L23 · Kerr lens and dispersion

Ti:Sapphire Ultrafast Oscillator

Advance a seeded cavity map, balance nonlinear phase against dispersion, and capture an intensity autocorrelation. Remove the Kerr aperture to test whether a localized pulse survives.

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}

Advance the balanced preset twice, capture its correlation, then remove the aperture. A short main peak alone does not prove an isolated stationary pulse.

A deterministic scalar complex-envelope teaching map uses a seeded Gaussian pulse, algebraically saturated gain, a Gaussian gain spectrum, fixed passive loss, an effective fast Kerr aperture, self-phase modulation, group-delay dispersion and an output coupler. The held carrier is set by an ideal wavelength selector; its gain follows an illustrative broad envelope centered at 800 nanometres. Gain and nonlinear parameters are illustrative, not a fitted Ti:sapphire device. The Kerr aperture stands for intensity-dependent spatial clipping: no transverse cavity, thermal lens or physical aperture radius is solved. The 1024-point, 2.048-picosecond periodic window represents a local pulse envelope, not the full cavity period. Edge energy warns when that approximation fails. The map starts from a 120-femtosecond, one-nanojoule seed; it does not establish self-starting from spontaneous noise. Each round trip applies a fixed-energy gain estimate, symmetric gain/dispersion steps, instantaneous Kerr loss and phase, then output coupling. There is no energy normalization. Field change removes only global phase; a small residual is a stationary state of this map, not a general stability certificate. Spectral edge energy warns of frequency-grid stress; fewer than eight samples across the main peak also flags sampling. Peak FWHM can miss satellites; inspect RMS duration, full window and spectrum. The acquired intensity autocorrelation is a non-wrapping integral of computed data. The zero-phase duration is a separate ideal reference, not a reconstruction. Repetition rate, gain recovery, quantum noise, CEP and astigmatism are omitted.

Start with the essentials

Focus question: What keeps an ultrashort pulse alive after every round trip?
One-sentence intuition: A stationary pulse is a balance of gain, loss, filtering, nonlinear phase and dispersion.

Core mathematical model

Gain depends on energy and 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. Gain is held fixed during one trip; this is not an inversion-recovery solve.

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 surround the instantaneous nonlinear element. The Gaussian gain spectrum has a user-set frequency FWHM; passive log 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}

Aperture saturation power is fifty kilowatts. This temporal surrogate represents spatial clipping; it does not compute a physical Kerr-lens radius.

Extract energy after the cavity map

\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 the post-coupler intracavity plane. Useful escape and stored energy obey the same coupling ledger.

Capture the computed intensity overlap

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

The field is zero outside the recorded interval. Autocorrelation broadens a Gaussian by a factor of the square root of two, but arbitrary pulses have no unique inversion.

Check the local-window boundary

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

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

Common difficulties

A short peak proves success

Typical misconceptionMain-peak width alone establishes pulse quality.

Better mental modelInspect satellites, RMS duration, numerical boundaries and the separate ideal reference.

Run the experiment

01
Compare states
Advance balanced and narrow-gain presets by two thousand trips.
What to observe: Compare duration, energy, spectrum and map residual.
02
Break localization
Remove the Kerr aperture and advance the map.
What to observe: Inspect the whole time window and edge-energy fraction.
03
Acquire a diagnostic
Capture autocorrelation after evolution pauses.
What to observe: Compare its width with the field width without assuming a Gaussian inversion.