Broadband birefringence spectroscopy with sub-kHz precision

TL;DR

The paper tackles the challenge of birefringence-induced noise in high-finesse crystalline mirror coatings by developing broadband cavity-mode dispersion spectroscopy (CMDS) using an optical frequency comb (OFC). It provides a theoretical framework linking cavity-mode positions to Gouy phase, mirror dispersion, and intracavity phases, and derives expressions for birefringent mode splitting Δν_{bir} and the corresponding refractive-index difference Δn_{bir}. Experimentally, it demonstrates CMDS on a 300 mm ultra-stable cavity with GaAs/AlGaAs crystalline mirrors, achieving a fractional frequency sensitivity of about $5e-14$ and measuring static birefringent splitting of ~169–163 kHz across two temperatures, corresponding to Δn_{bir} ≈ 305 ppm and 294 ppm over ~30 nm. The work highlights the potential of CMDS to map and understand birefringent noise and proposes a dual-comb approach to access frequency-noise characteristics across the optical spectrum, aiming to further improve fractional frequency stability in precision metrology.

Abstract

Although current amorphous high-reflective mirror coatings have had tremendous success in metrology applications, they are inherently limited by thermal fluctuations in their coating structure. Alternatively, crystalline coating technology has demonstrated superior thermal noise performance. However, recent studies have revealed birefringent noise sources, raising questions about the limits of frequency stability of high-finesse cryogenic silicon cavities with crystalline mirror coatings. Here, we show the applicability of cavity-mode dispersion spectroscopy to measure birefringent cavity mode splitting. We measured birefringence induced cavity mode splitting by probing the resonance frequencies of a high-finesse, ultra-low expansion glass cavity with all-crystalline mirror coatings, reaching fractional frequency sensitivity of \SI{5e-14}{} utilizing an optical frequency comb for two orthogonal polarizations. Subsequently, we calculated the static birefringent splitting of the refractive index for \SI{23.8}{\celsius} and \SI{31.3}{\celsius} on the order of \SI{305 \pm 3}{ppm} and \SI{294 \pm 3}{ppm} over \SI{30}{nm} respectively. Furthermore, we propose measurements of dispersive birefringent noise based on optical frequency combs. Our results not only extend the use of optical frequency combs to measure static birefringence, but also implicate a possibility to further study spectrally dependent frequency noise.

Figures (7)

• Figure 1: a) Schematic of the setup used for CMDS. AOM - acousto optic modulator, EOM - electro optic modulator, BS - beam splitter, DG - diffraction grating, DetPDH - Detector for PDH-lock, WP - half-waveplate, FTS - Fourier transform spectrometer, PD - photo detector. b) Locking scheme for CW and OFC. The CW laser (dashed green) is directly locked to a cavity mode, while the OFC (solid red) is directly phase-locked to the CW laser. c) Sampling schematic for an individual cavity mode (dashed blue). The comb mode corresponding to integer $n$ is translated by $\frac{n}{n_\mathrm{ref}}\Delta f_\mathrm{beat}$ when changing the beat note at mode number $n_\mathrm{ref}$. Measuring and interleaving the OFC spectrum (points orange) for each step returns the sampled cavity mode.
• Figure 2: Exemplary data for p-polarization at 23.8. Sampled cavity mode with Lorentzian fit and residuals, relative to the central beat note of the OFC and the CW laser. The central beat note was chosen, such as to maximize the transmission of the OFC through the cavity. The baseline correction of the fit is not included.
• Figure 3: Cavity mode width spectrum from Lorentzian fits, the inset shows the one-sigma uncertainty of the fitted mode width. The arrows depict outliers in the mode spectrum, which have increased FWHM. While the leftmost outlier could be linked to a systematic measurement artifact where the mode spectrum was distorted, the outliers at 1536.7nm show genuinely higher mode width, suggesting residual absorption inside the cavity. The right most outliers stem from spectral overlap with the CW laser, leaking into the measurement.
• Figure 4: Dispersive mode shift for two polarization states compared to the optical frequency comb. The inset shows the one-sigma uncertainty in the measured mode shift. Note that all curves cross each other at the wavelength of the CW laser close to 1542nm. While the FWHM is distorted for some measurements, the retrieved mode position is robust against measurement artifacts and outliers.
• Figure 5: a) Static birefringent cavity mode splitting after correcting for cavity drifts, b) resulting change in refractive index. The shaded areas correspond to the 4-sigma uncertainty of each measurement. Note that the increased uncertainty for the lowest wavelength at 23.8 stems from the fit, as the cavity mode was close to outside the sampled range.