Theory of Semiconductor Lasers: From Basis of Quantum Electronics to Analyses of the Mode Competition Phenomena and Noise

298 Pages·2014·7.1 MB·English

by  Minoru Yamada (auth.)

Springer Series in Optical Sciences 185
Minoru Yamada
Theory of
Semiconductor
Lasers
From Basis of Quantum Electronics
to Analyses of the Mode Competition
Phenomena and Noise
Springer Series in Optical Sciences
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Minoru Yamada
Theory of Semiconductor
Lasers
From Basis of Quantum Electronics
to Analyses of the Mode Competition
Phenomena and Noise
123
Minoru Yamada
Faculty of Electrical and Computer
Engineering
Kanazawa University
Kanazawa
Japan
ISSN 0342-4111 ISSN 1556-1534 (electronic)
ISBN 978-4-431-54888-1 ISBN 978-4-431-54889-8 (eBook)
DOI 10.1007/978-4-431-54889-8
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Preface
‘‘Laser’’ is a coined word, formed by taking the initials of words in the phrase
‘‘Light Amplification by Stimulated Emission of Radiation.’’ The laser was
invented in 1960 and was a ruby laser in which optical light was emitted and
amplified in a ruby crystal. The semiconductor laser was invented in 1962.
Semiconductorlasersarebeingusedasprincipaldevicesinthetechnicalfields
ofoptoelectronics,suchastheopticalfibercommunicationsystem,theopticaldisc
system, and the optical precise measuring system. Under certain operating situa-
tions, however, semiconductor lasers often show undesired performance charac-
teristics such as instability of lasing mode and generation of extra noise.
Understanding of these complex characteristics is not easy to achieve, because
semiconductor lasers have specific physics that do not play a part in other elec-
trical and optical devices.
Thisbookiswrittenforstudents,engineers,andresearcherswhowishtostudy
a unified theoretical account of the operating principles of semiconductor lasers
based on theoretical physics rather than on phenomenological treatments. The
above-mentioned complex operating characteristics are logically analyzed in this
book, which consists offour major sections.
In Part I, ‘‘Fundamental Theory,’’ descriptions from quantum electronics
focusing specifically on the semiconductor laser are provided from the broader
perspective ofquantum mechanics.The basic subject matterrequired toconstruct
a standard semiconductor laser theory is treated here. The author explains the
theoretical developments of semiconductor laser theory in a concise but self-
contained manner, relying on the Appendices to provide help in understanding
related principles and theorems.
The required background material for readers includes classical electromag-
netictheory,quantummechanics,andsemiconductorphysics.Familiaritywiththe
contentofthesefieldsisassumed.Readerswishingtoskipthederivationoflasing
gainandgodirectlytoastudyofthemechanismoflaserinstabilityandnoisecan
jump to Chap. 9 in Part II and omit the fundamental theory in Part I.
Although laser quality is evaluated with a view toward achieving a lower
threshold current level, higher efficiency, and higher optical power, properties of
the lasing modes, and noise problems are also subjects required in the evaluation
oflaserquality.Criteriaforthesepropertiesrequireanunderstandingofnonlinear
phenomena and other complex physics in semiconductor lasers.
v
vi Preface
One example, namely, whether we can achieve single-mode operation, is
analyzed from the perspective of nonlinear gain and discussion of mode compe-
tition phenomena. Another example, which is a serious issue for semiconductor
lasers, is so-called feedback noise, induced by reflections of output light from
connecting optical devices such as optical lenses, optical fibers, or optical discs.
Themechanismforfeedbacknoisegenerationandthetechnologytosuppressitare
explained in terms of nonlinear properties and quantum mechanical fluctuations.
Theoreticaltreatmentsofnonlinearproperties,modecompetitionphenomena,and
noise are given in Part II, ‘‘Advanced Theory for Mode Competition and Noise.’’
The standard structures used in semiconductor lasers are the double hetero-
junction, the stripe structure, and the Fabry–Perot cavity made of cleaved facets.
Besides these standard structures, many other structures have been developed to
achieve more superior operating characteristics and wider applications.
In Part III, ‘‘Structures for Superior Characteristics in the Semiconductor
Lasers,’’ the quantum well structure, distributed feedback and mode-selective
lasers, and surface-emitting lasers are introduced.
TheAppendices,comprisingthefourthmajorsection,presentrelatedprinciples
andtheoremstofacilitateunderstandingofthebasisofthetheoriescoveredinthe
book, making it unnecessary to draw upon other sources to achieve that
understanding.
Kanazawa, Japan Minoru Yamada
Contents
Part I Fundamental Theory
1 Overview of the Operating Principles of Lasers. . . . . . . . . . . . . . 3
1.1 Optical Emission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Coherent Light. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Fundamental Structure of Laser . . . . . . . . . . . . . . . . . . . . . . 7
1.4 Typical Structure of Semiconductor Laser. . . . . . . . . . . . . . . 10
1.5 Material. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.6 Injection Mechanism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.7 Output Light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.8 Lasing Mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 The Photon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Analytical Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Principles of Classical Mechanics. . . . . . . . . . . . . . . . . . . . . 18
2.3 Principles of Quantum Mechanics . . . . . . . . . . . . . . . . . . . . 22
2.4 Procedure for Quantization . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5 Classical Form of Maxwell’s Equations . . . . . . . . . . . . . . . . 24
2.6 Obtaining Quantum Mechanical Operators
from Classical Physical Quantities . . . . . . . . . . . . . . . . . . . . 26
2.7 Quantization of the Optical Wave. . . . . . . . . . . . . . . . . . . . . 27
2.8 Remark on the Photon. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.9 The Uncertainty Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.10 Coherent State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.11 Interaction Between the Optical Wave
and a Charged Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.12 Analysis of Electron Transition Using a Fully
Quantized Treatment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Laser Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1 Representation of Optical Amplification
with Laser Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
vii
viii Contents
3.2 Conditions for Laser Oscillation in Traveling Wave Model. . . 38
3.3 Amplification and Lasing Conditions in Standing
Wave Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4 Note on Laser Oscillation . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.5 Modal Power and Optical Spectrum . . . . . . . . . . . . . . . . . . . 47
4 Waveguides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.1 Method to Analyze Two-Dimensional Cross-Section. . . . . . . . 49
4.2 Guiding Modes Taking into Account Gain and Loss. . . . . . . . 53
4.3 Stripe Structures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5 Density Matrix of a Semiconductor Material. . . . . . . . . . . . . . . . 65
5.1 Quantum Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.2 Dynamic Equation of the Density Matrix . . . . . . . . . . . . . . . 67
5.3 Energy State in Semiconductor Material . . . . . . . . . . . . . . . . 68
5.4 Interaction Hamiltonian. . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.5 Dipole Moment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.6 Additional Phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.7 Intraband Relaxation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6 Gain Coefficient and Rate Equation . . . . . . . . . . . . . . . . . . . . . . 81
6.1 Laser Polarization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
6.2 Amplification Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.3 Representation by Continuous Energy Levels
in Semiconductor Material. . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.4 Deviation of Refractive Index . . . . . . . . . . . . . . . . . . . . . . . 94
6.5 Plasma Effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.6 Rate Equation for the Electron Density. . . . . . . . . . . . . . . . . 96
6.7 Spontaneous Emission and Electron Life Time . . . . . . . . . . . 98
6.8 Rate Equation for the Photon Number Including
Spontaneous Emission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
7 Typical Operating Characteristics. . . . . . . . . . . . . . . . . . . . . . . . 105
7.1 Threshold Current. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
7.2 Quantum Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
7.3 Direct Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Contents ix
Part II Advanced Theory for Mode Competition and Noise
8 Nonlinear Gain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8.1 Redefinition of the Gain Coefficient. . . . . . . . . . . . . . . . . . . 115
8.2 Perturbation Expansion of Density Matrix. . . . . . . . . . . . . . . 117
8.3 Nonlinear Polarization and Gain Coefficient . . . . . . . . . . . . . 122
8.4 Effect from Higher Order Transverse Mode. . . . . . . . . . . . . . 128
8.5 Variation of Electron Density. . . . . . . . . . . . . . . . . . . . . . . . 130
8.6 Frequency Beating of the Injected Electron Density
with Lasing Modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
8.7 Asymmetric Gain Saturation Coefficient. . . . . . . . . . . . . . . . 135
8.8 Relation Among Nonlinear Gain Coefficients . . . . . . . . . . . . 139
8.9 Improved Rate Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
9 Mode Competition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
9.1 Two Modes Competition with Symmetric Gain Saturation . . . 143
9.2 Mode Competition with Asymmetric Gain Saturation. . . . . . . 147
9.3 Single Mode Operation by Help of Mode Selective Cavity . . . 153
Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
10 Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
10.1 Classification of Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
10.2 Introduction of the Noise Sources. . . . . . . . . . . . . . . . . . . . . 159
10.2.1 Noise Sources for Photon Number
and Electric Phase . . . . . . . . . . . . . . . . . . . . . . . . . 159
10.2.2 Noise Source for the Electron Density . . . . . . . . . . . 163
10.3 Frequency Components of Noise Sources . . . . . . . . . . . . . . . 164
10.4 Noise Analysis Under Stable Single Mode Operation . . . . . . . 165
10.4.1 Using Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . 165
10.4.2 Intensity Noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
10.4.3 Frequency Noise and Spectrum Linewidth. . . . . . . . . 171
10.5 Mode Competition Noise. . . . . . . . . . . . . . . . . . . . . . . . . . . 173
10.5.1 Mode Competition due to Symmetric
Gain Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
10.5.2 Mode Competition Including Asymmetric
Gain Saturation . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
10.6 Optical Feedback Noise. . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
10.6.1 Model of Optical Feedback . . . . . . . . . . . . . . . . . . . 180
10.6.2 Calculated Examples. . . . . . . . . . . . . . . . . . . . . . . . 183
10.6.3 Mechanism of Noise Generation. . . . . . . . . . . . . . . . 184
10.7 Noise Reduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
10.7.1 Superposition of High Frequency Current . . . . . . . . . 188
10.7.2 Use of Self-Pulsation Lasers . . . . . . . . . . . . . . . . . . 192