We present a perturbation analysis that describes the effect of third-order dispersion on the similariton pulse solution
of the nonlinear Schr¨odinger equation in a fibre gain medium. The theoretical model predicts with sufficient
accuracy the pulse structural changes induced, which are observed through direct numerical simulations.
Recent theoretical investigations have demonstrated that the stability of mode-locked solution of multiple frequency
channels depends on the degree of inhomogeneity in gain saturation. In this paper, these results are
generalized to determine conditions on each of the system parameters necessary for both the stability and existence
of mode-locked pulse solutions for an arbitrary number of frequency channels. In particular, we find
that the parameters governing saturable intensity discrimination and gain inhomogeneity in the laser cavity also
determine the position of bifurcations of solution types. These bifurcations are completely characterized in terms
of these parameters. In addition to influencing the stability of mode-locked solutions, we determine a balance
between cubic gain and quintic loss, which is necessary for existence of solutions as well. Furthermore, we determine
the critical degree of inhomogeneous gain broadening required to support pulses in multiple frequency
channels.
We consider experimentally and theoretically a refined parameter space near the transition to multi-pulse modelocking.
Near the transition, the onset of instability is initiated by a Hopf (periodic) bifurcation. As cavity
energy is increased, the band of unstable, oscillatory modes generates a chaotic behavior between single- and
multi-pulse operation. Both theory and experiment are in good qualitative agreement and they suggest that the
phenomenon is of a universal nature in mode-locked lasers at the onset of multi-pulsing from N to N + 1 pulses
per round trip. This is the first theoretical and experimental characterization of the transition behavior, made
possible by a highly refined tuning of the gain pump level.
We present a theoretical description of the generation of ultra-short, high-energy pulses in two laser cavities
driven by periodic spectral filtering or dispersion management. Critical in driving the intra-cavity dynamics is
the nontrivial phase profiles generated and their periodic modification from either spectral filtering or dispersion
management. For laser cavities with a spectral filter, the theory gives a simple geometrical description of the
intra-cavity dynamics and provides a simple and efficient method for optimizing the laser cavity performance.
In the dispersion managed cavity, analysis shows the generated self-similar behavior to be governed by the
porous media equation with a rapidly-varying, mean-zero diffusion coefficient whose solution is the well-known
Barenblatt similarity solution with parabolic profile.
A comprehensive theoretical treatment is given of the mode-locking dynamics produced by the intensity discrimination
(saturable absorption) generated by the nonlinear mode-coupling in a waveguide array. Emphasis
is placed on the mode-locking stability as a function of the critical physical parameters in the laser cavity. The
theoretical characterization of the laser cavity's stability and dynamics allows for a comprehensive optimization
of the laser cavity parameters towards achieving high peak-power, high-energy pulses in both the anomalous and
normal dispersion regimes.
Current optical fiber-communication networks increasingly rely on wavelength-division multiplexing (WDM)
technologies in conjunction with optical time-division multiplexing (OTDM) of individual WDM channels. The
combination of high-repetition-rate data streams with a large number of WDM channels has pushed transmission
rates to nearly 1 TB/s, creating a demand for all-optical transmission sources that can generate pico-second modelocked
pulses at various wavelengths. Through nonlinear mode-coupling in a wave-guide array and a periodically
applied multi-notch frequency filter, robust multi-frequency mode-locking can be achieved in a laser cavity in
both the normal and anomalous dispersion regimes. We develop a theoretical description of this multiplewavelength
mode-locking, and characterize the mode-locked solutions and their stability for an arbitrary number
of frequency channels. The theoretical investigations demonstrate that the stability of the mode-locked pulse
solutions of multiple frequency channels depends on the degree of inhomogenity in gain saturation. Specifically,
only a small amount of inhomogeneous gain-broadening is needed for multi-frequency operation in the laser.
In this presentation, the conditions on the system parameters necessary for generating stable mode-locking is
explored for arbitrary number of frequency channels. The model suggests a promising source for multi-frequency
photonic applications.
Self-similarity is a ubiquitous concept in the physical sciences used to explain a wide range of spatial- or temporalstructures
observed in a broad range of applications and natural phenomena. Indeed, they have been predicted
or observed in the context of Raman scattering, spatial soliton fractals, propagation in the normal dispersion
regime with strong nonlinearity, optical amplifiers, and mode-locked lasers. These self-similar structures are
typically long-time transients formed by the interplay, often nonlinear, of the underlying dominant physical
effects in the system. A theoretical model shows that in the context of the universal Ginzburg-Landau equation
with rapidly-varying, mean-zero dispersion, stable and attracting self-similar pulses are formed with parabolic
profiles: the zero-dispersion similariton. The zero-dispersion similariton is the final solution state of the system,
not a long-time, intermediate asymptotic behavior. An averaging analysis shows the self-similarity to be governed
by a nonlinear diffusion equation with a rapidly-varying, mean-zero diffusion coefficient. Indeed, the leadingorder
behavior is shown to be governed by the porous media (nonlinear diffusion) equation whose solution
is the well-known Barenblatt similarity solution which has a parabolic, self-similar profile. The alternating
sign of the diffusion coefficient, which is driven by the dispersion fluctuations, is critical to supporting the
zero-dispersion similariton which is, to leading-order, of the Barenblatt form. This is the first analytic model
proposing a mechanism for generating physically realizable temporal parabolic pulses in the Ginzburg-Landau
model. Although the results are of restricted analytic validity, the findings are suggestive of the underlying
physical mechanism responsible for parabolic (self-similar) pulse formation in lightwave transmission and observed
in mode-locked laser cavities.
We present a theoretical description of the generation of ultra-short, high-energy pulses in an all-normal dispersion
laser cavity with spectral filtering. A reduced variational model based upon the Haus master mode-locking
equations with quintic saturation is shown to characterize the experimentally observed dynamics. Critical in
driving the intra-cavity dynamics is the nontrivial phase profiles generated and their periodic modification from
the spectral filter. The theory gives a simple geometrical description of the intra-cavity dynamics and possible
operation modes of the laser cavity. Further, it provides a simple and efficient method for optimizing the laser
cavity performance.
Current optical fiber-communication networks increasingly rely on wavelength-division multiplexing (WDM)
technologies in conjunction with optical time-division multiplexing (OTDM) of individual WDM channels. The
combination of high-repetition-rate data streams with a large number of WDM channels has pushed transmission
rates to nearly 1 TB/s, creating a demand for all-optical transmission sources that can generate pico-second modelocked
pulses at various wavelengths. Through nonlinear mode-coupling in a wave-guide array and a periodically
applied multi-notch frequency filter, robust multi-frequency mode-locking can be achieved in a laser cavity in
both the normal and anomalous dispersion regimes. We develop a theoretical description of this multiplewavelength
mode-locking, and characterize the mode-locked solutions and their stability for an arbitrary number
of frequency channels. The theoretical investigations demonstrate that the stability of the mode-locked pulse
solutions of multiple frequency channels depends on the degree of inhomogenity in gain saturation. Specifically,
only a small amount of inhomogeneous gain-broadening is needed for multi-frequency operation in the laser.
In this presentation, the conditions on the system parameters necessary for generating stable mode-locking is
explored for arbitrary number of frequency channels. The model suggests a promising source for multi-frequency
photonic applications.
Self-similarity is a ubiquitous concept in the physical sciences used to explain a wide range of spatial- or temporalstructures
observed in a broad range of applications and natural phenomena. Indeed, they have been predicted
or observed in the context of Raman scattering, spatial soliton fractals, propagation in the normal dispersion
regime with strong nonlinearity, optical amplifiers, and mode-locked lasers. These self-similar structures are
typically long-time transients formed by the interplay, often nonlinear, of the underlying dominant physical
effects in the system. A theoretical model shows that in the context of the universal Ginzburg-Landau equation
with rapidly-varying, mean-zero dispersion, stable and attracting self-similar pulses are formed with parabolic
profiles: the zero-dispersion similariton. The zero-dispersion similariton is the final solution state of the system,
not a long-time, intermediate asymptotic behavior. An averaging analysis shows the self-similarity to be governed
by a nonlinear diffusion equation with a rapidly-varying, mean-zero diffusion coefficient. Indeed, the leadingorder
behavior is shown to be governed by the porous media (nonlinear diffusion) equation whose solution
is the well-known Barenblatt similarity solution which has a parabolic, self-similar profile. The alternating
sign of the diffusion coefficient, which is driven by the dispersion fluctuations, is critical to supporting the
zero-dispersion similariton which is, to leading-order, of the Barenblatt form. This is the first analytic model
proposing a mechanism for generating physically realizable temporal parabolic pulses in the Ginzburg-Landau
model. Although the results are of restricted analytic validity, the findings are suggestive of the underlying
physical mechanism responsible for parabolic (self-similar) pulse formation in lightwave transmission and observed
in mode-locked laser cavities.
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