L29 · Let reflection correct frequency

Pound–Drever–Hall Frequency Lock

Can reflection tell a laser which way to tune? Yes. Add sidebands to the laser and reflect it off a reference cavity. The reflected sidebands beat against the carrier, and demodulating that beat gives an error signal whose sign tells you which side of resonance the laser is on. A quiet error signal means the laser follows the reference closely; it does not prove the reference itself is quiet.

Send phase-modulated light into a reference cavity, demodulate the reflected light and close the feedback loop. Record the laser pulling into lock, the noise being suppressed, the loop ringing and the lock being lost.

Elapsed record time $\text{—}$

Cavity power linewidth $\text{—}$

Assumed cavity sensor pole $\text{—}$

Laser minus cavity frequency $\text{—}$

Laser frequency from fixed origin $\text{—}$

Actual actuator correction $\text{—}$

Filtered factory-calibrated error $\text{—}$

End-of-record reflected power fraction $\text{—}$

Incident carrier power fraction $\text{—}$

Each first-sideband power fraction $\text{—}$

Slope relative to factory setting $\text{—}$

Local linear unity-gain frequency $\text{—}$

Local linear phase margin $\text{—}$

Separate linear tone suppression $\text{—}$

Captured frequency samples $\text{—}$

Captured PSD bin resolution $\text{—}$

Captured cavity-relative RMS $\text{—}$

Captured absolute frequency RMS $\text{—}$

End-of-record loop diagnostic $\text{—}$

L29

Can reflection tell a laser which way to tune?

Reference cavity length

L

20\,\mathrm{cm}

Cavity finesse

\mathcal F

10000

EOM modulation frequency

f_m

20\,\mathrm{MHz}

Phase-modulation depth

\beta

0.5\,\mathrm{rad}

Factory mixer phase

\theta

0\,\mathrm{{}^\circ}

Free laser frequency offset

\Delta f_0

120\,\mathrm{kHz}

Integral gain frequency setting

f_i

20\,\mathrm{kHz}

Proportional gain

K_p

0.6

Frequency actuator pole

f_a

20\,\mathrm{kHz}

Free laser disturbance amplitude

A_l

60\,\mathrm{kHz}

Disturbance tone frequency

f_d

5\,\mathrm{kHz}

Reference cavity wander amplitude

A_c

0\,\mathrm{kHz}

Feedback: open / closed

b

1

Detail: cavity / detector / servo

j

0

Compare the open loop with lock acquisition under the same disturbance. Then let the reference cavity wander: a small in-loop residual does not prove that the laser frequency is stable in absolute terms.

The reference is a lossless symmetric Fabry–Pérot cavity with exact Airy reflection. The phase modulation keeps orders from minus six through plus six; within the allowed modulation depth, the omitted orders carry less than one part per billion of the incident power. The optical carrier sits near one cavity resonance, and the RF sidebands are not assumed to be completely reflected. Detuning is the laser frequency minus the instantaneous cavity frequency, and a positive actuator correction raises the laser frequency. The mixer sign is chosen so that the default factory-calibrated error is positive for positive detuning. Demodulation phase and modulation depth change the real loop gain; normalizing the error does not retune the controller. The servo is a simplified baseband PI loop. The cavity acts as a first-order sensor with its pole at half the cavity power linewidth, and the actuator has its own adjustable pole. To show lock acquisition, the model also applies this near-resonance sensor approximation to the full nonlinear static error curve. It does not solve transient complex cavity fields, RF propagation delays or a real electronic circuit. The actuator command clips at six hundred kilohertz in either direction, with conditional integration as anti-windup. The disturbances are known sine tones, not random or measured device noise. The unity-gain frequency, phase margin and tone suppression come from a separate linear model that holds near carrier resonance without clipping. They do not guarantee nonlinear capture or the stability of real hardware. The run records five milliseconds of simulated frequencies at five hundred kilohertz. The last power-of-two block of samples gives the mean-removed Hann noise spectra and the frozen residual traces. The cavity-relative RMS includes the mean; the PSD removes it. Absolute laser frequency is measured from a fixed origin that does not move with the cavity. Opening the feedback keeps the actuator and integrator states, and the actuator then relaxes. Recording keeps the disturbance phase continuous. An ideal polarizing beam splitter and quarter-wave plate send the reflected light to the RF detector. Electrical RF and feedback wires are shown separately from the optical paths. Shot noise, residual amplitude modulation, laser amplitude and carrier dynamics, spatial mode mismatch, cavity thermal noise and electronics noise are left out. Distances, colors and moving markers are schematic.

Start with the essentials

Focus question: Can reflection tell a laser which way to tune?
One-sentence intuition: The servo holds the laser to the cavity, not to an absolute frequency; a quiet error signal cannot show that the cavity itself is quiet.

Core mathematical model

Phase modulation makes sidebands

\begin{aligned}E_{\rm in}&=E_0e^{i\omega t+i\beta\sin\Omega t}\\&=E_0\sum_n J_n(\beta)e^{i(\omega+n\Omega)t}\end{aligned}

Neighboring sidebands beat at the modulation frequency. The model keeps thirteen orders.

Cavity reflection at each frequency

\begin{aligned}r(\delta)&=\frac{r_m(1-e^{i\varphi})}{1-r_m^2e^{i\varphi}}\\\varphi&=2\pi\delta/f_{\rm FSR}\end{aligned}

On resonance, a lossless symmetric cavity reflects none of a single-frequency beam. The calculation still includes every sideband.

Demodulate the beat

\begin{aligned}a_n&=J_n(\beta)r(\delta+nf_m)\\C_1&=\sum_n a_{n+1}a_n^*\\\epsilon_0&=-2\operatorname{Im}C_1\\\epsilon_{90}&=2\operatorname{Re}C_1\end{aligned}

The mixer phase picks which quadrature of the beat you see. Shifting it by half a turn flips the sign of the feedback.

A simplified PI servo

\begin{aligned}\dot z&=2\pi f_i e\\\tau_a\dot u+u&=\operatorname{clip}(-K_pe-z)\\\delta&=f_{\rm free}+u-f_c\end{aligned}

The filtered error drives a proportional and an integral correction. Because the actuator command is clipped, the integrator needs anti-windup.

Linear loop near resonance

\begin{aligned}L(s)&=\left(K_p+\frac{2\pi f_i}{s}\right)GH_cH_a\\H_j(s)&=\frac{1}{1+s/(2\pi f_j)}\\S(s)&=\frac{1}{1+L(s)}\end{aligned}

This linear model holds only near carrier resonance and without clipping. A negative phase margin warns that the loop is unstable.

Relative and absolute frequency

\begin{aligned}f_{\rm laser}&=f_{\rm free}+u\\\delta&=f_{\rm laser}-f_c\\\delta f_{\rm bin}&=1/(N\Delta t)\end{aligned}

The residual trace and its Hann spectrum come from the same frozen frequency record. When the cavity wanders, the wander still shows up in the laser's absolute frequency.

Common difficulties

No reflected power means no error

Typical misconceptionWhen the carrier passes through the cavity, the detector has nothing useful to measure.

Better mental modelA small part of the carrier field is still reflected. The reflected sidebands beat against it, and the beat reveals the carrier's phase.

Low in-loop noise proves a stable laser

Typical misconceptionIf the error signal is quiet, the laser frequency is absolutely stable.

Better mental modelThe error signal measures how the laser moves relative to the cavity, and it relies on a model of the sensor. Checking absolute stability needs an independent reference.

Run the experiment

01
Check the error sign
Compare the Open loop and Acquire resonance presets, then reverse the mixer phase.
What to observe: With the right sign, the laser is pulled onto the carrier resonance; with the wrong sign, it is pushed away.
02
Find where the loop fails
Compare the Slow servo and Unstable loop presets, then press Kick laser frequency.
What to observe: A steep error slope is not enough. The loop must also be stable, and the actuator needs enough range to capture the laser.
03
Follow a moving reference
Capture the Wandering reference preset and compare the cavity-relative RMS with the absolute RMS.
What to observe: The laser follows the cavity as it moves, even while the in-loop residual stays small.