Ultrahigh continuous-wave intensities in high-NA optical cavities through suppression of the parametric oscillatory instability

Abstract

Ultrahigh continuous-wave intensities (>300 GW/cm$^2$) in high-NA optical cavities enable applications from phase-contrast electron microscopy to ultradeep dipole traps for molecules. However, the intensity can be limited by the parametric oscillatory instability (PI), where mirror vibrations scatter light from one cavity mode into another. We observe PI in a table-top Fabry-Pérot cavity, show that the mechanical modes are MHz-frequency bulk acoustic modes inside the mirrors, and measure their $Q$ factor. By using low-$Q$ mirrors, we achieve >500 GW/cm$^2$ intensities in an open, free-space cavity.

Figures (5)

• Figure 1: (a) Cross-sectional view of cavity mirror, held by flexure-mounted pads (hatched, one of three shown). The displacement $u_z$ of the $(m, n, p)=(1,0,10)$ mechanical mode, with a frequency of $\omega_\mathrm{m}/(2\pi) = 5.4MHz$, is shown in blue for positive and in red for negative displacements. The beatnote amplitude $F_0F_1^*$ of the driven and higher-order optical modes for transverse mode spacing $\Delta\nu_{01}=\omega_\mathrm{m}/(2\pi)$ is also shown (green/orange for positive/negative amplitude). (b) RF power spectrum of cavity transmission photodetector with circulating power below (blue) and above (orange) PI threshold. A weak laser sideband at frequency $f$ was applied to show the transverse mode spectrum of the (astigmatic) optical cavity. Data from ULE mirror set 2; 10kHz resolution bandwidth. (c) Calculated (blue line) and measured (black circles) mechanical mode radii $W_\mathrm{m}(z_S)$ and calculated optical mode radii $W(z_S)$ (orange line; for $\Delta\nu_{01}=\omega_\mathrm{m}/(2\pi)$) at mirror front surface $S$ and versus $\omega_\mathrm{m}/(2\pi)$. Error bars show 1$\sigma$ uncertainty.
• Figure 2: (a) A single step of a step-down measurement, where the total circulating power (power $P_{\mathrm{circ},0}$ in the $\mathrm{TEM}_{00,(q)}$ mode plus power in $\mathrm{TEM}_{10,(q-1)}$ mode, orange line) is first increased above the power threshold $P_\mathrm{th}$ (dashed orange line) and then reduced below it. The cavity transmission shows a beatnote, here at 15.1MHz and mixed-down for visibility (blue line), with exponentially rising and falling envelope when the power is varied. A fit to the latter gives the mechanical mode's exponential decay time $\tau_\mathrm{m}$ at the final circulating power $P_{\mathrm{circ},0}$. (b) A linear fit (black line) to the inverse decay times $1/\tau_\mathrm{m}$ (blue circles) at powers $P_{\mathrm{circ},0}$ gives the extrapolated power threshold $P_\mathrm{th}$ and $\tau_\mathrm{m}$ at zero power (\ref{['eq:DressedDecayTime']}). (c) Probe ring-down measurement, where the total circulating power (not shown) is reduced from above the power threshold $P_\mathrm{th}$ to zero (at time 0ms). The amplitude (blue circles) of the probe laser's reflection off the cavity at frequency $f_\mathrm{PI}$, proportional to the amplitude of mirror vibrations, decays exponentially. A fit (black line) gives $\tau_\mathrm{m}$. Data are average of 7 repetitions; error bars show 1$\sigma$ uncertainty.
• Figure 3: (a) Measured and calculated power thresholds $P_\mathrm{th}$ of the parametric instability at frequency $f$. Experimental data, for which $f$ is the observed PI oscillation frequency $f_\mathrm{PI}$, are shown for two different sets of ULE cavity mirrors (blue circles and diamonds). The calculated $\min_j P_{\mathrm{th},j}$ for $f \equiv \Delta\nu_{01}$ is shown as gray solid line (assuming the optical modes are centered on the mirrors). Envelopes of $\min_j P_{\mathrm{th},j}$ on resonance with $(1, 0)$ mechanical modes ($\Delta\nu_{01} = \omega_{\mathrm{m},j}/(2\pi)$) are shown for the optical modes centered (dashed black line; dotted black line shows $(3, 0)$ mechanical mode), radially offset by 1mm (dashed purple line), or rotated by $\pi/4$ (dashed green line). For the offset case, the result with the $(1, 1)$ mechanical mode on-resonance is also shown (dash-dotted purple line). Red triangles show the maximum circulating power demonstrated with Zerodur mirrors. PI was sporadically observed at higher powers, but not characterized (see text). Finesse is set to $\mathcal{F} = 28000.0$; data from mirror set 2 ($\mathcal{F} = 32000.0$) were scaled accordingly. The upper axis shows the optical cavity NA for $f = \Delta\nu_{01}$. Inset: $\min_j P_{\mathrm{th},j}$ with experimental $P_\mathrm{th}$, showing $P_\mathrm{th}$ clustering near the lower envelope; dashed and dotted lines mark $(1,0)$ and $(3,0)$ mechanical mode frequencies, respectively. Error bars show 1$\sigma$ statistical uncertainty; 3% power calibration uncertainty not included. (b) Mechanical $Q$ factors $Q_{\mathrm{m},j}$ (for mirror set 1 only) from step-down (blue circles) and probe ring-down (orange squares) measurements. Error bars show weighted standard deviation over data for step-down measurements and combined statistical and systematic 1$\sigma$ uncertainty for probe ring-down measurements.
• Figure 4: Vibration amplitude profiles of the mechanical modes in the cavity mirrors excited by the parametric instability, measured with the 852-nm probe laser. The contours show fits of Hermite-Gaussian modes with indices (a--c) $(m = 1, n = 0)$ and (d) $(1, 1)$. On average, including 4 additional measurements (at $f_\mathrm{PI} = 4.4MHz$ and 17.3MHz; not shown), the fitted mechanical mode radii $W_\mathrm{m}(z_S)$ agree within 4% with those expected from the mirror geometry (\ref{['fig:MirrorModeFigure']} (c)). Data from ULE mirror set 2.
• Figure 5: Numerical integration of \ref{['eq: D0 dot', 'eq: D1 dot', 'eq: X dot']}. (a) Stored energy in the $\mathrm{TEM}_{00,(q)}$, $\mathrm{TEM}_{10,(q-1)}$ and mechanical modes ($E_0$, $E_1$ and $E_\mathrm{m}$, respectively), relative to the PI threshold energy $E_\mathrm{th}=2LP_\mathrm{th}/c$. Due to the drastically different scales of stored mechanical and optical energy, we show $E_\mathrm{m}$ multiplied by $Q_0/Q_\mathrm{m}$, which has the same (average) magnitude as $E_1$. $E_0$ initially follows the drive value $E_{0,\mathrm{sp}}\propto|D_{0,\mathrm{sp}}|^2$ (black dashed line), before a spontaneous thermal fluctuation from $\hat{f}_n$ initiates oscillation through PI, and $E_0$ drops to the threshold value (gray dashed line). The setpoint energy is stepped down in the same manner as in the experiment (\ref{['fig:Signals']}). In the simulation, $Q_0=7.0e5\,Q_\mathrm{m}$, $\gamma_1 = \gamma_0$, $\gamma_\mathrm{m} = \gamma_0/400$ and $\Delta\omega = -\gamma_0$, which are typical values. (b) The beat signal $D_0D_1^*$ (in the rotating frame) is shown in blue oscillating at $\sigma \approx \Delta\omega$, and the amplitude $|D_0D_1^*|$ in orange. At each step-down of the drive field to a value below the threshold, an exponential decay is fit to the signal, shown in green. The inset shows the time constants of the fits against the circulating power after the step-down, in good agreement with the theoretical prediction of \ref{['eq:DressedDecayTime']}.