We present an analysis of a semiconductor laser subject to filtered optical feedback from two filtering elements (2FOF).
The motivation for this study comes from applications where two filters are used to control and stabilise the laser output.
Compared to a laser with a single filtered optical feedback loop, the introduction of the second filter significantly influences
the structure of the basic continuous-wave solutions, which are also known as external filtered modes (EFMs). We compute
and represent the EFMs of the underlying delay differential equation model as surfaces in the space of frequency <i>ω<sub>s</sub></i> and
inversion level <i>N<sub>s</sub></i> of the laser, and feedback phase difference <i>dC<sub>p</sub></i>. The quantity <i>dC<sub>p</sub></i> is a key parameter since it is
associated with interference between the two filter fields and, hence, controls the effective feedback strength. We further
show how the EFM surface in (<i>ω<sub>s</sub></i>, <i>dC<sub>p</sub></i>, <i>N<sub>s</sub></i>)-space changes upon variation of other filter parameters, in particular, the two
delay times. Overall, the investigation of the EFM-surface provides a geometric approach to the multi-parameter analysis
of the 2FOF laser, which allows for comprehensive insight into the solution structure and dynamics of the system.
We study a semiconductor laser subject to filtered optical feedback from two separate filters. This work is motivated by an
application where two fiber gratings are used to stabilize the output of a laser source. Specifically, we consider the structure
of the external filtered modes (EFMs), which are the basic cw-states of the system. The system is modelled by a set of
four delay differential equations with two delays that are due to the travel times of the light in each of the external cavities.
Here, each filter is approximated by a Lorentzian and we assume that there is no interaction between the two filters.
We derive a transcendental equation for the EFMs as a function of the widths, detunings and the feedback strengths
of the two filters. With continuation techniques we investigate how the number of EFMs changes with parameters. In
particular, we consider the equation for its envelope. This allows us to determine regions in the plane of the two detunings
that correspond to one, two or three EFM components - disjoint closed curves that are traced out by the EFMs as a
function of the feedback phase.