Fredholm Alternative for Periodic-Dirichlet Problems for Linear Hyperbolic Systems

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.Comment: 20 page

Traveling wave modeling of dynamics in semiconductor ring lasers

We use the traveling wave model for simulating and analyzing nonlinear dynamics of complex semiconductor ring laser devices. This modeling allows to consider temporal-spatial distributions of the counter-propagating slowly varying optical fields and the carriers, what can be important when studying non-homogeneous ring cavities, propagation of short pulses or fast switching. By performing numerical integration of the model equations we observe several dynamic regimes as well as transitions between them. The computation of ring cavity modes explains some peculiarities of these regimes

Numerical bifurcation analysis of traveling wave model of multisection semiconductor lasers

Traveling wave equations are used to model the dynamics of multisection semiconductor lasers. To perform a bifurcation analysis of this system of 1-D partial differential equations its low dimensional approximations are constructed and considered. Along this paper this analysis is used for the extensive study of the pulsations in a three section distributed feedback laser. Namely, stability of pulsations, different bifurcation scenaria, tunability of the pulsation frequency and its locking by the frequency of electrical modulation are considered. All these pulsation qualities are highly important when applying lasers in optical communication systems

Modeling and efficient simulations of broad-area edge-emitting semiconductor lasers and amplifiers

We present a (2+1)-dimensional partial differential equation model for spatial-lateral dynamics of edge-emitting broad-area semiconductor devices and several extensions of this model describing different physical effects. MPI-based parallelization of the resulting middlesize numerical problem is implemented and tested on the blade cluster and separate multi-core computers at the Weierstrass Institute in Berlin. It was found, that an application of 25-30 parallel processes on all considered platforms was guaranteeing a nearly optimal performance of the algorithm with the speedup around 20-25 and the efficiency of 0.7-0.8. It was also shown, that a simultaneous usage of several in-house available multi-core computers allows a further increase of the speedup without a significant loss of the efficiency. Finally, an importance of the considered problem and the efficient numerical simulations of this problem were illustrated by a few examples occurring in real world applications

A multi-mode delay differential equation model for lasers with optical feedback

In this paper, we discuss the relations between the spatially-distributed traveling wave, Lang-Kobayashi, and a new multi-mode delay differential equation models for Fabry-Perot type semiconductor diode lasers with an external optical feedback. All these models govern the dynamics of the slowly varying complex amplitudes of the optical fields and carrier density. To compare the models, we calculate the cavity modes determined by the threshold carrier density and optical frequency of the steady states in all three models. These calculations show that the Lang-Kobayashi type model is in good agreement with the traveling wave model only for the small feedback regimes, whereas newly derived multi-mode delay differential equation model remains correct even at moderate and large optical feedback regimes

Traveling wave modeling of nonlinear dynamics in multisection semiconductor lasers

A hierarchy of 1 (time) + 1 (space) dimensional first-order partial differential equation (traveling wave) models is used for a description of dynamics in individual semiconductor lasers, various multisection semiconductor lasers, and coupled laser systems. Consequent modifications of the basic traveling wave model allow for taking into account different physical effects such as the gain dispersion, the thermal detuning, the spatial hole burning of carriers, the nonlinear gain saturation, or various carrier exchange processes in quantum dot lasers. For illustration, the model was applied for simulations of dynamics in complex ring laser with four branches of filtered feedback. Finally, several advanced techniques for model analysis such as calculation of instantaneous optical modes, finding of steady states, and numerical continuation and bifurcation analysis of the model equations were discussed and illustrated by example simulations

Sampling techniques applicable for the characterization of the quality of self pulsations in semiconductor lasers

The aim of the presented report is to demonstrate how the sampling techniques can be used to characterize the quality of self pulsations in a multi-section semiconductor laser and the synchronization of self pulsations with an optical or electrical periodically modulated signal. The developed tools are described and some examples are given

Calculation of steady states in dynamical semiconductor laser models

We discuss numerical challenges in calculating stable and unstable steady states of widely used dynamical semiconductor laser models. Knowledge of these states is valuable when analyzing laser dynamics and different properties of the lasing states. The example simulations and analysis mainly rely on 1(time)+1(space)-dimensional traveling-wave models, where the steady state defining conditions are formulated as a system of nonlinear algebraic equations. The per- formed steady state calculations reveal limitations of the Lang-Kobayashi model, explain nontrivial bias threshold relations in lasers with several electrical contacts, or predict and explain transient dynamics when simulating such lasers

Calculation of the steady states in dynamic semiconductor laser models

We discuss numerical challenges in calculating stable and unstable steady states of widely used dynamic semiconductor laser models. Knowledge of these states is valuable when analyzing laser dynamics and different properties of the lasing states. The example simulations and analysis mainly rely on 1(time)+1(space)-dimensional traveling-wave models, where the steady state defining conditions are formulated as a system of nonlinear algebraic equations. The performed steady state calculations reveal limitations of the Lang-Kobayashi model, explain nontrivial bias threshold relations in lasers with several electrical contacts, or predict and explain transient dynamics when simulating such lasers

Longitudinal modes of multisection ring and edge-emitting semiconductor lasers

We use the traveling wave model for simulating and analyzing nonlinear dynamics of multisection ring and edge-emitting semiconductor laser devices. We introduce the concept of instantaneous longitudinal optical modes and present an algorithm for their computation. A semiconductor ring laser was considered to illustrate the advantages of the mode analysis