Phase noise of self-sustained optomechanical oscillators

In this paper we present a theory that predicts the phase noise characteristics of self-sustained optomechanical oscillators. By treating the cavity optomechanical system as a feedback loop consisting of an optical cavity and a mechanical resonator, we analytically derive the transfer functions relating the amplitude and phase noise of all the relevant dynamical quantities from the quantum Langevin equations, and obtain a closed-form expression for the phase noise spectral densities contributed from thermomechanical noise, photon shot noise, and low-frequency technical laser noise. We numerically calculate the phase noise for various situations and perform a sample calculation for an experimentally demonstrated system. We also show that the presented model reduces to the well-known Leeson's phase noise model when the amplitude noise and the amplitude to phase noise intertransfers are ignored. This paper not only addresses the practical questions about oscillator phase noise characteristics but also presents a theoretical framework for studying cavity optomechanical systems with large oscillation amplitude.

Figure 1

(a) Schematics of a Fabry-Pérot cavity with a movable mirror. The output field ${\widehat{s}}_{\mathrm{out}}\left(t\right)$ represents the reflected light from the cavity. (b) Schematics of a microdisk resonator with movable boundary. The output field ${\widehat{s}}_{\mathrm{out}}\left(t\right)$ represents the transmitted light in the thru-port.

Figure 2

(a) Feedback loop of a cavity optomechanical system. (b) Feedback loop of a typical electrical oscillator. (c) Feedback loop for the small signals in a cavity optomechanical system.

Figure 3

(a) Illustration of the spectral-filter decomposition described in Eq. (10). (b) Illustration of the amplitude and phase quadrature described in Eq. (11).

Figure 4

A qualitative illustration of the frequency spectra of all the relevant input and output quantities. The arrows represent the $\delta$-function spectrum of the coherent amplitude and the shaded areas represent the spectral densities of the noise. Dashed lines are used to separate different regions under the spectral filter decomposition. The quantities ${\widehat{s}}_{\mathrm{in}},\delta {\widehat{s}}_{\mathrm{vac}},\widehat{a}$, and ${\widehat{s}}_{\mathrm{out}}$ are expressed in the nonrotating frame to emphasize the original frequency scale. The three spectra of interest for characterizing the oscillator performance are highlighted by the red bounding box (${\widehat{n}}_1\left[\omega \right],{\widehat{P}}_{\mathrm{out},1}\left[\omega \right]$, and ${\widehat{x}}_1\left[\omega \right]$).

Figure 5

Relative magnitude of the higher harmonic components of (a) $a_m$ and (b) $P_{\mathrm{out},m}$ plotted against index $m$ at different $\kappa /\Omega$ and $\tilde{x}$. In all the calculations, the optimal detuning described by the white dashed line in Fig. 6 is used. For the calculation of $P_{\mathrm{out},m}$, critical coupling condition ${\kappa }_e={\kappa }_i$ is assumed.

Figure 6

(a) Contour plot of (upper) $-{\gamma }_{\mathrm{om}}(\tilde{x}=0)/\gamma C$ and (lower) $-{\nu }_{\mathrm{om}}(\tilde{x}=0)/\gamma C$ as functions of $\Delta /\Omega$ and $\kappa /\Omega$. The white dashed lines indicate the optimal detuning where the optomechanical antidamping at zero displacement $-{\gamma }_{\mathrm{om}}(\tilde{x}=0)$ is maximum. (b) (upper) $-{\gamma }_{\mathrm{om}}/\gamma C$ and (lower) $-{\nu }_{\mathrm{om}}/\gamma C$ plotted as functions of $\tilde{x}$ at different $\kappa /\Omega$. (c) (upper) The limit-cycle amplitude $\tilde{x}$ and (lower) frequency shift $\nu /\gamma$ plotted as functions of $\kappa /\Omega$. (d) (upper) The limit-cycle amplitude $\tilde{x}$ and (lower) the normalized output power $|P_{\mathrm{out},1}/P_{\mathrm{in}}|$ plotted as functions of $C$. In the calculations involved in (b), (c), and (d), optimal detuning described by the white dashed line of (a) is used.

Figure 7

Schematic illustrating the adiabaticity condition. The area shaded with the grey color indicates the frequency range $\omega <\kappa$ where adiabatic assumption is valid. The dashed line and the arrow indicate the frequency range $\omega <\Omega /2$ where $\omega$ is defined. In the unresolved sideband regime the adiabatic assumption is always valid.

Figure 8

(a) (upper) ${\gamma }_{\mathrm{om}}^{\prime }/\gamma$ and (lower) ${\nu }_{\mathrm{om}}^{\prime }/\gamma$ plotted as a function of $\kappa /\Omega$. (b) (upper) $-{\eta }_R$ and (lower) $-{\eta }_I$ plotted as a function of $\kappa /\Omega$.

Figure 9

Phase noise spectral densities of (a) $\delta {\widehat{x}}_1$ and (b) $\delta {\widehat{n}}_1$ and $\delta {\widehat{P}}_{\mathrm{out},1}$ due to thermomechanical noise. Solid lines are used for the RSR and dashed lines are used for the USR. (c) ${\nu }_{\mathrm{om}}^{\prime 2}/{\gamma }_{\mathrm{om}}^{\prime 2}$ plotted against $\kappa /\Omega$. (d) Normalized phase diffusion constant ${\mathcal{D}}_{\mathrm{th}}/{\mathcal{D}}_{\mathrm{th}}^{\left(0\right)}(\kappa /\Omega \to 0)$ plotted against $\kappa /\Omega$. ${\mathcal{D}}_{\mathrm{th}}^{\left(0\right)}$ denotes the diffusion constant for the limit-cycle with the smallest displacement when multiple solutions exist.

Figure 10

Correlation matrix elements of the optical shot noise force [Eq. (46)] in the adiabatic limit plotted against $\kappa /\Omega$.

Figure 11

Phase noise spectral densities of (a) $\delta {\widehat{x}}_1$, (b) $\delta {\widehat{n}}_1$, and (c) $\delta {\widehat{P}}_{\mathrm{out},1}$ due to photon shot noise. $\gamma /\Omega ={10}^{-4}$ and ${\kappa }_e={\kappa }_i$ is assumed in the calculation. Solid lines are used for the RSR and dashed lines are used for the USR.

Figure 12

Noise floor of (a) ${\mathcal{S}}_{\delta {\widehat{n}}_1^{\varphi }}^{SS}$ and (b) ${\mathcal{S}}_{\delta {\widehat{P}}_{\mathrm{out},1}^{\varphi }}^{SS}$ plotted against $\kappa /\Omega$. (c) Normalized phase diffusion constant ${\mathcal{D}}_{\mathrm{vac}}/{\mathcal{D}}_{\mathrm{vac}}^{\left(0\right)}(\kappa /\Omega \to 0)$ plotted against $\kappa /\Omega$. ${\mathcal{D}}_{\mathrm{vac}}^{\left(0\right)}$ denotes the diffusion constant for the limit cycle with the smallest displacement when multiple solutions exist.

Figure 13

Phase noise spectral densities of (a) $\delta {\widehat{x}}_1$ and (b) $\delta {\widehat{n}}_1$ and $\delta {\widehat{P}}_{\mathrm{out},1}$ due to low-frequency RIN noise. Solid lines are used for the RSR and dashed lines are used for the USR.

Figure 14

(a) Phase noise spectrum calculated for the optomechanical oscillator reported in Ref. [11]. (b) Phase noise at 100 kHz plotted against $\Delta /2\kappa$. (c) Phase noise at 100 kHz plotted against $P_{\mathrm{in}}/P_{\mathrm{th}}$. (a), (b), and (c) are for comparison with Figs. 7, 9, and 9 of Ref. [11].

Figure 15

(a) Simplified oscillator feedback loop in the phase noise space. (b) Phase noise transfer function with respect to input 1 and input 2.