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.

Compensated central RMS width

\sigma_T

$0.50$

Total modal group-velocity dispersion

\beta_2

$50\%$

Output / input integrated energy

E_{\rm out}/E_{\rm in}

$0.00\pi$

Model regime $\text{valid model regime}$

Model checked against analytic results

Fiber Pulse Propagation

Physical fiber length

L

15\,\mathrm m

Vacuum carrier wavelength

\lambda_0

1550\,\mathrm{nm}

Transform-limited Gaussian amplitude width

T_0

250\,\mathrm{fs}

Input spectral chirp; positive is red first

C

0

Actual core radius at fixed numerical aperture

a

3\,\mu\mathrm m

Ideal output compensator group-delay dispersion

G

0\,\mathrm{ps}^2

Inspection position along the fiber

z/L

60\%

A pulse can spread out without losing energy, and it can compress when its colors arrive together. Send one complex spectrum through the step-index fiber mode for the chosen core radius, move the zero-dispersion point and capture the compensated output. Even where one derivative of the phase is zero, the higher-order phase is still there.

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

BackgroundA 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 mattersHow 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 misconceptionAt the zero-dispersion wavelength every frequency has the same group delay.

Better mental modelOnly 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

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.
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.
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.

Fiber Pulse Propagation

Audit the physics of Fiber Pulse Propagation

Core mathematical model

One dispersion relation, two contributions

Sign convention, group delay and the moving time frame

Compression depends on the chirp sign and has a lower limit

Read energy and width from the samples

Common difficulties

Does zero dispersion mean no distortion?

Run the experiment

Scene 1: Material dispersion

Scene 2: Chirped-pulse compensation

Scene 3: Zero-dispersion crossing

Related experiments and articles

Controlled light and precision measurement