---
title: "Gain Curve Meets Longitudinal Modes Interactive Lab | Sandbox Physics"
description: "Which cavity modes survive after they saturate the gain?"
canonical_url: "https://sandboxphysics.com/labs/gain-longitudinal-modes/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/gain-longitudinal-modes/"
last_modified: "2026-10-27"
---

L09 · Modes and cavity design

# Gain Curve Meets Longitudinal Modes

**Which cavity modes survive after they saturate the gain?** If all modes draw on the same gain, usually only the one with the lowest threshold survives; it pulls the gain below what the others need. Hole burning breaks that sharing. When modes feed on different groups of atoms in the spectrum, or on different spots along the cavity, several can lase at once. A filter that adds loss at the rivals' frequencies raises their thresholds and leaves one.

Slide the cavity's mode comb across a gain line and see which modes survive. Compare shared homogeneous gain, separate spectral classes of atoms, standing-wave hole burning and a frequency-selective loss, then record a mode hop.

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

This is a simplified class-A competition model with 31 axial modes. The inversion is eliminated adiabatically, and mode intensities stay positive. Homogeneous gain has a Lorentzian line. Inhomogeneous gain averages Lorentzian classes of 80 MHz half-width over a Gaussian distribution. The gain width control sets the homogeneous half-width in the first case and the Gaussian standard deviation in the second. The spectral overlaps and whole-cavity spatial overlaps are exact. They feed a first-order saturation closure, which is a teaching approximation away from threshold. In a uniformly filled cavity, standing waves make self-saturation stronger than cross-saturation. All modes start from equal small seeds as classical initial conditions. Intensity is measured in units of the saturation intensity and time in photon lifetimes, so the model predicts no watts, beat phases, relaxation oscillations or linewidths. The frequency-pulling readout is a separate linear estimate for a single line at threshold. It is switched off for inhomogeneous, standing-wave or filtered cases, and it does not shift the competition spectrum.

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

## Threshold candidates compete for a finite gain

**Background** A cavity allows only a set of discrete frequencies. The gain line and any frequency-dependent loss decide which of these modes can grow from a weak start. Once the gain saturates, the outcome can change.

**Why it matters** Several modes can start above threshold, yet the steady state need not keep them all. What happens depends on whether the modes draw on the same atoms, on groups of atoms with different transition frequencies, or on different regions of a standing wave.

Start with the essentials

- **Focus question**: Why do identical cavity modes behave differently when the gain broadening or spatial overlap changes?
- **One-sentence intuition**: Threshold is judged with unsaturated gain, but survival depends on how each mode saturates its own gain and that of the others. Homogeneous, spatially uniform gain favors the lowest-threshold mode unless two thresholds are exactly equal; spectral or spatial hole burning lets several modes coexist.

### Core mathematical model

#### Cold-cavity comb

$$
\begin{aligned}\mathrm{FSR}&=\frac{c}{2L}\\\nu_j-\nu_a&=(j+\delta)\mathrm{FSR}\end{aligned}
$$

The optical length of the cavity sets the mode spacing. The offset slides the whole comb relative to the atomic line, and each mode keeps its relative order number while you tune.

#### Spectral overlap

$$
\begin{aligned}H_{jk}&=\frac{1}{1+[(\nu_j-\nu_k)/\gamma_h]^2}\\h_j&=\frac{\sum_kw_kH_{jk}}{\mathcal N}\end{aligned}
$$

A single class of atoms gives a homogeneous Lorentzian line. A Gaussian spread of classes gives a Voigt profile. The normalization makes the unsaturated gain at line center equal to the pump setting.

#### Whole-cavity spatial overlap

$$
\langle f_jf_l\rangle=\begin{cases}3/2&j=l\\1&j\ne l\end{cases}
$$

With intensities normalized to a mean of one, a standing wave overlaps more with itself than with the standing wave of a different axial order. This assumes gain spread evenly through the whole cavity; a short crystal needs a different overlap integral.

#### Simplified saturation and competition

$$
\begin{aligned}C_{jl}&=\frac{\sum_kw_kH_{jk}H_{lk}}{\mathcal N}\langle f_jf_l\rangle\\G_j&=\frac{ph_j}{1+\sum_lC_{jl}I_l/h_j}\end{aligned}
$$

The overlaps are exact for the chosen numerical integration and geometry. The saturation formula extends the first-order expansion of local saturation in a form that stays positive; it is not a full spatial Maxwell–Bloch solution.

#### Mode growth and frequency-dependent loss

$$
\begin{aligned}u&=t/\tau_p\\\frac{dI_j}{du}&=(G_j-\ell_j)I_j+\epsilon\\\ell_j&=1+f\left(\frac{\nu_j-\nu_f}{\mathrm{FSR}}\right)^2\end{aligned}
$$

The mode with the smallest ratio of loss to gain profile has the lowest unsaturated threshold. A tiny classical floor lets suppressed modes grow back after tuning. A mode with more than one percent of the total intensity counts as active.

#### Frequency pulling, estimated separately

$$
\begin{aligned}\nu_L&=\frac{\gamma_h\nu_c+\gamma_c\nu_a}{\gamma_h+\gamma_c}\\\gamma_c&=\frac{\kappa}{4\pi}\end{aligned}
$$

This linear estimate for a single line at threshold pulls the lasing frequency toward the atomic resonance. The cold-cavity linewidth comes from a base logarithmic round-trip power loss of 0.08. The competition spectrum stays at the cold-cavity frequencies, and the estimate is switched off when its assumptions do not hold.

### Common difficulties

#### Above threshold does not mean surviving

**Typical misconception** Every mode whose unsaturated gain exceeds its loss shows up in the final spectrum.

**Better mental model** Threshold is checked before the gain saturates. The final gain has to be worked out with all the competing modes present.

#### Multimode does not mean mode locked

**Typical misconception** Several surviving lines automatically form short pulses.

**Better mental model** This model tracks no mode phases, so it cannot predict beating, phase locking, pulse formation or linewidth. Later Labs cover those.

#### Uniform gain does not mean spatially uniform saturation

**Typical misconception** A homogeneously broadened standing-wave laser must have one mode.

**Better mental model** Different standing-wave patterns can use up the gain in different places. The travelling-wave reference removes this spatial effect, and even then modes with exactly equal thresholds can share the output.

### Run the experiment

1. 01

   #### Predict candidates, then watch selection

   Choose Homogeneous competition and note how many modes are threshold candidates. Advance three times, then compare that number with the active modes and the side-mode suppression.

   **What to observe:** Several modes initially qualify, but the mode with the smallest threshold depletes the shared gain and suppresses its neighbors.

2. 02

   #### Separate two kinds of hole burning

   Compare Spectral hole burning and Standing-wave hole burning at the same pump and cavity length. Advance each three times.

   **What to observe:** In spectral hole burning, the modes draw on different classes of atoms, so they saturate each other less. In standing-wave hole burning, each mode overlaps most with itself, which leaves gain for neighboring orders. The two mechanisms are different.

3. 03

   #### Force one mode to win

   Load Filtered inhomogeneous gain, then move the filter center. Inspect both loss and saturated gain before and after advancing.

   **What to observe:** Modes near the filter center see less loss. A wide gain line need not give broad output once the competing thresholds are raised.

4. 04

   #### Record a mode hop

   Load Near a mode hop and advance, then use Tune right and evolve twice. Watch the dominant mode order and the cold-cavity offset.

   **What to observe:** The winning mode changes when the comb moves past half a mode spacing. Tuning keeps the existing intensities, so the takeover can take time.

Physical conclusion

A frequency-selective loss raises the thresholds of competing modes. Shifting the cold-cavity comb can hand the win to a different axial mode; frequency pulling is a separate, smaller shift within one mode and has a different physical cause.

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)