Mode structure of a semiconductor laser with feedback from two external filters

Pages: 519 - 586,
Volume 20,
Issue 2,
March
2015 doi:10.3934/dcdsb.2015.20.519

Piotr Słowiński - Mathematics Research Institute, CEMPS, University of Exeter, North Park Road, Exeter EX4 4QF, United Kingdom (email)

Bernd Krauskopf - Department of Mathematics, The University of Auckland, Private Bag 92019, Auckland 1142, New Zealand (email)

Sebastian Wieczorek - Department of Applied Mathematics, University College Cork, Western Gateway Building, Cork, Ireland (email)

Abstract:
We investigate the solution structure and stability of a semiconductor
laser receiving time-delayed and frequency-filtered optical feedback
from two external filters. This system is referred to as
the 2FOF laser, and it has been used as pump laser in optical
telecommunication and as light source in sensor applications.
The underlying idea is that the two filter loops provide a means of
stabilizing and controling the laser output. The mathematical
model takes the form of delay differential equations
for the (real-valued) population inversion of the laser active medium
and for the (complex-valued) electric fields of the laser cavity
and of the two filters. There are two time delays, which are the
travel times of the light from the laser to each of the filters and
back.

Our analysis of the 2FOF laser focuses on the basic solutions,
known as continuous waves or external filtered modes (EFMs), which
correspond to laser output with steady amplitude and frequency.
Specifically, we consider
the EFM-surface in the $(\omega_s,\,N_s,\,dC_p)$-space of steady frequency
$\omega_s$, the corresponding steady population inversion $N_s$, and
the feedback phase difference $dC_p$. This surface emerges as the
natural object for the study of the 2FOF laser
because it conveniently catalogues information
about available frequency ranges of the EFMs. We identify five
transitions, through four different singularities and a cubic tangency, which
change the type of the EFM-surface locally and
determine the EFM-surface bifurcation diagram in the
$(\Delta_1,\,\Delta_2)$-plane. In this way, we classify the
possible types of the EFM-surface, which consist of a combination of
bands (covering the entire $dC_p$-range) and islands (covering only a
finite range of $dC_p$).

We also investigate the stability of the EFMs, where we focus on
saddle-node and Hopf
bifurcation curves that bound regions of stable EFMs on the
EFM-surface. It is shown how these stability regions evolve when
parameters are
changed along a chosen path in the $(\Delta_1,\,\Delta_2)$-plane.
From a viewpoint of practical
interests, we find various bands and islands of stability on the
EFM-surface that may be accessible experimentally.

Beyond their relevance for the 2FOF laser system, the results
presented here also showcase how advanced tools from
bifurcation theory and singularity
theory can be employed to uncover and represent the complex
solution structure of a delay differential equation model that
depends on a considerable number of input parameters, including two
time delays.

Keywords: Delay differential equation model, bifurcation analysis,
laser with filtered optical feedback, classification of external cavity
modes, singularities of surfaces.

Mathematics Subject Classification: 37M20, 37G10, 78A60.

Received: April 2014;
Revised:
May 2014;
Available Online: January 2015.

References