## Department of Mathematical Modelling

Technical University of Denmark

Building 321/305

DK-2800 Lyngby

Denmark

Telephone: +45 4588 1433

Telefax : +45 4588 2673

E-mail : imm@imm.dtu.dk

## Modelling colliding-pulse mode-locked semiconductor lasers

## Svend Bischoff

For a copy of this paper, either## Introduction

It is the goal of applied mathematics to study mathematical models of physical systems to elucidate their operation principles. The development of simple theoretical models requires a strong interaction between experimentalists and theoreticians. especially in the case of phenomenological models, where parameter values in a propsed model are fitted, so that the model output is in good agreement with experiments. In general many parameter values entering a derived or proposed model are determined by experiments while the theoretical models are typically used to improve the device design or to determine the optimum operation conditions.The purpose of this thesis is to elucidate some of the physics of interest in the field of semiconductor laser modelling, semiconductor optics and fiber optics. To be more specific we will investigate: The Colliding-Pulse Mode-Locked (CPM) Quantum Well (QW) laser diode; the excitonic semiconductor response for varying material thickness in the case of linear optics; and modulational instability of electromagnetic waves in media with spatially varying non-linearity.

## The CPM laser diode

Edge emitting semiconductor lasers are extensively used in today's optical communication networks due to their compactness, reliability and high speed modulation properties . Furthermore, semiconductor lasers can be designed to operate at a wavelength where the fiber loss is minimal. The direct modulation speed of semiconductor lasers can be improved by using an active layer made up of quantum wells (two-dimensional structure) instead of a bulk layer (three-dimensional structure). The improvement in modulation speed is attributed to the difference in the density of states function for two-dimensional (2-D) and three-dimensional (3-D) structures which results in a higher differential gain for the two-dimensional structures.Semiconductors can, by simply modulating the gain current, be modulated at frequencies of up-to 10-30 GHz. Another promising device is the electro-absorption modulator where a multiple QW semiconductor waveguide is reverse biased. The intensity modulation of the electro-absorption modulators is based on the Quantum Confined Stark Effect (QCSE). QW electro-absorption modulators have been monolithically integrated with Distributed Feed Back (DFB) lasers. These devices have been used in multi-gigabit transmission systems (10 GHz). Furthermore, one has good control of the lasing wavelength of DFB lasers through the device design which makes the monolithic integrated DFB QW electro-absorption modulator laser a suitable device for Wavelength Division Multiplexed (WDM) optical networks.

The generation of optical pulses in excess of 50 GHz is presently not possible with electrically modulated semiconductor lasers. However, pulse repetition rates beyond 50 GHz can be generated by monolithical Colliding-Pulse Mode-Locked (CPM) QW laser diodes. The CPM laser diodes have been demonstrated to produce pico- to sub-picosecond pulses at repetition frequencies from 16 to 350 GHz and have been proposed as a promising pulse source for Time Division Multiplexed (TDM) optical networks.

The monolithic CPM semiconductor waveguide is a three contact laser diode, where the two outer sections are forward biased and provide gain, while a short center section acts as a saturable absorber (reverse bias). A schematic plot of a CPM laser is shown in fig. 2.1. Passive mode-locking occurs by applying an appropriate DC-current value to the gain sections and an appropriate DC-reverse bias to the saturable absorber section. Steady state mode-locked operation is characterized by two counter propagating pulses which collide in the absorber and thus help each other to bleach the saturable absorber. The pulse repetition frequency of the CPM laser is determined by the cavity length.

In chapter 2 we have derived a large signal model for the CPM QW laser. The derived large signal model is quite general and may also be used to simulate electro-absorption modulators or other multi-contact lasers. However, in chapter 2 the large signal model has been used to elucidate the operating principles of the "standard" (three contact) CPM laser diode. The pulse-shaping mechanisms are attributed to an interplay of the material dynamics in the gain and absorber regions. Furthermore, the temporal and spectral properties of the CPM pulses are investigated for varying current and bias conditions, which show a good agreement between theory and measurements. We also study hybridly mode-locked CPM lasers where the reverse bias to the absorber section is modulated.

In App. A the derivation of the rate equation model describing the material dynamics is shown in detail. Furthermore, we discuss the CPM pulse width for different cavity and saturable absorber length in App. B.

## Excitonic semiconductor response for varying material thickness

Vertical Cavity Surface Emitting Laser (VCSEL) diodes have in recent years attracted a lot of attention. The modelling of VCSELs is normally performed under the assumptions of an ideal two-dimensional or three-dimensional active layer. However, it has been shown that propagation effects through the active layer result in significant changes in the transmission spectrum. (Here, the transmission spectrum denotes the in experimental measurements so called absorption spectrum, since propagation effects through the sample are also taken into account).In chapter 3 we present calculated transmission spectra for the case of linear optics showing that the approximation of only one quantized state in a QW only holds for QW widths of less than 8 nm. For a material thickness of 80 nm we are close to the three dimensional limit.

## Modulational instability of electromagnetic waves in media with periodic or random perturbations

The growth of an initially small perturbation of the steady state solution resulting from the joint action of nonlinear and dispersive effects is often referred to as Modulational Instability (MI). MI is a phenomena, which exists for a wide class of nonlinear systems such as in the field of plasma physics, deep water waves, optical waveguides etc.In this thesis the process of MI is investigated for an optical fiber, where the fiber core is periodically or randomly varied. The propagation of electromagnetic waves in optical fibers is theoretically described by the Non-Linear Schroedinger Equation (NLSE) where perturbation terms for the processes of loss, gain, self-steepening etc. can be added The process of MI is investigated and shown to exist for both normal and anomalous dispersion. Furthermore, the influence of the self-steepening effect on MI is discussed. It is shown that the steady state solution of the NLSE is unstable for any perturbation in the case of random fluctuations of the nonlinearity in the NLSE.

## IMM Ph.d thesis 28, 1997

Last modified May 16, 1997Go back

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