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We study both theoretically and experimentally typical operation
regimes of 40 GHz monolithic mode-locked lasers. The underlying Traveling Wave Equation model reveals quantitative agreement for characteristics of the fundamental mode-locking as pulse width and repetition frequency tuning, as well as qualitative agreement with the experiments for other dynamic regimes. Especially the appearance of stable harmonic mode-locking at 80 GHz
has been predicted theoretically and confirmed by measurements.
Furthermore, we derive and apply a simplified Delay-Differential-Equation model
which guides us to a qualitative analysis of bifurcations responsible for the appearance
and the breakup of different mode-locking regimes. Higher harmonics of mode-locking are predicted by this model as well.
Physics of short optical pulses is an important and active research area in nonlinear optics. In what follows we theoretically consider the most extreme representatives of short pulses that contain only several oscillations of electromagnetic field. Description of such pulses is traditionally based on envelope equations and slowly varying envelope approximation, despite the fact that the envelope is not “slow” and, moreover, there is no clear definition of such a “fast” envelope. This happens due to another paradoxical feature: the standard (envelope) generalized nonlinear Schrödinger equation yields very good correspondence to numerical solutions of full Maxwell equations even for few-cycle pulses, a thing that should not be. In what follows we address ultrashort optical pulses using Hamiltonian framework for nonlinear waves. As it appears, the standard optical envelope equation is just a reformulation of general Hamiltonian equations. In a sense, no approximations are required, this is why the generalized nonlinear Schrödinger equation is so effective. Moreover, the Hamiltonian framework greatly contributes to our understanding of “fast” envelope, ultrashort solitons, stability and radiation of optical pulses. Even the inclusion of dissipative terms is possible making the Hamiltonian approach an universal theoretical tool also in extreme nonlinear optics.
We derive an optoelectronic model based on a gradient formulation for the relaxation of electron-, hole- and photon-densities to their equilibrium state. This leads to a coupled system of partial and ordinary differential equations, for which we discuss the isothermal and the non-isothermal scenario separately.
We consider systems of reaction-diffusion equations as gradient systems with respect to an entropy functional and a dissipation metric given in terms of a so-called Onsager operator, which is a sum of a diffusion part of Wasserstein type and a reaction part. We provide methods for establishing geodesic $\lambda$-convexity of the entropy functional by purely differential methods, thus circumventing arguments from mass transportation. Finally, several examples, including a drift-diffusion system, provide a survey on the applicability of the theory.
We describe a new software package for the numerical solution of general linear or nonlinear switched differential-algebraic equations (DAEs).
The package embeds the DAE solvers GELDA and GENDA into a hybrid mode controller that
determines the switch points, organizes the mode switching, and provides consistent initial values to restart the integration method at the switch point.
It can deal with systems of arbitrary index and with linear systems that do not have unique solutions or inconsistencies in the initial values or the inhomogeneity.
Nonuniqueness and inconsistencies are treated in a least square sense.
The package includes the possibility of sliding mode simulation that allows an efficient treatment of chattering behavior during the simulation of a hybrid system.
We give explicit descriptions how to use the package and include a numerical example.
The Modified Nodal Analysis leads to differential algebraic equations
with properly stated leading terms. In this article a special structure of the DAEs
modelling electrical circuits is exploited in order to derive a new decoupling for
nonlinear index-2 DAEs. This decoupling procedure leads to a solvability result and
is also used to study general linear methods, a class of numerical schemes that covers
both Runge-Kutta and linear multistep methods. Convergence for index-2 DAEs is
proved.
We consider the behavior of a modulated wave solution to
an $\mathbb{S}^1$-equivariant autonomous system of differential equations under an external
forcing of modulated wave type. The modulation frequency of the forcing is assumed to be close to the
modulation frequency of the modulated wave solution, while the wave frequency of the forcing is supposed to be far
from that of the modulated wave solution. We describe the domain in the three-dimensional
control parameter space (of frequencies and amplitude of the forcing)
where stable locking of the modulation frequencies of the forcing and the modulated wave solution
occurs.
Our system is a simplest case scenario for the behavior of self-pulsating lasers under the influence of external
periodically modulated
optical signals.
This paper concerns $n\times n$ linear one-dimensional hyperbolic systems of the type
$$
\om\partial_tu_j + a_j(x)\partial_xu_j + \sum\limits_{k=1}^nb_{jk}(x)u_k = f_j(x,t),\; j=1,\ldots,n,
$$
with periodicity conditions in time and reflection boundary conditions in space.
We state sufficient conditions on the data $\om, a_j$ and $b_{jk}$
such that the system has a Fredholm like solvability behavior.
Moreover, we state sufficient conditions on the data such that for any right hand side there exists exactly
one solution, that the solution survives under small perturbations of the data, and that the corresponding
data-to-solution-map
is smooth with respect to appropriate function space norms. In particular, those sufficient conditions
imply that no small denominator effects occur. Moreover, we show that perturbations of the coefficients $\om$ and $a_j$ lead to
essentially different results than perturbations of the coefficients $b_{jk}$, in general.
This paper concerns hyperbolic systems of two linear first-order PDEs in one space dimension with periodicity conditions in time and reflection boundary conditions in space.
The coefficients of the PDEs are supposed to be time independent, but allowed to be discontinuous with respect to the space variable. We construct two scales of Banach spaces (for the solutions and for the right hand sides of the equations, respectively) such that the problem can be modeled by means of Fredholm operators of index zero between corresponding spaces of the two scales.
The main tools of the proofs are separation of variables, integral representation of the solutions of the corresponding boundary value problems of the ODE systems and an abstract criterion for Fredholmness which seems to be new.
We discuss the nonstandard problem of using the finite difference
method to solve numerically a partial differential equation posed on
an unbounded domain. We propose different strategies to construct
so-called discrete articial boundary conditions (ABCs) and present
an efficient implementation by the sum-of-exponential ansatz. The
derivation of the ABCs is based on the knowledge of the exact solution,
the construction of asymptotic solutions or the usage of a continued
fraction expansion to a second-order difference equation. Our approach
is explained by means of three different types of partial differential
equations arising in option pricing, in quantum mechanics and
in (underwater) acoustics. Finally, we conclude with an illustrating
numerical example from underwater acoustics showing the superiority
of our new approach.
In this paper we study the destabilization mechanism in a ring of unidirectionally coupled oscillators. We derive an amplitude equation of Ginzburg-Landau type that describes the destabilization of the stationary state for systems with a large number of oscillators. Based on this amplitude equation, we are able to provide an explanation for the fast transition to chaos (or hyperchaos)
that can be observed in such systems. We show that the parameter interval, where the transition from a stable periodic state to chaos occurs, scales like the inverse
square of the number of oscillators in the ring. In particular, for a sufficiently large
number of oscillators a practically immediate transition to chaos can be observed.
The results are illustrated by a numerical study of a system of unidirectionally
coupled Duffing oscillators.
This paper deals with the numerical solution of the time{dependent Schroedinger-
Poisson system in the spherically symmetric case. Since the problem is posed on an
unbounded domain one has to introduce artificial boundary conditions to confine
the computational domain. The main topic of this work is the construction of a
so-called discrete transparent boundary condition (TBC) for a Crank-Nicolsontype
predictor-corrector scheme for solving the Schroedinger-Poisson system. This
scheme has the property of mass and energy conservation exactly on the discrete
level. We propose different strategies for the discrete TBC and present an efficient
implementation. Finally, a numerical example illustrate the findings and shows the
comparison results between the different approaches.
We investigate a semiconductor laser with delayed optical feedback due
to an external cavity formed by a regular mirror. We discuss
similarities and differences of the well-known Lang-Kobayashi delay
differential equation model and the traveling wave partial
differential equation model. For comparison we locate the continuous
wave states in both models and analyze their stability.
We derive formulas for the minimal positive solution of a
particular non-symmetric Riccati
equation arising in transport theory. The formulas are based
on the eigenvalues of an
associated matrix. We use the formulas to explore some new
properties of the minimal positive solution and to derive
fast and highly accurate numerical methods. Some numerical tests
demonstrate the properties of the new methods.
In this paper, we empirically investigate the NP-hard problem of finding sparsest solutions to linear equation systems, i.e., solutions with as few nonzeros as possible. This problem has received considerable interest in the sparse approximation and signal processing literature, recently. We use a branch-and-cut approach via the maximum feasible subsystem problem to compute optimal solutions for small instances and investigate the uniqueness of the optimal solutions. We furthermore discuss five (modifications of) heuristics for this problem that appear in different parts of the literature. For small instances, the exact optimal solutions allow us to evaluate the quality of the heuristics, while for larger instances we compare their relative performance. One outcome is that the so-called basis pursuit heuristic performs worse, compared to the other methods. Among the best heuristics are a method due to Mangasarian and a bilinear approach.
Enforcing solvability of a nonlinear matrix equation and estimation of multivariate ARMA time series
(2013)
The matrix equation $X+AX^{-1}A^T=B$, arising in parameter estimation of certain time series models,
is solvable only for certain values of the matrices $A,B$.
We present a numerical method to modify $A,B$ in order to make the matrix equation solvable.
Since solvability depends on the location of the eigenvalues of the palindromic matrix polynomial $\lambda^2 A+\lambda B+A^T$,
our method works by moving those eigenvalues to specified locations using first order spectral perturbation theory.
The method is heuristic but works in practice, as is supported by several compelling numerical examples.
These examples arise from parameter estimation of a common time series model, the multivariate ARMA(1,1).
Electronic structure and optoelectronic properties of strained InAsSb/GaSb multi quantum-wells
(2008)
A study of the optical properties of a set of InAsxSb1-x/Al0.15In0.85As0.77Sb0.23/GaSb multiple quantum-wells (for x between 0.82 and 0.92) with build-in strains in the -0.62% to +0.05%-range is presented. The energy of the lowest quantum-confined optical transition is calculated by kp perturbation theory and experimentally determined by absorption measurements. Stokes shift of photoluminescence, photocurrent and of the emission from light emitting devices against the absorption edge of the quantum-well are quantified. The impact of the decreasing carrier confinement in the InAsxSb1-x quantum well system with increasing mole fraction is analyzed theoretically, and experimentally demonstrated by photoluminescence measurement. Our results allow for the improvement of optoelectronic devices, in particular for tailoring emission spectra of light emitting diodes.
In semiconductor devices one basically distinguishes three spatial scales: The atomistic scale of the bulk semiconductor materials (sub-Angstroem), the scale of the interaction zone at the interface between two semiconductor materials together with the scale of the resulting size quantization (nanometer) and the scale of the device itself (micrometer). The paper focuses on the two scale transitions inherent in the hierarchy of scales in the device. We start with the description of the band structure of the bulk material by kp Hamiltonians on the atomistic scale. We describe how the envelope function approximation allows to construct kp Schroedinger operators describing the electronic states at the nanoscale which are closely related to the kp Hamiltonians. Special emphasis is placed on the possible existence of spurious modes in the kp Schroedinger model on the nanoscale which are inherited from anomalous band bending on the atomistic scale. We review results of the mathematical analysis of these multi-band kp Schroedinger operators. Besides of the confirmation of the main facts about the band structure usually taken for granted, key results are conditions on the coefficients of the kp Schroedinger operator for the nanostructure, which exclude spurious modes and an estimate of the size of the band gap. Using these results, we give an overview of properties of the electronic band structure of strained quantum wells. Further, the assumption of flat-band conditions across the nanostructure allows for upscaling of quantum calculations to state equations for semi-classical models. We demonstrate this approach for parameters such as the quantum corrected band-edges, the effective density of states, the optical response, and the optical peak gain. Further, we apply the kp Schroedinger theory to low gap quantum wells, a case where a proper rescaling of the optical matrix element is necessary to avoid spurious modes. Finally, we discuss the application of the kp Schroedinger models to biased quantum wells, the operation mode of electro-optic modulators.
We study the perturbation theory of structured matrices under
structured rank one perturbations, and then focus on several classes of complex
matrices. Generic Jordan structures of perturbed matrices are identified.
It is shown that the perturbation
behavior of the Jordan structures
is substantially different from the corresponding theory for unstructured generic
rank one perturbations.