A generalized master mode-locking model is presented to capture the periodic transmission created by a series
of waveplates and polarizer in a mode-locked ring laser cavity, and the equation is referred to as the sinusoidal
Ginzburg-Landau equation (SGLE). Numerical comparisons with the full dynamics show that the SGLE is able
to capture the essential mode-locking behaviors including the multi-pulsing instability observed in the laser cavity
and does not have the drawbacks of the conventional master mode-locking theory. The SGLE model supports
high energy pulses that are not predicted by the master mode-locking theory, thus providing a platform for
optimizing the laser performance.
The onset of multi-pulsing, a ubiquitous phenomenon in laser cavities, imposes a fundamental limit on the maximum
energy delivered per pulse. Managing the nonlinear penalties in the cavity becomes crucial for increasing
the energy and suppressing the multi-pulsing instability. A Proper Orthogonal Decomposition (POD) allows for
the reduction of governing equations of a mode-locked laser onto a low-dimensional space. The resulting reduced
system is able to capture correctly the experimentally observed pulse transitions. Analysis of these models is
used to explain the the sequence of bifurcations that are responsible for the multi-pulsing instability in the master
mode-locking and the waveguide array mode-locking models. As a result, the POD reduction allows for simple
and efficient way to characterize and optimize the cavity parameters for achieving maximal energy output.
The mode-locking of dissipative soliton fiber lasers using large mode area fiber supporting multiple transverse
modes is studied experimentally and theoretically. Experiments using large core step-index fiber, photonic crystal
fiber, and chirally-coupled core fiber show that when the higher order mode content exceeds -27 dB, the maximum
stable single-pulse energy is significantly reduced. The averaged mode-locking dynamics in a multi-mode fiber are
studied using a distributed model. The co-propagation of multiple transverse modes is governed by a system of
coupled Ginzburg-Landau equations (CGLEs). Simulations show that stable and robust mode-locked pulses can
be produced. The maximum stable single pulse energy is found to increase with higher order mode filtering. This
work demonstrates that mode-locking performance is very sensitive to the presence of multiple waveguide modes
when compared to systems such as amplifiers and continuous-wave lasers, and gives a quantitative estimate of
what constitutes effectively single-mode operation. Robust, distributed higher order mode filtering is necessary
to maximize single-pulsing energy.
An iterative method is developed to characterize the mode-locking dynamics in a laser cavity mode-locked with
a combination of waveplates and a passive polarizer. The method explicitly accounts for an arbitrary alignment
of the fast- and slow-axes of the fiber with the waveplates and polarizer, fiber birefringence and saturating
gain dynamics. The general averaging scheme results in the cubic-quintic complex Ginzburg-Landau equation
(CQGLE), and an extensive comparison shows the agreement between the full model and the CQGLE and allows
for a characterization of the stability and operating regimes of the laser cavity.
The averaged mode-locking dynamics in a multi-mode fiber is studied. The transverse mode structures of the
electric field are determined from a linear eigenvalue problem, and the co-propagation of the corresponding mode
envelopes is governed by a system of coupled Ginzburg-Landau equations (CGLEs) which accounts explicitly for
bandwidth-limited saturable gain as well as saturable absorption. Simulations show that stable and robust modelocked
pulses with high energy can be produced. The maximum pulse energy is simulated as a function of the
linear coupling and coiling loss. The present work provides for an excellent tool for characterizing mode-locking
performance.
A low-dimensional model is constructed via a variational formulation which characterizes the mode-locking
dynamics in a laser cavity with a passive polarizer. The theoretical model accounts explicitly for the effects of
the passive polarizer with a Jones matrix. In combination with the nonlinear interaction of the orthogonally
polarized electromagnetic fields, the evolution of the mode-locked state reduces to the nonlinear interaction of
the amplitude, width and phase chirp. This model allows for an explicit analytic prediction of the steady-state
mode-locked state (fixed point) and its corresponding stability. The stability analysis requires a center manifold
reduction which reveals that the solution decays to the mode-locked state on a timescale dependent on the
gain bandwidth and the net cavity gain. Quantitative and qualitative agreement is achieved between the full
governing model and the low-dimensional model, thus providing for an excellent design tool for characterizing
and optimizing mode-locking performance.
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