L17 · Interference becomes a pulse

Mode Locking

Edit spectral phase, then evolve a pulse through saturated gain, filtering, dispersion and active or passive loss. Measure the resulting field in time and frequency.

Round trips $\text{—}$

Cavity repetition frequency $\text{—}$

Full-period intracavity energy $\text{—}$

Period-average useful power $\text{—}$

Useful peak power $\text{—}$

Peak to average ratio $\text{—}$

Strongest peak intensity FWHM $\text{—}$

Measured spectral intensity FWHM $\text{—}$

Measured time-bandwidth product $\text{—}$

Local peaks above half maximum $\text{—}$

L17

Can the same modal energy make a much higher peak?

Initial phase alignment

\eta_\phi

0\,\mathrm{\%}

Passive absorber: off or on

b

0

Small-signal logarithmic gain

g_0

0.5

Spectral filter scale

B

40\,\mathrm{GHz}

Loss modulation depth

q_0

0.6

Absorber saturation power

P_a

20\,\mathrm{W}

Group-delay dispersion per trip

\Phi_2

0\,\mathrm{ps^2}

Active gate drift per trip

\epsilon

0

First compare random and flat initial phases before advancing. The passive absorber ignores active gate drift; it uses the intensity of the evolving field.

A deterministic scalar complex envelope circulates in a uniform one GHz cavity with a one nanosecond round-trip window. A 1024-point periodic Fourier grid connects the displayed field and spectrum. Initial modal intensity has a Gaussian forty gigahertz full width at half maximum by default; a fixed random phase seed is blended toward flat phase without changing energy. Each lumped round trip applies energy-saturated logarithmic gain, a Gaussian spectral filter, quadratic spectral phase, then intensity-dependent or synchronous loss. Gain saturation energy is one nanojoule. Base logarithmic intensity loss is 0.08 per trip, including five percent useful output; powers refer to the field arriving at that output tap. Active detuning is the loss-minimum shift in fractions of one period per trip. It has no role in passive operation, where an instantaneous absorber depends on actual cavity power. Dispersion is group-delay dispersion in square picoseconds; the filter scale is in gigahertz. This reduced discrete map is inspired by envelope master equations; it is not a calibrated material model or a full continuous Haus-equation solver. Pulse width uses interpolated half-maximum crossings of the strongest connected periodic peak. Strong secondary peaks are counted, and period energy is not mislabeled as an isolated pulse energy. Spectral width is measured from the same field; a Gaussian transform-limited pulse has time-bandwidth product about 0.441, while other shapes need not. The flat-phase action is a deliberate external phase edit, not automatic laser dynamics. There is no ongoing spontaneous seed, optical carrier resolution, self-phase modulation, gain recovery, spatial structure, thermal physics or quantum noise. Physical changes reset evolution. Apparatus dimensions and brightness are schematic.

Start with the essentials

Focus question: When does interference persist as a pulse?
One-sentence intuition: The spectral field and time envelope are one Fourier pair. Gain, filtering, dispersion and loss must support the result together.

Core mathematical model

Same modes, different interference

\begin{aligned}A(t)&=\sum_m a_m e^{i(2\pi m f_{\rm rep}t+\phi_m)}\\P(t)&=|A(t)|^2\end{aligned}

Changing phase changes the peak, while the period integral is fixed by modal powers.

Saturated round-trip gain

\begin{aligned}E&=\int_0^{T_R}|A(t)|^2\,dt\\g(E)&=\frac{g_0}{1+E/E_s}\end{aligned}

The same evolving envelope sets gain saturation.

Spectral filtering and dispersion

\begin{aligned}\widetilde A_m^{+}&=\widetilde A_m\exp\!\left[\frac{g-\ell}{2}-\frac{\nu_m^2}{2B^2}\right]\\&\quad\times\exp\!\left[-\frac{i\Phi_2(2\pi\nu_m)^2}{2}\right]\end{aligned}

The filter removes energy; dispersion changes phase. Frequency units must match square-picosecond dispersion.

Two ways to favor a pulse

\begin{aligned}q_{\rm active}(t,j)&=q_0[1-\cos(2\pi(t/T_R-\epsilon j))]\\q_{\rm passive}(t)&=\frac{q_0}{1+|A(t)|^2/P_a}\\A^{+}(t)&=A(t)e^{-q(t)/2}\end{aligned}

Active loss favors a moving time window; a fast absorber favors existing intensity.

Measure compatible widths

\begin{aligned}\mathrm{TBP}&=\Delta t_{1/2}\Delta\nu_{1/2}\\\mathrm{TBP}_{\rm Gaussian}&=\frac{2\ln2}{\pi}\simeq0.441\end{aligned}

Both widths are intensity FWHM. The Gaussian number is a shape-specific transform limit.

Equal-amplitude analytic limit

\begin{aligned}|A(t)|^2&=|a|^2\frac{\sin^2(N\pi f_{\rm rep}t)}{\sin^2(\pi f_{\rm rep}t)}\\\frac{P_{\rm peak}}{\langle P\rangle}&=N\end{aligned}

With flat phase, interference enhances peak power by the number of equal-power modes without adding energy.

Common difficulties

One intensity spike is not a certificate

Typical misconceptionA random field with a bright maximum is already stable mode locking.

Better mental modelFollow repeated trips and count competing peaks; a transient spike is insufficient.

Shape-specific transform limit

Typical misconceptionEvery pulse must have Gaussian time-bandwidth product.

Better mental modelDifferent shapes have different transform limits; multi-peak widths need extra care.

Run the experiment

01
Hold modal energy fixed
Compare the random and coherent presets before advancing, or flatten phase once.
What to observe: The peak changes while the spectrum and period energy stay fixed.
02
Select a temporal window
Advance the active preset for 1000 trips; then try a detuned gate.
What to observe: Synchronous loss selects a pulse; a drifting gate can fail.
03
Let intensity reduce loss
Advance the passive preset and change absorber saturation power.
What to observe: The absorber reshapes the actual field instead of imposing its phase.
04
Check dispersion and width
Compare narrow and dispersive presets in both time and frequency.
What to observe: Phase curvature and bandwidth change pulse width and time-bandwidth product.