15839251
Boundary tracking algorithms for determining the stability of mode-locked pulses
We develop robust computational methods, referred to as boundary tracking algorithms, that can rapidly determine the existence and stability of pulses in passively mode-locked laser systems over a broad parameter range. Applying the boundary tracking algorithms to the cubic–quintic mode-locking equation, we find a rich dynamical structure.
43518800
Mode-locking of lasers
The evolution of the theory of mode-locking over the last three and a half decades is reviewed and some of the salient experiments are discussed in the context of the theory. The paper ends with two-cycle pulses of a mode-locked Ti:sapphire laser.
113986775
Few-Cycle Pulses Directly from a Laser
Advances in ultrashort-pulse laser technology have led to the genera- tion of few-cycle optical pulses directly from oscillators at high repetition rates, in contrast to external compression schemes. In this chapter, we review the technique of mode locking to obtain few-cycle pulses and describe the major developments leading to sub-two-cycle pulse generation from a Ti:sapphire laser. The master equation approach is used in a consistent way to analyze the nonlinear dynamics of such a laser as well as the noise and tuning properties of its frequency mode comb. Control over the mode combs of a few-cycle Ti:sapphire laser and a Cr:forsterite laser enables their coherent superposition in a long-term stable arrangement. An approach toward synthesizing single-cycle pulses using this technique is presented.
35353387
Simultaneous existence of a multiplicity of stable and unstable solitons in dissipative systems
Abstract We show that dissipative systems can have a multiplicity of stationary solutions in the form of both stable and unstable solitons. As a model equation, we use the complex cubic–quintic Ginzburg–Landau equation. For a given set of the equation parameters, this equation has many coexisting soliton solutions. Our stability results show that although most of them are unstable, they can have stable pieces. This partial stability leads to the phenomenon of soliton explosion.
5624576
Stability of the pulselike solutions of the quintic complex Ginzburg-Landau equation
We performed a detailed investigation of the stability of analytic pulselike solutions of the quintic complex Ginzburg–Landau equation that describes the dynamics of the field in a passively mode-locked laser. We found that in general they are unstable except in a few special cases. We also obtained regions in the parameter space in which stable pulse solutions exist. These stable solutions do not have analytical expressions and must be calculated numerically. We compared and connected the regions in which stable solitonlike solutions exist with the lines for which we had analytical solutions.
67812601
Roadmap to ultra-short record high-energy pulses out of laser oscillators
Highly-chirped dissipative solitons of the cubic–quintic Ginzburg–Landau equation found in this work may provide a roadmap to design passively mode-locked laser oscillators that generate pulses of extremely high energy. We provide a region in the space of the system parameters where high-energy dissipative solitons are found, along with their typical spectral and temporal features.
125464369
Differential Equations Dynamical Systems And An Introduction To Chaos
Thank you for reading differential equations dynamical systems and an introduction to chaos. Maybe you have knowledge that, people have look numerous times for their chosen novels like this differential equations dynamical systems and an introduction to chaos, but end up in malicious downloads. Rather than reading a good book with a cup of tea in the afternoon, instead they are facing with some harmful bugs inside their computer.
45614148
Computing spectra of linear operators using the Floquet-Fourier-Hill method
In order to establish the stability of an equilibrium solution U of an infinite-dimensional dynamical system u ? = X ( u ) , one is interested in the spectrum of the linear operator LU] obtained by linearizing the dynamical system around U. We use a spectrally accurate method for the computation of the spectrum of the maximal extension of the operator LU]. The method is particularly well-suited to the case of periodic U, although no periodic boundary conditions on the perturbations are imposed. By incorporating the fundamentals of Floquet theory, an almost uniform approximation to the entire spectrum of the maximal extension is obtained, as opposed to an approximation of a few selected elements. The numerical component of the method is limited to: (i) choosing the size of the matrices to be used; and (ii) an eigenvalue solver, such as the QR algorithm. Compared to often-used finite-difference approaches, the method is an order of magnitude faster for comparable accuracy. We illustrate that the method is efficiently extended to problems defined on the whole line.
33957048
Stability of passively mode-locked fiber lasers with fast saturable absorption.
The stability of passively mode-locked fiber lasers with fast saturable absorption is examined by a direct numerical approach. The laser operation is described by use of a standard model, closely related to the Ginzburg–Landau equation, which is valid when the change of the laser pulse during one round trip through the laser is small. This equation is then linearized around an equilibrium solution, and the eigenmodes and eigenvalues of the linearized equation are determined numerically. This approach retains the advantage of previous analytical research of permitting one to explore a wide range of parameter space while also permitting one to avoid approximations in the stability calculation. The stability of a figure-eight loop laser with the amplifier in the external loop is then studied by use of this approach.
16828448
Theory of mode locking with a fast saturable absorber
This paper presents a closed‐form analysis of saturable absorber mode locking of a homogeneously broadened laser. A solution is obtained for the case of a short relaxation time of the saturable absorber. This pulse is a hyperbolic secant as a function of time. For each choice of parameters two pulse widths are found. A stability analysis shows that the solution of greater width is stable. The requirements for achieving mode locking with a fast saturable absorber are stated. The effect of a time‐varying laser medium gain is investigated analytically.
122691948
Dissipative solitons in normal-dispersion fiber lasers
Mode-locked fiber lasers in which pulse shaping is based on filtering of a frequency-chirped pulse are analyzed with the cubic-quintic Ginzburg-Landau equation. An exact analytical solution produces a variety of temporal and spectral shapes, which have not been observed in any experimental setting to our knowledge. Experiments agree with the theory over a wide range of parameters. The observed pulses balance gain and loss as well as phase modulations, and thus constitute dissipative temporal solitons. The normal-dispersion fiber laser allows systematic exploration of this class of solitons.
10358153
Mode-Locked Soliton Lasers
A comprehensive treatment is given for the formation of mode-locked soliton pulses in optical fiber and solid state lasers. The pulse dynamics is dominated by the interaction of the cubic Kerr nonlinearity and chromatic dispersion with an intensity-dependent perturbation provided by the mode-locking element in the laser cavity. The intensity-dependent perturbation preferentially attenuates low intensity electromagnetic radiation which makes the mode-locked pulses attractors of the laser cavity. A review of the broad spectrum of mode-locked laser models, both qualitative and quantitative, is considered with the basic pulse formation phenomena highlighted. The strengths and weaknesses of each model are considered with two key instabilities studied in detail: Q-switching and harmonic mode-locking. Although the numerous mode-locking models are considerably different, they are unified by the fact that stable solitons are exhibited in each case due to the intensity discrimination perturbation in the laser cavity.
120590554
Role of dispersion in pulse emission from a sliding-frequency fiber laser
We present an experimental and theoretical investigation of the role of group-velocity dispersion in the generation of picosecond pulses from a sliding-frequency fiber loop laser.
120437400
Stability of pulses in the master mode-locking equation
The master mode-locking equation is a canonical model for mode locking in solid-state lasers. We consider the dynamics and stability of the localized pulse solutions that this equation admits of. We provide analytic proof of stable mode-locking behavior along with analysis that shows that mode locking can become destabilized as a result of either a radiation-mode or a saddle-node instability. This is to our knowledge the first analytic proof of the stability of the pulse solutions that takes the time-dependent gain saturation mechanism of mode-locked lasers into account.
15702241
The Evans function for nonlocal equations
In recent studies the master mode-locking equation, a model for solid-state cavity laser that includes nonlocal terms, bifurcations from stationary to seemingly time-periodic solitary waves have been observed. To decide whether the mechanism is a Hopf bifurcation of a bifurcattion from the essential spectrum, a general framework for the Evans function for equations with nonlocal terms is developed and applied to the master mode-locking model.
122519753
Soliton mode-locking with saturable absorbers
We investigate ultrashort pulse generation based on the fundamental soliton generation that is stabilized by a saturable absorber. The case of an absorber with a recovery time much longer than the pulsewidth of the generated soliton is investigated in detail. Based on soliton perturbation theory we derive equations for the soliton variables and the continuum generated in a mode-locked laser. Analytic criteria for the transition from stable to unstable soliton generation are derived. The results demonstrate the possibility of ultrashort pulse generation by a slow saturable absorber only. The theoretical results are compared with experiments. We generate pulses as short as 13 fs using only semiconductor saturable absorbers.
20564959
Edge Bifurcations for Near Integrable Systems via Evans Function Techniques
When studying the linearstability of waves for near integrable systems, a fundamental problem is the location of the point spectrum of the linearized operator. Internal modes may be created upon the perturbation, i.e., eigenvalues may bifurcate out of the continuous spectrum, even if the corresponding eigenfunction is not initially localized. This phenomenon is also known as an edge bifurcation. It has recently been shown that the Evans function is a powerful tool when one wishes to detect an edge bifurcation and track the resulting eigenvalues. It has been an open question as to the role played by the solutions to the Lax pair, associated with the integrable problem, in the construction of the Evans function and the detection of edge bifurcations. Using the Zakharov--Shabat eigenvalue problem and the massive Thirring model as illustrations, we show the connection between the inverse scattering formalism and the linear stability analysis of waves. In particular, we show a direct connection between the sca...
120077214
Understanding leaky modes: slab waveguide revisited
Computational methods for determining the complex propagation constants of leaky waveguide modes have become so powerful and so readily available that it is possible to use these methods with little understanding of what they are calculating. We compare different computational methods for calculating the propagation constants of the leaky modes, focusing on the relatively simple context of a W-type slab waveguide. In a lossless medium with infinite transverse extent, a direct determination of the leaky mode by using mode matching is compared with complete mode decomposition. The mode matching method is analogous to the multipole method in two dimensions. We then compare these results with a simple finite-difference scheme in a transverse region with absorbing boundaries that is analogous to finite-difference or finite-element methods in two dimensions. While the physical meaning of the leaky modes in these different solution methods is different, they all predict a nearly identical evolution for an initial, nearly confined mode profile over a limited spatial region and a limited distance. Finally, we demonstrate that a waveguide that uses bandgap confinement with a central defect produces analogous results.
120858011
Heavily-chirped solitary pulses in the normal dispersion region: New solutions of the cubic-quintic complex Ginzburg-Landau equation
A new type of the heavily-chirped solitary pulse solutions of the nonlinear cubic-quintic complex Ginzburg-Landau equation has been found. The methodology developed provides for a systematic way to find the approximate but highly accurate analytical solutions of this equation with the generalized nonlinearities within the normal dispersion region. It is demonstrated that these solitary pulses have the extra-broadened parabolic-top or fingerlike spectra and allow compressing with more than a hundredfold growth of the pulse peak power. The obtained solutions explain the energy scalable regimes in the fiber and solid-state oscillators operating within the normal dispersion region and promise to achieve microjoules femtosecond pulses at MHz repetition rates.
122134086
Chirped-pulse oscillators: theory and experiment
Theory of chirped-pulse oscillators operating in the positive dispersion regime is presented. It is found that the chirped pulses can be described analytically as solitary pulse solutions of the nonlinear cubic-quintic complex Ginzburg–Landau equation. Due to the closed form of the solution, basic characteristics of the regime under consideration are easily traceable. Numerical simulations validate the analytical technique and the chirped-pulse stability. Experiments with 10 MHz Ti:Sa oscillator providing up to 150 nJ chirped pulses, which are compressible down to 30 fs, are in agreement with the theory.
23891418
Chirped-pulse oscillators: a unified standpoint
A completely analytical and unified approach to the theory of chirped-pulse oscillators is presented. The approach developed is based on the approximate integration of the generalized nonlinear complex Ginzburg-Landau equation and demonstrates that a chirped-pulse oscillator is controlled by only two parameters. It makes it easy to trace spread of the real-world characteristics of both solid-state and fiber oscillators operating in the positive-dispersion regime.
18064581
Chirped dissipative solitons of the complex cubic-quintic nonlinear Ginzburg-Landau equation.
Approximate analytical chirped solitary pulse (chirped dissipative soliton) solutions of the one-dimensional complex cubic-quintic nonlinear Ginzburg-Landau equation are obtained. These solutions are stable and highly accurate under condition of domination of a normal dispersion over a spectral dissipation. The parametric space of the solitons is three-dimensional, that makes theirs to be easily traceable within a whole range of the equation parameters. Scaling properties of the chirped dissipative solitons are highly interesting for applications in the field of high-energy ultrafast laser physics.
120402000
Highly chirped dissipative solitons as a one-parameter family of stable solutions of the cubic–quintic Ginzburg–Landau equation
In this paper, the stability of the analytical solutions of the cubic–quintic Ginzburg–Landau equation (CQGLE) in the high-chirp approximation has been studied numerically. The existence domain for the stable solution in the CQGLE parameter set has been found. A temporal and spectral shape of the stable solution as dependent of the cavity parameters has been analyzed. Direct comparison of the spectra with numerical calculations has been performed, demonstrating 10−2–10−4 accuracy of the analytical solution for chirp parameter f>10. The stable solutions represent the dissipative soliton family with only one composite parameter. Inside this family, the pulse shape in the time domain evolves from the conventional soliton shape, sech−2, to a rectangular one in the opposite limit with a parabolic shape as an intermediate one. The obtained theoretical results make it possible to classify experimentally observed highly chirped pulses and to optimize experimental schemes with an all-normal-dispersion cavity.
122882989
Noise of mode-locked lasers
A theory of noise in mode-locked lasers is developed that applies to additive pulse mode-locked and Kerr lens mode-locked systems. Equations of motion are derived for pulse energy, carrier linewidth, frequency pulling, and timing jitter. The effect of gain fluctuations, mirror vibrations, and index fluctuations are determined. Measurements that can determine all four fluctuation spectra are described. Experimental data in the literature are compared with theory. >
18045043
The quantum-limited comb lineshape of a mode-locked laser: fundamental limits on frequency uncertainty.
We calculate the quantum-limited shape of the comb lines from a mode-locked Ti:sapphire laser using experimentally-derived parameters for the linear response of the laser to perturbations. The free-running width of the comb lines is found across the laser spectrum. By modeling the effect of a simple feedback loop, we calculate the spectrum of the residual phase noise in terms of the quantum noise and the feedback parameters. Finally, we calculate the frequency uncertainty in an optical frequency measurement if the limiting factor is quantum noise in the detection of the optical heterodyne beat.
15212331
Comparison of numerical methods for modeling laser mode locking with saturable gain
The widely used split-step Fourier method has difficulties when solving partial differential equations with saturable gain. Here, we describe a modified split-step Fourier method, and we compare it to several different algorithms for solving the Haus mode-locking equation and related equations that are used to model mode-locked lasers and other optical oscillators and amplifiers with saturable gain. These equations all include the product of a scalar nonlinearity and a stiff nonlinear operator. We find that a modified split-step method is the easiest to program with the same level of reliability and accuracy as the other methods that we investigated.
120918062
Noise of mode-locked lasers (Part I): numerical model
A numerical model for the calculation of noise spectra of actively or passively mode-locked lasers has been developed. Fluctuations not only of the timing error, but also of all other quantities of interest can be quantified. The model is based on a pulse propagation algorithm with quantum noise sources. It allows the study of a much wider class of phenomena than those accessible with analytical techniques, and it is useful for testing the validity limits of analytical results.
198120256
Numerical Optimization
Numerical Optimization presents a comprehensive and up-to-date description of the most effective methods in continuous optimization. It responds to the growing interest in optimization in engineering, science, and business by focusing on the methods that are best suited to practical problems. For this new edition the book has been thoroughly updated throughout. There are new chapters on nonlinear interior methods and derivative-free methods for optimization, both of which are used widely in practice and the focus of much current research. Because of the emphasis on practical methods, as well as the extensive illustrations and exercises, the book is accessible to a wide audience. It can be used as a graduate text in engineering, operations research, mathematics, computer science, and business. It also serves as a handbook for researchers and practitioners in the field. The authors have strived to produce a text that is pleasant to read, informative, and rigorous - one that reveals both the beautiful nature of the discipline and its practical side.
120353362
Operating regimes, split-step modeling, and the Haus master mode-locking model
We develop an iterative (averaging) method to characterize the mode-locking dynamics in a laser cavity mode locked with a combination of wave plates and a passive polarizer. The model explicitly accounts for the effects of self- and cross-phase modulation, an arbitrary alignment of the fast- and slow-axes of the fiber with the wave plates and polarizer, fiber birefringence, saturable gain, and chromatic dispersion. The general averaging scheme results in the cubic-quintic Ginzburg-Landau equation at the leading order and the Swift-Hohenberg equation at the next order. An extensive comparison between the full model and the averaged equations shows a quantitative agreement that allows for characterizing the stability and operating regimes of the laser cavity.
11377647
Essential instability of pulses and bifurcations to modulated travelling waves
Reaction-diffusion systems on the real line are considered. Localized travelling waves become unstable when the essential spectrum of the linearization about them crosses the imaginary axis. In this article, it is shown that this transition to instability is accompanied by the bifurcation of a family of large patterns that are a superposition of the primary travelling wave with steady spatially periodic patterns of small amplitude. The bifurcating patterns can be parametrized by the wavelength of the steady patterns; they are time-periodic in a moving frame. A major difficulty in analysing this bifurcation is its genuinely infinite-dimensional nature. In particular, finite-dimensional Lyapunov–Schmidt reductions or centre-manifold theory do not seem to be applicable to pulses having their essential spectrum touching the imaginary axis.
3239155
A MATLAB differentiation matrix suite
A software suite consisting of 17 MATLAB functions for solving differential equations by the spectral collocation (i.e., pseudospectral) method is presented. It includes functions for computing derivatives of arbitrary order corresponding to Chebyshev, Hermite, Laguerre, Fourier, and sinc interpolants. Auxiliary functions are included for incorporating boundary conditions, performing interpolation using barycentric formulas, and computing roots of orthogonal polynomials. It is demonstrated how to use the package for solving eigenvalue, boundary value, and initial value problems arising in the fields of special functions, quantum mechanics, nonlinear waves, and hydrodynamic stability.
14734955
Pulse dynamics in mode-locked lasers: relaxation oscillations and frequency pulling.
A theoretical description of the pulse dynamics in a modelocked laser including gain dynamics is developed. Relaxation oscillations and frequency pulling are predicted that influence the pulse parameters. Experimental observations of the response of a mode-locked Ti:sapphire laser to an abrupt change in the pump power confirm that the predicted behavior occurs. These results provide a framework for understanding the effects of noise on the spectrum of the laser.
17506810
Carrier-envelope phase locking of multi-pulse lasers with an intra-cavity Mach-Zehnder interferometer.
We propose the use of an intra-cavity Mach Zehnder interferometer (MZI), for increasing the repetition rate at which carrier-envelope phase-locked pulses are generated in passively mode-locked fiber lasers. The attractive feature of the proposed scheme is that light escaping through the open output ports of the MZI can be used as a monitor signal feeding a servo loop that allows multiple pulses to co-exist in the cavity, while rigidly controlling their separation. The proposed scheme enables in principle a significant increase in the pulse-rate with no deterioration in the properties of the generated pulses.
17304446
A covariance matrix method to compute bit error rates in a highly nonlinear dispersion-managed soliton system
We completely describe the covariance matrix method for the first time, and we use it to compute the noise evolution in a 10-Gb/s single-channel dispersion-managed soliton system propagating over 24000 km. The linearization assumption upon which the covariance matrix method is based breaks down, unless we explicitly separate the phase and timing jitter of each pulse from the noise. We describe a procedure for carrying out this separation.