Fundamental Thermal Noise Limits for Optical Microcavities

We present a joint theoretical and experimental analysis of thermorefractive noise in high-quality-factor ($Q$), small-mode-volume ($V$) optical microcavities. Analogous to well-studied stability limits imposed by Brownian motion in macroscopic Fabry-Perot resonators, we show that microcavity thermorefractive noise gives rise to a mode-volume-dependent maximum effective quality factor. State-of-the-art fabricated microcavities are found to be within one order of magnitude of this bound. By measuring the first thermodynamically limited frequency noise spectra of wavelength-scale high-$Q/V$ silicon photonic crystal cavities, we confirm the assumptions of our theory, demonstrate a broadband sub-$\mu \mathrm{K}/\sqrt{\mathrm{Hz}}$ temperature sensitivity, and unveil a new technique for discerning subwavelength changes in microcavity mode volumes. To illustrate the immediate implications of these results, we show that thermorefractive noise limits the optimal performance of recently proposed room-temperature, all-optical qubits using cavity-enhanced bulk material nonlinearities. Looking forward, we propose and analyze coherent thermo-optic noise cancellation as one potential avenue toward violating these bounds, thereby enabling continued development in quantum optical measurement, precision sensing, and low-noise integrated photonics.

Popular Summary

Calculating the energy and temperature fluctuations within a small volume is a classic problem in statistical mechanics. This fundamental “thermal noise” limits the stability of optical cavities, as the resonant frequency is affected by temperature-dependent optical properties of the confining structure (such as mirrors). In large optical cavities, such as those in gravitational-wave interferometers, temperature fluctuations are small, and the resulting bounds on cavity stability are well studied. Here, we extend these thermal noise investigations to wavelength-scale optical microcavities, where the amplitude and bandwidth of fundamental temperature fluctuations are significantly enhanced.

By comparing our measurements of frequency noise in microcavities to a general thermal noise model, we show that temperature fluctuations are indeed the dominant noise source and demonstrate the ability to distinguish subwavelength changes in the cavity volume. Our results suggest that temperature fluctuations in sufficiently small cavities can become large enough to substantially limit the cavity’s quality factor, thereby imposing a fundamental performance limit. We derive this corresponding bound and show that it lies within one order of magnitude of current experimental devices.

This limit has immediate implications for an array of applications. We specifically explore quantum information science and show that microcavity thermal noise limits the coherence of recently proposed all-optical qubits. To overcome these limitations, we propose and evaluate a noise cancellation technique that coherently suppresses thermally driven frequency fluctuations, thereby enabling continued development in quantum optics, precision sensing, and low-noise integrated photonics.

Figure 1

Comparison of thermorefractive noise (TRN) in macroscopic resonators (a) and microcavities (b). Mode-averaged temperature fluctuations $\delta \overline{T}$ in large cavities induce refractive index noise $\delta \overline{n}$ (and, thus, path length changes $\delta \overline{L}$) due to the mirrors’ nonzero thermo-optic coefficient ${\alpha }_{\mathrm{TR}}=dn/dT$. The large mode volume $V$ reduces $\delta \overline{T}$, yielding a narrow-band resonant frequency noise spectrum $S_{\omega \omega }(\omega )$ and an rms resonant frequency fluctuation $\delta {\omega }_{\mathrm{rms}}\ll \mathrm{\Gamma }$, the cavity half-linewidth. This nondominant thermal noise inhomogeneously broadens the intracavity field spectrum $S_{aa}(\omega )$. Decreasing $V$ increases both the magnitude $\delta {\omega }_{\mathrm{rms}}$ and bandwidth ${\mathrm{\Gamma }}_T$ of TRN, while increasing the resonator quality factor $Q={\omega }_0/2\mathrm{\Gamma }$ causes both quantities to exceed $\mathrm{\Gamma }$. TRN, therefore, becomes a dominant source of homogeneous broadening in wavelength-scale high-$Q/V$ microcavities, leading to thermal dephasing and reduced resonant excitation efficiency of the cavity field $a(t)$.

Figure 2

Calibrated measurement of TRN in high-$Q/V$ silicon PhC cavities. A shot-noise-limited, balanced homodyne detector (a) is locked to the phase quadrature of the cavity reflection signal and records the spectrum of resonant frequency fluctuations. The finite difference time domain (FDTD) simulated mode profiles, thermal mode volume ${\tilde{V}}_T$ [Eq. (6)], and thermal decay rate ${\mathrm{\Gamma }}_T$ [Eq. (3)] of the $L3$ and $L4/3$ devices tested are shown in (b). The radii of the green holes are increased by up to 5% to form superimposed gratings which improve vertical coupling efficiency. The measured spectral density of cavity resonant frequency and temperature noise—$S_{\mathrm{ff}}(f)$ and $S_{\mathrm{TT}}(f)$, respectively—(red) for $L3$ (c) and $L4/3$ (d) cavities are compared to noise from a specular reflection off the sample surface, finite-element method (FEM) simulations of cavity TRN, as well as single- and multimode fits. The listed multimode fit parameters agree with the predicted values in (b). Inset reflection spectra of each device reveal quality factors on the order of ${10}^5$, which compare favorably with the FDTD computed values ($1.7\times {10}^5$ and $1.6\times {10}^5$ for the $L4/3$ and $L3$, respectively). Micrographs of the fabricated designs with enlarged holes relative to the optimal designs in (b) are also inset.

Figure 3

Comparison of corrected $L4/3$ cavity frequency noise spectra at various input powers ${\tilde{P}}_{\mathrm{in}}$ normalized to the maximum value for the dataset. The rms fractional frequency fluctuation $\sqrt{⟨\delta {\omega }^2⟩}/{\omega }_0$ is computed from the integrated noise over the plotted measurement bandwidth and plotted in the inset as a function of ${\tilde{P}}_{\mathrm{in}}$. No significant power scaling or deviation from the mean value (black dashed line) is observed, indicating that noise contributions from nonlinear effects can be neglected.

Figure 4

Thermal-noise-limited room-temperature quality factor to mode volume ratios ($Q_{\mathrm{eff}}^{\max }/\tilde{V}$) for the materials considered in Appendix pp3 assuming a Gaussian-shaped mode at ${\lambda }_0=1550\mathrm{nm}$ admitting thermal diffusion in two or three dimensions (dash-dotted and solid lines, respectively). These limits are compared our devices as well as other fabricated and proposed microcavities. Insets illustrate typical confinement geometries for the range of $\tilde{V}$ listed (see Refs. [23, 53, 54, 64, 80, 82, 83, 84, 85, 86, 87, 88, 89]).

Figure 5

Performance of room-temperature all-optical qubits using bulk ${\chi }^{(3)}$ (left) or electric-field-induced ${\chi }^{(2)}$ (right) nonlinearities in silicon microcavities as a function of loaded cavity quality factor $Q$ at ${\lambda }_0=2.3\mu \mathrm{m}$ and the relevant normalized nonlinear mode volume $\tilde{V}$ (${\tilde{V}}_{\text{Kerr}}={\tilde{V}}_T$ and ${\tilde{V}}_{\mathrm{shg}}$, assumed to be equal to ${\tilde{V}}_T$, for ${\chi }^{(3)}$ and ${\chi }^{(2)}$, respectively). The figure of merit (FOM)—the ratio of qubit coupling rate $g$ to the composite decay and thermal dephasing rate $2{\mathrm{\Gamma }}_{\mathrm{eff}}={\omega }_0/Q_{\mathrm{eff}}$—is largest for strong coupling ($g/2\mathrm{\Gamma }\gg 1$) and weak dephasing ($Q_{\mathrm{eff}}\approx Q$). These competing characteristics yield an optimum quality factor $Q=Q_{\mathrm{opt}}$ for any $\tilde{V}$. Three-dimensional thermal diffusion in a homogeneous medium is assumed.

Figure 6

Athermal polymer (PMMA) microcavity design by coherent thermo-optic noise cancellation. The $Q$-optimized baseline ladder-type nanobeam cavity (a) with $\{a,a^{\prime },w_o,w_i,w_s\}=\{647\mathrm{nm},0.914a,3a,0.85w_o,0\}$ and thickness $t=3a$ is formed by quadratically tapering the lattice constant $a$ to $a^{\prime }$ over the inner 12 periods with a constant “rung” duty cycle ${\eta }_r=w_r/a=0.4$. This cavity is driven by anticorrelated volumetric and boundary heat sources corresponding to TR and TE noise, respectively, in FEM-based fluctuation-dissipation simulations. The composite TO frequency noise spectrum $S_{\mathrm{ff}}\propto [\nabla \delta T(\overrightarrow{r},f){]}^2$ is computed from the resulting harmonic temperature profiles $\delta T(\overrightarrow{r},f)$ (b). The TO noise level is bounded by the coherent ($-$) and incoherent ($+$) sum of TR and TE contributions (shaded region). The optimized parameters $\{a,a^{\prime },w_o,w_i,w_s,t,{\eta }_r\}=\{697\mathrm{nm},0.874a,3a,0.85w_o,10\mathrm{nm},0.8a,0.6\}$ provide broadband TON cancellation (e) as visually depicted by reduced $\delta T(\overrightarrow{r},f)$ at low frequencies (d). The color scale in (d) is equal to (b) to facilitate comparison.

Figure 7

Normalized noise spectra ${\tilde{S}}_{\omega \omega }(\omega )=I(\omega /{\mathrm{\Gamma }}_T)$ for single-mode [Eq. (a31)] and multimode [Eq. (a30)] TRN in an infinite slab of finite thickness.

Figure 8

The complete region of integration can be divided into three subspaces which yield different conditions for $k(t)$.

Figure 9

Schematic of the setup built to measure TRN in photonic crystal cavities. An amplified (PriTel PMFA) continuous wave laser (Santec TSL-710) is separated into a local oscillator and cavity signal by a polarizing beam splitter (PBS). The LO line is passively path-length matched to the cavity signal using a tunable retroreflector delay line. The cavity signal is combined with a linearly polarized (LP) white light source (LED) using a dichroic mirror (DM) and is reflected from a PhC cavity rotated 45° from the incident polarization (adjustable with a half-wave plate, $\lambda /2$), which allows the cavity signal to be isolated from the specular reflection using a PBS. A quarter-wave plate ($\lambda /4$) allows the specular reflection to be extracted for comparison to the cavity-only reflection. The reflected illumination light is separated and imaged onto a silicon CCD. The cavity signal can be directed with flip mirrors toward an IR camera for imaging, an IR avalanche photodetector (ThorLabs PDB410C 10 MHz InGaAs APD) to collect low-noise reflection spectra, or toward the balanced homodyne detector. For the latter, a balanced photodetector (ThorLabs PDB480C-AC 1.6 GHz InGaAs $p\text{−}i\text{−}n$ photodetector) measures the homodyne signal from the recombined cavity reflection and local oscillator, and the result is recorded on an electronic spectrum analyzer (ESA; Agilent N9010A EXA Signal Analyzer). The dc signal extracted from a low-pass filter (LPF) is used as the feedback signal for a digital PID controller which stabilizes the signal-LO phase difference by actuating a piezoactuated mirror. An EOM provides a known phase noise which can be used to calibrate the frequency noise of the PhC cavity. The sample stage is temperature stabilized to $\mathrm{\Delta }T<0.01\mathrm{K}$ using a Peltier plate and a feedback temperature controller.

Figure 10

Measurement of phase modulator modulation depth ${\varphi }_m(V_p)$ at $\lambda =1550\mathrm{nm}$. A balanced homodyne measurement is performed on the output of a Mach-Zehnder interferometer with the EOM in one arm, yielding spectra similar to that of (a). The sideband amplitudes are fitted to find the modulation depth at each peak drive voltage $V_p$, and a linear fit is applied to find the modulation efficiency (b). The measured value ${\varphi }_m/V_p=0.82\pm 0.01\mathrm{rad}/\mathrm{V}$ corresponds to a half-wave voltage $V_{\pi }=3.83$ V.

Figure 11

Approximate spectrum of microcavity noise sources for the experimental parameters in Table 1. Note that $S_{\mathrm{ff}}^{1/2}$ is plotted as a fractional stability (units $1/\sqrt{\mathrm{Hz}}$ to aid comparison with cavity stabilization literature). Noises from two-photon absorption (2PA) and self-phase modulation (SPM) at their respective nonlinear threshold powers—which approximates the maximum noise level—are still smaller than TRN.