---
title: "Pound–Drever–Hall Frequency Lock Interactive Lab | Sandbox Physics"
description: "Can reflection tell a laser which way to tune? Yes. Add sidebands to the laser and reflect it off a reference cavity."
canonical_url: "https://sandboxphysics.com/labs/pdh-frequency-lock/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/pdh-frequency-lock/"
last_modified: "2026-10-27"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

Laser foundations · Light, gain and feedback

[L01 · Compare sources](https://sandboxphysics.com/labs/laser-source-comparison/) [L02 · Count transitions](https://sandboxphysics.com/labs/light-matter-transitions/) [L03 · Build inversion](https://sandboxphysics.com/labs/population-inversion/) [L04 · Amplify once](https://sandboxphysics.com/labs/single-pass-amplifier/) [L05 · Add feedback](https://sandboxphysics.com/labs/passive-optical-resonator/) [L06 · Start from noise](https://sandboxphysics.com/labs/laser-noise-startup/) [L07 · Build and debug](https://sandboxphysics.com/labs/build-debug-laser/) [L08 · Measure and diagnose](https://sandboxphysics.com/labs/laser-diagnostic-bench/) [L09 · Select axial modes](https://sandboxphysics.com/labs/gain-longitudinal-modes/) [L10 · Confine the beam](https://sandboxphysics.com/labs/stable-resonator-geometry/) [L11 · Select spatial modes](https://sandboxphysics.com/labs/transverse-laser-modes/) [L12 · Propagate and measure](https://sandboxphysics.com/labs/gaussian-beam-quality/) [L13 · Align and match](https://sandboxphysics.com/labs/alignment-mode-matching/) [L14 · Optimize extraction](https://sandboxphysics.com/labs/output-coupler-optimization/) [L15 · Drive the dynamics](https://sandboxphysics.com/labs/laser-dynamics-relaxation/) [L16 · Store and release](https://sandboxphysics.com/labs/q-switched-giant-pulse/) [L17 · Lock phases into pulses](https://sandboxphysics.com/labs/mode-locking-frequency-comb/) [L18 · Calibrate the comb](https://sandboxphysics.com/labs/optical-frequency-comb/) [L19 · Inject carriers](https://sandboxphysics.com/labs/semiconductor-laser-diode/) [L20 · Compare gain media](https://sandboxphysics.com/labs/laser-medium-zoo/) [L21 · Distributed gain](https://sandboxphysics.com/labs/fiber-fbg-laser/) [L22 · Choose feedback](https://sandboxphysics.com/labs/laser-feedback-architectures/) [L23 · Balance ultrafast pulses](https://sandboxphysics.com/labs/ti-sapphire-oscillator/) [L24 · Stretch and compress](https://sandboxphysics.com/labs/chirped-pulse-amplification/) [L25 · Trap and extract](https://sandboxphysics.com/labs/regenerative-multipass-amplifier/) [L26 · Convert frequencies](https://sandboxphysics.com/labs/nonlinear-frequency-conversion/) [L27 · Scatter and cascade](https://sandboxphysics.com/labs/raman-brillouin-lasers/) [L28 · Synchronize a slave](https://sandboxphysics.com/labs/injection-locking/) [L29 · Lock to a reference](https://sandboxphysics.com/labs/pdh-frequency-lock/) [L30 · Measure phase noise](https://sandboxphysics.com/labs/laser-linewidth-phase-noise/) [L31 · Scatter for feedback](https://sandboxphysics.com/labs/random-laser/) [L32 · Diagnose thermal scaling](https://sandboxphysics.com/labs/thermal-lensing/) [L33 · Combine coherent fields](https://sandboxphysics.com/labs/coherent-beam-combining/) [L34 · Sustain a Kerr pulse](https://sandboxphysics.com/labs/microresonator-kerr-comb/) [L35 · Return a delayed field](https://sandboxphysics.com/labs/external-feedback-chaos/) [L36 · Scan without mode hops](https://sandboxphysics.com/labs/tunable-laser-scan/)

Physical basis: [MIT: coherence and spectrum](https://ocw.mit.edu/courses/res-6-006-video-demonstrations-in-lasers-and-optics-spring-2008/resources/coherence-length-and-source-spectrum/) · [Rate equations](https://ocw.mit.edu/courses/6-977-ultrafast-optics-spring-2005/resources/chapter2/) · [Pumping and gain media (§7.2)](https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/bf3bdeb971819ddc9c68719d28bb2745_classnotes.pdf) · [Laser dynamics](https://ocw.mit.edu/courses/6-977-ultrafast-optics-spring-2005/resources/chapter4/)

Physics tutorial

## Reflection contains a direction, not just a power

**Background** A narrow-linewidth cavity turns a small frequency offset into a phase shift of the light it reflects.

**Why it matters** Beating the reflected light against phase-modulation sidebands and mixing it down gives an error signal with a sign. That signal can then steer the laser.

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 misconception** When the carrier passes through the cavity, the detector has nothing useful to measure.

**Better mental model** A 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 misconception** If the error signal is quiet, the laser frequency is absolutely stable.

**Better mental model** The 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

1. 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.

2. 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.

3. 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.

Physical conclusion

Ask three separate questions: does the error signal point the right way, is the linearized loop stable near resonance, and does the recorded nonlinear run actually acquire lock?

Waves & optics · Starter path

## From one ray to interference measurement

Start with refraction, find where the ray picture stops working, then use interference to measure tiny distances.

[Start here → **Optical Boundary Explorer** Turn the incident angle and watch reflected and transmitted power trade off. Then switch to the critical-angle scene. Observe · 11 min](https://sandboxphysics.com/labs/optical-boundary-explorer/)

1. [1 Optical Boundary Explorer](https://sandboxphysics.com/labs/optical-boundary-explorer/)
2. [2 Rainbow Caustic Observatory](https://sandboxphysics.com/labs/rainbow-laboratory/)
3. [3 Camera Obscura](https://sandboxphysics.com/labs/camera-obscura/)
4. [4 Coherent Wave Composer](https://sandboxphysics.com/labs/coherent-wave-composer/)
5. [5 Young Double-Slit Bench](https://sandboxphysics.com/labs/young-double-slit-bench/)
6. [6 Michelson Precision Lab](https://sandboxphysics.com/labs/michelson-precision-lab/)

[View the full path →](https://sandboxphysics.com/topics/waves-and-optics/#starter-path)