Using Allan Deviation to Characterize Frequency Lock Performance

Overview

In the field of precision measurement, saying a laser is "locked" is only the beginning of the conversation. The critical question is: How stable is the lock?

While a simple oscilloscope trace of the error signal can show you if the system is holding resonance, it cannot easily quantify long-term drift or distinguish between different types of underlying noise. To do that, we use the Allan Deviation (ADEV).

In this post, we'll look at how we compute Allan deviation from our FPGA-based controller data and how we use it to diagnose the health of our optical frequency comb locks.

What Allan Deviation Measures

Allan deviation $\sigma_y(\tau)$ is a measure of frequency stability in the time domain. Unlike a standard deviation, which will diverge for many types of oscillator noise (like random walk), the Allan deviation remains convergent.

It is defined based on the average of the fractional frequency difference between adjacent measurements over an averaging time $\tau$:

$$\sigma_y^2(\tau) = \frac{1}{2(M-1)} \sum_{i=1}^{M-1} (\bar{y}_{i+1} - \bar{y}_i)^2$$

By plotting $\sigma_y(\tau)$ against the averaging time $\tau$ on a log-log scale, we can identify the "noise floor" of our system and see how stability improves (or degrades) over time.

Noise Types and Slope Identification

The power of the Allan deviation lies in its ability to identify the Power Spectral Density (PSD) of the noise by the slope of the plot:

Diagnosing Lock Problems

When we look at a real-time Allan deviation plot in our Web UI, we look for three specific features:

1. •The "Servo Bump": A localized peak at short $\tau$ (high frequency). This indicates that the PID loop is near instability or has a resonance that needs further filtering.
2. •The Flicker Floor: The flat region where averaging longer no longer improves stability. This is the ultimate limit of your local oscillator or your reference.
3. •Long-term Drift: If the plot starts to turn upward at $\tau > 10$ seconds, it usually points to a physical environment problem—a laboratory air conditioner cycling, or a thermal drift in the optical table.

Real-Time Display in Quanta Control

Our architecture computes the Allan deviation directly from the streamed error signal data. In the Web UI, we use a sliding window to provide a live ADEV plot.

Because our FPGA decimates the signal to 1 MSPS before streaming, we have the bandwidth to characterize stability from $1 \mu s$ up to hours of continuous locking. This real-time feedback is invaluable during the "tuning" phase of a new frequency comb, allowing us to see immediately if a change in $K_i$ gain improves long-term stability or just adds high-frequency jitter.

Optical Frequency-Comb Stability Example

In a typical Optical Frequency-Comb stabilization experiment, we might see the Allan deviation of the repetition rate $f_{rep}$ start at $10^{-10}$ at $\tau = 1 ms$ and follow a $-1/2$ slope down to $10^{-13}$ at $\tau = 1 s$. If it hits a floor there, we know that our 14-bit ADC noise or the reference clock is the limiting factor, and we might need to graduate to the product-grade custom hardware we discussed previously.

Summary

The Allan deviation is more than just a metric; it is a diagnostic window into the physics of your system. By integrating real-time ADEV computation into our open-source control stack, we provide researchers with the tools they need to move beyond "it's locked" to "it's characterized."

With the completion of this series, we have covered the entire chain of a precision FPGA feedback system—from hardware selection to digital filtering, PID implementation, and final stability analysis.

Explore the full source code and let us know how we can help with your next precision control challenge.