---
title: "Laser Dynamics & Relaxation Oscillation Interactive Lab | Sandbox Physics"
description: "Why does a laser overshoot before it settles?"
canonical_url: "https://sandboxphysics.com/labs/laser-dynamics-relaxation/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/laser-dynamics-relaxation/"
last_modified: "2026-10-27"
---

L15 · Population, photons and response

# Laser Dynamics & Relaxation Oscillation

**Why does a laser overshoot before it settles?** Energy sloshes back and forth between the population inversion, which responds slowly, and the light in the cavity, which responds fast. The light overshoots and drains the inversion, then dips until the inversion recovers, in smaller and smaller swings. In class B lasers this shows up as damped relaxation oscillations. Class A lasers have no slow energy store and settle smoothly.

Turn a laser on, nudge its pump or modulate it, and watch the inversion and the photons trade energy. Compare how class A and class B lasers respond to turn-on, to a small pump step and to a measured sinusoidal pump modulation.

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

The model is a uniform, four-level, single-mode intensity model. It assumes the polarization relaxes faster than both the population and the photons. Inversion is measured in units of its threshold value without modulation, and photon density is scaled so that its steady value equals the pump minus one. Lifetimes are in microseconds. The pump is an injection rate in units of its threshold value, not an optical or electrical power in watts. The threshold inversion stores an illustrative ten microjoules, and sixty percent of the base photon loss leaves through the output mirror. The run starts from a finite photon seed, but there is no ongoing spontaneous emission and nothing random. The two rate equations are integrated together with an adaptive, step-doubling Runge–Kutta method, using the logarithm of the photon density. In the class A limit the population relaxes much faster than the photons decay; in class B the order is reversed. The eigenvalues keep the damping exactly. A damped frequency of zero means the poles are real, not that the laser does not respond. The small-signal curves are a separate linearization about the unmodulated state above threshold. Measured points come from projecting the output directly onto a sinusoid after transients die out, with a finite modulation depth. Reset or any change to a physical control clears the measurements. The five-percent step starts from steady state, sets the modulation to zero and never takes the pump above its maximum. Complete startup advances at least thirty population lifetimes. The apparatus is schematic, and the marker fractions and logarithmic brightness are only for display. Spatial variation, coherent polarization dynamics, multimode competition, heating and detector noise are left out.

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

## Two reservoirs make a laser ring after a disturbance

**Background** The population stores the pump energy, and the photons in the cavity carry it away. How fast each one responds decides whether the output settles smoothly or oscillates.

**Why it matters** A steady laser can react strongly to a weak pump ripple at a particular frequency.

Start with the essentials

- **Focus question**: When does a disturbance in the output die away with oscillations?
- **One-sentence intuition**: In a class B laser, energy swings back and forth between the slowly responding population and the fast-decaying photons. In class A the population relaxes so fast that it no longer acts as a slow store.

### Core mathematical model

#### Normalized rate equations for inversion and photons

$$
\begin{aligned}\dot D&=\frac{r(t)-D-DS}{\tau_L}\\\dot S&=\frac{(D-1)S}{\tau_p}\end{aligned}
$$

The same stimulated-emission term removes population and adds photons. The normalized photon variable is not a power in watts.

#### Steady state above and below threshold

$$
\begin{aligned}D_s&=1,\quad S_s=r-1\quad(r>1)\\D_s&=r,\quad S_s=0\quad(r\le1)\end{aligned}
$$

This steady state assumes a constant pump and ignores the ongoing seeding by spontaneous emission.

#### Poles of the linearized equations

$$
\begin{aligned}s_\pm&=-\gamma\pm\sqrt{\gamma^2-\omega_0^2}\\\gamma&=\frac{r}{2\tau_L}\\\omega_0^2&=\frac{r-1}{\tau_L\tau_p}\end{aligned}
$$

Real poles mean the laser relaxes without oscillating; complex poles mean a damped oscillation. Both statements hold above threshold.

#### Ringing frequency and the two laser classes

$$
\begin{aligned}f_d&=\frac{\sqrt{\max(0,\omega_0^2-\gamma^2)}}{2\pi}\\\text{A: }\tau_L&\ll\tau_p\\\text{B: }\tau_p&\ll\tau_L\end{aligned}
$$

Both limits assume the polarization relaxes much faster still. A large enough pump can overdamp even a class B laser.

#### Response to a small pump modulation

$$
\begin{aligned}H(s)&=\frac{\delta S/S_s}{\delta r/r}\\&=\frac{r/(\tau_L\tau_p)}{s^2+(r/\tau_L)s+\omega_0^2}\\H(0)&=\frac{r}{r-1}\end{aligned}
$$

The response is the fractional change in output divided by the fractional change in pump. Near threshold its low-frequency value is not one.

#### Converting to energy and power

$$
\begin{aligned}E_g&=E_0D\\U&=E_0\frac{\tau_p}{\tau_L}S\\P_{\rm out}&=\eta\frac{U}{\tau_p}\\\dot E_g+\dot U&=\frac{E_0}{\tau_L}(r-D-S)\end{aligned}
$$

The illustrative threshold energy sets the scale in watts. The pump input, minus population decay and photon escape, equals the rate at which the stored energy changes.

### Common difficulties

#### The response peak is not a cavity mode

**Typical misconception** The peak in the response must be one of the cavity's longitudinal optical resonances.

**Better mental model** This frequency comes from energy swapping between population and photons. It lies far below the optical frequency.

#### Laser class depends on time scales

**Typical misconception** Every solid-state laser is automatically class B.

**Better mental model** The class comes from the order of the relaxation times in the model, and both classes assume the polarization is already fast.

#### Measured points need not sit on the curve

**Typical misconception** A measurement with finite modulation depth must land exactly on the linear curve.

**Better mental model** The curve assumes an infinitely small modulation. The measurements come from integrating the full nonlinear equations.

### Run the experiment

1. 01

   #### Watch the laser start

   Choose Class B storage and press Complete startup. Compare this first turn-on with the later pump-step test.

   **What to observe:** Starting from a small seed, the light waits while the population builds up, then overshoots as stimulated emission drains the inversion.

2. 02

   #### Disturb a steady laser

   Press Apply five-percent pump step, then repeat with Class A fast population.

   **What to observe:** The class B output rings. The class A output follows the step with no relaxation resonance.

3. 03

   #### Measure near resonance

   Choose Modulate near relaxation and press Measure modulation. Compare the amber point with the curve.

   **What to observe:** A small pump modulation can make the output swing by a much larger fraction.

4. 04

   #### Go beyond the small-signal limit

   Raise the modulation depth, choose Near threshold, or choose Modulate too fast.

   **What to observe:** The simulated response can drift away from the linear prediction, and fast modulation is strongly filtered out.

Physical conclusion

Check the simulated transient and the measured modulation response against the separately computed poles and transfer function.

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)