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Annual Meeting Optical Society of America

Coherent optical interactions in semiconductor lasers

Lloyd W. Hillman

Published Jan 1, 1987• 0 citations
Publisher

Abstract

Fluctuations in the amplitude, frequency, and polarization of the light emitted by semiconductor laser diodes limit the performance and reliability of many optical systems, such as communication links, optical radar, and optical disk read/write memories. Although it seems logical to explain this noise as random or stochastic, such erratic behavior can arise from completely deterministic models of lasers that retain the dynamics of the coherently induced polarization. This has been shown to be true for the simple model of a laser based on a two-level resonance, which results in the coupled Maxwell-Bloch equations. These equations are termed coherent because the Bloch equations describe, not only the inversion, but also the induced polarization of the gain media. The Maxwell-Bloch equations predict an array of instabilities and novel operation conditions for a laser: the output light may oscillate regularly or pulsate chaotically. Such behavior is not found in the incoherent population rate equation that adiabatically eliminated the dynamics of the polarization. However, the Maxwell-Bloch equations apply only to dilute materials in which the total gain is described by an ensemble average over many noninteracting molecules, such as gas lasers.

DOI
10.1364/oam.1987.wo10

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Fluctuations in the amplitude, frequency, and polarization of the light emitted by semiconductor laser diodes limit the performance and reliability of many optical systems, such as communication links, optical radar, and optical disk read/write memories. Although it seems logical to explain this noise as random or stochastic, such erratic behavior can arise from completely deterministic models of lasers that retain the dynamics of the coherently induced polarization. This has been shown to be true for the simple model of a laser based on a two-level resonance, which results in the coupled Maxwell-Bloch equations. These equations are termed coherent because the Bloch equations describe, not only the inversion, but also the induced polarization of the gain media. The Maxwell-Bloch equations predict an array of instabilities and novel operation conditions for a laser: the output light may oscillate regularly or pulsate chaotically. Such behavior is not found in the incoherent population rate equation that adiabatically eliminated the dynamics of the polarization. However, the Maxwell-Bloch equations apply only to dilute materials in which the total gain is described by an ensemble average over many noninteracting molecules, such as gas lasers.
A generalized Bloch-Maxwell formulation for laser-field-coupled semiconductors is derived from a two-band model which includes direct Coulomb interactions. The momentum-dependent, microscopic, electron-hole equations of motion in the time-dependent Hartree-Fock approximation and neglecting interband exchange interactions form the starting point for the formulation. A self-consistent set of coupled equations in four dynamical variables for the medium, together with the electric-field amplitude coupled through the Maxwell wave equation in a semiclassical approximation, are obtained to lowest order in the coherent Coulomb exchange interaction and the density-of-states distribution. Intrinsic optical bistability is predicted in a steady state due to a carrier density-dependent redshift of the band edge due explicitly to coherent Coulomb exchange. Integration of the dynamical equations for conditions which correspond to the ultrafast time regime exhibit intrinsic adiabatic inversion, adiabatic following, anomalous Rabi cycling, and unique, as well as fast, optical switching, all of which depend upon the coherent Coulomb exchange interaction.
This chapter begins with a heuristic derivation of atomic population rate equations which include changes to level populations due to stimulated transitions as well as level relaxations and pump. These equations are used to study the saturation of population inversion with increasing light intensity and its dependence on the type of gain medium (three-level gain media vs. four-level gain media), lineshape and nature of line broadening (homogeneous vs inhomogeneous broadening). How saturation affects lineshape and thus gain and absorption giving rise to phenomena like power broadening and spectral hole burning is also discussed. The chapter summarizes the equations of motion for atomic polarization, inversion and the cavity field describing their nonlinear dynamical coupling. These equations, known as the Maxwell-Bloch equations, provide a complete description of atom-field interaction in a laser. Approximations of these equations, including the rate equation approximation, to deal with different laser classes are briefly described. These will be used in the next few chapters to explore and understand the rich and varied behavior of lasers. The chapter ends with a summary of select gain media for different types of commonly used lasers.
A numerical study is presented of random lasers as a function of the pumping rate above the threshold for lasing. Depending on the leakiness of the system resonances, which is typically larger in random lasers compared to conventional lasers, we observe that the stationary lasing regime becomes unstable above a second threshold. Coherent instabilities are observed as self pulsation at a single frequency of the output intensity, population inversion, as well as the atomic polarization. We find these Rabi oscillations have the same frequency everywhere in the random laser despite the fact that the field intensity strongly depends on the spatial location.
This book is an edited version of lectures given by the authors at the 1985 Summer School on Lase Physics. The School was held in Jilin University in Changchun, Peoples' Republic of China. Contents: The book is divided into 2 parts, the first is ''An Outline of Laser Physics and Laser Instabilities'': An Introduction to the Theory of the Laser: The Rate Equations; The Interaction of Light and Matter: The Classical Picture; The Basic Laser Equations: Homogeneously Broadened Systems; Steady State Behavior of a Homogeneously Broadened Ring Laser; Linear Stability Analysis: The Resonant Case; The Adiabatic Elimination of the Fast Variables; Linear Stability Analysis: The General Case; The Single Mode Laser: Deterministic Chaos; Dynamical Behavior of the Maxwell-Bloch Equations; The Spectral Analysis of the Output Field; Theory of a Laser with Inhomogeneous Broadening. The second part is on the ''principles of Laser Physics and Laser Instabilities'': Background Notes for the Discussion of Experimental Laser Systems; Amplified Spontaneous Emission; The Two-Level Atom; Laser Cavities and Modes; Types of Laser Systems and Classification of Lasers; the Xe 3.51 ..mu..m Laser; Chaos in Single Mode Lasers; Forced Laser Systems, Modulated Lasers and Lasers with an Injected Signal; Dynamical Phenomena in Multimode Laser Systems.
We summarize the basic problems in laser coherence and the concepts and techniques needed to solve them. Special techniques needed to handle partition noise in semiconductor lasers will be touched on. We address the problem of how a self-sustained oscillator works. Why is the laser line so narrow? Why does a small amount of noise produce a broadening of the line as opposed to additive side bands? What is the role of zero point oscillations and spontaneous emission in determining the line width? In what sense can the transition to a lasing state be described as a phase transition? The concepts used to deal with the response of a laser system involve a quasi-Markoffian description. This in turn validates a quantum regression theorem which relates the spectrum of the noise to the transport response even in a system far from equilibrium. A significant simplification is made by introducing non-commuting (quantum) noise sources. The techniques required in the solution of laser noise problems involve a multi-time correspondence between quantum variables and c-number random variables that reduces a quantum stochastic problem to a classical stochastic problem. Adiabatic elimination is used to eliminate the fast variables to obtain a reduced problem for the fields alone or the fields and some total atomic variables (total inversion and polarization). Reduction to a rotating wave version of the van der Pol oscillator and numerical solution of the Fokker-Planck equation yields (a) the line shape (b) the spectrum of intensity fluctuations and (c) the distribution of photocounts observed over some time interval. Comparison of theoretical results with later experiments will be presented.