Sensor limitations often result in devices with particularly high spatial-imaging resolution or high sampling rates but not both concurrently. Adaptive optics control mechanisms, for example, rely on high-fidelity sensing technology to predictively correct wavefront phase aberrations. We propose fusing these two categories of sensors: those with high spatial resolution and those with high temporal resolution. As a prototype, we first sub-sample simulations of the Kuramoto-Sivashinsky equation, known for its chaotic flow from diffusive instability, and build a map between such simulated sensors using a Shallow Decoder Neural Network. We then examine how to fuse the merits of a common sensor in aero-optical sensing, the Shack-Hartmann wavefront sensor, with the increased spatial information of a Digital Holography wavefront sensor, training on supersonic wind-tunnel wavefront data provided by the Aero-Effects Laboratory at the Air Force Research Laboratory Directed Energy Directorate. These maps merge the high-temporal and high-spatial resolutions from each respective sensor, demonstrating a proof-of-concept for wavefront sensor fusion for adaptive optical applications.
Aero-optical beam control relies on the development of low-latency forecasting techniques to quickly predict wavefronts aberrated by the turbulent boundary layer around an airborne optical system, and its study applies to a multidomain need from astronomy to microscopy for high-fidelity laser propagation. We leverage the forecasting capabilities of the dynamic mode decomposition (DMD) — an equation-free, data-driven method for identifying coherent flow structures and their associated spatiotemporal dynamics — to estimate future state wavefront phase aberrations to feed into an adaptive optic control loop. We specifically leverage the optimized DMD (opt-DMD) algorithm on a subset of the Airborne Aero-Optics Laboratory-Transonic experimental dataset, characterizing aberrated wavefront dynamics for 23 beam propagation directions via the spatiotemporal decomposition underlying DMD. Critically, we show that opt-DMD produces an optimally debiased eigenvalue spectrum with imaginary eigenvalues, allowing for arbitrarily long forecasting to produce a robust future state prediction, while exact DMD loses structural information due to modal decay rates.
KEYWORDS: Digital micromirror devices, Control systems, Data modeling, Wavefronts, Systems modeling, Machine learning, Transient nonlinear optics, Near field optics, Mathematical modeling, Algorithm development
We demonstrate the use of physics-informed machine learning algorithms for the adaptive, real-time characterization of aero-optical systems. From deep learning algorithms to nonlinear control methods, the optical sciences are an ideal platform for integrating data-driven control and machine learning for robust characterization and system identification. For the specific case of aero-optics, the ability to extract dominant coherent structures, transients and turbulent behaviors is critical for a diverse number of applications, including the complex and dynamic aero-optic effects on airborne-based laser platforms. Specifically, aero-optical beam control relies on the development of low-latency predictors that can quickly predict aberrated wavefronts to feed into an adaptive optic control loop. We propose develop a number of data-driven methods, including the dynamic mode decomposition (DMD), for real-time forecasting and control.
We demonstrate the use of machine learning for adaptive control of nonlinear optical systems. From deep learning algorithms to nonlinear control methods, the optical sciences are an ideal platform for integrating data-driven control and machine learning for robust, self-tuning operation. For the specific case of mode-locked lasers, commercially available servo-controllers enact a training and execution software module capable of self tuning the laser cavity even in the presence of mechanical and/or environmental perturbations and discrepancies, thus providing algorithmic stabilization of mode-locking performance. The execution stage quickly stabilizes optimal mode-locking using various algorithmic innovations including (i) extremum seeking control, (ii) deep reinforcement learning and (iii) deep model predictive control. The demonstrated methods are robust and equation-free, thus requiring no detailed or quantitatively accurate model of the physics.
Advanced methods in data science are driving the characterization and control of nonlinear dynamical systems in optics. In this work, we investigate the use of machine learning, sparsity methods and adaptive control to develop a self-tuning fiber laser, which automatically learns and adapts to maintain high-energy ultrashort pulses. In particular, a two-stage procedure is introduced consisting of a machine learning algorithm to recognize different dynamical regimes with distinct behavior, followed by an adaptive control algorithm to reject disturbances and track optimal solutions despite stochastically varying system parameters. The machine learning algorithm, called sparse representation for classification, comes from machine vision and is typically used for image recognition. The adaptive control algorithm is extremum-seeking control, which has been applied to a wide range of systems in engineering; extremum-seeking is beneficial because of rigorous stability guarantees and ease of implementation.
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.
We propose a new model which is valid for ultra-fast pulse propagation in a mode-locked laser cavity in the
few femtosecond to hundreds of attoseconds pulse regime, thus deriving the equivalent of the master mode-locking
equation for ultra-short pulses that has dominated mode-locking theory for two decades. The short pulse
equation with dissipative gain and loss terms allows for the generation of stable ultra-short optical pulses from
initial white-noise,thus providing the first theoretical framework for quantifying the pulse dynamics and stability
as pulseswidths approach the attosecond regime.
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.
Non-adiabatic pulse compression of cascaded higher-order optical soliton is investigated. We demonstrate high degree
compression of pulses with soliton order N=2, 3, 4 and 5 in two or three nonlinear fibers with different second-order
dispersion coefficients. Each fiber length is shorter than half of its soliton period. This compression technique has
significant advantages over the widely reported adiabatic and higher-order soliton compression.
The manipulation and routing of light-bullets in a VCSEL-like planar waveguide array is studied numerically. By
partitioning the gold contact layer used for current injection into discrete and individually addressable segments,
an electronically controllable and non-uniform gain profile is created. Light-bullets typically follow the gradient
of the gain and are therefore completely controllable by manipulating the gain profile. In addition, by exploiting
gain-mediated interactions between nearby light-bullets, the NAND and NOR gates are also constructed. Therefore,
planar waveguide arrays with addressable gain profiles appear to be an ideal technology for optical routing
applications as well as for photonic logic devices.
We develop a simple iterative model to simulate a laser with homogeneous gain and intensity dependent loss. Simulation
results show that a laser with homogenous gain can operate at multiple wavelengths if the intensity-dependent loss
exhibits saturable transmitter characteristics. Our results also show that for nonlinear losses that have both saturable
transmitter and saturable absorber characteristics, such as arises from nonlinear optical loss mirrors (NOLM) or
nonlinear polarization rotation (NPR), the multiwavelength output power spectrum can become very flat. The laser can
also exhibit periodic and chaotic behaviors. We find that the same theoretical model can also be used to describe multipulsing
dynamics of mode-locked lasers when the cavity energy increases. Near the multi-pulsing transitions, both
periodic and chaotic behavior can be observed as operating states of the laser cavity. Our iterative model provides a
simple geometrical description of the entire multi-pulsing transition behavior as a function of increasing cavity energy.
The model captures all the key features observed in experiments, including the periodic and chaotic mode-locking
regions, and further provides valuable insight into laser cavity engineering for maximizing performance.
A theoretical proposal is presented for the generation of mode-locked light-bullets in planar waveguide arrays,
extending the concept of time-domain mode-locking in waveguide arrays to spatial (transverse) mode-locking
in slab waveguides. The model presented yields three-dimensional localized states that act as global attractors
to the waveguide array system. Single pulse stationary and time-periodic solutions as well as the transition to
multi-pulse solutions as a function of gain are observed to be stabilized in such a system.
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.
A new method is proposed and considered theoretically for using phase-sensitive amplification as the intensitydiscrimination
(saturable absorption) element in a laser cavity to generate stable and robust mode-locking. The
phase-sensitive amplifier acts as a phase-filter for selecting the specific intensity dependent phase-rotation of the
mode-locked pulse that locks the phase to the amplifier pump phase. The nonlinear phase-rotation is analogous
to the nonlinear polarization rotation which is used with passive polarizers for mode-locking. It is demonstrated
that the phase-sensitive amplification mechanism can indeed result in stable mode-locking. An average cavity
model explicitly calculates the stability of the mode-locked pulses.
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.
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.
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.
An averaged evolution equation is presented and its dynamics studied for a mode-locked laser where the intensitydiscrimination
(saturable absorption) in the cavity is provided by phase-senstitive amplification. The phasesensitive
amplifier acts as a phase-filter for selecting the specific intensity dependent phase-rotation of the modelocked
pulse that locks the phase to the amplifier pump phase. The resulting averaged equation is a Swift-
Hohenberg type model which is a fourth-order diffusion equation with cubic-quintic nonlinearities. Additionally,
the governing evolution has a new linear growth term which couples to the nonlocal cavity energy. This parameter
is a standard bifurcation parameter in Swift-Hohenberg models and is controlled by the cavity saturable gain.
Such a modification to the governing evolution is the first of its kind to be considered theoretically in the context
of the Swift-Hohenberg equation, and its significant impact on the mode-locked pulse dynamics and multi-pulsing
behavior is explored.
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.
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.
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.
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.
We theoretically demonstrate X-waves as global attractors that enable mode-locking of a laser cavity operating
in the normal dispersion regime. This result is based upon a fully comprehensive physical model of the laser
cavity, where the nonlinear discrete diffraction dynamics of a waveguide array mediates the spontaneous periodic
generation of spatio-temporal X-waves.
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.
The intensity dynamics of a five-emitter laser array subject to a linearly decreasing injection current are examined numerically.
We have matched the results of the numerical model to an experimental AlGaAs quantum-dot array laser and
have achieved the same robust oscillatory power output with a nearly π phase shift between emitters that was observed in
experiments. Due to the linearly decreasing injection current, the output power of the waveguide decreases as a function of
waveguide number. For injection currents ranging from 380 to 500 mA, the oscillatory behavior persists with only a slight
change in phase difference. However, the fundamental frequency of oscillation increases with injection current, and higher
harmonics as well as some fine structures are produced.
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.
Photonic packet switching for all-optical networks is a rapidly developing technology since it circumvents many of the traditional bottlenecks created by the use of electronics. All-optical networking has application to both long-haul communications systems and high-performance computing systems. In each case, all-optical technologies are responsible for the routing, switching and logic decisions of the network. Characterizing the performance of a network includes calculating the latency and scalability of a given architecture assuming ideal behavior of its physical components. However, the physical layer ultimately determines the feasibility of data transmission. Thus accurately calculating the accumulated bit-error-rate (BER) is fundamental to evaluating the optical network as a whole, regardless of the network architecture. A new simulation technique, which is based upon experimental findings, is introduced which characterizes the physical layer performance of a given network architecture known as the Data Vortex. Experiments show that almost all the physical layer penalty is generated by the nodes which are used for switching and routing. Specifically, at each node data packets are amplified by a semiconductor optical amplifier so that coupling and routing losses are compensated. In this process, the data packets receive a noise penalty which results primarily from amplified spontaneous emission and in small part from spectral broadening. By using a phenomenological approach to modeling the noise penalties, the performance of the network nodes can be characterized. The modeling allows for a comprehensive understanding of the network and is a highly efficient computational tool for evaluating performance when compared to conventional time-domain techniques.
Recently there has been interest in producing "cubic-like" effects, such as self-focusing, in materials engineered to have a rapidly oscillating quadratic nonlinearity. If the nonlinearity oscillates on a fast enough scale, the governing quadratic equations can be effectively averaged to give cubic equations. We propose a multiple scales approach in which diffraction is neglected at leading order. In doing so, we obtain exact solutions to the leading order. In doing so, we obtain exact solutions to the leading order system and solvability conditions on the slow evolution and transverse spatial dependence which, ensure that the higher order corrections are periodic. Using a variational approach, dynamics and stability of the solutions to the slow evelope equations are described.
We consider the mean field model of the optical parametric oscillator (OPO) when the second harmonic of the OPO is driven externally by a spatially periodic pump field. Exact solutions for the first and second harmonics can be derived using Jacobi elliptic functions. These solutions can describe behavior that is both sinusoidally varying as well as front and pulse-like in the transverse direction. Numerical simulations show that for a wide range of parameter space these solutions are stabilized transverse field structures. Bifurcations can also occur which result in new nontrivial, but periodic, spatial structures.
We consider a new model for the active modulation component of a modelocked laser cavity which allows for the construction of exact pulse train solutions. The model begins with the nonlinear Schrodinger equation for propagation in the laser cavity which is influenced by chromatic dispersion and Kerr induced self-phase modulation. Additionally, a bandwidth limited gain term is included to capture the amplification process in the cavity. The active modelocking element allows for periodically spaced regions of preferential gain. Thus a modelocked pulse train will align itself under the peaks of the gain region while radiation energy outside this region is attenuated. We consider a novel form of the periodic, active modelocking element by making use of the Jacobi elliptic functions. Two families of pulse train solutions are generated: one in which neighboring pulses are in-phase, and a second in which neighboring pulses are out-of-phase. The model predicts that only out-of-phase pulse train solutions can be stabilized. Under large perturbation, the pulse train is often stabilized to a two-pulse per round trip configuration. All in-phase solutions are unstable and are destroyed. Further, for the out-of-phase solutions, if the pulse spacing is not sufficiently far, then the nearest neighbor interactions can dominate and lead to Q-switching behavior. For short cavities, this Q-switching can result in quasi-periodic behavior of the pulse train. For long cavities, the resulting Q-switching is chaotic in nature.
We consider modelocking in an optical fiber laser cavity using a passive long period fiber grating. The grating as a passive modelocking element arises from the the nonlinear mode-coupling which occurs between co-propagating core and cladding modes. The underlying concept is as follows: a resonant and linear mode-coupling interaction transfers energy periodically between the core and cladding modes. Nonlinearity, however, can be used to detune the resonant interaction by shifting the propagation constant of each mode via self-phase and cross-phase modulation. Thus the low intensity parts of a pulse which propagate through the grating can be coupled out to the cladding and attenuated while high intensity portions are detuned and transmitted through the grating with minimal loss. This intensity discrimination when acting in combination with chromatic dispersion, self-phase modulation, and a bandwidth limited gain, can lead to stable modelocking operation in a optical fiber laser cavity. Stable pulses are generated for a wide variety of nonlinear coupling strengths between core and cladding modes. Further, dispersive radiation can be completely attenuated while generating the stable pulse trains. Self-starting in the cavity along with stability of the pulse trains under perturbation are considered. In conclusion, the long period fiber grating provides a simple, compact passive component for a modelocked laser source which is robust and efficient.
Passively modelocked fiber lasers operating in the soliton regime can generate pulses at multi-gigahertz harmonic repetition rates. The lasers are modelocked with an ultrafast saturable absorber and the low loss cavities support the formation of multiple equally spaced soliton pulses. These sources are potentially attractive for applications in high speed fiber optic communications systems. The design and construction of these laser sources as well as their application to spectrally sliced wavelength division multiplexed transmission is described.
Erbium-doped fibers are ideal as the basic components for lasers and amplifiers operating near 1550 nm. Consequently, considerable interest and research in the past few years has focused on the use of Erbium-doped fibers in actively and passively modelocked fiber lasers operating in both the normal and anomalous dispersion regimes. This paper presents a variety of analytic and numerical techniques which can be used to quantitatively describe the dynamic pulse formation in a passive laser cavity which includes dispersion, nonlinearity, loss, and bandwidth limited gain-saturation. The mechanisms for modelocking (e.g. quantum-well semiconductor structures, APM action, etc.) are incorporated into the modelocking models via various appropriate approximations. The models capture the fundamental intensity-dependent loss required to achieve stable modelocked operation. This generic feature of modelocked lasers will be discussed in detail and illustrated in various modelocking devices and configurations. Additionally, the characteristic differences between modelocking in the normal dispersion regime, for which highly chirped pulses are observed, and the anomalous dispersion regime, for which soliton pulses can be achieved, will be discussed. Finally, stability of the modelocked pulses will be investigated. Here we will emphasize the range of stable one-pulse per round trip operation and the instability to multi-pulse operation. For multi-pulse operation, mechanisms for harmonic modelocking such as gain- relaxation, dispersive radiation, and the acoustic effect will be discussed. In most instances, analytic methods can be utilized to gain significant insight into the laser operation and stability.
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