resonators are shown in Fig. 1. The optical field inside each resonator is perturbed by the mechanical motion and consequently exerts a modulated optical force on the mechanical mode. The modulated force has two components: one is in phase of the motion, which corresponds to the optical spring effect and changes the mechanical resonance frequency; the other is out of phase of the motion, which modifies the mechanical damping. Both negative and positive damping can be obtained by detuning the input light with respect to the optical resonance.

Fig. 1 The schematics and sideband picture of the optomechanical system based on (a) a single resonator; (b) two coupled resonators. The mechanical motion modifies the optical resonance frequency.

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The dynamic back-action can be also simply understood by using the sideband picture [1]. In Fig. 1(a), the input light with the angular frequency ω is coupled to the resonator. The optical field inside the cavity is modulated by the mechanical motion with the mechanical resonance frequency Ω_m and this modulation generates two sidebands at ω ± Ω_m. If the input frequency is not equal to the optical resonance frequency, the resonator will favor one sideband whose frequency is closer to the resonance. Thus the two sidebands will have different amplitudes. Because of the different photon energies of the two sidebands, the total optical energy inside the resonator will not be conserved, corresponding to an energy transfer between the optical field and the mechanical motion. If the input light is red (blue) detuned from the resonance, energy is transferred from (to) the mechanical motion and the input light produces a damping (amplification) force on the mechanical motion.

Based on the sideband picture, the origin of the mechanical damping/amplification is the imbalanced amplitudes of sidebands. The input light is required to be detuned from the cavity resonance to generate asymmetric sidebands in a single resonator optomechanical system. In other words, the input light is not resonance-enhanced. In the resolved-sideband regime, where the mechanical resonance frequency is much larger than the cavity linewidth (Γ), only a small fraction of the input power can enter the cavity and interact with the mechanical motion.

The motivation of introducing coupled resonator system is to eliminate the detuning requirement of the input light. A conceptual model of optomechanical systems based on two coupled resonators is shown in Fig. 1(b). The first resonator is evanescently coupled to the bus waveguide and the second resonator. The system can exhibit two split resonances in the transmission spectrum when the two resonators are strongly coupled. By controlling the coupling strength between the resonators, we can set the spacing between two split resonance frequencies close to the mechanical frequency. Therefore, if the frequency of the input light equals one optical resonance frequency, both the input light and one of its sidebands can be simultaneously resonance-enhanced. Note that here the first order sidebands are mainly generated by phase modulation of the optical field. The large asymmetry of sideband amplitudes will increase the back-action force, leading to an enhanced optomechanicam interaction. In the following sections, by using a perturbation method, we demonstrate that the coupled resonator system can provide much larger optomechanical interaction compared to the single resonator system.

3. Quantitative formalism and simulation results

As shown in Fig. 1(b), the optomechanical system consists of two optical resonators. The optical field inside each resonator is modulated by one mechanical mode. Here we assume the two mechanical modes have no direct coupling. The dynamic response of the system can be described using the coupled equations:

\frac{d a_{1}}{d t} = (i Δ_{1} - Γ_{1} / 2 - i g_{1} x_{1}) a_{1} + i \sqrt{Γ_{e}} A_{in} + i κ a_{2}

\frac{d a_{2}}{d t} = (i Δ_{2} - Γ_{2} / 2 - i g_{2} x_{2}) a_{2} + i κ a_{1}

\frac{d^{2} x_{i}}{d t^{2}} + Γ_{m_{i}} \frac{d x_{i}}{d t} + Ω_{m_{i}}^{2} x_{i} = \frac{F_{i}}{m_{i}} + \frac{F_{th}}{m_{i}}

where A_in is the power amplitude of the input light; a_i (i = 1, 2) is the energy amplitude of the optical field inside each resonator; κ is the decay rate for the coupling between the resonators; Γ₁, Γ_e are the loaded decay rate and the external decay rate of the first resonator; Γ₂ is the intrinsic decay rate of the second resonator; x_i, Γ_mi, Ω_mi, F_i, F_th, m_i (i = 1, 2) are the displacement, decay rate, resonance frequency, optically induced force, thermally induced force and effective mass of each mechanical mode. In the general case, the two resonators have different resonance frequencies ω₁ and ω₂, and the frequency detuning of the input light from the resonance frequency is Δ_i = ω − ω_i. The optomechanical coupling coefficient g_i is defined as g_i = dω_i/dx_i. The above equations can be solved in the steady-state for optical field amplitudes and mechanical displacements:

{\bar{a}}_{1} = \frac{- i \sqrt{Γ_{e}} A_{in}}{i {\bar{Δ}}_{1} - Γ_{1} + \frac{κ^{2}}{i {\bar{Δ}}_{2} - Γ_{2} / 2}}

{\bar{a}}_{2} = \frac{- i κ {\bar{a}}_{1}}{i {\bar{Δ}}_{2} - Γ_{2} / 2}

{\bar{x}}_{i} = - \frac{g_{i} {| {\bar{a}}_{i} |}^{2}}{m_{i} Ω_{m_{i}}^{2} ω}

We assume the optomechanical coupling is not too strong to generate bistability and the input optical power is small so that the modified detuning, Δ̄_i = Δ_i − g_ix̄_i, is approximately equal to Δ_i. The thermal forces are neglected as well for the equilibrium solution. The system provides two types of optical transmission based on the optical coupling rate (κ) between two resonators, as shown in Fig. 2.

Fig. 2 Optical field energies of the two resonators in the: (a) weak coupling regime, κ = Γ₂; (b) strong coupling regime, κ = 10Γ₂. The intrinsic Q-factors are Q_i₁ = Q_i₂ = 10⁶, and the external Q-factor of the first resonator is Q_e₁ = 10⁵. The input optical power is 1mW. For simplicity, the two resonators are assumed to have the same intrinsic resonance, Δ = Δ₁ = Δ₂.

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In the weak coupling regime (Fig. 2(a)), because the second resonator has a larger loaded Q-factor, the circulating power inside the first resonator is absorbed into the second resonator in the vicinity of the resonance. In the strong coupling regime (Fig. 2(b)), because the coupling rate between two resonators is larger than their linewidth, the system exhibits two split resonances. The frequency spacing between the split resonances is approximately equal to 2κ for κ ≫ Γ₂.

Using the perturbation method we can obtain the dynamic equations for perturbed optical field amplitudes δa_i and mechanical displacements δx_i (i = 1, 2)

\begin{array}{l} \frac{d δ a_{1}}{d t} = (i \bar{Δ} - \frac{Γ_{1}}{2}) δ a_{1} - i g_{1} {\bar{a}}_{1} δ x_{1} + i κ δ a_{2} + i \sqrt{Γ_{e}} δ A_{in} \\ \frac{d δ a_{2}}{d t} = (i \bar{Δ} - \frac{Γ_{2}}{2}) δ a_{2} - i g_{2} {\bar{a}}_{2} δ x_{2} + i κ δ a_{1} \\ \frac{d^{2} δ x_{i}}{d t^{2}} + Γ_{m i} \frac{d δ x_{i}}{d t} + Ω_{m i}^{2} δ x_{i} = - \frac{g_{i}}{m_{i} ω} ({\bar{a}}_{i} δ a_{i}^{*} + {\bar{a}}_{i}^{*} δ a_{i}) + \frac{F_{th}}{m_{i}} \end{array}

Based on the source of the driven force, optomechanical systems can be categorized into: (1) thermal force driven system (2) external modulation driven system. For the thermally driven system, the mechanical oscillation is driven by the thermal reservoir and the amplitude of the input light is constant. The system response can be found by solving Eq. (7) in the frequency domain with δA_in = 0. For the external modulation driven system, the input optical field is modulated near the mechanical resonance. The back-action force generated by the modulated optical field is much larger than the thermal forces, which means we can neglect F_th and solve Eq. (7) with a time-dependent δA_in. Next we solve these equations in frequency domain under three different situations.

3.1. Strong coupling under thermal force driven

First we consider that the optical resonators are strongly coupled with each other, which creates two split resonances as shown in Fig. 2(b). We set δA_in = 0 for the thermally driven system. The optomechanical coupling is assumed to occur only within the first resonator (δx₂ = 0). Then Eq. (7) lead to a modified equation for the mechanical oscillation

[(Ω_{m_{1}}^{2} - Ω^{2} - i Γ_{m_{1}} Ω) + \frac{i g_{1}^{2} {| {\bar{a}}_{1} |}^{2}}{m_{1} ω} [f^{+} (\bar{Δ}, Ω) + f^{-} (\bar{Δ}, Ω)]] δ {\tilde{x}}_{1} (Ω) = \frac{{\tilde{F}}_{th} (Ω)}{m_{1}}

where the cavity factors f^±(Δ̄, Ω) are defined as

\begin{array}{l} f^{+} (\bar{Δ}, Ω) = {[i (\bar{Δ} + Ω) - Γ_{1} / 2 + \frac{κ^{2}}{i (\bar{Δ} + Ω) - Γ_{2} / 2}]}^{- 1} \\ f^{-} (\bar{Δ}, Ω) = {[i (\bar{Δ} - Ω) + Γ_{1} / 2 + \frac{κ^{2}}{i (\bar{Δ} - Ω) + Γ_{2} / 2}]}^{- 1} \end{array}

The dressed mechanical motion exhibits the modified resonance frequency Ω′_m₁ and damping rate Γ′_m₁

{(Ω_{m_{1}}^{'})}^{2} = Ω_{m_{1}}^{2} + ℜ {\frac{i g_{1}^{2} {| {\bar{a}}_{1} |}^{2}}{m_{1} ω} [f^{+} (\bar{Δ}, Ω) + f^{-} (\bar{Δ}, Ω)]}

Γ_{m_{1}}^{'} = Γ_{m_{1}} - ℑ {\frac{i g_{1}^{2} {| {\bar{a}}_{1} |}^{2}}{m_{1} ω Ω} [f^{+} (\bar{Δ}, Ω) + f^{-} (\bar{Δ}, Ω)]}

The results are similar to the previous report [1] except for different expressions of the cavity factors. If the frequency spacing between two optical resonances equals the mechanical resonance frequency, the input light and one of its sidebands can be simultaneously on resonance for Δ̄ ≈ ±Ω_m/2. We plot the modified mechanical damping rate and resonance frequency in Fig. 3 for κ = 0.5Ω_m. For the single resonator system (Fig. 3(a) and 3(c)), the modification of the mechanical damping rate is maximized at Δ̄ = ±Ω_m, and the maximum optical spring effect is obtained around Δ̄ ≈ ±0.4Ω_m. While for the coupled resonator system (Fig. 3(b) and 3(d)), both the mechanical damping modification and optical spring effect are maximized when the input light is on resonance with one of the split resonances. The results agree with previous analysis using the sideband picture. Because of the enhancement of both the input light and its sideband, the modifications of the damping rate and resonance frequency for the mechanical oscillation increase by a percentage of about 100% and 75%, respectively.

Fig. 3 Calculated (a)(b) optomechanical amplification (Γ′_m₁ < Γ_m₁) or cooling (Γ′_m₁ > Γ_m₁) and (c)(d) optical spring effect for (1) single optical resonator with Q_e = Q_i = 10⁶, Ω_m = 2π × 500MHz; (2) coupled resonator with Q_e₁ = Q_i₁ = Q_i₂ = 10⁶, κ = 0.5Ω_m. The blue curves denote the normalized optical energy inside the resonators. The input power is 1mW. The optomechanical coupling coefficient is g_i = 10GHz/nm. The effective mass and Q-factor of the mechanical oscillator are set at m_i = 10pg and Q_m=1000, respectively.

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3.2. Strong coupling under external modulation driven

For the external modulation driven system, the Optomechanically Induced Transparency (OMIT) effect has been reported for the single resonator optomechanical system, where a control light induces a narrow transparency window for a weak, probe light [3, 21]. To investigate the OMIT effect in the coupled resonator system, the control light (A_in) is set at one of the resonances with smaller frequency. A frequency modulation of the control light generates two sidebands, one of which serves as the probe light. The modulation frequency is then scanned to obtain the transmission spectrum of the probe light. We also assume that the mechanical oscillation only occurs in the first resonator (δx₂ = 0). Because the back-action force generated by the modulated optical field is much larger than the thermal force, we can neglect F_th and solve Eq. (7) in the frequency domain:

δ {\tilde{a}}_{1} (Ω) = \frac{- i \sqrt{Γ_{e}} f^{+} (Ω) {δ {\tilde{A}}_{in} (Ω) + i g_{1}^{2} ω^{- 1} χ (Ω) {\bar{a}}_{1} f^{-} (Ω) [{\bar{a}}_{1}^{*} δ {\tilde{A}}_{in} (Ω) - {\bar{a}}_{1} δ {\tilde{A}}_{in}^{*} (- Ω)]}}{1 + i g_{1}^{2} ω^{- 1} χ (Ω) {| {\bar{a}}_{1} |}^{2} [f^{+} (Ω) + f^{-} (Ω)]}

where δÃ_in is the power amplitude of the probe light, δã₁(Ω) is the optical field amplitude at the probe frequency inside the first resonator, and the tilde symbol denotes the Fourier transform of optical fields. The cavity factors f^±(Δ̄, Ω) are defined in Eq. (9). The mechanical susceptibility χ(Ω) is defined as

χ (Ω) = {[m (Ω_{m}^{2} - Ω^{2} - i Γ_{m} Ω)]}^{- 1}

.

From Eq. (12), the transmission of the probe light can be calculated as a function of its frequency detuning from the control light. When the detuning gets close to the mechanical frequency, the intra-cavity field of the probe light experiences destructive interference with the anti-Stokes sideband of the control light generated by the mechanical oscillation. Therefore, a narrow transparency window arises, as shown in Fig. 4(a), corresponding to the OMIT effect [3, 21]. If the mechanical frequency is much larger than the linewidth of the split resonances, one of the cavity factors is much smaller than the other and can be neglected consequently, leading to a similar expression as derived in [3]. In Fig. 4(b), we calculate the OMIT window peak transmission with different control light powers. The peak transmission is normalized as t = (t_p − t_r)/(1 − t_r) for both the single resonator system and the coupled resonator system, where t_p is the peak value of the transmission of the transparency window, and t_r is the residue transmission in the absence of the control light. Because of the resonance-enhancement of the control light, the pump power efficiency is greatly increased for generating a transparency window in the coupled resonator optomechanical system.

Fig. 4 (a) Normalized power transmission in the coupled resonator system for OMIT effect. The control light is on resonance with one of the optical resonances with smaller frequency. The frequency detuning of the probe light from the control light is scanned to obtain its transmission spectrum. The input power of control and probe light are 1mW and 10μW, respectively. Q_i₁ = Q_i₂ = Q_e₁ = 10⁶, m₁ = 10pg, g₁ = 20GHz/nm, Q_m = 1000, Ω_m = 2π × 500MHz, κ = 0.5Ω_m. (b) Comparison of the normalized peak transmission for OMIT effect in a single resonator and coupled resonator system. The single resonator system has the same optical decay rates and properties of mechanical oscillation as the first resonator in the coupled resonator system.

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3.3. Weak coupling under thermal force driven

In the weak coupling regime, when the frequency of the input light is scanned near the intrinsic resonance, the optical energy inside of the first (second) resonator decreases (increases) when reducing the frequency detuning, as shown in Fig. 2(a). This indicates that the dynamic back action force will cool the mechanical mode of one resonator but simultaneously amplify the mechanical mode of the other one (here we consider that there is a mechanical mode coupled to the optical resonance in both resonators).

We assume that both optical resonators are perturbed by mechanical motions with harmonic functions, i.e. δx_i(t) = δx_i cos(Ω_{m_i}t). The perturbation of optical fields inside each resonator can be calculated by solving Eq. (7). For simplicity, we assume the mechanical modes have the same resonance frequency Ω_m

\frac{d δ a_{1}}{d t} = (i Δ_{1} - Γ_{1}) δ a_{1} + i κ δ a_{2} - i g_{1} x_{1} cos (Ω_{m} t) {\bar{a}}_{1}

\frac{d δ a_{2}}{d t} = (i Δ_{2} - Γ_{2}) δ a_{2} + i κ δ a_{1} - i g_{2} x_{2} cos (Ω_{m} t) {\bar{a}}_{2}

Rewrite

δ a_{i} = δ a_{i}^{+} e^{i Ω_{m} t} + δ a_{i}^{-} e^{- i Ω_{m} t}

and plug δa_i back into the equations above. We obtain

δ a_{1}^{\pm} = \frac{1}{2} \frac{i g_{1} x_{1} {\bar{a}}_{1} [i (Δ_{2} \mp Ω_{m}) - Γ_{2} / 2] + g_{2} x_{2} {\bar{a}}_{2} κ}{[i (Δ_{1} \mp Ω_{m}) - Γ_{1} / 2] [i (Δ_{2} \mp Ω_{m}) - Γ_{2} / 2] + κ^{2}}

δ a_{2}^{\pm} = \frac{1}{2} \frac{i g_{2} x_{2} {\bar{a}}_{2} [i (Δ_{1} \mp Ω_{m}) - Γ_{1} / 2] + g_{1} x_{1} {\bar{a}}_{1} κ}{[i (Δ_{1} \mp Ω_{m}) - Γ_{1} / 2] [i (Δ_{2} \mp Ω_{m}) - Γ_{2} / 2] + κ^{2}}

For small perturbation, the dynamic back-action force generated by the optical field is equal to F = −2g𝔕(ā^*δa_i)/ω, and contains two components:

F_{I_{i}} = - \frac{2 g_{i}}{ω_{i}} ℜ [{\bar{a}}_{i}^{*} (δ a_{i}^{+} + δ a_{i}^{-})]

F_{Q_{i}} = \frac{2 g_{i}}{ω_{i}} ℑ [{\bar{a}}_{i}^{*} (δ a_{i}^{+} - δ a_{i}^{-})]

The in-phase component F_I (proportional to cos(Ω_mt)) relates to the optical spring effect that changes the mechanical frequency, while the quadrature component F_Q (proportional to sin(Ω_mt)) relates to the change of damping rate of the mechanical motion. The net power transfered from the optical field to the mechanical motion equals 〈P〉 = 〈Fẋ(t)〉 = −F_Qδxsin(Ω_mt). Therefore, positive (negative) F_Q corresponds to a cooling (amplifying) process of the mechanical modes. During the calculation, the properties of the mechanical modes are set at m_i = 10pg, g_i = 10GHz/nm, Q_{m_i} = 1000, Ω_{m_i} = 2π × 200MHz, and we assume the two resonators have the same intrinsic resonance, Δ₁ = Δ₂.

First, we consider that the loaded Q-factor of the first resonator is much smaller than the second resonator. Fig. 5(a) shows the optical energy inside each resonator depending on the normalized detuning of the input light. The calculated F_{Q_i} are shown in Fig. 5(b). The red and blue curves represent the quadrature component of the force inside the first and second resonator, respectively. For red (blue) detuning, the optical fields amplify (cool) the mechanical mode in the first resonator and cool (amplify) the mechanical mode in the second resonator. We further compare the results to the cases where only one mechanical motion exists in the system, as shown in Fig. 5(b). The optical force in the first resonator (F_Q₁) is greatly enhanced with the existence of the mechanical motion (δx₂) in the second resonator, while F_Q₂ changes very little with or without δx₁. In other words, because most of the optical energy accumulates in the second resonator around the resonance, δx₁(t) is controlled by δx₂(t) via the optomechanical interaction.

Fig. 5 (a) The optical energy inside the first (red) and second (blue) resonator. (b) The calculated quadrature force component generated by optical field in the first (red) and second (blue) resonator. The solid lines are calculated when mechanical oscillation exists in both resonators. The dotted lines are calculated when only one resonator has mechanical motion. The Q-factors of the two resonators are Q_i₁ = Q_i₂ = 10⁶, Q_e = 10⁵, Q_κ = 10⁶.

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When the loaded Q-factor of the two resonators are similar, the maximum optical energies in both resonators are on the same level near the resonance (Fig. 6(a)). Thus, the optical forces in both resonators are influenced with the existence of mechanical motion of the other, as shown in Fig. 6(b). When the input light is red detuned from the resonance, both the optical field and the mechanical motion δx₁(t) generate additional damping on δx₂(t). In Fig. 6(b), we also compare the calculated backaction forces with the ones where only one mechanical mode exists. The result indicates that the optical field transfers part of the energy in δx₂(t) to δx₁(t) when Δ̄ ≈ −Ω_m. The energy transfer process is reversed for blue detuning. Although the two mechanical modes do not have direct coupling, the input optical field serves as an optical spring that facilitates the energy transfer process. With proper optimization, this phenomenon can be used to manipulate phonon energy among different mechanical oscillators, and provides the potential for employing optomechanical systems for signal storage and processing.

Fig. 6 (a) The optical energy and (b) the calculated quadrature force component generated by optical field in the first (red) and second (blue) resonator. The Q-factors of the two resonators are: Q_i₁ = Q_κ = 10⁶, Q_i₂ = Q_e = 5 × 10⁵,.

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To form a complete analysis, we also investigate the coupled resonator system under external modulation in the weak coupling regime. The external modulation is usually used to drive the mechanical motion or to generate the OMIT effect. However, due to the lack of mode splitting in the weak coupling regime, the coupled resonator system does not exhibit noticeable advantages over a single resonator with respect to the enhancement of optomechanical force or power efficiency of OMIT effect.

4. Summary

Based on the calculation of the dynamic back-action in two coupled resonators, we show that building optomechanical systems based on the compound resonator system can control and enhance the optomechanical interaction. For the modulation driven system, we show that the power efficiency of the control light to obtain OMIT effect can be greatly increased. For the thermally driven system, the optomechanical interaction is enhanced compared to the single resonator system. With properly choosing the parameters of the coupled optomechanical system, an optically controlled mechanical energy transfer between different mechanical oscillators is possible.

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Enhanced optomechanical interaction in coupled microresonators

Abstract

1. Introduction

2. Conceptual model

3. Quantitative formalism and simulation results

3.1. Strong coupling under thermal force driven

3.2. Strong coupling under external modulation driven

3.3. Weak coupling under thermal force driven

4. Summary

References and links

Cited By

Figures (6)

Equations (18)

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