---
title: "Mode Locking Interactive Lab | Sandbox Physics"
description: "Can a laser reach a much higher peak power without any more energy? Yes, if the cavity modes are locked in phase."
canonical_url: "https://sandboxphysics.com/labs/mode-locking-frequency-comb/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/mode-locking-frequency-comb/"
last_modified: "2026-10-28"
---

L17 · Interference becomes a pulse

# Mode Locking

**Can a laser reach a much higher peak power without any more energy?** Yes, if the cavity modes are locked in phase. Their fields then add up in one short burst per round trip, so the same total energy arrives as a far taller peak. The pulse survives only if gain, filtering, dispersion and loss all support it together.

The page opens on a recorded start-up: 400 round trips from random mode phases. Drag the output coupler to change the cavity length, and compare the pulse with its comb teeth.

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

A deterministic scalar complex envelope circulates in a linear standing-wave cavity. The round-trip window is twice the mirror spacing divided by the speed of light, so a 15 cm cavity repeats at about 999 MHz and the comb teeth sit at that spacing. The displayed field and spectrum are linked on a 1024-point periodic Fourier grid. By default, the initial mode intensities follow a Gaussian with a forty gigahertz full width at half maximum. The phases start from a fixed random seed and can be blended toward flat phase without changing the energy. Each round trip is lumped into steps: energy-saturated logarithmic gain, a Gaussian spectral filter, quadratic spectral phase, then intensity-dependent or synchronous loss. The gain saturation energy is one nanojoule. The base logarithmic intensity loss is 0.08 per trip and includes five percent useful output; all powers refer to the light reaching that output tap. Active detuning is how far the loss minimum shifts each trip, in fractions of one period. It plays no role in passive operation, where an instantaneous absorber responds to the actual cavity power. Dispersion is group-delay dispersion in square picoseconds, and the filter scale is in gigahertz. This reduced discrete map is inspired by envelope master equations; it is not a calibrated material model or a full continuous Haus-equation solver. Pulse width comes from interpolated half-maximum crossings of the strongest connected peak in the period. Strong secondary peaks are counted, and the energy in a period is not reported as the energy of one isolated pulse. Spectral width is measured from the same field. A transform-limited Gaussian pulse has a time-bandwidth product of about 0.441; other shapes can differ. The analytic reference is the steady Gaussian of the continuous active master equation with a parabolic loss minimum; it is shown only for active locking without gate drift and is not a second simulation. The carrier-envelope offset readout is bookkeeping, the repetition rate times the chosen phase slip per round trip; the envelope map does not resolve the carrier. The model has no continuing spontaneous seed, no resolved optical carrier, and no self-phase modulation, gain recovery, spatial structure, thermal physics or quantum noise. Changing a physical setting records a new 400 round-trip evolution from the same random phases. Mirror spacing and the pulse length in the scene share one scale; component sizes are schematic and brightness is linear with a fixed full scale.

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

## Watch the phases lock into a pulse

**Background** The longitudinal modes of a cavity interfere, and their phases decide where they add up. The right kind of loss can favor the brief moment when many modes add in phase.

**Why it matters** A tall, short peak does not mean more energy. The record shows whether the cavity itself keeps selecting one pulse round trip after round trip, starting from random phases.

Start with the essentials

- **Focus question**: When does interference persist as a pulse?
- **One-sentence intuition**: The spectrum and the pulse in time are the same field, linked by a Fourier transform. A pulse lasts only if gain, filtering, dispersion and loss all support it together.

### Core mathematical model

#### Same modes, different interference

$$
\begin{aligned}A(t)&=\sum_m a_m e^{i(2\pi m f_{\rm rep}t+\phi_m)}\\P(t)&=|A(t)|^2\end{aligned}
$$

Changing the phases changes the peak. The energy in one period stays fixed, because it depends only on the power in each mode.

#### Saturated round-trip gain

$$
\begin{aligned}E&=\int_0^{T_R}|A(t)|^2\,dt\\g(E)&=\frac{g_0}{1+E/E_s}\end{aligned}
$$

The energy of the circulating envelope in one period sets how strongly the gain saturates.

#### Spectral filtering and dispersion

$$
\begin{aligned}\widetilde A_m^{+}&=\widetilde A_m\exp\!\left[\frac{g-\ell}{2}-\frac{\nu_m^2}{2B^2}\right]\\&\quad\times\exp\!\left[-\frac{i\Phi_2(2\pi\nu_m)^2}{2}\right]\end{aligned}
$$

The filter removes energy, and dispersion changes the phase. The frequency units must match the dispersion, which is given in square picoseconds.

#### Two ways to favor a pulse

$$
\begin{aligned}q_{\rm active}(t,j)&=q_0[1-\cos(2\pi(t/T_R-\epsilon j))]\\q_{\rm passive}(t)&=\frac{q_0}{1+|A(t)|^2/P_a}\\A^{+}(t)&=A(t)e^{-q(t)/2}\end{aligned}
$$

Active modulation lowers the loss in a time window that can drift from trip to trip. A fast absorber lowers the loss wherever the light is already intense.

#### Repetition rate and the active pulse width

$$
\begin{aligned}f_{\rm rep}&=\frac{c}{2L},\quad \omega_m=2\pi f_{\rm rep}\\\tau_a^4&=\frac{D_f}{M_s},\quad M_s=\frac{q_0\,\omega_m^2}{4}\\\Delta t_{1/2}&=2\sqrt{\ln 2}\,\tau_a\end{aligned}
$$

The comb teeth are spaced by the repetition rate. Near the loss minimum the modulator acts like a parabola, and its balance with filtering gives a Gaussian steady state. The dashed reference curve uses this result; the filter scale sets the filter curvature.

#### Time-bandwidth product

$$
\begin{aligned}\mathrm{TBP}&=\Delta t_{1/2}\Delta\nu_{1/2}\\\mathrm{TBP}_{\rm Gaussian}&=\frac{2\ln2}{\pi}\simeq0.441\end{aligned}
$$

Both widths are intensity FWHM. The Gaussian value is the transform limit for that one shape.

#### Equal-amplitude modes with flat phase

$$
\begin{aligned}|A(t)|^2&=|a|^2\frac{\sin^2(N\pi f_{\rm rep}t)}{\sin^2(\pi f_{\rm rep}t)}\\\frac{P_{\rm peak}}{\langle P\rangle}&=N\end{aligned}
$$

With flat phase, interference raises the peak power above the average by a factor equal to the number of equal-power modes. No energy is added.

### Common difficulties

#### One bright spike does not prove mode locking

**Typical misconception** If a random field has one bright maximum, the laser is already stably mode locked.

**Better mental model** Watch many round trips and count the competing peaks. A spike that comes and goes is not enough.

#### The transform limit depends on pulse shape

**Typical misconception** Every pulse must have the Gaussian time-bandwidth product.

**Better mental model** Each pulse shape has its own transform limit. When the field has several peaks, any single width needs extra care to interpret.

### Run the experiment

1. 01

   #### Watch the record form a pulse

   Keep the default preset, Modulator locks the teeth, and watch the replay count up the recorded round trips.

   **What to observe:** The glow along the cavity gathers into one short packet. The peak-to-average ratio rises while the energy per round trip settles first.

2. 02

   #### Compare a cavity that does not lock

   Load Free running: no locking.

   **What to observe:** The light stays spread around the cavity and the peak is only a few times the average, even though the gain and filter are the same.

3. 03

   #### Drag the output coupler

   Drag the output coupler to lengthen the cavity, or load Longer cavity, more teeth.

   **What to observe:** The repetition rate drops, more teeth fit under the filter, and the pulse grows longer roughly as the square root of the round-trip time, as the dashed reference predicts.

4. 04

   #### Change the locking element and add dispersion

   Load Absorber shortens the pulse, then Dispersion chirps the pulse, and open the tooth phase panel.

   **What to observe:** The absorber gives a shorter pulse. With dispersion the tooth phases bend into a parabola and the time-bandwidth product rises above 0.441.

Physical conclusion

Before you call it stable mode locking, check the energy, peak power, pulse width and any secondary pulses.

## Related experiments and articles

Probes and measurement · Reading path

### Controlled light and precision measurement

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Frequency combs measure both the spacing and the offset**](https://sandboxphysics.com/stories/probe-evidence/frequency-comb-self-reference/)
2. [Lab **Microresonator Kerr Comb** Compare a circulating pulse with its comb](https://sandboxphysics.com/labs/microresonator-kerr-comb/)
3. [Lab **Optical Frequency Comb** Check how a comb is calibrated](https://sandboxphysics.com/labs/optical-frequency-comb/)
4. [Lab **Nonlinear Frequency Conversion** See why the doubling crystal needs phase matching](https://sandboxphysics.com/labs/nonlinear-frequency-conversion/)

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)