Model of a distributed-feedback fiber laser with a multiple-π-shift cavity

Advances in technology made it possible to form complex cavities including a set of Bragg gratings and \ensuremath{\pi} shifts for distributed feedback (DFB) fiber lasers. This allows for precise control of the mode structure of the cavity. In such lasers, the input pump power defines the gain distribution along the fiber, leading to different lasing regimes. Direct accounting for these effects leads to highly resource-intensive space-time laser models, either to models based on modal decomposition of the field and population inversion associated with a problem of closing the system of variables. We propose a hybrid model that solves the mentioned problems by considering the spatial distribution of the population inversion combined with the field modal decomposition. The spatial field distribution within a single Bragg grating period and corresponding hole burning in the population inversion is characterized by two additional variables describing the population inversion spatial grating. We illustrate the application of the model on an example of $\mathrm{E}{\mathrm{r}}^{3+}$/$\mathrm{Y}{\mathrm{b}}^{3+}$ laser with a cavity including two \ensuremath{\pi} shifts. We study the resulting mode dynamics and demonstrate hysteretic behavior of lasing curves. We show that in our example, mode coexistence is observed, whereas an exclusion of the population inversion grating changes the lasing regime to single-mode lasing with one of the modes being suppressed. We hope that the proposed model will find numerous applications in modeling DFB fiber lasers.