Skip to main content
Sandbox Physics

L29 · Let reflection correct frequency

Pound–Drever–Hall Frequency Lock

Send phase-modulation sidebands into a reference cavity, demodulate the reflection and close the servo. Capture acquisition, suppression, ringing and loss of lock.

Interactive modelPound–Drever–Hall Frequency Lock
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{—}
Reflected fraction of incident power—\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{—}
Loop diagnostic—\text{—}

Physics tutorial

Reflection contains a direction, not just a power

BackgroundA narrow cavity converts a frequency offset into the phase of its reflected field.

Why it mattersMix that field’s sideband beating down to a signed error, then use it to tune the laser.

Start with the essentials

Focus question
Can reflection tell a laser which way to tune?
One-sentence intuition
The servo suppresses motion relative to its reference; a quiet error signal cannot establish an absolutely quiet reference.

Core mathematical model

Create phase sidebands

Ein=E0eiωt+iβsin⁡Ωt=E0∑nJn(β)ei(ω+nΩ)t\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.

Reflect each frequency

r(δ)=rm(1−eiφ)1−rm2eiφφ=2πδ/fFSR\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}

The lossless symmetric cavity has zero monochromatic reflection on resonance. Sidebands remain in the calculation.

Demodulate the beat

an=Jn(β)r(δ+nfm)C1=∑nan+1an∗ϵ0=−2Im⁡C1ϵ90=2Re⁡C1\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}

Mixer phase selects a quadrature. Rotating it by half a turn reverses the sign of feedback.

Close the reduced servo

z˙=2πfieτau˙+u=clip⁡(−Kpe−z)δ=ffree+u−fc\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}

A filtered error drives proportional and integral correction. Command clipping requires anti-windup.

Predict only the local loop

L(s)=(Kp+2πfis)GHcHaHj(s)=11+s/(2πfj)S(s)=11+L(s)\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 reference assumes operation near carrier resonance with no clipping; negative phase margin warns of instability.

Do not confuse relative and absolute motion

flaser=ffree+uδ=flaser−fcδfbin=1/(NΔt)\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 same frozen frequency record supplies the residual trace and its Hann spectrum. Cavity wander remains visible in absolute frequency.

Common difficulties

No reflected power means no error

Typical misconceptionThe detector cannot make a useful error when the carrier is transmitted.

Better mental modelReflected sidebands beat against a small reflected carrier field and reveal its phase.

Low in-loop noise proves a stable laser

Typical misconceptionA quiet error signal proves absolute frequency stability.

Better mental modelIt measures laser motion relative to the reference and depends on the sensor model; an independent reference is needed.

Run the experiment

  1. 01

    Acquire a sign

    Compare open loop and acquisition, then reverse mixer phase.

    What to observe: A correct sign attracts the carrier resonance; an incorrect sign repels it.
  2. 02

    Find the dynamical limit

    Compare a slow servo with an unstable loop, then kick the free laser.

    What to observe: Static error slope alone does not guarantee stable capture or enough actuator range.
  3. 03

    Follow a moving reference

    Capture the wandering-reference preset and compare relative and absolute RMS.

    What to observe: The laser follows reference motion even while the in-loop residual is small.