Progress in Optics

14: The Lugiato–Lefever equation for the Fabry–Perot cavity

The temporal-longitudinal version of the equation introduced in 1987 by Lugiato and Lefever (LLE) (Lugiato & Lefever, 1987) reads, in the case of anomalous dispersion (Brambilla, Castelli, Gatti, Lugiato, & Prati, 1993; Haelterman, Trillo, & Wabnitz, 1992)

$\begin{array}{l}\hfill \frac{\partial E}{\partial \overline{t}}=E_I-E_I-i\theta E+i|E{|}^{2}E-i\overline{\eta }\frac{{\partial }^{2}E}{\partial {\overline{\tau }}^2},\end{array}$

  (127)

where $\overline{t}=kt$, $\overline{\tau }$ is also dimensionless and proportional to the retarded time τ = t − z/v_g