Fiber Optic Communications
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"The detailed, worked examples and first-principlesderivations of key results are helpful pedagogical features.Students seeking their first exposure to this field who also wishto learn about advanced topics will find their requirements met bythis book." (Optics and Photonics News, 28August 2014)More details
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Acknowledgments xvii
1 Electromagnetics and Optics 1
1.1 Introduction 1
1.2 Coulomb's Law and Electric Field Intensity 1
1.3 Ampere's Law and Magnetic Field Intensity 3
1.4 Faraday's Law 6
1.4.1 Meaning of Curl 7
1.4.2 Ampere's Law in Differential Form 9
1.5 Maxwell's Equations 9
1.5.1 Maxwell's Equation in a Source-Free Region 10
1.5.2 Electromagnetic Wave 10
1.5.3 Free-Space Propagation 11
1.5.4 Propagation in a Dielectric Medium 12
1.6 1-Dimensional Wave Equation 12
1.6.1 1-Dimensional Plane Wave 15
1.6.2 Complex Notation 16
1.7 Power Flow and Poynting Vector 17
1.8 3-Dimensional Wave Equation 19
1.9 Reflection and Refraction 21
1.9.1 Refraction 22
1.10 Phase Velocity and Group Velocity 26
1.11 Polarization of Light 31
Exercises 31
Further Reading 34
References 34
2 Optical Fiber Transmission 35
2.1 Introduction 35
2.2 Fiber Structure 35
2.3 Ray Propagation in Fibers 36
2.3.1 Numerical Aperture 37
2.3.2 Multi-Mode and Single-Mode Fibers 39
2.3.3 Dispersion in Multi-Mode Fibers 39
2.3.4 Graded-Index Multi-Mode Fibers 42
2.4 Modes of a Step-Index Optical Fiber* 44
2.4.1 Guided Modes 46
2.4.2 Mode Cutoff 51
2.4.3 Effective Index 52
2.4.4 2-Dimensional Planar Waveguide Analogy 53
2.4.5 Radiation Modes 54
2.4.6 Excitation of Guided Modes 55
2.5 Pulse Propagation in Single-Mode Fibers 57
2.5.1 Power and the dBm Unit 60
2.6 Comparison between Multi-Mode and Single-Mode Fibers 68
2.7 Single-Mode Fiber Design Considerations 68
2.7.1 Cutoff Wavelength 68
2.7.2 Fiber Loss 69
2.7.3 Fiber Dispersion 74
2.7.4 Dispersion Slope 76
2.7.5 Polarization Mode Dispersion 78
2.7.6 Spot Size 79
2.8 Dispersion-Compensating Fibers (DCFs) 79
2.9 Additional Examples 81
Exercises 89
Further Reading 91
References 91
3 Lasers 93
3.1 Introduction 93
3.2 Basic Concepts 93
3.3 Conditions for Laser Oscillations 101
3.4 Laser Examples 108
3.4.1 Ruby Laser 108
3.4.2 Semiconductor Lasers 108
3.5 Wave-Particle Duality 108
3.6 Laser Rate Equations 110
3.7 Review of Semiconductor Physics 113
3.7.1 The PN Junctions 118
3.7.2 Spontaneous and Stimulated Emission at the PN Junction 120
3.7.3 Direct and Indirect Band-Gap Semiconductors 120
3.8 Semiconductor Laser Diode 124
3.8.1 Heterojunction Lasers 124
3.8.2 Radiative and Non-Radiative Recombination 126
3.8.3 Laser Rate Equations 126
3.8.4 Steady-State Solutions of Rate Equations 128
3.8.5 Distributed-Feedback Lasers 132
3.9 Additional Examples 133
Exercises 136
Further Reading 138
References 138
4 Optical Modulators and Modulation Schemes 139
4.1 Introduction 139
4.2 Line Coder 139
4.3 Pulse Shaping 139
4.4 Power Spectral Density 141
4.4.1 Polar Signals 142
4.4.2 Unipolar Signals 142
4.5 Digital Modulation Schemes 144
4.5.1 Amplitude-Shift Keying 144
4.5.2 Phase-Shift Keying 144
4.5.3 Frequency-Shift Keying 145
4.5.4 Differential Phase-Shift Keying 146
4.6 Optical Modulators 149
4.6.1 Direct Modulation 149
4.6.2 External Modulators 150
4.7 Optical Realization of Modulation Schemes 158
4.7.1 Amplitude-Shift Keying 158
4.7.2 Phase-Shift Keying 160
4.7.3 Differential Phase-Shift Keying 162
4.7.4 Frequency-Shift Keying 163
4.8 Partial Response Signals* 163
4.8.1 Alternate Mark Inversion 169
4.9 Multi-Level Signaling* 172
4.9.1 M-ASK 172
4.9.2 M-PSK 174
4.9.3 Quadrature Amplitude Modulation 178
4.10 Additional Examples 182
Exercises 185
Further Reading 186
References 187
5 Optical Receivers 189
5.1 Introduction 189
5.2 Photodetector Performance Characteristics 190
5.2.1 Quantum Efficiency 193
5.2.2 Responsivity or Photoresponse 197
5.2.3 Photodetector Design Rules 199
5.2.4 Dark Current 200
5.2.5 Speed or Response Time 201
5.2.6 Linearity 202
5.3 Common Types of Photodetectors 202
5.3.1 pn Photodiode 203
5.3.2 pin Photodetector (pin-PD) 203
5.3.3 Schottky Barrier Photodetector 204
5.3.4 Metal-Semiconductor-Metal Photodetector 204
5.3.5 Photoconductive Detector 206
5.3.6 Phototransistor 206
5.3.7 Avalanche Photodetectors 207
5.3.8 Advanced Photodetectors* 212
5.4 Direct Detection Receivers 219
5.4.1 Optical Receiver ICs 220
5.5 Receiver Noise 224
5.5.1 Shot Noise 224
5.5.2 Thermal Noise 226
5.5.3 Signal-to-Noise Ratio, SNR 227
5.6 Coherent Receivers 227
5.6.1 Single-Branch Coherent Receiver 228
5.6.2 Balanced Coherent Receiver 232
5.6.3 Single-Branch IQ Coherent Receiver 234
5.6.4 Balanced IQ Receiver 237
5.6.5 Polarization Effects 239
Exercises 242
References 244
6 Optical Amplifiers 247
6.1 Introduction 247
6.2 Optical Amplifier Model 247
6.3 Amplified Spontaneous Emission in Two-Level Systems 248
6.4 Low-Pass Representation of ASE Noise 249
6.5 System Impact of ASE 251
6.5.1 Signal-ASE Beat Noise 253
6.5.2 ASE-ASE Beat Noise 256
6.5.3 Total Mean and Variance 256
6.5.4 Polarization Effects 258
6.5.5 Amplifier Noise Figure 260
6.5.6 Optical Signal-to Noise Ratio 262
6.6 Semiconductor Optical Amplifiers 263
6.6.1 Cavity-Type Semiconductor Optical Amplifiers 264
6.6.2 Traveling-Wave Amplifiers 268
6.6.3 AR Coating 270
6.6.4 Gain Saturation 271
6.7 Erbium-Doped Fiber Amplifier 274
6.7.1 Gain Spectrum 274
6.7.2 Rate Equations* 275
6.7.3 Amplified Spontaneous Emission 280
6.7.4 Comparison of EDFA and SOA 281
6.8 Raman Amplifiers 282
6.8.1 Governing Equations 283
6.8.2 Noise Figure 287
6.8.3 Rayleigh Back Scattering 287
6.9 Additional Examples 288
Exercises 298
Further Reading 300
References 300
7 Transmission System Design 301
7.1 Introduction 301
7.2 Fiber Loss-Induced Limitations 301
7.2.1 Balanced Coherent Receiver 306
7.3 Dispersion-Induced Limitations 313
7.4 ASE-Induced Limitations 315
7.4.1 Equivalent Noise Figure 317
7.4.2 Impact of Amplifier Spacing 318
7.4.3 Direct Detection Receiver 319
7.4.4 Coherent Receiver 322
7.4.5 Numerical Experiments 326
7.5 Additional Examples 327
Exercises 333
Further Reading 334
References 334
8 Performance Analysis 335
8.1 Introduction 335
8.2 Optimum Binary Receiver for Coherent Systems 335
8.2.1 Realization of the Matched Filter 342
8.2.2 Error Probability with an Arbitrary Receiver Filter 345
8.3 Homodyne Receivers 345
8.3.1 PSK: Homodyne Detection 347
8.3.2 On-Off Keying 349
8.4 Heterodyne Receivers 350
8.4.1 PSK: Synchronous Detection 351
8.4.2 OOK: Synchronous Detection 353
8.4.3 FSK: Synchronous Detection 356
8.4.4 OOK: Asynchronous Receiver 359
8.4.5 FSK: Asynchronous Detection 364
8.4.6 Comparison of Modulation Schemes with Heterodyne Receiver 367
8.5 Direct Detection 368
8.5.1 OOK 368
8.5.2 FSK 371
8.5.3 DPSK 374
8.5.4 Comparison of Modulation Schemes with Direct Detection 379
8.6 Additional Examples 381
Exercises 387
References 388
9 Channel Multiplexing Techniques 389
9.1 Introduction 389
9.2 Polarization-Division Multiplexing 389
9.3 Wavelength-Division Multiplexing 391
9.3.1 WDM Components 394
9.3.2 WDM Experiments 401
9.4 OFDM 402
9.4.1 OFDM Principle 402
9.4.2 Optical OFDM Transmitter 406
9.4.3 Optical OFDM Receiver 407
9.4.4 Optical OFDM Experiments 408
9.5 Time-Division Multiplexing 409
9.5.1 Multiplexing 409
9.5.2 Demultiplexing 410
9.5.3 OTDM Experiments 412
9.6 Additional Examples 413
Exercises 415
References 416
10 Nonlinear Effects in Fibers 419
10.1 Introduction 419
10.2 Origin of Linear and Nonlinear Refractive Indices 419
10.2.1 Absorption and Amplification 423
10.2.2 Nonlinear Susceptibility 424
10.3 Fiber Dispersion 426
10.4 Nonlinear Schrödinger Equation 428
10.5 Self-Phase Modulation 430
10.6 Combined Effect of Dispersion and SPM 433
10.7 Interchannel Nonlinear Effects 437
10.7.1 Cross-Phase Modulation 438
10.7.2 Four-Wave Mixing 448
10.8 Intrachannel Nonlinear Impairments 454
10.8.1 Intrachannel Cross-Phase Modulation 454
10.8.2 Intrachannel Four-Wave Mixing 455
10.8.3 Intra- versus Interchannel Nonlinear Effects 457
10.9 Theory of Intrachannel Nonlinear Effects 457
10.9.1 Variance Calculations 463
10.9.2 Numerical Simulations 466
10.10 Nonlinear Phase Noise 471
10.10.1 Linear Phase Noise 471
10.10.2 Gordon-Mollenauer Phase Noise 474
10.11 Stimulated Raman Scattering 478
10.11.1 Time Domain Description 481
10.12 Additional Examples 483
Exercises 491
Further Reading 493
References 493
11 Digital Signal Processing 497
11.1 Introduction 497
11.2 Coherent Receiver 497
11.3 Laser Phase Noise 498
11.4 IF Estimation and Compensation 501
11.5 Phase Estimation and Compensation 503
11.5.1 Phase Unwrapping 505
11.6 CD Equalization 506
11.6.1 Adaptive Equalizers 510
11.7 Polarization Mode Dispersion Equalization 513
11.8 Digital Back Propagation 516
11.8.1 Multi-Span DBP 521
11.9 Additional Examples 522
Exercises 524
Further Reading 525
References 525
AppendixA 527
Appendix B 533
Index 537
Chapter 1
Electromagnetics and Optics
1.1 Introduction
In this chapter, we will review the basics of electromagnetics and optics. We will briefly discuss various laws of electromagnetics leading to Maxwell's equations. Maxwell's equations will be used to derive the wave equation, which forms the basis for the study of optical fibers in Chapter 2. We will study elementary concepts in optics such as reflection, refraction, and group velocity. The results derived in this chapter will be used throughout the book.
1.2 Coulomb's Law and Electric Field Intensity
In 1783, Coulomb showed experimentally that the force between two charges separated in free space or vacuum is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. The force is repulsive if the charges are alike in sign, and attractive if they are of opposite sign, and it acts along the straight line connecting the charges. Suppose the charge is at the origin and is at a distance as shown in Fig. 1.1. According to Coulomb's law, the force on the charge is
1.1where is a unit vector in the direction of and is called the permittivity that depends on the medium in which the charges are placed. For free space, the permittivity is given by
1.2For a dielectric medium, the permittivity is larger than . The ratio of the permittivity of a medium to the permittivity of free space is called the relative permittivity, ,
1.3It would be convenient if we could find the force on a test charge located at any point in space due to a given charge . This can be done by taking the test charge to be a unit positive charge. From Eq. (1.1), the force on the test charge is
1.4The electric field intensity is defined as the force on a positive unit charge and is given by Eq. (1.4). The electric field intensity is a function only of the charge and the distance between the test charge and .
Figure 1.1 Force of attraction or repulsion between charges.
For historical reasons, the product of electric field intensity and permittivity is defined as the electric flux density ,
1.5The electric flux density is a vector with its direction the same as the electric field intensity. Imagine a sphere of radius around the charge as shown in Fig. 1.2. Consider an incremental area on the sphere. The electric flux crossing this surface is defined as the product of the normal component of and the area .
1.6where is the normal component of . The direction of the electric flux density is normal to the surface of the sphere and therefore, from Eq. (1.5), we obtain . If we add the differential contributions to the flux from all the incremental surfaces of the sphere, we obtain the total electric flux passing through the sphere,
1.7Since the electric flux density given by Eq. (1.5) is the same at all points on the surface of the sphere, the total electric flux is simply the product of and the surface area of the sphere ,
1.8Thus, the total electric flux passing through a sphere is equal to the charge enclosed by the sphere. This is known as Gauss's law. Although we considered the flux crossing a sphere, Eq. (1.8) holds true for any arbitrary closed surface. This is because the surface element of an arbitrary surface may not be perpendicular to the direction of given by Eq. (1.5) and the projection of the surface element of an arbitrary closed surface in a direction normal to is the same as the surface element of a sphere. From Eq. (1.8), we see that the total flux crossing the sphere is independent of the radius. This is because the electric flux density is inversely proportional to the square of the radius while the surface area of the sphere is directly proportional to the square of the radius and therefore, the total flux crossing a sphere is the same no matter what its radius is.
Figure 1.2 (a) Electric flux density on the surface of the sphere. (b) The incremental surface on the sphere.
So far, we have assumed that the charge is located at a point. Next, let us consider the case when the charge is distributed in a region. The volume charge density is defined as the ratio of the charge and the volume element occupied by the charge as it shrinks to zero,
1.9Dividing Eq. (1.8) by where is the volume of the surface and letting this volume shrink to zero, we obtain
1.10The left-hand side of Eq. (1.10) is called the divergence of and is written as
1.11Eq. (1.11) can be written as
1.12The above equation is called the differential form of Gauss's law and it is the first of Maxwell's four equations. The physical interpretation of Eq. (1.12) is as follows. Suppose a gunman is firing bullets in all directions, as shown in Fig. 1.3 [1]. Imagine a surface that does not enclose the gunman. The net outflow of the bullets through the surface is zero, since the number of bullets entering this surface is the same as the number of bullets leaving the surface. In other words, there is no source or sink of bullets in the region . In this case, we say that the divergence is zero. Imagine a surface that encloses the gunman. There is a net outflow of bullets since the gunman is the source of bullets and lies within the surface , so the divergence is not zero. Similarly, if we imagine a closed surface in a region that encloses charges with charge density , the divergence is not zero and is given by Eq. (1.12). In a closed surface that does not enclose charges, the divergence is zero.
Figure 1.3 Divergence of bullet flow.
1.3 Ampere's Law and Magnetic Field Intensity
Consider a conductor carrying a direct current . If we bring a magnetic compass near the conductor, it will orient in the direction shown in Fig. 1.4(a). This indicates that the magnetic needle experiences the magnetic field produced by the current. The magnetic field intensity is defined as the force experienced by an isolated unit positive magnetic charge (note that an isolated magnetic charge does not exist without an associated ), just like the electric field intensity is defined as the force experienced by a unit positive electric charge.
Figure 1.4 (a) Direct current-induced constant magnetic field. (b) Ampere's circuital law.
Consider a closed path or around the current-carrying conductor, as shown in Fig. 1.4(b). Ampere's circuital law states that the line integral of about any closed path is equal to the direct current enclosed by that path,
1.13The above equation indicates that the sum of the components of that are parallel to the tangent of a closed curve times the differential path length is equal to the current enclosed by this curve. If the closed path is a circle () of radius , due to circular symmetry, the magnitude of is constant at any point on and its direction is shown in Fig. 1.4(b). From Eq. (1.13), we obtain
1.14or
1.15Thus, the magnitude of the magnetic field intensity at a point is inversely proportional to its distance from the conductor. Suppose the current is flowing in the -direction. The -component of the current density may be defined as the ratio of the incremental current passing through an elemental surface area perpendicular to the direction of the current flow as the surface shrinks to zero,
1.16The current density is a vector with its direction given by the direction of the current. If is not perpendicular to the surface , we need to find the component that is perpendicular to the surface by taking the dot product
1.17where is a unit vector normal to the surface . By defining a vector , we have
1.18and the incremental current is given by
1.19The total current flowing through a surface is obtained by integrating,
1.20Using Eq. (1.20) in Eq. (1.13), we obtain
1.21where is the surface whose perimeter is the closed path .
In analogy with the definition of electric flux density, magnetic flux density is defined as
1.22where is called the permeability. In free space, the permeability has a value
1.23In general, the permeability of a medium is written as a product of the permeability of free space and a constant that depends on the medium. This constant is called the relative permeability ,
1.24The magnetic flux crossing a surface can be obtained by integrating the normal component of magnetic flux...
