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Allan Deviation & Laser Frequency Stability
1. Fractional frequency
For a nominal optical frequency $\nu_0$, define the fractional frequency fluctuation
\[y(t)=\frac{\nu(t)-\nu_0}{\nu_0}.\]Allan deviation is usually reported as the dimensionless fractional instability $\sigma_y(\tau)$.
The corresponding absolute frequency instability is
\[\boxed{\sigma_\nu(\tau)=\nu_0\,\sigma_y(\tau).}\]This conversion is important for optical systems because the carrier frequency is hundreds of terahertz. A fractional instability that looks extremely small can still correspond to measurable hertz-level frequency motion.
2. Allan variance and Allan deviation
Divide a frequency record into adjacent intervals of duration $\tau$. Let $\bar y_i(\tau)$ be the average fractional frequency in the $i$th interval. The two-sample Allan variance is
\[\boxed{ \sigma_y^2(\tau)=\frac{1}{2(M-1)}\sum_{i=1}^{M-1} \left[\bar y_{i+1}(\tau)-\bar y_i(\tau)\right]^2 }\]and the Allan deviation is
\[\boxed{\sigma_y(\tau)=\sqrt{\sigma_y^2(\tau)}.}\]Unlike ordinary variance, Allan variance remains useful for several oscillator-noise processes for which conventional variance does not converge conveniently.
3. Allan deviation from phase or time-error data
If the measured quantity is phase expressed as time error $x_i$ at equally spaced times separated by $\tau$, then
\[\boxed{ \sigma_y(\tau)= \sqrt{\frac{1}{2(N-2)\tau^2} \sum_{i=1}^{N-2} \left(x_{i+2}-2x_{i+1}+x_i\right)^2} }.\]The second difference suppresses a constant phase offset and converts the phase/time record into a frequency-stability statistic.
4. Why use Allan deviation instead of ordinary standard deviation?
Oscillators often contain power-law noise such as flicker frequency noise and random-walk frequency noise. For these processes, simply calculating the standard deviation of a long frequency record can give a result that depends strongly on record length and drift.
Allan deviation was developed specifically for oscillator and frequency-standard characterization. It simultaneously provides
- a stability number;
- an averaging-time dependence;
- clues about the dominant noise process;
- a natural way to compare oscillators, lasers, cavities, clocks and frequency combs.
5. Reading an Allan-deviation plot
Allan deviation is normally plotted on logarithmic axes:
\[\log \sigma_y(\tau)\quad\text{versus}\quad\log \tau.\]A common laser-stability curve falls with $\tau$ at short averaging times, reaches a minimum, then rises at long $\tau$ as drift or random-walk processes dominate.
The averaging time at the minimum is often experimentally important: averaging beyond that point may no longer improve the frequency estimate.
6. Noise type from Allan-deviation slope
For common power-law oscillator noise processes, the approximate slope of $\sigma_y(\tau)$ on a log-log plot is informative.
| Dominant process | Approximate Allan-deviation scaling | Log-log slope |
|---|---|---|
| white phase modulation (PM) | $\sigma_y\propto\tau^{-1}$* | $-1$* |
| flicker phase modulation | approximately $\tau^{-1}$* | approximately $-1$* |
| white frequency modulation (FM) | $\sigma_y\propto\tau^{-1/2}$ | $-1/2$ |
| flicker FM | $\sigma_y\approx\text{constant}$ | $0$ |
| random-walk FM | $\sigma_y\propto\tau^{+1/2}$ | $+1/2$ |
| deterministic linear frequency drift | $\sigma_y\propto\tau$ | $+1$ |
*The exact PM behavior depends on measurement bandwidth, and ordinary Allan deviation does not cleanly distinguish white PM from flicker PM. Modified Allan deviation is useful for that purpose.
7. Overlapping Allan deviation
The basic non-overlapping estimator discards many possible samples. The overlapping Allan deviation reuses all valid overlapping windows and therefore improves statistical confidence without changing the underlying Allan-variance definition.
For time-error data $x_i$ sampled at interval $\tau_0$, with $\tau=m\tau_0$,
\[\sigma_y^2(\tau)= \frac{1}{2m^2\tau_0^2(N-2m)} \sum_{i=1}^{N-2m} \left(x_{i+2m}-2x_{i+m}+x_i\right)^2.\]For experimental laser data, overlapping ADEV is generally the more useful default because long averaging times otherwise contain very few independent points.
8. Modified Allan deviation (MDEV)
Modified Allan deviation adds an additional averaging operation before forming the Allan statistic. It is especially useful for separating white and flicker phase-modulation noise and for characterizing systems dominated by high-frequency phase noise.
MDEV is commonly written
\[\mathrm{MDEV}(\tau)=\mathrm{mod}\,\sigma_y(\tau).\]It should not be mixed numerically with ordinary ADEV: the two statistics have different transfer functions and can report different values from the same data.
For precision laser and comb work, it is good practice to state explicitly whether a plot shows
- ADEV;
- overlapping ADEV;
- MDEV;
- overlapping MDEV.
9. Hadamard deviation and deterministic drift
Long-term laser records often contain linear frequency drift from reference-cavity aging or temperature drift. Ordinary Allan deviation responds strongly to deterministic linear drift.
The Hadamard deviation uses a higher-order difference and is insensitive to linear frequency drift, making it useful when the scientific question concerns stochastic stability in the presence of deterministic drift.
Another common approach is to fit and remove a known linear drift before calculating ADEV. If this is done, report both
- the fitted drift rate, for example Hz/s;
- whether the Allan-deviation data are raw or detrended.
Removing drift without reporting it can make the long-term stability appear artificially better.
10. Linewidth versus Allan deviation
These quantities answer different questions.
| Metric | Main question | Typical domain |
|---|---|---|
| optical linewidth $\Delta\nu$ | how broad is the optical spectrum? | frequency domain |
| frequency-noise PSD $S_\nu(f)$ | at what Fourier frequencies does frequency noise occur? | frequency-noise spectrum |
| phase-noise PSD $S_\phi(f)$ | how does oscillator phase fluctuate versus Fourier frequency? | frequency domain |
| Allan deviation $\sigma_y(\tau)$ | how stable is average frequency versus averaging time? | time domain |
A cavity-locked laser can therefore have a sub-hertz central linewidth yet show a rising Allan deviation at long $\tau$ because the cavity resonance drifts thermally.
11. Worked optical-frequency example
For a 509 nm coupling laser,
\[\nu_0=\frac{c}{\lambda}\approx5.89\times10^{14}\ \text{Hz}.\]If
\[\sigma_y(1\ \text{s})=10^{-12},\]then the corresponding absolute 1-second instability is
\[\sigma_\nu(1\ \text{s}) =\nu_0\sigma_y \approx589\ \text{Hz}.\]If cavity locking improves the fractional instability to
\[\sigma_y(1\ \text{s})=10^{-14},\]the corresponding absolute instability is only
\[\sigma_\nu(1\ \text{s})\approx5.9\ \text{Hz}.\]This is why fractional units are convenient for comparing optical oscillators at different wavelengths.
12. Allan deviation of a cavity-locked laser
A typical cavity-stabilized-laser measurement is
The resulting curve can reveal several regions:
- short $\tau$: residual fast laser noise, counter noise or servo limitations;
- intermediate $\tau$: cavity thermal-noise or technical stability floor;
- long $\tau$: temperature drift, cavity aging, vibration environment or reference drift.
A low linewidth does not guarantee a low long-term ADEV, and a good long-term ADEV does not by itself prove a narrow instantaneous linewidth.
13. Two-laser beat measurements
If two independent lasers of similar stability are compared, their frequency fluctuations add in variance. With both fractional instabilities referred to approximately the same optical carrier frequency,
\[\sigma_{y,\mathrm{beat}}^2\approx \sigma_{y,1}^2+\sigma_{y,2}^2.\]For two statistically independent lasers with equal stability,
\[\boxed{\sigma_{y,\mathrm{single}}\approx \frac{\sigma_{y,\mathrm{beat}}}{\sqrt{2}}.}\]For three independent references, a three-cornered-hat analysis can estimate the individual oscillator instabilities without assuming two equal sources, provided correlations are treated carefully.
14. Allan deviation through second-harmonic generation
For ideal SHG,
\[\nu_{2\omega}=2\nu_\omega, \qquad \delta\nu_{2\omega}=2\delta\nu_\omega.\]Therefore the fractional frequency fluctuation is unchanged:
\[\frac{\delta\nu_{2\omega}}{\nu_{2\omega}} = \frac{\delta\nu_\omega}{\nu_\omega}.\]Hence, in the absence of additional noise from the amplifier, nonlinear crystal, enhancement cavity or delivery optics,
\[\boxed{\sigma_{y,2\omega}(\tau)=\sigma_{y,\omega}(\tau).}\]The absolute Allan deviation in hertz doubles:
\[\boxed{\sigma_{\nu,2\omega}(\tau)=2\sigma_{\nu,\omega}(\tau).}\]This is the time-domain counterpart of the SHG frequency-noise relation $S_{\nu,2\omega}=4S_{\nu,\omega}$.
15. What an Allan-deviation measurement can diagnose in the lab
ADEV is particularly useful for identifying the timescale on which a stabilization system stops improving. Examples include
- insufficient slow integrator gain in the laser lock;
- cavity temperature excursions;
- HVAC or acoustic cycles;
- vibration sensitivity;
- wavemeter drift;
- atomic-reference drift;
- optical-fiber path-length changes;
- SHG oven-temperature fluctuations;
- frequency-counter reference instability.
Periodic disturbances often appear more clearly in the raw frequency record or PSD than in ADEV, so inspect all three when troubleshooting.
16. Measurement practice
For a defensible Allan-deviation result, report
- nominal optical frequency or wavelength;
- measured quantity: absolute frequency, beat frequency, phase or time error;
- counter/phase-recorder model and measurement mode;
- sample interval and gate time;
- total record duration;
- presence of dead time;
- ADEV estimator used: non-overlapping, overlapping or modified;
- whether linear drift was removed;
- reference-oscillator contribution;
- confidence intervals or equivalent degrees of freedom when relevant.
17. Connection to Rydberg EIT and spectroscopy
In an atomic experiment, Allan deviation can quantify how the laser detuning changes on timescales much longer than the instantaneous optical coherence time.
For a coupling laser with detuning $\Delta_c(t)$, slow frequency instability can produce
- motion of the EIT resonance center;
- apparent broadening after repeated scans;
- reduced averaging benefit;
- drift in extracted Autler-Townes splittings or Stark shifts;
- systematic variation in fitted line centers.
This makes ADEV useful alongside linewidth and $S_\nu(f)$ when deciding whether a measured EIT linewidth is limited by atomic physics or by the laser system.
A useful characterization set is
\[\boxed{ \Delta\nu_{laser} +S_\nu(f) +\sigma_y(\tau) +\text{servo error spectrum} }\]rather than a single quoted linewidth.
18. References
- W. J. Riley and D. A. Howe, Handbook of Frequency Stability Analysis, NIST Special Publication 1065. NIST publication page
- D. W. Allan, “The measurement of frequency and frequency stability of precision oscillators,” NBS Technical Note 669 (1975). DOI
- D. W. Allan, “Statistics of atomic frequency standards,” Proceedings of the IEEE 54, 221–230 (1966). DOI
- NIST, “Time and Frequency from A to Z — Allan Deviation.” NIST reference
Related: Laser Systems · Optics & Photonics · Measurements & Instruments · Theory ↔ Experiment · Hydrogen Maser