---
title: "Fiber Pulse Propagation Interactive Lab | Sandbox Physics"
description: "How does dispersion reshape an ultrashort pulse as it travels along a fiber? Group-velocity dispersion makes different frequencies travel at different speeds."
canonical_url: "https://sandboxphysics.com/labs/fiber-pulse-propagation/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/fiber-pulse-propagation/"
last_modified: "2026-10-29"
---

Optics 084 · Waveguides, structured light, and modern optics

# Fiber Pulse Propagation

**How does dispersion reshape an ultrashort pulse as it travels along a fiber?** Group-velocity dispersion makes different frequencies travel at different speeds. A pulse with no chirp spreads out as it goes, and its frequencies get sorted along it. A pulse launched with the opposite chirp first gets shorter as the fiber undoes that sorting, then spreads again. Near the zero-dispersion wavelength, higher-order dispersion takes over and distorts the pulse shape.

Propagate a pulse in 3D through a fiber whose silica dispersion follows the Sellmeier equation and whose fundamental mode is found by boundary matching. Track the full spectral phase, compress a chirped pulse, apply ideal compensation and freeze complex samples.

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

The fiber is lossless, linear and scalar, with an assumed fixed numerical aperture and infinite cladding. Pulse energy is normalized to one. Nonlinear effects are not modeled, and no calibrated device is reproduced. Geometric sizes and pulse timing in the scene are rescaled for display.

Physics tutorial

## Audit the physics of Fiber Pulse Propagation

**Background** A short pulse contains many frequencies. They all travel in the same guided mode, but each picks up a different phase. In a frame moving with the pulse, the spectrum evolves as $\widetilde A(z,\Omega)=\widetilde A(0,\Omega)e^{i[\beta(\omega_0+\Omega)-\beta_0-\beta_1\Omega]z}$. The waveform is the Fourier sum of this propagated complex spectrum.

**Why it matters** How does dispersion reshape an ultrashort pulse as it travels along a fiber?

Start with the essentials

- **Focus question**: How does dispersion reshape an ultrashort pulse as it travels along a fiber?
- **One-sentence intuition**: A different core radius confines the light differently, and that changes how propagation depends on frequency. The second-order dispersion has two parts: $\beta_2=\beta_{2,\rm material}+\beta_{2,\rm waveguide}$. Where this sum is zero, the quadratic phase term cancels, but the higher-order terms remain. The Lab keeps the full modal phase at every sampled frequency.

### Core mathematical model

#### One dispersion relation, two contributions

$$
\begin{aligned}n_1^2-1&=\sum_{j=1}^3\frac{B_j\lambda^2}{\lambda^2-C_j^2}\\n_2^2&=n_1^2-\mathrm{NA}^2\\\beta(\omega)&=\frac{\omega}{c}\sqrt{n_2^2+b(V)\mathrm{NA}^2}\\V&=\frac{a\omega}{c}\mathrm{NA}\end{aligned}
$$

The core uses the published coefficients for fused silica. The cladding relation is assumed, not fitted. Matching ordinary and modified Bessel functions at the core boundary gives the normalized propagation constant. Waveguide dispersion is defined by subtracting the bulk-core contribution from the propagation constant before differentiating.

#### Sign convention, group delay and the moving time frame

$$
\begin{aligned}E^{(+)}&=A(z,T)e^{i\beta_0z-i\omega_0t}\\T&=t-\beta_1z,\quad \beta_j=\frac{d^j\beta}{d\omega^j}\\v_g&=\beta_1^{-1},\quad \tau_g=\beta_1L\\A(z,T)&=\frac1{2\pi}\int\widetilde A(z,\Omega)e^{-i\Omega T}\,d\Omega\end{aligned}
$$

This complex convention is the conjugate of the one in the MIT notes, and every sign in the spectral propagation is converted to match. A positive local frequency offset equals minus the derivative of the envelope phase. Removing the carrier phase and the group delay only changes coordinates; it does not change the energy or the broadening.

#### Compression depends on the chirp sign and has a lower limit

$$
\begin{aligned}\widetilde A(0,\Omega)&\propto e^{-T_0^2\Omega^2/2}e^{iCT_0^2\Omega^2/2}\\\sigma_T^{(2)}(L)&=\frac{T_0}{\sqrt2}\sqrt{1+\left(C+\frac{\beta_2L+G}{T_0^2}\right)^2}\\\sigma_{T,\rm TL}&=T_0/\sqrt2\end{aligned}
$$

Here the chirp is the spectral quadratic phase divided by the square of the transform-limited width. It is not the time-domain chirp parameter defined at a fixed input intensity width. This expression is a separate reference that keeps only the quadratic phase; it is not the waveform solver. The ideal output compensator adds group-delay dispersion without changing the spectrum, and the higher-order residual remains.

#### Read energy and width from the samples

$$
\begin{aligned}E&\approx\sum_k|A_k|^2\Delta T\\\langle T\rangle&=E^{-1}\sum_kT_k|A_k|^2\Delta T\\\sigma_T^2&=E^{-1}\sum_k(T_k-\langle T\rangle)^2|A_k|^2\Delta T\end{aligned}
$$

Parseval's theorem links the sampled complex spectrum to the time record of the same pulse. No curve has its intensity rescaled on its own. The finite time window is sized for the worst-case width, and the energy near its edges is checked; if too much wraps around, the model flags the result instead of presenting it as a valid pulse.

### Common difficulties

#### Does zero dispersion mean no distortion?

**Typical misconception** At the zero-dispersion wavelength every frequency has the same group delay.

**Better mental model** Only the local slope of the group delay is zero there; the third and higher derivatives remain. Tune to the model's zero, use a shorter pulse, and look at the lopsided residual distortion. A compensator that adds only quadratic phase generally cannot remove it.

### Run the experiment

1. 01

   #### Scene 1: Material dispersion

   Start with material dispersion and increase the length. The peak drops as the pulse spreads, while the total energy and the spectrum stay the same.

   **What to observe:** The fiber runs along the lab X axis. Distance is in metres, retarded time in picoseconds and carrier wavelength in nanometres. The duration control sets the amplitude width of the transform-limited Gaussian, not the duration of the chirped pulse. Width readouts use the RMS width of the intensity about its center, which works for non-Gaussian outputs too.

2. 02

   #### Scene 2: Chirped-pulse compensation

   Choose chirped-pulse compensation. The input pulse has red at the front and blue at the back, and anomalous total dispersion brings the frequencies back together. Try the opposite chirp and compare. Then cancel the quadratic phase at the output.

   **What to observe:** The core index follows Malitson's data for bulk fused silica at twenty degrees Celsius. The cladding is assumed to have a constant reduction in squared index, which gives a numerical aperture of 0.12. The scalar fundamental mode is solved separately at every retained frequency. This is neither a measured doped fiber nor a full-vector solution. The carrier range stays below the cutoff of the first higher-order scalar mode.

3. 03

   #### Scene 3: Zero-dispersion crossing

   Tune to the model's zero-dispersion point. Change the core radius and watch the zero move. Drag the propagation map to look at the chirp partway along the fiber. Then acquire output samples and change the settings: the recorded samples stay frozen.

   **What to observe:** The 3D apparatus squeezes metres into scene units and enlarges the micrometre-sized core. The packet's speed and length on screen are display choices, not measurements of group velocity. The quantitative timing is in the separate retarded-time map and the delay readout. Normalizing the energy to one is a mathematical convention. Nonlinear power effects, loss, the finite cladding and the detector response are left out.

Physical conclusion

Dispersion changes phase, not energy. The right chirp can make a pulse shorter, but zero quadratic dispersion does not mean every phase error is gone.

## 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 **Kao: guiding light was only half the problem**](https://sandboxphysics.com/stories/probe-evidence/kao-loss-and-fibre-communication/)
2. [Lab **Optical Fiber: Rays to Modes** See how a fibre confines light](https://sandboxphysics.com/labs/optical-fiber-rays-to-modes/)
3. [Lab **Gradient-Index Optics Observatory** Watch steep rays catch up in a graded slab](https://sandboxphysics.com/labs/gradient-index-worlds/)

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)