L17 · Interference becomes a pulse

Mode Locking

Can a laser reach a much higher peak power without any more energy? Yes, if the cavity modes are locked in phase. Their fields then add up in one short burst per round trip, so the same total energy arrives as a far taller peak. The pulse survives only if gain, filtering, dispersion and loss all support it together.

The page opens on a recorded start-up: 400 round trips from random mode phases. Drag the output coupler to change the cavity length, and compare the pulse with its comb teeth.

Round trips $\text{—}$

Repetition rate and tooth spacing $\text{—}$

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 $\text{—}$

Teeth above half the strongest $\text{—}$

Carrier-envelope offset frequency $\text{—}$

L17

Can a laser reach a much higher peak power without any more energy?

Initial phase alignment

\eta_\phi

0\,\mathrm{\%}

Locking element

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

Mirror spacing

L

15\,\mathrm{cm}

Carrier-envelope phase slip per trip

\Delta\varphi_{\mathrm{ce}}/2\pi

0.25

Every preset starts from the same random phases and records 400 round trips. Initial phase alignment only changes the starting field; the passive absorber ignores the active gate drift.

A deterministic scalar complex envelope circulates in a linear standing-wave cavity. The round-trip window is twice the mirror spacing divided by the speed of light, so a 15 cm cavity repeats at about 999 MHz and the comb teeth sit at that spacing. The displayed field and spectrum are linked on a 1024-point periodic Fourier grid. By default, the initial mode intensities follow a Gaussian with a forty gigahertz full width at half maximum. The phases start from a fixed random seed and can be blended toward flat phase without changing the energy. Each round trip is lumped into steps: energy-saturated logarithmic gain, a Gaussian spectral filter, quadratic spectral phase, then intensity-dependent or synchronous loss. The gain saturation energy is one nanojoule. The base logarithmic intensity loss is 0.08 per trip and includes five percent useful output; all powers refer to the light reaching that output tap. Active detuning is how far the loss minimum shifts each trip, in fractions of one period. It plays no role in passive operation, where an instantaneous absorber responds to the actual cavity power. Dispersion is group-delay dispersion in square picoseconds, and 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 comes from interpolated half-maximum crossings of the strongest connected peak in the period. Strong secondary peaks are counted, and the energy in a period is not reported as the energy of one isolated pulse. Spectral width is measured from the same field. A transform-limited Gaussian pulse has a time-bandwidth product of about 0.441; other shapes can differ. The analytic reference is the steady Gaussian of the continuous active master equation with a parabolic loss minimum; it is shown only for active locking without gate drift and is not a second simulation. The carrier-envelope offset readout is bookkeeping, the repetition rate times the chosen phase slip per round trip; the envelope map does not resolve the carrier. The model has no continuing spontaneous seed, no resolved optical carrier, and no self-phase modulation, gain recovery, spatial structure, thermal physics or quantum noise. Changing a physical setting records a new 400 round-trip evolution from the same random phases. Mirror spacing and the pulse length in the scene share one scale; component sizes are schematic and brightness is linear with a fixed full scale.

Start with the essentials

Focus question: When does interference persist as a pulse?
One-sentence intuition: The spectrum and the pulse in time are the same field, linked by a Fourier transform. A pulse lasts only if gain, filtering, dispersion and loss all support it 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 the phases changes the peak. The energy in one period stays fixed, because it depends only on the power in each mode.

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 energy of the circulating envelope in one period sets how strongly the gain saturates.

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, and dispersion changes the phase. The frequency units must match the dispersion, which is given in square picoseconds.

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 modulation lowers the loss in a time window that can drift from trip to trip. A fast absorber lowers the loss wherever the light is already intense.

Repetition rate and the active pulse width

\begin{aligned}f_{\rm rep}&=\frac{c}{2L},\quad \omega_m=2\pi f_{\rm rep}\\\tau_a^4&=\frac{D_f}{M_s},\quad M_s=\frac{q_0\,\omega_m^2}{4}\\\Delta t_{1/2}&=2\sqrt{\ln 2}\,\tau_a\end{aligned}

The comb teeth are spaced by the repetition rate. Near the loss minimum the modulator acts like a parabola, and its balance with filtering gives a Gaussian steady state. The dashed reference curve uses this result; the filter scale sets the filter curvature.

Time-bandwidth product

\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 value is the transform limit for that one shape.

Equal-amplitude modes with flat phase

\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 raises the peak power above the average by a factor equal to the number of equal-power modes. No energy is added.

Common difficulties

One bright spike does not prove mode locking

Typical misconceptionIf a random field has one bright maximum, the laser is already stably mode locked.

Better mental modelWatch many round trips and count the competing peaks. A spike that comes and goes is not enough.

The transform limit depends on pulse shape

Typical misconceptionEvery pulse must have the Gaussian time-bandwidth product.

Better mental modelEach pulse shape has its own transform limit. When the field has several peaks, any single width needs extra care to interpret.

Run the experiment

01
Watch the record form a pulse
Keep the default preset, Modulator locks the teeth, and watch the replay count up the recorded round trips.
What to observe: The glow along the cavity gathers into one short packet. The peak-to-average ratio rises while the energy per round trip settles first.
02
Compare a cavity that does not lock
Load Free running: no locking.
What to observe: The light stays spread around the cavity and the peak is only a few times the average, even though the gain and filter are the same.
03
Drag the output coupler
Drag the output coupler to lengthen the cavity, or load Longer cavity, more teeth.
What to observe: The repetition rate drops, more teeth fit under the filter, and the pulse grows longer roughly as the square root of the round-trip time, as the dashed reference predicts.
04
Change the locking element and add dispersion
Load Absorber shortens the pulse, then Dispersion chirps the pulse, and open the tooth phase panel.
What to observe: The absorber gives a shorter pulse. With dispersion the tooth phases bend into a parabola and the time-bandwidth product rises above 0.441.

Mode Locking

Watch the phases lock into a pulse

Core mathematical model

Same modes, different interference

Saturated round-trip gain

Spectral filtering and dispersion

Two ways to favor a pulse

Repetition rate and the active pulse width

Time-bandwidth product

Equal-amplitude modes with flat phase

Common difficulties

One bright spike does not prove mode locking

The transform limit depends on pulse shape

Run the experiment

Watch the record form a pulse

Compare a cavity that does not lock

Drag the output coupler

Change the locking element and add dispersion

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