The beatnote phase exposes differential locking error
The paper asks whether the frequency spacing between longitudinal modes of an optical cavity can be measured more precisely than the beatnote of two stabilized lasers alone allows. The obstacle is that each laser is locked to a cavity resonance through a Pound-Drever-Hall (PDH) loop, and sensing noise, control noise, or finite loop gain detune each laser from its resonance. Those detunings appear in the transmitted beatnote and masquerade as changes in the mode spacing.
The scheme uses two resonant lasers, labeled a and b, stabilized to resonances separated by one free spectral range (FSR) of a 19 m cavity. Their interference beatnote is measured twice: at photodetector PDi before the light enters the cavity, and at PDt in transmission. The cavity-induced phase shift that PDH exploits in reflection also appears on the transmitted beatnote, so any differential error in the two locking points produces a differential phase between the incident and transmitted beatnotes.
Because that differential phase is dynamic, it can be converted into a differential frequency error using only the cavity storage time. Subtracting this error from the measured transmitted beatnote frequency in post-processing yields the corrected quantity, the change in frequency spacing between the cavity resonances. In the ideal case where the two lasers sit on resonances differing only in longitudinal mode order, this corrected signal equals the change in the cavity FSR.
Why the differential phase scales to frequency
The conversion rests on the cavity storage time, reported as 1.308 ± 0.010 ms. The dynamic differential frequency error is the difference between the transmitted and incident beatnote phase fluctuations divided by that storage time. The static part of the phase difference, which includes the optical path length between the cavity mirrors and the photodetectors, drops out because it does not fluctuate; the paper notes that changes in such parameters couple inversely with the beatnote wavelength, which is on the order of the cavity length, so their effect on the measured phase is negligible.
This is the mechanism that separates two things readers could easily conflate. The transmitted beatnote frequency fluctuation is not the same quantity as the change in cavity mode spacing. The former contains both the differential locking error and any real change in the resonance spacing; the latter is what remains after the differential error is subtracted. The paper's corrected spectrum is an estimate of the mode-spacing change, not a direct measurement of the raw beatnote.
What the corrected spectrum actually improves
The measurement used two Nd:YAG non-planar ring oscillators at 1064 nm, both phase-modulated by electro-optic modulators for PDH sensing, with error signals fed to PID controllers driving piezo and temperature actuators on the laser crystals. The lasers were stabilized to resonances separated by one FSR, producing a beatnote at 7.89165 ± 0.00003 MHz. The beatnote frequency could be read with sub-hertz precision but drifted by roughly 30 Hz over the measurement because the cavity length changed. Spatial overlap between the lasers and the fundamental cavity mode was 97 ± 1%.
In the noise spectra, above 1 mHz the measured frequency detuning agrees exceptionally well with the transmitted beatnote frequency noise. Below 1 mHz, cavity length noise becomes the dominant source of beatnote frequency noise. This is the paper's evidence that, across most of the spectrum, the leading cause of beatnote frequency noise is residual noise in the laser frequency stabilization systems rather than the cavity itself.
The corrected spectrum suppresses the estimated residual differential frequency noise by nearly an order of magnitude between 10 mHz and 10 Hz, and by up to a factor of 30 at some frequencies in that range. Below 10 mHz and from 8 Hz to 50 Hz, the corrected signal resolves the frequency changes between cavity resonances that are due to cavity length noise. Between roughly 100 mHz and 5 Hz, excess noise keeps the corrected spectrum above the cavity length noise and limits sensitivity; the paper states the exact origin is unknown but attributes it to stray light creating small vector phase noise at the photodetectors.
Injected signals show the noise removal directly
A second test injected sinusoids into the error points of the Laser A and Laser B PDH locks at 12 Hz and 14 Hz, with amplitudes well above the out-of-loop noise of the stabilization systems. In the transmitted beatnote and in the laser-to-resonance frequency detuning, the injected lines are clearly visible. After the corrected beatnote frequency is computed, the lines nearly disappear.
This is the paper's cleanest demonstration that the technique reduces estimated residual frequency noise by more than two orders of magnitude at those frequencies. It also shows the method is not merely smoothing the spectrum: it removes a known, deliberately added disturbance that the raw beatnote faithfully reports.
Integration time decides which signal to trust
The practical comparison in the paper is between the corrected and uncorrected beatnote frequency noise spectra as a function of integration time. The corrected data resolves the cavity length noise for integration times less than 100 s, while the uncorrected frequency data requires at least 1000 s to do so. That is roughly an order-of-magnitude difference in the averaging time needed to reach the same conclusion about cavity length noise.
The comparison also has a frequency boundary. The corrected signal is useful where it sits at or below the cavity length noise, which the paper reports below 10 mHz and from 8 Hz to 50 Hz. Between roughly 100 mHz and 5 Hz, the unexplained excess noise keeps the corrected spectrum above the cavity length noise, so the correction does not deliver the same sensitivity there. A reader deciding whether to adopt this scheme should check both the integration time and the Fourier-frequency band of interest against these two ranges.
A diagnostic as well as a correction
The paper presents the technique in two roles. First, it performs an out-of-loop measurement of the differential noise in the laser frequency control loops, which the authors describe as a valuable diagnostic tool for characterizing laser frequency stabilization systems. Second, it removes that noise in post-processing to improve the sensing precision of the cavity mode spacing.
The authors also argue that encoding the noise on the phase of a megahertz beatnote, which follows from using heterodyne sensing, makes the method robust for measuring low-frequency error-point noise. This is an author interpretation of why the approach works, not a separately measured comparison against other readout methods in this paper.
Where the method is shown and where it is not
The demonstration covers one 19 m cavity, two 1064 nm Nd:YAG lasers, and longitudinal modes of identical polarization separated by one FSR, with the corrected signal interpreted as the change in cavity FSR. The paper states that the technique can be expanded to other mode spacings, such as those between higher-order modes or polarization eigenmodes, and suggests it could be applied more broadly to resonant media even without an optical cavity by measuring the beatnote between two lasers before and after passing through the medium. Those extensions are proposed, not demonstrated here.
The unexplained excess noise between roughly 100 mHz and 5 Hz is the main measured limitation on sensitivity in this configuration, and the paper attributes it tentatively to stray-light-induced vector phase noise at the photodetectors. The 30 Hz beatnote drift over the measurement is a separate practical constraint on the raw beatnote, and it is the kind of drift the correction is designed to work around rather than eliminate at the source.
The priority is the frequency band, not the headline factor
The most decision-relevant result is not the single largest suppression factor of 30. It is the pairing of two boundaries: the corrected signal resolves cavity length noise below 10 mHz and from 8 Hz to 50 Hz, and it does so for integration times under 100 s, while the uncorrected beatnote needs at least 1000 s. Between roughly 100 mHz and 5 Hz, excess noise keeps the corrected spectrum above the cavity length noise, so the correction does not buy the same sensitivity there.
For a reader weighing this scheme, the first question should therefore be which Fourier-frequency band and which integration time the intended measurement occupies. If it falls in the resolved bands and can tolerate sub-100 s averaging, the corrected beatnote is the signal to use; if it falls in the 100 mHz to 5 Hz window, the paper's own data show the correction is limited by noise whose origin is not yet identified. The technique's demonstrated scope is longitudinal modes of identical polarization separated by one FSR in a 19 m cavity, and its extension to other mode spacings or cavity-free resonant media remains a proposal.
Sources used in this answer
Heterodyne-Enhanced Sensing of Cavity Mode Spacing
Comparing the interference beatnote phase of two resonant lasers before and in transmission of a 19 m cavity measures the differential locking error between them; subtracting it in post-processing suppressed residual differential frequency noise by nearly an order of magnitude between 10 mHz and 10 Hz and up to a factor of 30 at some frequencies, and reduced injected 12 Hz and 14 Hz calibration signals by more than two orders of magnitude.