Mechanical quality factor estimation from a nonlinear optical cavity response

arXiv:2609.07393 · physics.optics, physics.app-ph, physics.space-ph, quant-ph · Submitted 2026-09-07 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "Mechanical quality factor estimation from a nonlinear optical cavity response".

Mira: Here's the gist of the paper. The gist: This work introduces an all-optical ringdown method in the unresolved-sideband regime that exploits nonlinear, time-averaged optical response induced by strong mechanical motion to estimate intrinsic damping rate and quality factor (Q) without resolving the mechanical oscillation Analytical Model The paper derives an analytical model for the time-averaged optical cavity response probed by a weak frequency-scanning laser, which is subject to sinusoidal frequency modulation The power spectral density of a resonant system usually follows a Lorentzian lineshape L(ω, ω0) = 1/(1 +

2(ω − ω0)/2: 2), characterized by a linewidth κ and a resonant frequency ω0.

Kai: The time-averaged response Λ(ω, α) is described by an integral involving the Lorentzian lineshape modulated by sinusoidal frequency modulation This integral is rewritten in a more general form after normalizing the detuning ν = 2(ω − ω0)/κ and the amplitude A = 2α/κ.

Mira: The closed-form, real-valued expression for the solution is given by Λ(ν, A) = s√η+η− + ηA/(2η+η−), where ηA = 1 + A squared − ν squared and η± = 1 + (A ± ν) squared.

Kai: Amplitude Estimation of Modulation The analysis distinguishes between two regimes based on the ratio between modulation amplitude (A) and the resonance’s linewidth (κ), showing a simple Lorentzian approximation for weak modulation where the sensitivity S = ∂Λ/∂A is maximized by measuring the variation of the in-resonance response.

Title and authors: Kai: So, we're looking at this paper titled "Mechanical quality factor estimation from a nonlinear optical cavity response," and the authors are Ali, Tomasella, Requena, Huck, and Lund Andersen. They are focusing on figuring out intrinsic damping rate and quality factor Q in optomechanical systems.

Mira: That sounds like they’re tackling a problem where measuring those Q factors is hard because probe-induced dynamical backaction messes things up. The paper suggests that we can get those key parameters from just looking at the time-averaged optical cavity response instead of needing to resolve the mechanical oscillation itself.

Lev: I’m still stuck on the core idea here, though. If you don't resolve the oscillation, how do you know what's actually happening mechanically in that system?

Kai: Well, they introduce an all-optical ringdown method in the unresolved-sideband regime that exploits how strong mechanical motion modulates the optical response into a double-horned profile. It’s about using that time-averaged optical response to estimate the intrinsic damping rate and quality factor Q without having to resolve the mechanical oscillation itself <ref:2609.07393#pg1>.

Mira: It sounds like they are bypassing the need for that high-bandwidth measurement, which is usually what limits us in these high-Q systems when dealing with those tricky devices. They derive an analytical model for the time-averaged optical cavity response probed by a weak frequency-scanning laser <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1,the time-averaged optical cavity response probed by a weak frequency-scanning>.

Lev: And they’re saying this method is robust against probe-induced dynamical backaction, which is something we always worry about when we push the probe power of our measurement setup.

Kai: Exactly, that's one of the main points because they show that by continuously scanning the laser across the optical resonance, you get a low-bandwidth optical readout that removes the need to continuously resolve the mechanical spectrum <ref:two thousand six hundred nine point zero seven three nine three#pg2 <ref:2609.07393#pg1>.

Mira: And they’re providing a closed-form expression for this time-averaged response, which is pretty powerful because it lets us analyze it in different regimes based on the modulation amplitude A and resonance linewidth kappa <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1>.

Lev: I wonder about the assumptions behind that closed form. What are the underlying physical constraints they're working with for that equation to be valid?

Mira: That closed form is the core of their analytical model, and it’s what allows them to distinguish between different regimes based on the ratio between modulation amplitude A and the resonance linewidth kappa <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1>.

Lev: What about when you have a strong coherent drive, where A is much greater than one? How does that look in their model?

Kai: They used a membrane-in-the-middle, or MIM system, which is a phononic-crystal nanomechanical resonator made of silicon nitride placed between two highly reflective mirrors patterned with a phononic shield <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1>.

Lev: The numbers there are important for us to see how practical it is for real hardware characterization. They have the mechanical frequency m = one point one three four MHz and the optical resonance linewidth kappa/two pi = thirty-seven point eight MHz <ref:two thousand six hundred nine point zero seven three nine three#pg1.

The paper's summary: Kai: So, to recap, this paper "Mechanical quality factor estimation from a nonlinear optical cavity response" shows how you can figure out the damping rate and quality factor Q of a mechanical resonator just by looking at how light responds when you push it hard enough to make it move <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1,Mechanical quality factor estimation from a nonlinear optical cavity response>.

Mira: Yeah, they’re using this all-optical ringdown method to get those key parameters without having to actually resolve the mechanical oscillation itself <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1>. It’s about exploiting that nonlinear, time-averaged optical response induced by strong mechanical motion <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1,nonlinear, time-averaged optical response induced by strong mechanical motion>.

Kai: They show that you get this time-averaged response probed by a weak frequency-scanning laser <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1,response probed by a weak frequency-scanning laser>. The real kicker is their analytical model for this response, which they write down as a closed form <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1>.

Mira: This closed form lets them distinguish between different regimes based on the modulation amplitude A and the resonance linewidth kappa <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1>. In the weak modulation limit, it’s basically a simple Lorentzian approximation, (nu, A) about L one(nu, A), which they show has a sensitivity S = d /d A maximized by measuring the variation of that in-resonance response <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1>.

Kai: But when you push it hard—when the drive amplitude A is large, say bigger than one—that resonance profile changes from a simple Lorentzian to a double-horned shape <ref:two thousand six hundred nine point zero seven three nine three#pg2 <ref:2609.07393#pg1>.

Mira: And this double-horned profile actually admits an asymptotically affine relation between the peak positions and that modulation amplitude A, which is nu A about A - one/sqrt three as A gets big <ref:two thousand six hundred nine point zero seven three nine three#pg2.

Kai: That linear growth with A for the peak separation means you can track the driving strength really clearly, even when you’re in that strong modulation regime <ref:two thousand six hundred nine point zero seven three nine three#pg2 <ref:2609.07393#pg1>.

Mira: Experimentally, they built this on a membrane-in-the-middle system, which is a phononic-crystal nanomechanical resonator made of silicon nitride <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1>. They set the mechanical frequency m = one point one three four MHz and the optical linewidth kappa/two pi = thirty-seven point eight MHz <ref:two thousand six hundred nine point zero seven three nine three#pg1.

Kai: And they extracted the quality factor Q m from the exponential decay of that mechanical amplitude, where Q m = m / m <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1>. They found a quality factor around one hundred eight point four times ten six which is consistent with other measurements <ref:two thousand six hundred nine point zero seven three nine three#pg2 <ref:2609.07393#pg1>.

Mira: What really matters for me is that this method provides an in situ diagnostic that reduces the impact of probe-induced dynamical backaction, or DBA <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1>. They do this by continuously scanning the laser across the optical resonance, which keeps your time-averaged detuning near zero <ref:two thousand six hundred nine point zero seven three nine three#pg2 <ref:2609.07393#pg1>.

Kai: That continuous scanning is what bypasses the need to resolve fast dynamics or demodulate mechanical oscillations, which are usually huge headaches in these setups <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1>.

Mira: It's a big win because it means you get intrinsic damping rate and quality factor Q are key parameters in optomechanical systems determined from that time-averaged optical cavity response <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1,intrinsic damping rate and quality factor Q are key parameters in optomechanical>.

Lev: So, to wrap up on "Mechanical quality factor estimation from a nonlinear optical cavity response," this paper shows how we can estimate intrinsic damping rate and quality factor Q in optomechanical systems by exploiting that nonlinear, time-averaged optical response induced by strong mechanical motion <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1,Mechanical quality factor estimation from a nonlinear optical cavity response>.

Mira: This approach enables parameter estimation without resolving the fast dynamics or demodulating the mechanical oscillations, which is usually required in conventional approaches <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1>.

Lev: We get to know that intrinsic damping rate and quality factor Q are key parameters in optomechanical systems determined from a time-averaged optical cavity response <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1,intrinsic damping rate and quality factor Q are key parameters in optomechanical>.

The paper's improvements: Kai: So, we’ve covered how this paper "Mechanical quality factor estimation from a nonlinear optical cavity response" uses nonlinear optical responses to estimate mechanical quality factor Q in optomechanical systems <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1,Mechanical quality factor estimation from a nonlinear optical cavity response>.

Mira: The key improvement they suggest is that by using the closed form solution, we can analyze it across different regimes based on the modulation amplitude A and resonance linewidth kappa <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1>.

Kai: In the weak modulation limit, it’s basically a simple Lorentzian approximation, L one(nu, A), which they show has a sensitivity S = d /d A maximized by measuring the variation of that in-resonance response <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1>.

Kai: So to sum up, this paper gives us a way to estimate intrinsic damping rate and quality factor Q in optomechanical systems by exploiting that nonlinear optical response induced by strong mechanical motion <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1,intrinsic damping rate and quality factor Q>.

Lev: The mathematical derivations show that the two maxima of the spectrum are separated by a distance nu = two nu A about two(A - one/sqrt three), which grows linearly with A <ref:two thousand six hundred nine point zero seven three nine three#pg3.

Kai: That linear growth with A is what makes the measurement so sensitive to the driving strength <ref:two thousand six hundred nine point zero seven three nine three#pg3 <ref:2609.07393#pg1>.

Mira: This method lets you get those key parameters without having to resolve the actual mechanical oscillation itself <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1>.

Lev: For our hardware building, it means we have a cleaner way to characterize the damping dynamics of these systems <ref:two thousand six hundred nine point zero seven three nine three#pg2 <ref:2609.07393#pg1>.

Conclusion: Kai: So, we've covered how this paper "Mechanical quality factor estimation from a nonlinear optical cavity response" uses time-averaged optical responses to figure out damping rate and quality factor Q <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1,Mechanical quality factor estimation from a nonlinear optical cavity response>.

Mira: Yeah, the big idea is getting those key mechanical parameters without having to resolve the actual mechanical oscillation itself <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1>. They show that by continuously scanning the laser across that optical resonance, you get this low-bandwidth readout that cuts out a lot of the probe-induced dynamical backaction <ref:two thousand six hundred nine point zero seven three nine three#pg2 <ref:2609.07393#pg1>.

Lev: That low bandwidth is what would actually matter for running this on real hardware; we can't afford to lose signal in a noisy environment <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1>.

Mira: They give us a closed form solution for that response, and they show that even when the modulation is strong, like in the double-horned profile case, there's an asymptotically affine relation between peak positions and modulation amplitude A <ref:two thousand six hundred nine point zero seven three nine three#pg2 <ref:2609.07393#pg1>.

Kai: That means you can track how much you’re driving it really clearly by looking at the shift of those peaks, even when the signal is strong <ref:two thousand six hundred nine point zero seven three nine three#pg2 <ref:2609.07393#pg1>.

Lev: For error correction hardware, knowing that Q factor estimation is robust against these kinds of probe effects is important because DBA usually lands right where your error correction overhead lands <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1>.

Mira: Exactly, and they even give you bounds on the deviations, which is good because it tells you how accurate that approximation stays <ref:two thousand six hundred nine point zero seven three nine three#pg3 <ref:2609.07393#pg1>.

Kai: So what this means for us is we can get an in situ diagnostic that lets us measure Q without needing the high bandwidth or complex demodulation usually required <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1>.

Lev: It simplifies things a lot; less data acquisition time and less noise injection from the measurement itself <ref:two thousand six hundred nine point zero seven three nine three#pg2 <ref:2609.07393#pg1>.

Mira: It’s applicable beyond just damping rate estimation, too; they mention using this model to characterize magnetic fields in NV center coupling or even modeling redshift in astrophysics <ref:two thousand six hundred nine point zero seven three nine three#pg2 <ref:2609.07393#pg1>.

Kai: So we're looking at the paper "Mechanical quality factor estimation from a nonlinear optical cavity response," and it proves you can get intrinsic damping rate and quality factor Q are key parameters in optomechanical systems determined from a time-averaged optical cavity response <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1,intrinsic damping rate and quality factor Q are key parameters in optomechanical>.

Lev: It’s about getting those key parameters without resolving the mechanical oscillation, which avoids demanding bandwidth and data-acquisition requirements typically imposed by conventional reconstruction methods <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1>.

Mira: And this approach enables parameter estimation without resolving the fast dynamics or demodulating the mechanical oscillations, which is usually required in conventional approaches <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1>.

Kai: It's a powerful tool for getting parameters without the bandwidth demands of traditional methods <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1>.

Lev: The mathematical derivations show that the two maxima of the spectrum are separated by a distance delta nu equal to two nu A, which grows linearly with A <ref:two thousand six hundred nine point zero seven three nine three#pg3 <ref:2609.07393#pg1>.

Kai: So we've covered how this paper uses nonlinear optical responses to estimate mechanical quality factor Q in optomechanical systems <ref:two thousand six hundred nine point zero seven three nine three#pg1 <ref:2609.07393#pg1>.

Center for Macroscopic Quantum States (bigQ) · James C. Wyant College of Optical Sciences, The University of Arizona

physics.optics, physics.app-ph, physics.space-ph, quant-ph

Submitted: 2026-09-07

Updated: 2026-10-20

Comments: 23 pages and 4 Figures

License: http://creativecommons.org/licenses/by-nc-nd/4.0/

Importance score: 83/100

The gist: The gist: This work introduces an all-optical ringdown method in the unresolved-sideband regime that exploits nonlinear, time-averaged optical response induced by strong mechanical motion to estimate

Key concepts

Time-Averaged Optical Response: This refers to the average optical signal measured when a laser probes a resonant system that is being modulated by strong mechanical motion. Instead of measuring instantaneous fluctuations, this technique captures the steady-state response averaged over time, which reveals information about the system's intrinsic properties like damping rate and quality factor.
Double-Horned Profile: When subjected to strong coherent driving (modulation amplitude A >> 1), the optical resonance profile changes from a simple Lorentzian shape to a double-horned pattern. This specific shape is key because the separation between the two peaks in this profile follows an asymptotically affine relation with the modulation amplitude, providing a measurable signature.
Dynamical Backaction (DBA): DBA occurs when the measurement process itself disturbs the system being measured. The paper shows that continuous frequency scanning across the optical resonance helps reduce DBA by averaging out probe-induced effects. This makes the estimation of mechanical parameters more accurate and less dependent on resolving fast dynamics.
Quality Factor (Q): The quality factor (Q) is a measure of how underdamped a mechanical resonator is, essentially quantifying its energy storage capacity relative to its dissipation. In this study, Q is estimated by analyzing the decay of the mechanical amplitude over time, derived from the optical response measurements.

Summary

The gist: This work introduces an all-optical ringdown method in the unresolved-sideband regime that exploits nonlinear, time-averaged optical response induced by strong mechanical motion to estimate intrinsic damping rate and quality factor (Q) without resolving the mechanical oscillation

Analytical Model

The paper derives an analytical model for the time-averaged optical cavity response probed by a weak frequency-scanning laser, which is subject to sinusoidal frequency modulation The power spectral density of a resonant system usually follows a Lorentzian lineshape L(ω, ω0) = 1/(1 + [2(ω − ω0)/2]2), characterized by a linewidth κ and a resonant frequency ω0. The time-averaged response Λ(ω, α) is described by an integral involving the Lorentzian lineshape modulated by sinusoidal frequency modulation This integral is rewritten in a more general form after normalizing the detuning ν = 2(ω − ω0)/κ and the amplitude A = 2α/κ. The closed-form, real-valued expression for the solution is given by Λ(ν, A) = s√η+η− + ηA/(2η+η−), where ηA = 1 + A squared − ν squared and η± = 1 + (A ± ν) squared.

Amplitude Estimation of Modulation

The analysis distinguishes between two regimes based on the ratio between modulation amplitude (A) and the resonance’s linewidth (κ), showing a simple Lorentzian approximation for weak modulation where the sensitivity S = ∂Λ/∂A is maximized by measuring the variation of the in-resonance response. When a strong coherent drive (A ≫ 1) is present, the resonance changes from an approximated Lorentzian to a double-horned profile. A more robust approach for strong modulation exploits the separation between the two off-center maxima 2νA. The asymptotic behavior for strong modulation is described by νA ≃ A − 1/√3 + O(A−1), indicating an asymptotically affine relation between peak positions and the modulation amplitude.

Experimental Methodology

The experimental setup utilizes a membrane-in-the-middle (MIM) system, which is a phononic-crystal nanomechanical resonator made of silicon nitride placed between two highly reflective mirrors patterned with a phononic shield. The mechanical frequency is omegam = 1.134 MHz while the optical resonance linewidth is κ/2π = 37.8 MHz. The measurement involves exciting the mechanics with a blue-detuned laser, and then scanning the laser frequency with a triangular wave to probe the two-horned resonance shape. The quality factor Qm is extracted from the exponential decay of the mechanical amplitude A(t) = A0 exp − t/τ, where τ = 2/Γm, and Qm = omegam/Γm.

Key Findings and Robustness

The method provides an in situ diagnostic that reduces the impact of probe-induced dynamical backaction (DBA). The measurement is performed by continuously scanning the laser across the optical resonance, which allows for a low-bandwidth optical readout that removes the need to continuously resolve the mechanical spectrum. The estimated quality factor is 108.4 ± 0.7 × 106, which is consistent with independent measurements. Furthermore, the signal-to-noise ratio (SNR) scaling for the decay rate improves with the square root of the number of scans, making it robust to unstable and noisy environments. The analysis shows that deviations from the approximated result are bounded by O(A4 + min A2ν2 + ν4, A2ν2 + ν4).

Applications and Extensions

The derived model can be applied to other strongly modulated resonant systems, such as characterizing the strength of a driving magnetic field in NV center coupling or modeling redshift in astrophysics. The double-horned profile also appears in other fields because it can arise from the convolution between a Lorentzian line shape and an averaged oscillating quantity. The methodology is generally applicable for estimating the amplitude of external modulation or any physical derivative of the given response. This approach enables an in situ diagnostic and reduces effects of dynamical backaction by continuously scanning the probe across the optical resonance, such that the time-averaged detuning is ideally zero. The proposed method does not require resolving the fast dynamics or demodulating the mechanical oscillations, as is typically required in conventional approaches.

Mathematical Derivations

The analytical solution of the integral is obtained using complex analysis and contour integration around a closed path z = 1. The final result for the integral is written as Λ(ν, A) = s√η+η− + ηA/(2η+η−). The approximation in the weak modulation limit is given by Λ(ν, A) ≈ L1(ν, A):= 1 − A2/2 / (1 + ν2). The strong modulation regime yields an asymptotic relation for the peak separation as νA ∼ sA − 1/√3 + O(A−1), which grows linearly with A. The analysis confirms that the Lorentzian peak splits for A > √2, and the separation between the two new maxima grows monotonically with A.

Numerical Simulations

Numerical simulations follow a Hamiltonian describing an optical cavity mode coupled to a mechanical resonator. The time-domain simulations were performed in Python 3.12 using the SciPy RK45 integrator. When increasing the power (Pin = 100 µW), the direction of the probe’s scan becomes a factor, as the mechanical oscillator is periodically heated and cooled by the scanning probe laser. At higher input powers, optomechanical gain exceeds intrinsic mechanical dissipation, resulting in negative effective damping (Γeff < 0). The threshold for parametric instability constrains the quantity g20ncav, allowing estimation of the single-photon optomechanical coupling rate g0. The effective damping is given by Γeff = Γm + Γopt = 0 at the instability threshold.

Supplementary Mathematical Results

The analysis involves several propositions, such as Proposition 1, which states that η+η− can be rewritten in terms of ηA and ην as η+η− = η2A + 4ν2, proving the relationship between the quantities. The final approximation for the peaks' positions is derived from a depressed cubic equation, where only one solution respects the condition 0 < νA < A, leading to an asymptote as νA = A − 1/√3. This confirms that the two maxima of the spectrum are separated by a distance ∆ν = 2νA ∼ 2(A − 1/√3), which grows linearly with A. The analysis proves that the approximation is valid for each value of A > √2, where an exact solution exists. The final solution that satisfies 0 < νA < A has an asymptote as νA = A − 1/√3. This means that the two maxima of the spectrum are separated by a distance ∆ν = 2νA ∼ 2(A − 1/√3), which grows linearly with A. The final solution that satisfies 0 < νA < A has an asymptote as νA = A − 1/√3.

Improvements for AI systems

Bold header: Enhanced Mechanical Quality Factor (Q) Estimation in High-Q Systems

This method allows for intrinsic damping rate and quality factor (Q) are key parameters in optomechanical systems to be determined from a time-averaged optical cavity response, providing an in situ diagnostic that is robust against probe-induced dynamical backaction (DBA).

Bold header: Robust Parameter Estimation Under Fast Coherent Modulation

The system can estimate the amplitude of fast coherent driving, as the paper shows that for strong modulation, the double-horned profile that emerges admits an asymptotically affine relation between the peak positions and the modulation amplitude.

Bold header: Reduced Measurement Requirements for High-Q Resonators

The technique enables parameter estimation without resolving the mechanical oscillation, thereby avoiding demanding bandwidth and data-acquisition requirements typically imposed by conventional reconstruction methods.

Abstract

Intrinsic damping rate and quality factor (Q) are key parameters in optomechanical systems. However, in high-Q devices, their measurement is often challenging because probe-induced dynamical backaction (DBA) and the need to resolve and accurately track the mechanical resonance limit the performance of state-of-the-art techniques. Here, we introduce an all-optical ringdown method in the unresolved-sideband regime that exploits the nonlinear, time-averaged optical response induced by strong mechanical motion. We derive an analytical model of the time-averaged optical cavity response probed by a weak frequency-scanning laser. During the evolution of the mechanical oscillator, the excited motion modulates the Lorentzian lineshape into a double-horned profile whose peak separation enables an accurate estimation of the modulation amplitude. Tracking this response during the mechanical ringdown allows the intrinsic damping rate to be determined without resolving the mechanical oscillation. We apply the method to a density-modulated phononic crystal membrane placed inside a high-finesse cavity and obtain mechanical quality factors consistent with independent measurements on the same platform. Our method requires a minimal optical setup, while offering an in situ diagnostic and reduced impact from probe-induced DBA and robustness against measurement noise. Finally, the model can describe other resonant systems dispersively coupled to coherent frequency modulations, providing new insights and measurement strategies that expand beyond the optomechanical platform.