Distributed Feedback Semiconductor Lasers
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Concentrating on presenting a thorough analysis of DFB lasers from a level suitable for research students, this book emphasises and gives extensive coverage of computer aided modelling techniques.
Inspec keywords: optical losses; spontaneous emission; distributed feedback lasers; optical design techniques; laser feedback; semiconductor lasers
Other keywords: spontaneous emissions; distributed feedback semiconductordiode lasers; optical energy exchange; optical design; optical loss; DFB laser
Subjects: Design of specific laser systems; Lasing action in semiconductors; Semiconductor lasers
 Book DOI: 10.1049/PBCS010E
 Chapter DOI: 10.1049/PBCS010E
 ISBN: 9780852969175
 eISBN: 9781849191661
 Page count: 440
 Format: PDF

Front Matter
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1 The semiconductordiode laser
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This chapter has set the scene for the operation and fabrication of semiconductor lasers, emphasising why the authors see lasers with grating reflectors as so important. The earliest lasers were examined, showing that a key requirement is to confine the photons and electrons to the same physical region. A geometrical factor Γ, known as the confinement factor, expresses the degree to which this has achieved. The simplest requirements for lasing were put forward so that the reader understands the importance, just as in electronic feedback oscillators, of the roundtrip gain and phase in achieving a stable oscillation. The importance of singlemode lasers for modern optical communication along silica fibres over many tens of kilometres was noted. A variety of different types of laser were outlined, and in particular it was observed that to gain a stable single mode the favoured technology is to incorporate some form of Bragg grating. The fundamentals of gratings were introduced and the elements of different laser designs with gratings buried in their structures were presented.
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2 Gain, loss and spontaneous emission
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This chapter discusses the spontaneous emission and optical gain in semiconductor lasers. Spontaneous emission which initiates lasing, optical gain which is essential to achieve lasing, and other processes involved in lasing all use quantum processes at the level of single atoms and electrons within the lasing material. Without simplifications, the physics and mathematics necessary to describe such atomic systems fully is too complicated, certainly for the level of this book. The simplification and approximations must, however, be done in a way which is adapted to the requirements of semiconductor lasers, and the limitations must be understood. The results and implications of quantum physics are discussed here but there are only illustrative outlines of any derivations.
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3 Principles of modelling guided waves
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Principles of modelling guided waves is presented. The mechanisms and mathematics required for optical guiding in semiconductorslab guides is discussed. Many modes are permitted and the effective refractive index changes with increasing mode number from a value close to the largest value of refractive index down to a value close to the smallest value of refractive index of the different layers.
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4 Optical energy exchange in guides
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This chapter presents two basic classical methods of modelling mathematically the operation of semiconductor lasers and shows that they are, indeed, just different aspects of the same physics of energy conservation and are wholly compatible with one another. The first method applies the concepts of photon/electron particle exchange where one discusses the rate of absorption and emission of photons along with the rate of recombination of holes and electrons, ensuring at each stage that there is a detailed balance between photon generation and electron/hole recombination leading to particle conservation and energy conservation. This is the standard rate equation approach which is robust and well researched but can be difficult to apply when there are strong nonuniformities, and even more difficult when the phase of the electromagnetic field is important. For distributedfeedback lasers, both the phase of the field and nonuniformities are important and so one has to abandon the photonrate equation in favour of an approach based on interactions between electromagnetic fields and the electric dipoles in an active optical medium. The electromagnetic field analysis is essential when the refractive index/permittivity changes periodically inside the laser. The chapter then concludes with an analysis of the coupledmode equations which determine how a fraction of the forwardtravelling field is coupled into the reversetravelling field with a medium which has a periodic permittivity, i.e. the waveguide contains a Bragg grating.
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5 Basic principles of lasers with distributed feedback
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This chapter starts with a more physical derivation of the coupledmode equations and moves on to new features such as the eigenmodes for the analytic solutions, the influence of grating parameters on the dispersion diagram and the 'stopband' of nonpropagating frequencies.
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6 More advanced distributed feedback laser design
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These more advanced problems of the static design of lasers are outlined here and the chapter ends with a discussion on some results of modelling the dynamic performance of DFB lasers, considering problems associated with carrier transport into quantum wells. The dynamic performance of DFB lasers highlights yet further the problems that have already been met with a uniform grating in a uniform DFB laser.
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7 Numerical modelling for DFB lasers
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This chapter presents the basics of largesignal timedomain modelling using the travellingwave time/distance nonlinear partial differential equations of the laser. The field patterns and electron densities in the laser are computed permitting electronphoton interactions to be visualised. The lasers are excited by random 'spontaneous noise' which leads to outputs which are never precisely the same from run to run. However, the random output is not normally a major drawback but the timedomain modelling of low frequency noise can require excessively long times of computation.
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8 Future devices, modelling and systems analysis
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This chapter discusses mathematical modeling as a key component in the design of devices and systems in optoelectronics projects. One thrust of the book has been to lay out the physical and mathematical modelling techniques for the electromagnetic and electronic interactions within distributed feedback lasers to give better explanations and to facilitate new designs. Optical systems where arrays of devices may be interconnected are one such important area and are discussed. Novel concepts such as that of the pushpull laser, tunable lasers with Bragg gratings, and surfaceemitting lasers are all candidate areas for applications of the modelling principles given in this text. The future for mathematical modelling, coupled with a sound physical under standing, is bright, extensive and assured.
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Appendix 1: Maxwell, plane waves and reflections
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This appendix provides a summary of planewave interactions at dielectric interfaces and a summary of special cases which are of importance in laserdiode design, in particular, indicating one reason why TE modes are slightly more strongly reflected from a cleaved facet than TM modes.
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Appendix 2: Algorithms for the multilayer slab guide
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This appendix gives a systematic approach to providing a program for solving propagation, confinement factor and farfield emission when electromagnetic waves are guided by slab waveguides. Almost arbitrary numbers of layers can be computed with complex refractive indices. Here, by examining a fivelayer system, the way forward to a semiautomated method for an arbitrary number of layers with complex refractive indices is demon strated. Multilayer guides are particularly relevant in discussing DFB lasers where the longitudinally periodic profile of the permittivity has lateral variations across the waveguide which can be taken into account by having a series of layers with different patterns of permittivity, giving an average permittivity and an average periodic component which can be calculated using the programs.
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Appendix 3: Group refractive index of laser waveguides
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The group velocity is the velocity of a wave packet (i.e. the velocity of energy) that is centred on a central carrier frequency f = ω/2π. The group velocity is different from the phase velocity for two reasons: (i) the waveguide changes the propagation coefficient as a function of frequency; and (ii) the material permittivity changes with frequency so that the propagation coefficient in the material changes with frequency. This appendix illustrates these two roles using a symmetric threelayer waveguide so that one can appreciate the physics through putting numbers into an analytic solution.
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Appendix 4: Smallsignal analysis of singlemode laser
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This appendix provides a more detailed account of the classic rate equation analyses of appropriately uniform lasers and shows the firstorder effects of carrier transport into the active quantumwell region, and the effects of spontaneous emission and gain saturation on damping of the photonelectron resonance. The appendix ends with largesignal rate equations showing the influence of four key parame ters which shape the main features of the largesignal response.
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Appendix 5: Electromagnetic energy exchange
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Electromagnetic energy exchange phenomena is presented. The rate equations derived from the particle balance are consistent with Maxwell's equations and the group velocity appears in the travellingfield equations.The parameter χ is called the susceptibility with l+χ=ε_{r} giving the relative permittivity. In electrooptic material, it is more correcdy considered as a tensor so that the polarisation is not necessarily in exactly the same direction as the applied field, but in this work it is adequate to treat χ as a scalar. The random polarisation P_{spont} gives rise to spontaneous polarisation currents.
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Appendix 6: Pauli equations
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This appendix gives the detailed calculations for finding the steady state fields in a uniform DFB laser with uniform gain, and thereby finding the threshold conditions. The results are essential if one wishes to make comparisons with the numerical algorithms to estimate the accuracy of these algorithms. The appendix also provides a tutorial on the use of Pauli matrices for coupled differential equations.
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Appendix 7: KramersKrönig relationships
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The KramersKrδnig relationships provide fundamental rules for the relationships between the real and imaginary parts of the spectrum of any real physically realisable quantity when expressed as a complex function of frequency. One cannot, for example, design the optical gain spectrum to be any desired function of frequency without discovering that the phase spectrum is then closely prescribed. Frequently, such connections appear to be abstract and mathemat ically based. This appendix looks at three different ways of discovering these relationships which should help the reader to understand the fundamental nature and physics of the KramersKronig relationships. The appendix includes at the end a collection of approximations for the real refractive indices of relevant laser materials.
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Appendix 8: Relativeintensity noise (RIN)
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The relativeintensity noise (RIN) is an important quantity in determining whether lasers are acceptable for use in opticalcommu nication systems. Its analytic study can require extensive algebra, but in this appendix the emphasis is on the physical significance of RIN and simulations using timedomain modelling to estimate its value for DFB lasers.
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Appendix 9: Thermal, quantum and numerical noise
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This appendix first considers thermal and quantum noise and the ideal signaltonoise power ratio that can be measured using a 100%efficient photodetector. An ideal optical amplifier followed by an ideal detector is then considered. At the output of this ideal amplifier, the signaltonoisepower ratio depends on the spontaneous emission and it is argued that this ratio has to be the same as the signal tonoisepower ratio for the ideal direct detection of optical fields at the input.
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Appendix 10: Laser packaging
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Laser optoelectronic packaging of DFBlaser chips is reported. Thermal properties, laser monitoring by photodiode, packagerelated backreflection and fiber coupling is presented.
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Appendix 11: Tables of device parameters and simulated performance for DFB laser structures
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This appendix contains tables to provide summary of material and device parameters used for largesignal dynamic modelling of uniformgrating DFB laser.
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Appendix 12: About MATLAB programs
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Programs written in MATLAB 4 that has been designed. Before obtaining the programs from the World Wide Web, users are recommended to create, within their main MATLAB directory, a subdirectory (or folder), say laser, with further subdirectories dfb, diff, fabpero, fftexamp, filter, grating, slab, spontan. Within MATLAB, the effective initiation file is the Mfile called matlabrc.m and this should be backed up for future reference into a second file, say matlabrb.m.
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Back Matter
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