The master equation for passive modelocking

Passive modelocking (PML) of lasers is pivotal in modern science and industry. Here, solving a half-century-long challenge, we present the first master equation (ME) describing PML on all time scales, from Q-switched to fundamental and harmonic modelocking, valid for both slow and fast saturable absorption, short and long cavities. The proposed ME should become the workhorse for analytical and numerical studies of ultrafast lasers in both the photonics engineering and laser physics communities.