---
title: "Ti:Sapphire Ultrafast Oscillator Interactive Lab | Sandbox Physics"
description: "What keeps an ultrashort pulse alive after every round trip? Gain must replace the energy lost each trip, and the pulse must regain its shape."
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page_type: "lab"
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  zh-CN: "https://sandboxphysics.com/zh/labs/ti-sapphire-oscillator/"
last_modified: "2026-10-27"
---

L23 · Kerr lens and dispersion

# Ti:Sapphire Ultrafast Oscillator

**What keeps an ultrashort pulse alive after every round trip?** Gain must replace the energy lost each trip, and the pulse must regain its shape. A Kerr lens and a suitably placed aperture can favor intense peaks; dispersion and intensity-dependent phase then shape the circulating pulse. Their balance can support a localized pulse.

Compare a prepared short pulse with narrow gain, positive dispersion and removal of the effective Kerr aperture. The same computed field supplies its spectrum, output envelope and intensity autocorrelation.

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

This is a deterministic teaching model of a scalar complex envelope. It includes a seeded Gaussian pulse, gain with algebraic saturation, a Gaussian gain spectrum, fixed passive loss, an effective fast Kerr aperture, self-phase modulation, group-delay dispersion and an output coupler. An ideal wavelength selector holds the carrier; the gain at that carrier follows an illustrative broad envelope centered at 800 nanometres. The gain and nonlinear parameters are illustrative and are not fitted to a real Ti:sapphire device. The Kerr aperture stands in for intensity-dependent spatial clipping; the model does not solve the transverse cavity, a thermal lens or a physical aperture radius. The periodic window has 2048 points with one-femtosecond spacing and spans 2.048 picoseconds. It holds a local pulse envelope, not the whole cavity period, and the edge energy warns when that approximation fails. The map starts from a 120-femtosecond, one-nanojoule seed, so it does not show that the laser starts itself from spontaneous noise. Each round trip applies a gain estimate at fixed energy, symmetric gain and dispersion steps, instantaneous Kerr loss and phase, and then output coupling. The energy is never renormalized. The field change ignores only the global phase. A small residual means a stationary state of this map, not a general proof of stability. Spectral edge energy warns when the frequency grid is strained, and fewer than eight samples across the main peak also raises a sampling flag. The peak FWHM can miss satellite pulses, so check the RMS duration, the full window and the spectrum. The captured intensity autocorrelation is a non-wrapping integral of the computed data. The zero-phase duration is a separate ideal reference, not a reconstruction. Repetition rate, gain recovery, quantum noise, CEP and astigmatism are left out. Each selection computes two thousand trips and captures the correlation; replay only moves schematic transport. The output-arm volume samples the post-coupler intensity and scales it by the coupler transmission ratio. Its retarded-time length is compressed nonlinearly, and its brightness uses a fixed logarithmic map. The ring transport does not depict local pulse shapes inside every element. The ideal output monitor does not model detector bandwidth, saturation or voltage response.

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

## A pulse must survive the cavity

**Background** A Ti:sapphire oscillator sends light around a resonator repeatedly. Gain replaces energy lost at the mirrors and aperture; intensity-dependent loss and phase help select the pulse shape.

**Why it matters** A large spectrum permits a short pulse, but its frequencies also have to arrive together. The same envelope must recover after every trip.

Start with the essentials

- **Focus question**: What keeps an ultrashort pulse alive after every round trip?
- **One-sentence intuition**: A pulse returns unchanged after each round trip only when gain, loss, filtering, nonlinear phase and dispersion balance one another.

### Core mathematical model

#### Gain depends on pulse energy and the held carrier

$$
\begin{aligned}E&=\int |A(t)|^2\,dt\\ g(E)&=\frac{g_0h}{1+E/E_s}\\ h&=\exp[-4\ln 2\,(d/b)^2]\\ d&=\nu_0-\nu_{800}\\ b&=100\,\mathrm{THz}\end{aligned}
$$

The saturation energy is ten nanojoules. The gain stays fixed during each round trip; the model does not solve for inversion recovery.

#### Linear half step

$$
\begin{aligned}\widetilde A\prime&=\widetilde A\,e^{u-iv}\\u&=\frac{g(E)H(\nu)-\ell}{4}\\v&=\frac{\phi_2\Omega^2}{4}\end{aligned}
$$

Two half steps sit on either side of the instantaneous nonlinear element. You set the frequency FWHM of the Gaussian gain spectrum; the passive logarithmic intensity loss is 0.08.

#### Effective Kerr aperture and phase

$$
\begin{aligned}A\prime&=A\,e^{-q(P)/2-i\gamma_\lambda P}\\q(P)&=\frac{q_0}{1+P/P_a}\\\gamma_\lambda&=\gamma\,\frac{800\,\mathrm{nm}}{\lambda_0}\end{aligned}
$$

The aperture saturation power is fifty kilowatts. This loss acts in time as a stand-in for spatial clipping; the model does not compute a physical Kerr-lens radius.

#### Output coupling after each round trip

$$
\begin{aligned}A_{n+1}&=\sqrt{1-T}\,A_{\rm before\,coupler}\\ E_{\rm out}&=\frac{T}{1-T}E_{n+1}\end{aligned}
$$

The plotted envelope is taken inside the cavity just after the output coupler. The coupler transmission sets both the energy that leaves and the energy that stays.

#### Intensity autocorrelation of the computed pulse

$$
C(\tau)=\int P(t)P(t+\tau)\,dt
$$

The field is zero outside the recorded window. For a Gaussian pulse the autocorrelation is wider by a factor of the square root of two, but for an arbitrary pulse the autocorrelation cannot be inverted uniquely.

#### Energy near the window edges

$$
\epsilon_{\rm edge}=\frac{\int_{\rm outer\,10\%}P(t)\,dt}{E}
$$

Edge energy, RMS duration and the phase-aligned field change help tell an isolated stationary pulse apart from a short peak in a field that is spreading or splitting.

### Common difficulties

#### A short main peak is not the whole story

**Typical misconception** If the main peak is short, the pulse must be good.

**Better mental model** Also check the satellite pulses, the RMS duration, the energy near the edges of the numerical grid and the separate ideal reference.

### Run the experiment

1. 01

   #### Compare prepared pulses

   Choose Balanced pulse, then Narrow gain. Each preset computes two thousand round trips and captures its own intensity autocorrelation automatically.

   **What to observe:** The narrower gain spectrum supports a broader pulse. Compare the main width and spectrum with the recorded formation history below.

2. 02

   #### Remove pulse selection

   Choose Remove Kerr aperture. Open Advanced controls and enable Show full time window.

   **What to observe:** Energy spreads across the local periodic window. The edge fraction distinguishes this field from the localized default pulse, even if one peak looks short.

3. 03

   #### Separate replay from a new calculation

   Pause replay without changing parameters. In Advanced controls, choose Compute 1000 more trips.

   **What to observe:** Pausing freezes only the schematic transport. Explicit further computation advances the map and captures a new correlation; its width still requires a pulse-shape assumption before conversion to a duration.

Physical conclusion

A short peak, a localized pulse and a stationary cavity state are three different checks. Inspect the full window and the formation history together.

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)