Stability of resonant opto-mechanical oscillators

A. B. Matsko; A. A. Savchenkov; L. Maleki

doi:10.1364/OE.20.016234

1. Introduction

Opto-mechanical oscillators (OMOs) generate spectrally pure radio frequency (RF) signals [1–8] with ponderomotive interaction between photons and phonons. Phase noise and linewidth are the main characteristics determining the performance of an oscillator and its practical usefulness. It was shown that the OMO linewidth can be small enough to be ultimately described by a Schawlow-Townes-like formula [9]. A Leeson model [10] of the phase noise far from the OMO carrier was presented in [11]. In this paper we derive a generalized formula for the OMO phase noise that takes into account the noise of the light pumping the resonator. Using the formula, we analyze the linewidth of the oscillator. The predicted performance of the OMO is then compared with the performance of existing electronic oscillators.

Resonant opto-mechanical oscillation can be described as a strongly nondegenerate parametric process in which a pump photon is transformed to a photon of lower frequency (Stokes photon) and a phonon [12]. This occurs when the pump power exceeds a certain threshold determined by the loss of the system, and by the coupling efficiency between the light and the mechanical modes. In the case of microcavity-based OMO, the process is allowed if the resonator has at least two optical modes with a frequency difference equal to the frequency of a mechanical mode, a condition that satisfies energy conservation. The momentum conservation (phase matching of the parametric interaction) is met by the requirement of spatial overlap between the optical and the mechanical modes to ensure that the convolution integral of electric fields and mechanical displacement, as well as mechanical strain, is nonzero [13].

Generally, laser radiation is assumed to be phase insensitive. It means that the phase of the emitted light in a laser does not depend on the phase of the pump, and that there is no phase matching conditions (preferred directionality) in the system. An OMO, even one based on Brillouin lasing [14–16], has different properties. The mechanical quality factor is high enough to endow the generated phonons with some well defined phase. The phase of the Stokes light emitted by the OMO depends on the phonon phase, so the amplification of the Stokes wave becomes phase sensitive and the phase fluctuation of the optical pump leaks to the phase fluctuation of the Stokes light. This phase sensitivity can be suppressed if the bandwidth of the optical modes is many orders of magnitude larger than the bandwidth of the mechanical mode [17–20]. In what follows we develop a model for an OMO and find the phase noise of the generated signal for an arbitrary ratio between the quality factors of the optical and mechanical modes involved in the oscillation process.

It was shown that the linewidth of an OMO increases if generation of an anti-Stokes optical sideband is allowed by the system architecture [9, 12]. In this paper we are interested in the ultimate performance of the oscillator, so we analyze an ideal case where only two high quality (Q-) factor optical modes and a single mechanical mode interact. The optical pump is resonant with the higher frequency optical mode and the Stokes optical sideband is generated in the lower frequency mode. The bandwidth of the optical modes is assumed to be much smaller than the mechanical frequency. This kind of interaction has been realized in oscillators based on stimulated Brillouin scattering [14, 15] and surface acoustic waves [21–25].

A limitation of an OMO is that the frequency stability of the mechanical mode is determined in the same way as in electronic quartz oscillators. The fluctuations of the thermal bath as well as the thermodynamic noise and drift of the mechanical mode limit the frequency stability. The pumping light is used as the power source in an OMO, similar to the electric power source in electronic oscillators. Optical pumping is usually more noisy than electronic one. Therefore, it is unlikely that the OMO will outperform its electronic analogs. Our calculations confirm this qualitative conclusion. An advantage of an OMO, on the other hand, is in its potential long term stabilization via stability transfer from the optical frequency domain. Another advantage of the oscillator, confirmed by our calculations, is that the generated Stokes light can have significantly lower phase noise compered with the phase noise of the pump light [17–20].

The paper is organized as follows. The equations are presented in Section II, and their solutions are described in Section III. The OMO phase noise and linewidth are analyzed and compared to similar parameters of existing quartz oscillators in Section IV.

2. Basic equations

The triply-resonant opto-mechanical interaction is described by equations

\dot{A} = - Γ_{A} A - i g C B + F_{A},

\dot{B} = - Γ_{B} B - i g C^{†} A + F_{B},

\dot{C} = - Γ_{C} C - i g B^{†} A + F_{C} .

where A, B, and C are the slowly-varying amplitudes the pump (optical), the Stokes (optical mode red shifted with respect to the pump), and the signal (mechanical) fields; Γ_A, Γ_B, and Γ_C are the linear resonant terms of optical and mechanical modes respectively

\begin{array}{l} Γ_{A} = i (ω_{a} - ω_{0}) + γ + γ_{c a}, \\ Γ_{B} = i (ω_{b} - ω_{-}) + γ + γ_{c b}, \\ Γ_{C} = i (ω_{c} - ω_{M}) + γ_{M}, \end{array}

γ and γ_M are the intrinsic decay rates of the optical and mechanical modes, γ_ca and γ_cb are optical loading (coupling) rates (the loading of the optical modes can be different because modes belong to different families); g is the opto-mechanical coupling constant,

g = ω_{0} \sqrt{\frac{K_{ε} \bar{h}}{2 m^{*} L^{2} ω_{c}}},

K_ε is the correction coefficient showing that radiation pressure results not only in a change in the size of the resonator, but also in its index of refraction through strain, m^* is an effective mass of the mechanical mode, L is an effective spatial parameter of the mode.

The terms F_A and F_B represent Langevin forces with two uncorrelated parts arising from the internal and coupling loss of the modes

F_{A} = 〈 F_{A} 〉 + F_{c A} + F_{r A}, 〈 F_{A} 〉 = e^{i ϕ F_{A}} \sqrt{\frac{2 P γ_{c a}}{\bar{h} ω_{0}}},

〈 F_{c A} (t) F_{c A}^{†} (t^{'}) 〉 = 2 γ_{c a} δ (t - t^{'}),

〈 F_{r A} (t) F_{r A}^{†} (t^{'}) 〉 = 2 γ δ (t - t^{'}),

F_{B} = 〈 F_{B} 〉 + F_{c B} + F_{r B}, 〈 F_{B} 〉 = 0,

〈 F_{c B} (t) F_{c B}^{†} (t^{'}) 〉 = 2 γ_{c b} δ (t - t^{'}),

〈 F_{r B} (t) F_{r B}^{†} (t^{'}) 〉 = 2 γ δ (t - t^{'}),

where 〈...〉 stands for ensemble averaging, and P is the power of the external optical pump of the mode A.

The Langevin force describing the thermal fluctuations of the mechanical system is defined in the similar way

〈 F_{C} 〉 = 0,

〈 F_{C} (t) F_{C}^{†} (t^{'}) 〉 = 2 γ_{M} ({\bar{n}}_{t h} + 1) δ (t - t^{'})

where n̄_th = [exp(h̄ω_c/k_BT) − 1]⁻¹ is the initial averaged number of thermal phonons in the mechanical mode, T is the ambient temperature, and k_B is the Boltzmann constant. The actual number of thermal phonons in the mode changes due to the interaction with light.

We present the slowly-varying complex amplitudes of the optical and mechanical modes as

A = | A | e^{i ϕ_{A}},

B = | B | e^{i ϕ_{B}},

C = | C | e^{i ϕ_{C}},

and derive two sets of equations for the amplitude and phase parts of the complex amplitudes from Eqs. (1)–(3)

| \dot{A} | = - (γ + γ_{c a}) | A | - g | C | | B | sin ϕ + | 〈 F_{A} 〉 | cos (ϕ_{F_{A}} - ϕ_{A}) + F_{A r},

| \dot{B} | = - (γ + γ_{c b}) | B | + g | C | | A | sin ϕ + F_{B r},

| \dot{C} | = - γ_{M} | C | + g | B | | A | sin ϕ + F_{C r};

and

{\dot{ϕ}}_{A} = - (ω_{a} - ω_{0}) - g \frac{| C | | B |}{| A |} cos ϕ + \frac{| 〈 F_{A} 〉 |}{| A |} sin (ϕ_{F_{A}} - ϕ_{A}) + \frac{F_{A i}}{| A |},

{\dot{ϕ}}_{B} = - (ω_{b} - ω_{-}) - g \frac{| C | | A |}{| B |} cos ϕ + \frac{F_{B i}}{| B |},

{\dot{ϕ}}_{C} = - (ω_{c} - ω_{M}) - g \frac{| B | | A |}{| C |} cos ϕ + \frac{F_{C i}}{| C |};

where

ϕ = ϕ_{A} - ϕ_{B} - ϕ_{C},

F_{A r} = \frac{1}{2} (F_{c A} e^{- i ϕ_{A}} + F_{c A}^{†} e^{i ϕ_{A}}) + \frac{1}{2} (F_{r A} e^{- i ϕ_{A}} + F_{r A}^{†} e^{i ϕ_{A}}),

F_{B r} = \frac{1}{2} (F_{c B} e^{- i ϕ_{B}} + F_{c B}^{†} e^{i ϕ_{B}}) + \frac{1}{2} (F_{r B} e^{- i ϕ_{B}} + F_{r B}^{†} e^{i ϕ_{B}}),

F_{C r} = \frac{1}{2} (F_{C} e^{- i ϕ_{C}} + F_{C}^{†} e^{i ϕ_{C}}),

F_{A i} = \frac{1}{2 i} (F_{c A} e^{- i ϕ_{A}} - F_{c A}^{†} e^{i ϕ_{A}}) + \frac{1}{2 i} (F_{r A} e^{- i ϕ_{A}} - F_{r A}^{†} e^{i ϕ_{A}}),

F_{B i} = \frac{1}{2 i} (F_{c B} e^{- i ϕ_{B}} - F_{c B}^{†} e^{i ϕ_{B}}) + \frac{1}{2 i} (F_{r B} e^{- i ϕ_{B}} - F_{r B}^{†} e^{i ϕ_{B}}),

F_{C i} = \frac{1}{2 i} (F_{C} e^{- i ϕ_{C}} - F_{C}^{†} e^{i ϕ_{C}}) .

Equations (16)–(21) completely describe the triply-resonant opto-mechanical interaction.

3. Solution

Using the set of equations for the phase [Eqs. (19)–(21)] we get the equation for phase difference ϕ

\begin{array}{l} \dot{ϕ} = - (ω_{a} - ω_{b} - ω_{c}) + g cos ϕ (\frac{| B | | A |}{| C |} + \frac{| C | | A |}{| B |} - \frac{| C | | B |}{| A |}) + \\ \frac{| 〈 F_{A} 〉 |}{| A |} sin (ϕ_{F_{A}} - ϕ_{A}) + \frac{F_{A i}}{| A |} - \frac{F_{B i}}{| B |} - \frac{F_{C i}}{| C |}, \end{array}

We introduce ψ = ϕ − π/2 and rewrite Eq. (29) as

\begin{array}{l} \dot{ψ} = - (ω_{a} - ω_{b} - ω_{c}) - g sin ψ (\frac{| B | | A |}{| C |} + \frac{| C | | A |}{| B |} - \frac{| C | | B |}{| A |}) \\ + \frac{| 〈 F_{A} 〉 |}{| A |} sin (ϕ_{F_{A}} - ϕ_{A}) + \frac{F_{A i}}{| A |} - \frac{F_{B i}}{| B |} - \frac{F_{C i}}{| C |} . \end{array}

Next, we introduce an expectation time-independent value for the phase difference, 〈ψ〉, and fluctuational time-dependent part, δϕ, so that ψ = 〈ψ〉 + δϕ, and get

(ω_{a} - ω_{b} - ω_{c}) + g sin 〈 ψ 〉 (\frac{| B | | A |}{| C |} + \frac{| C | | A |}{| B |} - \frac{| C | | B |}{| A |}) = \frac{| 〈 F_{A} 〉 |}{| A |} sin (ϕ_{F_{A}} - ϕ_{A}),

\begin{array}{l} δ \dot{ϕ} + g δ ϕ cos 〈 ψ 〉 (\frac{| B | | A |}{| C |} + \frac{| C | | A |}{| B |} - \frac{| C | | B |}{| A |}) = \\ δ [- g sin 〈 ψ 〉 (\frac{| B | | A |}{| C |} + \frac{| C | | A |}{| B |} - \frac{| C | | B |}{| A |}) + \\ \frac{| 〈 F_{A} 〉 |}{| A |} sin (ϕ_{F_{A}} - ϕ_{A})] + \frac{F_{A i}}{| A |} - \frac{F_{B i}}{| B |} - \frac{F_{C i}}{| C |}, \end{array}

where δ[...] means a deviation from the expectation value.

Using Eqs. (16)–(21) we find a set of relationships for the expectation values of the oscillator parameters in the steady state

\frac{{| B |}^{2}}{{| C |}^{2}} = \frac{γ_{M}}{γ + γ_{c b}},

{| A |}^{2} = \frac{γ_{M}}{γ + γ_{c b}} \frac{{| Γ_{B} |}^{2}}{g^{2}}

\frac{ω_{b} - ω_{-}}{ω_{c} - ω_{M}} = \frac{γ + γ_{c b}}{γ_{M}},

e^{i 〈 ψ 〉} = e^{i ϕ_{Γ_{B}}} .

The expectation value of the amplitude of the electric field in the pumped mode increases, while below the oscillation threshold, with increase of the pump power, in accordance with expression |A| = |F_A|/|Γ_A|, and then stays constant, in accordance with Eq. (34).

For the sake of simplicity, we assume that the Stokes sideband has much lower power compared with the pump, |A| ≫ |B|, and that the system is triply-resonant (the expectation values of the frequencies are resonant with corresponding modes). We note that

\frac{| B | | A |}{| C |} + \frac{| C | | A |}{| B |} - \frac{| C | | B |}{| A |} = - \frac{g}{| Γ_{B} |} {| B |}^{2} + (\sqrt{\frac{γ_{M}}{γ + γ_{c b}}} + \sqrt{\frac{γ + γ_{c b}}{γ_{M}}}) \sqrt{\frac{γ_{M}}{γ + γ_{c b}}} \frac{| Γ_{B} |}{g},

so that for the case of relatively weak Stokes sideband and all-resonant tuning

g (\frac{| B | | A |}{| C |} + \frac{| C | | A |}{| B |} - \frac{| C | | B |}{| A |}) ≃ γ_{M} + γ + γ_{c b} .

Using the assumptions and Eq. (38) we find

δ [- g sin 〈 ψ 〉 (\frac{| B | | A |}{| C |} + \frac{| C | | A |}{| B |} - \frac{| C | | B |}{| A |}) + \frac{| 〈 F_{A} 〉 |}{| A |} sin (ϕ_{F_{A}} - ϕ_{A})] \approx (δ ϕ_{F_{A}} - δ ϕ_{A}) \frac{| 〈 F_{A} 〉 |}{| A |},

where we took into account that, in accordance with Eq. (31) and Eq. (36), 〈ϕ_{F_A} − ϕ_A〉 = 0 for the resonant tuning.

We obtain a set of linear equations consisting of reduced Eq. (19) and Eq. (32)

δ \dot{ϕ} + (γ_{M} + γ + γ_{c b}) δ ϕ = (δ ϕ_{F_{A}} - δ ϕ_{A}) \frac{| 〈 F_{A} 〉 |}{| A |} + \frac{F_{A i}}{| A |} - \frac{F_{B i}}{| B |} - \frac{F_{C i}}{| C |},

δ {\dot{ϕ}}_{A} = (δ ϕ_{F_{A}} - δ ϕ_{A}) \frac{| 〈 F_{A} 〉 |}{| A |} + \frac{F_{A i}}{| A |} .

The condition of small Stokes sideband can be written in the form

\frac{| 〈 F_{A} 〉 |}{| A |} = γ + γ_{c a} .

Substituting Eq. (42) into Eq. (40) and Eq. (41) we derive

δ {\dot{ϕ}}_{A} + (γ + γ_{c a}) δ ϕ_{A} = \frac{F_{A i}}{| A |},

δ {\dot{ϕ}}_{B} + δ {\dot{ϕ}}_{C} + (γ_{M} + γ + γ_{c b}) (δ ϕ_{B} + δ ϕ_{C}) = (γ_{M} + γ + γ_{c b}) δ ϕ_{A} + \frac{F_{B i}}{| B |} + \frac{F_{C i}}{| C |} .

The third equation for phase deviations can be derived from Eq. (20) and Eq. (21):

δ {\dot{ϕ}}_{B} \sqrt{\frac{γ_{M}}{γ + γ_{c b}}} - δ {\dot{ϕ}}_{C} \sqrt{\frac{γ + γ_{c b}}{γ_{M}}} = \sqrt{\frac{γ_{M}}{γ + γ_{c b}}} \frac{F_{B i}}{| B |} - \sqrt{\frac{γ + γ_{c b}}{γ_{M}}} \frac{F_{C i}}{| C |} .

We solve the set of linear differential equations [Eqs. (43)–(45)] using Fourier transform, e.g.

F_{B i} = \int_{- \infty}^{\infty} f_{B i} (ω) e^{- i ω t} \frac{d ω}{2 π},

F_{C i} = \int_{- \infty}^{\infty} f_{C i} (ω) e^{- i ω t} \frac{d ω}{2 π},

where f̂_Bi(ω) and f̂_Ci(ω) are the Fourier components of the noise,

〈 F_{B i} (t) F_{B i} (t^{'}) 〉 = \frac{1}{2} (γ + γ_{c b}) δ (t - t^{'}),

〈 F_{C i} (t) F_{C i} (t^{'}) 〉 = γ_{M} ({\bar{n}}_{t h} + \frac{1}{2}) δ (t - t^{'}),

〈 f_{B i} (ω) f_{B i} (ω^{'}) 〉 = π (γ + γ_{c b}) δ (ω + ω^{'}),

〈 f_{C i} (ω) f_{C i} (ω^{'}) 〉 = 2 π γ_{M} ({\bar{n}}_{t h} + \frac{1}{2}) δ (ω + ω^{'}) .

Readout of the OMO signal is accomplished by tracking the frequency beat note produced by the optical pump and the generated sideband on a fast photodiode, so the phase and frequency of the measured oscillation signal are determined by the argument of the product of optical amplitudes (AB^*). The phase noise of the signal is given by difference δϕ_B − δϕ_A. On the other hand, the mechanical frequency can be read using electronics means, e.g. a capacitive displacement sensor. By neglecting the electronics back action, we can estimate the phase noise of the signal evaluating δϕ_C. We find the expressions for Fourier amplitudes of these parameters

δ ϕ_{A} (ω) = \frac{f_{A i}}{| A |} \frac{1}{- i ω + γ + γ_{c a}},

\begin{array}{l} δ ϕ_{A} (ω) - δ ϕ_{B} (ω) = \frac{- i ω + γ_{M}}{- i ω + γ_{M} + γ + γ_{c b}} δ ϕ_{A} (ω) - \\ \frac{- i ω + γ_{M}}{- i ω (- i ω + γ_{M} + γ + γ_{c b})} \frac{f_{B i}}{| B |} + \frac{γ + γ_{c b}}{- i ω (- i ω + γ_{M} + γ + γ_{c b})} \frac{f_{C i}}{| C |}, \end{array}

\begin{array}{l} δ ϕ_{C} (ω) = \frac{γ_{M}}{- i ω + γ_{M} + γ + γ_{c b}} δ ϕ_{A} (ω) - \\ \frac{γ_{M}}{- i ω (- i ω + γ_{M} + γ + γ_{c b})} \frac{f_{B i}}{| B |} + \frac{- i ω + γ + γ_{c b}}{- i ω (- i ω + γ_{M} + γ + γ_{c b})} \frac{f_{C i}}{| C |} . \end{array}

Equation (53) and Eq. (54) can be used to find the phase noise of the OMO. For example, single-sideband phase noise ℒ_c(ω) of the mechanical oscillation is defined as,

〈 δ ϕ_{C} (t) δ ϕ_{C} (t - τ) 〉 = \int_{- \infty} ℒ_{c} (ω) e^{i ω t} \frac{d ω}{2 π} .

Using definition

δ ϕ_{C} (t) = \int_{- \infty}^{\infty} δ ϕ_{C} (ω) e^{- i ω t} \frac{d ω}{2 π},

we find

\begin{array}{l} ℒ_{c} = \frac{γ_{M}^{2}}{ω^{2} + {(γ_{M} + γ + γ_{c b})}^{2}} \frac{γ_{c a}^{2}}{ω^{2} + {(γ + γ_{c a})}^{2}} ℒ_{i n} + \\ \frac{γ_{M}^{2}}{ω^{2} [ω^{2} + {(γ_{M} + γ + γ_{c b})}^{2}]} \frac{γ + γ_{c b}}{2 {| B |}^{2}} + \frac{ω^{2} + {(γ + γ_{c b})}^{2}}{ω^{2} [ω^{2} + {(γ_{M} + γ + γ_{c b})}^{2}]} \frac{γ_{M}}{{| C |}^{2}} ({\bar{n}}_{t h} + \frac{1}{2}), \end{array}

where ℒ_in stands for Fourier frequency dependent input phase noise of the pump laser. This value is usually much larger than the contribution from the quantum white noise of the laser.

Using similar reasoning we obtain

\begin{array}{l} ℒ_{a - b} = \frac{ω^{2} + γ_{M}^{2}}{ω^{2} + {(γ_{M} + γ + γ_{c b})}^{2}} \frac{γ_{c a}^{2}}{ω^{2} + {(γ + γ_{c a})}^{2}} ℒ_{in} + \\ \frac{ω^{2} + γ_{M}^{2}}{ω^{2} [ω^{2} + {(γ_{M} + γ + γ_{c b})}^{2}]} \frac{γ + γ_{c b}}{2 {| B |}^{2}} + \frac{{(γ + γ_{c b})}^{2}}{ω^{2} [ω^{2} + {(γ_{M} + γ + γ_{c b})}^{2}]} \frac{γ_{M}}{{| C |}^{2}} ({\bar{n}}_{t h} + \frac{1}{2}) . \end{array}

It is also useful to write an expression for the phase noise of the optical Stokes mode, determining the stability and spectral purity of the Brillouin laser [14]

\begin{array}{l} ℒ_{b} = \frac{{(γ + γ_{c b})}^{2}}{ω^{2} + {(γ_{M} + γ + γ_{c b})}^{2}} \frac{γ_{c a}^{2}}{ω^{2} + {(γ + γ_{c a})}^{2}} ℒ_{in} + \\ \frac{ω^{2} + γ_{M}^{2}}{ω^{2} [ω^{2} + {(γ_{M} + γ + γ_{c b})}^{2}]} \frac{γ + γ_{c b}}{2 {| B |}^{2}} + \frac{{(γ + γ_{c b})}^{2}}{ω^{2} [ω^{2} + {(γ_{M} + γ + γ_{c b})}^{2}]} \frac{γ_{M}}{{| C |}^{2}} ({\bar{n}}_{t h} + \frac{1}{2}) . \end{array}

Equations (57)–(59) can be simplified further using the ratio between the photon number in the Stokes mode and the phonon number in the mechanical mode [Eq. (33)], and the expression connecting the number of pump and Stokes photons and output power of the pump and Stokes light

{| A |}^{2} = \frac{2 γ_{c a}}{{(γ + γ_{c a})}^{2}} \frac{P}{\bar{h} ω_{0}}, {| B |}^{2} = \frac{P_{Bout}}{2 γ_{c b} \bar{h} ω_{0}},

where we assumed that the carrier frequency of the Stokes light is approximately equal to the frequency of the pump light. Finally we get

\begin{array}{l} ℒ_{a - b} = \\ \frac{ω^{2} + γ_{M}^{2}}{ω^{2} + {(γ_{M} + γ + γ_{c b})}^{2}} \frac{γ_{c a}^{2}}{ω^{2} + {(γ + γ_{c a})}^{2}} ℒ_{in} + \\ \frac{ω^{2} + 2 γ_{M}^{2} ({\bar{n}}_{t h} + 1)}{[ω^{2} + {(γ_{M} + γ + γ_{c b})}^{2}]} \frac{γ_{c b} (γ + γ_{c b})}{ω^{2}} \frac{\bar{h} ω_{0}}{P_{Bout}}, \end{array}

\begin{array}{l} ℒ_{c} = \\ \frac{γ_{M}^{2}}{ω^{2} + {(γ_{M} + γ + γ_{c b})}^{2}} \frac{γ_{c a}^{2}}{ω^{2} + {(γ + γ_{c a})}^{2}} ℒ_{i n} + \\ \frac{γ_{M}^{2} {{\bar{n}}_{t h} + 1 + ω^{2} / [2 {(γ + γ_{c b})}^{2}]}}{[ω^{2} + {(γ_{M} + γ + γ_{c b})}^{2}]} \frac{2 γ_{c b} (γ + γ_{c b})}{ω^{2}} \frac{\bar{h} ω_{0}}{P_{Bout}}, \end{array}

\begin{array}{l} ℒ_{b} = \\ \frac{{(γ + γ_{c b})}^{2}}{ω^{2} + {(γ_{M} + γ + γ_{c b})}^{2}} \frac{γ_{c a}^{2}}{ω^{2} + {(γ + γ_{c a})}^{2}} ℒ_{i n} + \\ \frac{ω^{2} + 2 γ_{M}^{2} ({\bar{n}}_{t h} + 1)}{[ω^{2} + {(γ_{M} + γ + γ_{c b})}^{2}]} \frac{γ_{c b} (γ + γ_{c b})}{ω^{2}} \frac{\bar{h} ω_{0}}{P_{Bout}} . \end{array}

Equation (61) gives a complete description of the phase noise characteristic of the radio frequency photonic oscillator based on demodulation of the light output of the OMO on a fast photodiode. Equation (62) shows the limiting phase noise of the oscillating mechanical mode that could be observed, if the mechanical oscillation is measured using an external devise that does not disturb the system. Equation (63) describes the phase noise of the SBS laser. These expressions can be used to find the linewidth and Allan variance of the corresponding OMO signals. By definition, the Allan variance of the frequency of the oscillator is given by

σ^{2} (τ) = \int_{0}^{\infty} \frac{4 ω^{2} ℒ}{ω_{0}^{2}} \frac{{sin}^{4} (ω τ / 2)}{{(ω τ / 2)}^{2}} \frac{d ω}{2 π} .

Let us consider the case where the OMO signal is retrieved by demodulation of the light leaving the resonator on a photodiode. The single sided power density of the phase noise is related to the linewidth of the oscillator Δν as ℒ(ω → 0) = 2πΔν/ω². We find for the cases of relatively low (γ_M ≫ γ) and relatively high-Q (γ_M ≪ γ) of the mechanical mode

{Δ ν_{a - b} |}_{γ_{M} ≫ γ} ≃ \frac{γ_{c a}^{2}}{{(γ + γ_{c a})}^{2}} Δ ν_{pump} + γ_{c b} (γ + γ_{c b}) ({\bar{n}}_{t h} + 1) \frac{\bar{h} ω_{0}}{π P_{Bout}},

{Δ ν_{a - b} |}_{γ_{M} ≪ γ} ≃ \frac{γ_{M}^{2}}{{(γ + γ_{c b})}^{2}} \frac{γ_{c a}^{2}}{{(γ + γ_{c a})}^{2}} Δ ν_{pump} + \frac{γ_{c b}}{γ + γ_{c b}} γ_{M}^{2} ({\bar{n}}_{t h} + 1) \frac{\bar{h} ω_{0}}{π P_{Bout}} .

Here we assume that γ ≈ γ_ca ≈ γ_cb. Therefore, in the case of low-Q mechanical mode, the linewidth of the radio frequency beat note generated by the OMO is determined by the linewidth of the pumping light; while in the case of high-Q mechanical mode, the linewidth is described by Schawlow-Townes-like formula [9] and the phase noise of the pumping light is suppressed.

It is also worth noting that in the case of high-Q mechanical mode the dependence of the phase noise of the Stokes optical component [Eq. (63)] on the phase noise of the pumping laser is suppressed rather significantly. This result supports the findings made for Brillouin fiber lasers, where the Stokes can be several orders of magnitude narrower than the linewidth of the pump laser [17–20].

4. Discussion

Let us estimate the phase noise of the oscillator and compare it to the phase noise of an electronic oscillator. We assume that Q_M = 10⁵, ω_M = 2π × 10⁸ rad/s, γ_M = 2π × 500 rad/s, γ = 2π × 10⁴ rad/s, γ_cb = γ_ca = 2π × 10⁵ rad/s, ω₀ = 2π × 2 × 10¹⁴ rad/s, Δν_pump = 1 kHz, P_Bout = 100 μW, and P = 1 mW.

The light escaping the cavity is demodulated on a photodiode to produce the radio frequency signal. The photodiode introduces thermal noise and frequency independent (white) shot noise

ℒ_{P D} = \frac{2 q R ρ P_{D} + k_{B} T}{P_{R F}}

in addition to the phase noise [Eq. (61)] coming from the opto-mechanical oscillator. Here ρ is the resistance of the photodiode, R = ηq/h̄ω₀ is the responsivity of the photodiode, η is quantum efficiency of the photodiode, q is the electron charge, and P_D is the total optical power reaching the photodiode. We assume that P_D = P. The thermal noise depends on the ambient temperature T and the power of the radio frequency signal leaving the photodiode P_RF = 2ρR²PP_Bout. To find its value we assume that the resistance at the output of the photodiode is ρ = 50 Ohm, the responsivity of the photodiode is 0.8 A/W, and the temperature is T = 300 K. For these parameters, the expected power of the radio frequency signal escaping the photodiode is P_RF = 6 μW.

Adding ℒ_PD and ℒ_a₋_b we plot the spectrum of the single sideband phase noise [line (1) in Fig. 1(a)]. It is dominated by the phase noise of the pump laser [line (2) in Fig. 1(a)]. If a very narrow linewidth laser is used instead, the phase noise of the OMO would be given by the convolution of curves (4) and (3) (phase diffusion of the oscillator as well as shot noise) in Fig. 1(a). In the same figure we also show the phase noise of a commercially available 100 MHz oven controlled quartz oscillator [line (5) in Fig. 1(a)]. Apparently, the electronic oscillator has much lower phase noise than the OMO, even though thermal drift resulting in the flicker noise is not taken into account in our analysis.

Fig. 1 (a) Phase noise of the 100 MHz opto-mechanical oscillator (1) versus offset frequency f = ω/2π characterized with parameters defined in the text. Curve (2) defines the contribution from the phase noise of the pumping laser. Curve (3) stands for the thermal and white shot noise defined in Eq. (67). Curve (4) shows phase diffusion of the opto-mechanical oscillator. Curve (5) stands for phase noise of a commercially available 100 MHz ovenized quartz oscillator. (b) Phase noise of the 1 GHz opto-mechanical oscillator (1) characterized with parameters defined in the text. Curve (2) defines the contribution from the phase noise of the pumping laser. Curve (3) stands for the thermal and white shot noise defined in Eq. (67). Curve (4) shows phase diffusion of the opto-mechanical oscillator. Curve (5) stands for phase noise of a commercially available 100 MHz ovenized quartz oscillator corrected by 20 dB to reflect the frequency difference.

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The relative performance of the opto-mechanical oscillator can be improved if higher frequency signals are generated and if the optical modes have comparably low quality factors. For example, assuming that Q_M = 10⁴, ω_M = 2π × 10⁹ rad/s, γ_M = 2π × 5 × 10⁴ rad/s, γ = 2π × 10⁴ rad/s, γ_cb = γ_ca = 2π × 10⁷ rad/s, ω₀ = 2π × 2 × 10¹⁴ rad/s, Δν_pump = 10 Hz, P_Bout = 1 mW, and P = 10 mW, we find that the oscillator has phase noise shown by line (1) in Fig. 1(b).

We have studied phase fluctuations of the OMO. In addition to phase fluctuations, the quality of an oscillator signal depends on frequency drift in the system. The drift results from stress release, thermal sensitivity, as well as other technical features of a particular oscillator structure. Let us evaluate the sources of frequency drift of the opto-mechanical oscillator. We note that, according to Eq. (35) and energy conservation law, ω₀ = ω₋ +ω_M,

ω_{0} - ω_{-} = ω_{M} = \frac{γ + γ_{c b}}{γ + γ_{c b} + γ_{M}} ω_{c} + \frac{γ_{M}}{γ + γ_{c b} + γ_{M}} (ω_{0} - ω_{b}) .

The OMO frequency drifts if the frequencies of the optical, ω_b, and mechanical, ω_c, modes drift.

It is possible to lock the frequency of the pump light to the frequency of the corresponding optical mode, ω₀ = ω_a. Then the OMO frequency becomes dependent on the parameters of the microcavity only: ω_a − ω_b as well as ω_c. This is similar to the case of a quartz oscillator, except the OMO frequency depends on both the eigenfrequency of the mechanical and optical modes. The dependence of the oscillation on the drift of the mechanical mode can be suppressed if the bandwidth of the optical mode is much smaller than the bandwidth of the mechanical mode.

A clear advantage of the OMO over an electronic oscillator is that it can be readily stabilized by locking the pump frequency to the frequency of the corresponding cavity mode, ω₀ = ω_a, and then locking the temperature of the cavity to a thermally insensitive optical reference line, for example, an atomic transition. Such a stabilization of the temperature will stabilize the long term drift of the oscillation [25].

5. Conclusion

We have theoretically studied the phase noise of a triply-resonant opto-mechanical oscillator based on an optical microcavity. The oscillator generates Stokes optical photons and mechanical phonons out of photons of a coherent pumping light. We have shown that the spectral purity of the opto-mechanical signal is primarily limited by the phase noise of the pump laser. The overall short term performance of the oscillator is expected to be worse than the performance of a conventional electronic oscillator. An important advantage of the opto-mechanical oscillator is the possibility to optically stabilize it. With a proper electronic locking scheme it is possible to transfer frequency stability from the optical domain to radio frequency domain. If stabilized to an atomic transition, an opto-mechanical oscillator can be made to have better long term stability compared to an oven controlled quartz oscillator.

Acknowledgments

Andrey Matsko acknowledges illuminating discussions with Prof. Kerry Vahala as well as helpful comments of Prof. Michail Gorodetsky, Prof. Sunil Bhave, Prof. Mani Hossein Zadeh, and Prof. Gaurav Bahl.

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Stability of resonant opto-mechanical oscillators

Abstract

1. Introduction

2. Basic equations

3. Solution

4. Discussion

5. Conclusion

Acknowledgments

References and links

Cited By

Figures (1)

Equations (69)

Optics Express