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.

Computed round trips $\text{—}$

Post-coupler intracavity energy $\text{—}$

Energy extracted per round trip $\text{—}$

Intracavity peak power $\text{—}$

Main peak FWHM $\text{—}$

Intensity RMS duration $\text{—}$

Zero-phase reference FWHM $\text{—}$

Phase-aligned field change $\text{—}$

Outer-window energy fraction $\text{—}$

Captured correlation samples $\text{—}$

Captured round trip $\text{—}$

Captured correlation FWHM $\text{—}$

Envelope state $\text{—}$

Spectral edge energy fraction $\text{—}$

L23

What keeps an ultrashort pulse alive after every round trip?

Show full time windowInspect satellites and boundary occupation without changing the field or its captured record.Small-signal log gain

g_0

0.38

Gain-spectrum FWHM

\Delta\nu_g

40\,\mathrm{THz}

Net group-delay dispersion

\phi_2

-80\,\mathrm{fs^2}

Nonlinear phase coefficient

\gamma

1.2\,\mathrm{\frac{rad}{MW}}

Kerr-aperture loss depth

q_0

0.12

Output transmission

T

0.05

Held carrier wavelength

\lambda_0

800\,\mathrm{nm}

Compare Balanced pulse and Narrow gain, then remove the aperture and inspect the full time window. The correlation is acquired automatically; explicit further computation is in Advanced controls.

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.

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 misconceptionIf the main peak is short, the pulse must be good.

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

Run the experiment

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.
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.
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.