Optical Frequency Averaging in Metrology

• Optical Frequency Averaging is a set of methods that reduce stochastic uncertainty in optical frequencies, improving stability, linewidth, and the interpretation of coherence.
• Techniques include direct arithmetic averaging, phase-locked loop mapping, and dual-comb coherent averaging to mitigate noise and systematic drift.
• Applications span optical clock comparisons, ultrastable frequency transfer over fiber, and integrated photonic implementations, often achieving noise reductions scaling as 1/√N.

Searching arXiv for the core paper and closely related work on optical frequency averaging and frequency transfer. Optical frequency averaging denotes a set of related operations in optics and frequency metrology in which averaging is applied to optical frequencies, optical phase, or frequency-estimation records in order to reduce stochastic uncertainty, narrow spectral linewidth, improve frequency stability, or clarify the meaning of measured coherence. In the recent literature, the term encompasses direct arithmetic averaging of independent optical frequencies to synthesize radiation at the mean frequency, averaging over the integration time $\tau$ used in Allan-deviation-based stability analysis, coherent averaging of periodic dual-comb signals, and the distinction between finite-time, ensemble, and frequency averages in optical coherence theory (Loh et al., 2023, Maurice et al., 2020, Veen et al., 2024).

1. Statistical principle and scope

The general statistical principle is that averaging reduces noise in statistically independent systems that exhibit similar levels of stochastic fluctuation. For $N$ statistically independent noise sources of similar magnitude, the noise in the mean is reduced by a factor of $1/\sqrt{N}$. In optical frequency averaging of light, this principle is applied not to optical power or intensity, but to frequency fluctuations themselves; in clock metrology it appears through the dependence of fractional frequency stability on the averaging time $\tau$; and in coherence theory it appears through distinct averaging domains that need not be equivalent (Loh et al., 2023, Maurice et al., 2020, Veen et al., 2024).

A central distinction is between averaging as an overview operation and averaging as a measurement operation. In the first case, one constructs an optical output at an arithmetic mean frequency from two or more optical sources. In the second, one estimates how fractional frequency fluctuations decrease with integration time. A further distinction arises in coherence theory, where finite time averaging by a detector and ensemble averaging over many realizations are different operations unless stationarity and ergodicity hold. This suggests that “optical frequency averaging” is not a single protocol but a family of procedures whose common feature is the controlled suppression or interpretation of fluctuations.

2. Direct averaging of optical frequencies

Directly averaging the frequency of light is nontrivial because summing optical frequencies would normally require nonlinear optics. A demonstrated solution is to map optical frequency fluctuations onto a voltage using a phase-locked loop (PLL), perform linear operations electronically, and convert the result back to a frequency using a voltage-controlled oscillator (VCO) and acousto-optic modulation (AOM) (Loh et al., 2023).

In the reported implementation, one seed laser is split into two optical paths, and each path is independently stabilized to its own fiber-based ultralow-noise reference cavity using Pound-Drever-Hall locking. An AOM in the second path imparts an independently controllable frequency offset, so that the two paths behave as independent lasers even though both are seeded from a single source. The beat note between the two outputs yields $f_2-f_1$; this signal is converted to a voltage via a PLL, divided by 2, and used to control a VCO that generates $(f_2-f_1)/2$, which drives another AOM. Recombination with one original optical signal synthesizes an output at the arithmetic mean frequency,

$f_{\mathrm{avg}}=\frac{f_1+n_1+f_2+n_2}{2},$

where $n_1$ and $n_2$ are the noise processes of the two paths. For uncorrelated noise,

$$\mathrm{Var}(f_{\mathrm{avg}})=\frac{1}{4}\left[\mathrm{Var}(n_1)+\mathrm{Var}(n_2)\right],$$

which for identical noises becomes

$N$0

Experimentally, two fiber-cavity-stabilized paths each exhibited frequency fluctuations of about $N$1 at $N$2, while the averaged output showed $N$3, consistent with the expected $N$4 reduction. At a $N$5 offset, the frequency-noise power spectral density decreased from about $N$6 per individual path to about $N$7 in the average, and the integrated linewidth narrowed from $N$8 to $N$9 (Loh et al., 2023).

The significance of the method is that it addresses linewidth constraints of a single laser, particularly when the limit is set by fundamental noise sources such as thermal noise, irrespective of the spectral shape of the noise. The same work states that the approach can be extended to more than two lasers, with reduction in noise scaling as $1/\sqrt{N}$0 for $1/\sqrt{N}$1 independent lasers, and identifies integrated photonics as a natural implementation domain (Loh et al., 2023).

3. Averaging time, Allan deviation, and optical standards

In optical frequency standards and optical clock comparisons, averaging is usually quantified through fractional frequency stability as a function of averaging time $1/\sqrt{N}$2. The standard metric is the Allan deviation,

$1/\sqrt{N}$3

where $1/\sqrt{N}$4 is the average fractional frequency over interval $1/\sqrt{N}$5 of duration $1/\sqrt{N}$6. Modified Allan deviation and overlapping Allan deviation are also used when white phase noise, dead-time-free counters, or long-record estimation are important (Maurice et al., 2020, Raupach et al., 2014).

A compact optical frequency standard based on the Doppler-free, two-photon transition of $1/\sqrt{N}$7Rb at $1/\sqrt{N}$8 reported

$1/\sqrt{N}$9

for averaging times $\tau$0. In that case, short-term behavior was limited by intermodulation noise of the DBR laser, and the benefit of longer averaging was ultimately constrained by temperature-dependent linear drift associated with misalignment of interrogation beams and by insufficient stabilization against environmental magnetic field changes (Maurice et al., 2020).

A direct optical clock ratio measurement between neutral ytterbium and strontium showed how averaging time can be compressed by synchronous interrogation and phase-locked frequency-comb transfer. The reported instability was

$\tau$1

and a frequency-ratio uncertainty of $\tau$2 was reached at an averaging time of $\tau$3. The measured ratio was

$\tau$4

with a fractional uncertainty of $\tau$5 (Nemitz et al., 2016).

For active optical frequency standards, the literature distinguishes two averaging regimes. At short averaging times, stability is limited by photon shot noise from limited emitted laser power; at long averaging times, it is limited by phase diffusion of the laser output. The total Allan deviation is given by

$\tau$6

with the short-term term scaling as $\tau$7 and the long-term term as $\tau$8. Using realistic numbers for an active clock with $\tau$9Sr, an optimized stability of

$f_2-f_1$0

was found to be achievable (Kazakov et al., 2022).

Taken together, these results show that optical frequency averaging in metrology is not merely “longer integration.” It is a balance among white frequency noise, white phase noise, intermodulation noise, photon shot noise, phase diffusion, and systematic drift. A common misconception is that longer averaging always yields continued $f_2-f_1$1 improvement; the cited standards explicitly identify regimes in which drift or diffusion arrests that behavior (Maurice et al., 2020, Kazakov et al., 2022).

4. Time averages, ensemble averages, and frequency filtering in coherence theory

Optical coherence theory introduces a distinct use of averaging. When stationarity and ergodicity become implicit assumptions, finite time averaging by a detector and ensemble averaging of the optical field can appear interchangeable, even though they are not generally the same. This distinction is central to the Magyar and Mandel-Wolf paradox (Veen et al., 2024).

For a field envelope $f_2-f_1$2, the ensemble average is written as $f_2-f_1$3, while a finite time average over detector integration time $f_2-f_1$4 is

$f_2-f_1$5

The photodetector output is

$f_2-f_1$6

so real measurements can contain both time averaging and ensemble averaging. With narrow frequency filtering,

$f_2-f_1$7

and for very narrow filters this simplifies to

$f_2-f_1$8

Wolf’s spectral coherence is

$f_2-f_1$9

The paradox is resolved by the statement that Magyar and Mandel defined the visibility as the intensity coherence while Wolf defined it as the amplitude coherence. A short detector window can show high single-shot fringe visibility for independent sources because the relative phase is approximately constant over that window, while the ensemble-averaged visibility can be low because the phase is random across repetitions. The consequence is methodological: one must specify whether “coherence,” “visibility,” or “frequency average” refers to a time average, an ensemble average, or a spectrally filtered quantity. Time average is not equal to ensemble average unless the field is stationary and ergodic (Veen et al., 2024).

5. Frequency transfer links and distributed averaging

In practical optical metrology, frequency averaging is often meaningful only if the transfer network contributes less instability than the clocks or oscillators under comparison. Long-haul fiber and free-space transfer experiments therefore report how Allan deviation and modified Allan deviation average down over seconds, hours, and days (Raupach et al., 2014, Sinclair et al., 2017).

A $(f_2-f_1)/2$0 loop of installed underground fiber demonstrated ultrastable transfer of an optical frequency using a remote fiber Brillouin amplifier as the only means of remote amplification. Distances of $(f_2-f_1)/2$1 and $(f_2-f_1)/2$2 were bridged between amplifications. Over several days of uninterrupted measurement, the instability of the frequency transfer, expressed as Allan deviation of $(f_2-f_1)/2$3-weighted data with $(f_2-f_1)/2$4 gate time, was around $(f_2-f_1)/2$5 and less for averaging times longer than $(f_2-f_1)/2$6. The modified Allan deviation reached $(f_2-f_1)/2$7 at $(f_2-f_1)/2$8, and for averaging times longer than $(f_2-f_1)/2$9 it fell into the $f_{\mathrm{avg}}=\frac{f_1+n_1+f_2+n_2}{2},$0 range. A conservative value of the overall accuracy was $f_{\mathrm{avg}}=\frac{f_1+n_1+f_2+n_2}{2},$1, and several days of uninterrupted, cycle-slip-free operation were achieved (Raupach et al., 2014).

In-line extraction over a $f_{\mathrm{avg}}=\frac{f_1+n_1+f_2+n_2}{2},$2 installed telecommunication-fiber link showed that the residual frequency noise at an extraction end can be noticeably below that at the main link output when the extraction is near the input end. The analytical relation

$f_{\mathrm{avg}}=\frac{f_1+n_1+f_2+n_2}{2},$3

quantifies this behavior. Relative frequency instabilities, expressed as overlapping Allan deviation, were $f_{\mathrm{avg}}=\frac{f_1+n_1+f_2+n_2}{2},$4 at $f_{\mathrm{avg}}=\frac{f_1+n_1+f_2+n_2}{2},$5 averaging time and a few $f_{\mathrm{avg}}=\frac{f_1+n_1+f_2+n_2}{2},$6 at $f_{\mathrm{avg}}=\frac{f_1+n_1+f_2+n_2}{2},$7 day, with systematic frequency shifts below $f_{\mathrm{avg}}=\frac{f_1+n_1+f_2+n_2}{2},$8 over $f_{\mathrm{avg}}=\frac{f_1+n_1+f_2+n_2}{2},$9 days (Bercy et al., 2014).

A branching fiber network with passive phase noise cancellation at each remote user used a single AOM per remote site as both frequency distinguisher and phase noise compensator. Measurements with a back-to-back system gave $n_1$0 at $n_1$1, reaching $n_1$2 after $n_1$3. After compensation on a $n_1$4 fiber link, the relative frequency instability was $n_1$5 at the $n_1$6 averaging time and $n_1$7 at $n_1$8 (Xue et al., 2021).

Carrier-phase optical two-way time-frequency transfer through a turbulent $n_1$9 free-space link achieved a residual instability of $n_2$0 at $n_2$1 and tracked the relative optical phase of distant optical oscillators to $n_2$2 at $n_2$3 averaging. Because the method continuously tracks relative optical phase, it directly supports optical frequency comparisons across air at a level suitable for advanced clock networking and distributed optical frequency averaging (Sinclair et al., 2017).

6. Frequency combs, coherent averaging, and optical frequency division

Frequency combs supply several architectures in which optical frequency averaging becomes a control, synchronization, or division problem rather than an arithmetic mean of two cw laser frequencies. In dual-comb spectroscopy, the main issue is periodicity: if the offset frequencies $n_2$4 of two combs jitter independently, the multiheterodyne signal is non-periodic, beatnotes broaden, and coherent averaging becomes difficult (Hillbrand et al., 2022).

Optical injection locking provides a remedy. A single comb line isolated via an optical Vernier filter serves as a master oscillator for injection locking, while electrical injection stabilizes the repetition rates. With $n_2$5 and constant $n_2$6, the dual-comb signal

$n_2$7

becomes strictly periodic. This enables coherent averaging using analog electronics, which increases the SNR and reduces the data size by one and three orders of magnitude, respectively, and the resulting dual-comb signal is periodic and stable over thousands of periods (Hillbrand et al., 2022).

A different comb-based role for averaging appears in transfer-oscillator frequency conversion. In a Cs-based optical frequency measurement of the $n_2$8 $n_2$9Yb$$\mathrm{Var}(f_{\mathrm{avg}})=\frac{1}{4}\left[\mathrm{Var}(n_1)+\mathrm{Var}(n_2)\right],$$0 transition, an Er-fiber optical frequency comb generator was configured as a transfer oscillator so that fluctuations of the pulse repetition rate and of the carrier offset frequency did not degrade the stability of the frequency conversion. For fountain operation with optical molasses loaded from a laser cooled atomic beam source, the stability corresponded to a fractional Allan deviation of $$\mathrm{Var}(f_{\mathrm{avg}})=\frac{1}{4}\left[\mathrm{Var}(n_1)+\mathrm{Var}(n_2)\right],$$1 (Tamm et al., 2013).

Microresonator combs have also been stabilized so that their line spacing itself exhibits metrologically useful averaging behavior. A mechanically actuated microrod-resonator microcomb reached a residual line-spacing stability of $$\mathrm{Var}(f_{\mathrm{avg}})=\frac{1}{4}\left[\mathrm{Var}(n_1)+\mathrm{Var}(n_2)\right],$$2 for $$\mathrm{Var}(f_{\mathrm{avg}})=\frac{1}{4}\left[\mathrm{Var}(n_1)+\mathrm{Var}(n_2)\right],$$3 averaging, and measurements of different spectral slices revealed less than $$\mathrm{Var}(f_{\mathrm{avg}})=\frac{1}{4}\left[\mathrm{Var}(n_1)+\mathrm{Var}(n_2)\right],$$4 variation among $$\mathrm{Var}(f_{\mathrm{avg}})=\frac{1}{4}\left[\mathrm{Var}(n_1)+\mathrm{Var}(n_2)\right],$$5 comb lines spanning $$\mathrm{Var}(f_{\mathrm{avg}})=\frac{1}{4}\left[\mathrm{Var}(n_1)+\mathrm{Var}(n_2)\right],$$6. The stabilization system exhibited a $$\mathrm{Var}(f_{\mathrm{avg}})=\frac{1}{4}\left[\mathrm{Var}(n_1)+\mathrm{Var}(n_2)\right],$$7 averaging behavior, characteristic of white frequency noise (Papp et al., 2012).

In 2-point optical frequency division, optical cavity modes anchor two spectral endpoints defined by frequency-comb lines, and the comb need not be self-referenced. The repetition rate is set by

$$\mathrm{Var}(f_{\mathrm{avg}})=\frac{1}{4}\left[\mathrm{Var}(n_1)+\mathrm{Var}(n_2)\right],$$8

where $$\mathrm{Var}(f_{\mathrm{avg}})=\frac{1}{4}\left[\mathrm{Var}(n_1)+\mathrm{Var}(n_2)\right],$$9 and $N$00 are optical reference frequencies and $N$01 is the comb mode-number separation. In the reported microcomb implementation, a frequency-agile single-mode dispersive wave defined one endpoint and a compact all-solid-state optical cavity with a measured $N$02-factor of $N$03 provided the optical reference. The description in the paper states that common-mode noise is rejected and the remaining noise is divided down by the large mode-number separation, transferring the optical cavity’s stability to the microwave repetition rate (Ji et al., 2024).

7. Limitations, misconceptions, and emerging directions

The limiting mechanisms identified across the literature are diverse. In direct optical frequency averaging of cavity-stabilized paths, the relevant floor is the thermal noise of the independent reference cavities (Loh et al., 2023). In compact optical standards, long-term averaging is limited by temperature-dependent linear drift from beam misalignment and by magnetic-field sensitivity (Maurice et al., 2020). In active optical frequency standards, short-term instability is set by photon shot noise and long-term instability by phase diffusion (Kazakov et al., 2022). In long-haul fiber links, differential temperature fluctuations between local and remote optical setups affect instability below $N$04 (Raupach et al., 2014). In free-space links, the measured instability can approach the fundamental limit set by atmospheric reciprocity breakdown (Sinclair et al., 2017). In microcomb stabilization, the present limitations arise mainly from pump laser noise and the microwave reference rather than the mechanical stabilization scheme (Papp et al., 2012).

Several common misconceptions are resolved by these results. One is that averaging is always equivalent to low-pass smoothing; in fact, direct arithmetic averaging of optical frequencies is an overview operation that can narrow linewidth beyond the limit of a single cavity-stabilized source when the constituent noises are statistically independent (Loh et al., 2023). A second is that all experimentally observed coherence corresponds to a single average; the coherence literature shows that amplitude coherence and intensity coherence are not interchangeable because time averaging and ensemble averaging are different operations (Veen et al., 2024). A third is that longer averaging is unconditionally beneficial; optical frequency standards and active clocks show explicit drift-limited or diffusion-limited regimes (Maurice et al., 2020, Kazakov et al., 2022).

The current trajectory suggests several converging directions. One is extension from two-source averaging to $N$05-source averaging with $N$06 scaling for independent systems (Loh et al., 2023). Another is compact implementation: portable optical standards on micro-optics breadboards, integrated dual-comb spectrometers enabled by coherent averaging, and chip-scale optical frequency division using microcombs and dispersive waves (Maurice et al., 2020, Hillbrand et al., 2022, Ji et al., 2024). A further implication is large-scale dissemination, since in-line extraction, passive branching compensation, remote Brillouin amplification, and carrier-phase two-way transfer all show that the transfer link need not be the limiting factor for modern optical clock comparison on practical averaging timescales (Bercy et al., 2014, Xue et al., 2021, Raupach et al., 2014, Sinclair et al., 2017).