Mode-Locked Laser Lab (2D)

Modes of the cavity

A laser cavity of length L only supports light whose round trip fits a whole number of wavelengths. The allowed frequencies are a comb with equal spacing f_rep = c / (2 L), the inverse of the round-trip time T = 2 L / c. The gain medium amplifies a band of width B, so about N = B / f_rep modes lase together.

Adding the modes

The field is E(t) = sum_k a_k exp[i (2 pi k f_rep t + phi_k)] and the intensity is |E|^2. With random phases the modes beat into a noisy, nearly constant output. If a saturable absorber or a Kerr lens ties all phases together (phi_k = 0), the modes add up in step once per round trip and cancel in between: one short pulse per T.

Pulse width and peak power

For a Gaussian spectrum of intensity width B the shortest (transform-limited) pulse has tau = 0.441 / B = 0.441 / (N f_rep). The peak intensity over the mean is T / (1.0645 tau) = 2.13 N. A real titanium-sapphire laser has about 10^5 modes; this page caps N at 180 so the sum stays instant, but the same formulas hold.

Dispersion

If the glass delays red and blue differently (group delay dispersion), the mode phases acquire a term phi_k proportional to k^2 and the pulse stretches to tau = tau0 sqrt(1 + xi^2), with xi = 4 ln2 GDD / tau0^2. A prism pair provides the opposite sign, which is why it sits in the cavity.

Frequency comb

Mode k sits at f_k = f_ceo + k f_rep. The offset f_ceo makes the carrier wave slip by 2 pi f_ceo / f_rep relative to the envelope from one pulse to the next; the graph draws an illustrative carrier under three pulses. Locking both f_rep and f_ceo turns the laser into a ruler for optical frequencies.

Simplifications

Linear two-mirror cavity, Gaussian gain spectrum, equal mode spacing (cavity dispersion enters only as the quadratic chirp), no gain dynamics or noise build-up, the mode locking is imposed rather than derived, and at most 801 modes are summed.

1. Start with Random phases: a noisy, quasi-constant intensity. Press Lock all modes and watch one narrow pulse per round trip appear.
2. Raise the number of modes: the pulse shortens as 1/N while the peak grows as N.
3. Shorten the cavity (3 cm) with 180 modes: sub-picosecond pulses.
4. Add chirp: the pulse stretches by sqrt(1 + xi^2); negative and positive chirp stretch equally.
5. Drag the locking slider between 0 and 1: a pulse grows out of the noise floor.
6. Change the carrier-envelope offset to see the carrier shift under the envelope.