The mechanical displacement x shifts the cavity resonance by G x (G = g0/xzpf). With direct detection, the transmitted power is modulated at Ωm. The transduction |H(Ωm)| (optical watts per metre) depends on where the laser sits relative to the cavity line and to the motional sidebands at ωl ± Ωm.
- Unresolved sideband (Ωm ≪ κ): the field follows the motion adiabatically, |H| ≈ G·|∂Pout/∂Δ|. The transduction is maximum on the steepest slope, Δ = ±κ/(2√3) ≈ ±0.29κ, and vanishes at Δ = 0.
- Resolved sideband (Ωm ≫ κ): the signal is the beat between the transmitted carrier and a sideband that is resonantly enhanced by the cavity, so the maxima move to Δ ≈ ±Ωm. In this side-coupled geometry, weaker maxima remain near Δ ≈ ±κ/2, where the carrier itself sits on the cavity slope.
- Bottom-right: the detected peak also includes dynamical backaction. The mode is amplified (Γeff < Γm, Teff > T) on the blue side and cooled on the red side. The ESA signal is therefore asymmetric in Δ even though |H| is symmetric. Raise Pin to see this grow.
Optical transmission vs detuning
Side-coupled waveguide (equivalently, reflection from a one-sided cavity). Orange: current Δ. Dotted: Δ = ±Ωm.
Transduction |H(Ωm)| vs detuning
Optical power modulation at the cavity output per unit displacement, without backaction.
Unresolved → resolved: normalized transduction
|H(Ωm)|² for several Ωm/κ, each normalized to its maximum (same κ). Black dashed: current device. Gray dotted: (∂T/∂Δ)², the adiabatic limit.
Detected ESA peak vs detuning
Peak of the ESA trace (dBm in the RBW) with and without dynamical backaction. Red bands: Γeff ≤ 0 (parametric instability).
Theory used in this tab
ā = √κe αin / (κ/2 − iΔ), αin = √(Pin/ħωl), ncav = |ā|², κ = κi + κe āout = αin − √κe ā, T(Δ) = |1 − κe/(κ/2 − iΔ)|² = [Δ² + (κi − κe)²/4] / [Δ² + κ²/4]Linearizing a = ā + δa with ωc(x) = ωc − Gx gives δa(ω) = i G ā χc(ω) x(ω), where χc(ω) = 1/[κ/2 − i(Δ+ω)]. The output power fluctuation is δPout(ω) = H(ω) x(ω), with
H(ω) = −i ħωl √κe G [ āout* ā χc(ω) − āout ā* χc*(−ω) ]For ω → 0 this reduces to |H| = G |∂Pout/∂Δ| (checked numerically). In the resolved regime the two terms are the beats of the transmitted carrier with the upper and lower motional sidebands; only the one that falls inside the cavity line survives.
The detected signal is |H|² × Sxx. The area under the mechanical peak is ∝ |H|²⟨x²⟩ with ⟨x²⟩ = kBTeff/(meffΩm²) = 2 neff xzpf². Because H ∝ G = g0/xzpf, the area scales as g0² neff, so meff drops out at fixed g0. Changing meff alone changes only the displacement calibration (tab 4), not the ESA trace. The peak height is area/Γeff.
- g0: area ∝ g0² (backaction also grows as g0²).
- Tbath: area ∝ nth ≈ kBT/ħΩm.
- Teff ≠ Tbath when backaction is present: neff = (Γmnth + A+)/(Γm + Γopt) ≈ nthΓm/Γeff. Area ∝ Teff; peak ∝ Teff/Γeff.
- Ωm: nth ∝ 1/Ωm, and |H| changes with Ωm/κ (tab 1).
- Pin: |H|² ∝ Pin² while shot noise ∝ Pin.
Use Hold trace, change a parameter, and compare.
Simulated ESA trace
Scaling with a single parameter
Theory used in this tab
Sxx(ω) = 4 kBTeff Γeff / { meff [ (Ωeff² − ω²)² + Γeff² ω² ] }, kBTeff ≡ ħΩm neff SPdet(ω) = η² |H(ω)|² Sxx(ω) + 2ħωlPdet + NEP² [W²/Hz], Pdet = η T Pin Sel(ω) = gdet² SPdet(ω)/RL + DANL [W/Hz], ESA reading = 10 log10(Sel·RBW / 1 mW) [dBm]gdet is the detector conversion gain (V/W) into the load RL = 50 Ω. Classical (high-temperature) thermal spectrum; zero-point motion and quantum backaction noise on the detector are neglected, which is fine for neff ≫ 1. Laser technical noise is not included.
Absorbed intracavity power heats the resonator. Through the thermo-optic effect and thermal expansion the resonance red-shifts: ωc = ωc0 − K·Pabs, with Pabs = ħωlκabsncav. The steady state solves a cubic equation, so above a power threshold it is bistable and the transmission depends on sweep direction.
- λ increasing (ωl decreasing, the usual direction): the heated cavity is dragged along with the laser. The effective detuning stays on the blue side (Δeff > 0) over a long range, giving the triangular "thermal broadening", then the system drops abruptly when the branch disappears.
- λ decreasing: the resonance runs away from the laser and the dip looks narrow.
- Consequence for transduction: plotted against laser wavelength, the mechanical signal is stretched over the blue flank and the red flank is nearly absent. The spectrum vs laser frequency is strongly asymmetric. The red side is thermally unstable, which is why thermal locking only works on the blue side (Carmon, Yang & Vahala, Opt. Express 12, 4742 (2004)).
- Turn on dynamical backaction to see the optical spring (curved Ωm in the map) and the blue-side amplification. The two effects add to the asymmetry.
Transmission during the sweep
Quasi-static thermal model (slow sweep).
Effective detuning Δeff/κ during the sweep
Thermal locking keeps Δeff > 0 (blue side). Dotted: Δeff = ±Ωm.
Mechanical peak on the ESA vs laser position
Peak ESA reading (dBm in RBW) at the mechanical resonance.
ESA map: RF frequency vs laser position
What a spectrogram looks like during the sweep (dBm in RBW). Where Γeff ≤ 0 (red bands on the left) the map uses a clamped linewidth, so it only marks where self-oscillation happens.
Theory used in this tab
Δeff = ΔL + K ħωl κabs ncav(Δeff), ncav(Δ) = κe|αin|²/(κ²/4 + Δ²) ⇒ (Δeff − ΔL)(κ²/4 + Δeff²) = K ħωl κabs κe |αin|² (cubic; up to 3 real roots)The sweep follows the root closest to the previous one (hysteresis). The thermal response is assumed much slower than Ωm, so it shifts the cavity but does not act on the mechanical mode, and much faster than the sweep. Wavelength axis: δλ = −λ²ΔL/(2πc). Thermal fluctuations of the cavity frequency (thermorefractive noise) are not modelled.
An ESA reports electrical power in a resolution bandwidth (dBm). Getting to displacement means walking the detection chain backwards. There are two common routes:
- (A) Model transduction: divide by the known RBW, load, detector gain, collection efficiency and |H|². This needs the absolute gain of every element and knowledge of Δ, κe, Pin and g0.
- (B) Thermal calibration (standard in practice): at low power, with no backaction and the mode thermalized, the area under the peak must equal ⟨x²⟩ = kBT/(meffΩm²). This fixes the calibration factor in W per m² without knowing the gains. With a known meff it also yields g0. The phase-modulation method of Gorodetsky et al., Opt. Express 18, 23236 (2010), gives g0 without needing meff or the gains.
Step-by-step chain at the mechanical peak (current simulation)
Calculator: your own measurement
Pre-filled from the simulation. Edit any field with your measured values.
Displacement spectrum inferred from the ESA trace (route A)
√Sxx obtained by dividing the full measured trace by the transduction. The flat part is the measurement imprecision floor, i.e., the noise floor expressed in m/√Hz.
The optomechanical coupling rate g = g0√ncav grows with power, so the dynamical backaction terms are linear in Pin (weak-coupling limit):
- Optical spring δΩm: stiffening (δΩ > 0) for blue detuning and softening for red, in the unresolved regime. The sign pattern changes in the resolved regime.
- Optical damping Γopt > 0 for red detuning (cooling and linewidth broadening) and < 0 for blue (amplification and narrowing). At Pthr = PinΓm/|Γopt| the effective damping crosses zero and the resonator self-oscillates (optomechanical oscillator, OMO). Beyond this point the linear model is not valid (shaded).
- The ESA peak grows as Pin² from transduction alone. Backaction bends this curve up (blue) or down (red).
Solid lines: current Δ. Dashed lines: mirrored detuning −Δ. Curves stop where the weak-coupling formulas break down (4g > κ, |Γopt| > 0.2κ, or |δΩm| > 0.3Ωm). The power axis spans Pin/1000 to 300·Pin around the sidebar value.
Mechanical frequency shift δΩm/2π vs Pin
y-range set for Pin ≤ 10× the sidebar value; double-click to autoscale.
Mechanical linewidth Γeff/2π vs Pin
Gray dotted: intrinsic Γm/2π. Black: zero (instability threshold).
ESA peak vs Pin
Phonon occupation neff vs Pin
ESA traces at several powers (current Δ)
Shows the shift, narrowing or broadening, and the growth of the peak together.
Theory used in this tab
A∓ = g² κ / [κ²/4 + (Δ ± Ωm)²], Γopt = A− − A+, g = g0√ncav δΩm = g² { (Δ − Ωm)/[κ²/4 + (Δ − Ωm)²] + (Δ + Ωm)/[κ²/4 + (Δ + Ωm)²] } neff = (Γm nth + A+) / (Γm + Γopt)These are the weak-coupling results of RMP Sec. VII (valid for g ≪ κ and Γopt ≪ κ). In strong coupling (4g ≳ κ) the mechanical and optical modes hybridize (normal-mode splitting).
- Dynamical backaction vs detuning at the current power. In the resolved regime the damping is concentrated at Δ = ±Ωm.
- Sideband cooling: neff vs Δ, and the quantum backaction limit nmin = A+/(A− − A+). This reaches (κ/4Ωm)² in the resolved regime and ≈ κ/(4Ωm) ≫ 1 in the unresolved regime. Ground-state cooling therefore requires resolved sidebands.
- OMIT / OMIA: a weak probe is swept across the cavity while the control laser sits at the current Δ (try Δ = −Ωm for transparency and +Ωm for amplification). A window of width ≈ Γm(1 + C) opens in the cavity dip, where C = 4g²/(κΓm) is the cooperativity (Weis et al., Science 330, 1520 (2010); Safavi-Naeini et al., Nature 472, 69 (2011)).
Optical spring δΩm/2π vs detuning
Optical damping Γopt/2π vs detuning
Phonon occupation vs detuning (current Pin)
Quantum limit of sideband cooling
nmin minimized over Δ, vs sideband resolution Ωm/κ. Orange: current device.
OMIT: probe transmission across the cavity
Control at the current Δ and Pin. Gray dashed: bare cavity (g = 0).
OMIT: zoom at two-photon resonance
Probe–control detuning minus Ωm.
Theory used in this tab
OMIT is computed from the full linearized equations, without the rotating-wave approximation, for a probe at ωl + ω:
tp(ω) = 1 − κe χc(ω) [ 1 − g² η(ω) χc(ω) / (1 + g² η(ω) (χc(ω) − χ̃c(ω))) ] χc(ω) = 1/[κ/2 − i(Δ+ω)], χ̃c(ω) = 1/[κ/2 + i(Δ−ω)], η(ω) = 1/[Γm/2 − i(ω−Ωm)] − 1/[Γm/2 − i(ω+Ωm)]With Δ = −Ωm in the resolved limit this reduces to the familiar tp = 1 − κe/[κ/2 − i(Δ+ω) + g²/(Γm/2 − i(ω − Ωm))].