This book is intended for scientists, engineers, and designers who would like to learn about optical communications and about the operation and service of optical wireless (atmospheric) and wired (fiber optic) communication links, laser beam systems, and fiber optic multiuser networks. It will be useful to undergraduate and postgraduate students alike and to practicing scientists and engineers.

Over the preceding forty years, many excellent books have been published about different aspects of optical waves and about laser beam propagation in both atmospheric links and within guiding structures, such as fiber optic cables. Wireless and wired communications have often been described separately without taking note of their similarities. In this monograph, we consider both media and describe techniques for transmitting information over such channels when the optical signals are corrupted by the fading that is typical of such communication links because of the existence of artificial (man‐made) and/or natural sources of fading.

This monograph methodically unifies the basic concepts and the corresponding mathematical models and approaches to describing optical wave propagation in material media, in waveguide structures and fiber optic structures, and in the troposphere (the lowest layer of the atmosphere). It describes their similarity to other types of electromagnetic waves, e.g. radio waves, from other regions of the electromagnetic spectrum.

Without entering into an overly deep and detailed description of the physical and mathematical fundamentals of the atmosphere as a propagation channel or of the fiber optic structure as a waveguide structure, this monograph focuses the reader's attention on questions related to the coding and decoding that is useful when using such channels. In particular, the monograph analyzes different types of fading and their sources and considers types of modulation that mitigate the effects of fading.

The monograph briefly describes several sources of optical radiation, such as lasers, and presents several particularly relevant optical signal detectors.

The monograph contains material about the atmospheric communication channel, including the effects of atmospheric turbulence and different kinds of hydrometeors, such as aerosols, rain, snow, and clouds, on optical wave propagation in an atmospheric link. The principal goal of this book is to explain the effects of fading and energy loss in information‐carrying optical signals. We consider the various situations that occur in the atmospheric link and, finally, show how to mitigate the effects of natural phenomena such as turbulence and hydrometeors that affect the propagation of optical rays and laser beams through the atmosphere.

This book introduces the reader to fading and describes its dispersive nature. It considers fading of optical waves propagating in the irregular turbulent atmosphere in close proximity to the ground surface and elucidates its relation to similar signal dispersive fading phenomena that occur in fiber optic channels where there is a wired link.

The book is organized as follows. Part I consists of two chapters. Chapter 1 describes the fundamental aspects of optical wireless and wired communication links and of the spectrum of optical waves. It also provides a description of the evolution of optical networks (from first to fifth generation networks). End‐to‐end descriptions of optical channels are provided. Block diagrams of the receiver (the detector of optical waves) and the transmitter (the radiator of optical waves) are given, and information transfer though the channel is described. In Chapter 2, the similarities between radio and optical waves are described. The description makes use of some of the fundamental notions of wave electrodynamics. In particular, the differential and integral presentation of optical waves is developed from Maxwell's equations. Maxwell's equations are also presented in the form of phasors. The principal features of optical wave propagation in material media, both dielectric and conductive, are described. Finally, the reflection and refraction of optical waves from the boundary of two media is described via the introduction of the parameters of refraction (instead of the dielectric and magnetic parameters of the medium), and the effect of total internal reflection, one the main features in any guiding structure (including a fiber optic cable), is considered. There are exercises at the end of Chapter 2.

Part II, which describes the fundamentals of optical communication, consists of six chapters. The first chapter, Chapter 3, describes types of optical signals propagating through wireless or wired communication links. Both continuous and discrete channels are considered, and the relation between them is described. In Chapter 3, we show that for non‐correlated optical waves or signals, the average powers of a continuous signal and of a discrete signal (e.g. pulse) are equivalent. The reader is then asked to consider the bandwidth of the signals and to note that one is narrowband and the other wideband. A mathematical/statistical framework is then established for considering these signals in the space, time, and frequency domains. Chapter 4 presents the fundamental principles of discrete signal coding and decoding. The effects of white Gaussian noise on such signals are described briefly and both linear and nonlinear codes are considered. The error probability when decoding such codes is considered for a variety of decoding algorithms for cyclic codes, Reed–Solomon codes, etc. Finally, a general scheme for decoding cyclic codes is developed. In Chapter 5, we apply what we have learned about coding and decoding to optical communication links. Low density parity check codes are considered in detail. Finally, the coding process in optical communication links is described and a comparative analysis of different codes is presented. Chapter 6 considers the effects of fading as it occurs in real optical communication and describes how it is caused by various sources of multiplicative noise. It is shown that by considering signal parameters (pulse duration and bandwidth) and parameters related to channel coherency (in time and frequency), fading phenomena can be described as flat or frequency selective, as slow (in the time domain) or large scale (in the space domain), or as fast (in the time domain) or small scale (in the space domain). Next, mathematical descriptions of fast and slow fading are provided by using the Rayleigh or Rice distribution, gamma‐gamma distribution, and the Gaussian distribution. Chapter 7 deals with the modulation of optical signals in wireless and wired communication links. It starts by describing types of analog modulation: analog amplitude modulation and analog phase and frequency modulation, considering them as two types of a general angle modulation of continuous optical signals. The relation between the spectral bandwidths of the two last types of modulation, phase and frequency modulation, is considered, and their signal‐to‐noise (SNR) ratio is analyzed. Finally, several types of digital signal modulation are presented briefly: amplitude shift keying, phase shift keying, and frequency shift keying. There are several exercises at the end of Chapter 7.

In Chapter 8, optical sources and detectors are described. A brief description of the fundamentals of emission and absorption of optical waves is given. Then the operational characteristics of laser sources and diodes, as well as other types of photodiodes, are briefly described, and several types of modulation schemes that can be used with lasers are demonstrated. Finally, the operational characteristics of photodiodes are presented, and a clear description of the relations between the optical and electrical parameters of typical diode‐based schemes is given.

Part III consists of two chapters. In Chapter 9, guiding structures related to fiber optical ones are briefly described. The reader is shown two types of fiber optic structures: step‐index fiber and graded‐index fiber. Their parameters are determined and described. Next, the propagation of optical waves in fiber optic structures is analyzed, and it is shown that frequency dispersion is an issue when dealing with multimode propagation in such guiding structures. These dispersion properties are examined in Chapter 10, where the corresponding multimode dispersion parameters are presented for the two types of fiber optic cables described in Chapter 9: step‐index and graded‐index fiber. This modal dispersion is compared with material dispersion caused by the inhomogeneous structure of the material along the length of the fiber. The data loss caused by these two types of dispersion for two kinds of codes – non‐return‐to‐zero (NRZ) codes and return‐to‐zero (RZ) codes – mentioned in Chapter 1 is described.

Part IV consists of a single chapter. Chapter 11 describes the propagation of optical waves in the atmosphere, considered as an inhomogeneous gaseous structure, and briefly describes the main parameters used to describe the atmosphere. The content of the atmosphere is presented briefly. In particular, in Chapter 11 the structure of aerosols and their dimensions, concentration, spatial distribution of aerosols' sizes, and their spectral extinction and altitude localization are briefly presented. Then, the existence of various water and ice particles, called hydrometeors, in the inhomogeneous atmosphere, their spatial and altitudinal distribution, size distribution, and their effects on optical wave propagation are briefly discussed. Atmospheric turbulent structures caused by temperature and humidity fluctuations combined with turbulent mixing by wind and convection‐induced random changes in the air density of the atmosphere (as an irregular gaseous medium) are briefly discussed. Next, the scintillation phenomenon caused by an optical wave passing through the turbulent atmosphere is analyzed. The corresponding formulas for the scintillation index of signal intensity variation are presented as the main parameter of signal fading in the turbulent atmosphere caused by scattering phenomena from turbulent structures. Finally, the corresponding functions used to describe such scattering are described so that the relation between the scintillation index and the fading parameters can be elucidated.

Part V, concerning signal data flow transmission in wireless and fiber optic communication links, consists of one chapter. Chapter 12 starts with definitions related to the characteristics of a communication link: capacity, spectral efficiency, and bit error rate (BER). These important, well‐known parameters are presented in a unified manner both for atmospheric and fiber optic channels via the fading parameter, K. Use is made of its relation to the scintillation index that was described in the previous chapter. The relation between the characteristic parameters of the communication link and the fading parameter are described by unified unique formulas and corresponding algorithms. Our understanding of these quantities allows us to perform relevant computations and present clear graphical illustrations for both NRZ and RZ signals.

This book provide a synthesis of several physical and mathematical models in order to present a broad and unified approach for the prediction of data stream parameters for various types of codes used with optical signals traversing optical channels, whether atmospheric or fiber optic, having similar fading time/dispersive effects caused by a variety of sources. In the atmosphere, scattering is due to turbulent structures and hydrometeors; in fiber optic structures, it is due to multimode effects and inhomogeneities in the cladding or core.

The authors would like to thank their colleagues for many helpful discussions. They also have the pleasure of acknowledging the computational work of their students – work that led to graphics describing data stream parameters for various situations occurring in wireless atmospheric and wired fiber optic communication channels.

The authors would like to thank the staff at Wiley, the reviewers, and the technical editors for their help in making this book as clear and precise as possible. They would also like to thank Brett Kurzman, Steven Fassioms, and Amudhapriya Sivamurthy of Wiley for their help in bringing this book to market.

The authors are pleased to acknowledge their debt to their families for providing the time, atmosphere, and encouragement that made writing this book such a pleasant undertaking.

• A – arbitrary vectors of electromagnetic field
• B – vector of induction of magnetic field
• E – vector of electric field component of the electromagnetic wave
• E(z, t) – 2‐D vector of electrical component of the electromagnetic wave
• – phasor of the electrical component of the electromagnetic wave
• D – vector of electric field displacement or vector induction of electric field
• H – vector of magnetic field component of the electromagnetic wave
• H(z, t) – 2‐D vector of magnetic field component of the electromagnetic wave
• – phasor of the magnetic field component of the electromagnetic wave
• j – vector of electric current density
• J – vector of the full current in medium/circuit
• j _c – conductivity current density
• j _d – displacement current density
• k – wave vector
• M – momentum of the magnetic ambient source
• P – vector of polarization
• A _c – amplitude of carrier signal
• A _m – amplitude of modulated signal
• B _c – coherence bandwidth
• B _D – Doppler spread bandwidth
• B _f – maximum bandwidth of the modulating signal
• B _F – equivalent RF bandwidth of the bandpass filter
• B _ω – detector bandwidth
• B _Ω – bandwidth of multiplicative noise
• C _BSC = 1 − η(p) – capacity of binary symmetric channel
• p – probability of 0 or 1
• f _c – frequency of carrier signal
• f _D – Doppler shift
• f _m – frequency of modulated signal
• B _S – signal bandwidth
• C – channel capacity
• C(D) – effective cross‐section of rain drops as function of their diameter D
• C ₁ – square ( k × k )‐matrix
• C ₂ – matrix of dimension ( k × n − k )
• – product of inverse C ₁ matrix and regular C ₂ matrix
• C – (m × m)‐cyclic permutation matrix
• – velocity of light in free space
• – refraction structure parameter
• – capacity per length l of fiber for propagation of NRZ pulses
• C _RZ × l = 0.875 (Mbit/c) × km – capacity per length l of fiber for propagation of RZ pulses
• d l – differential of the vector of a line l
• d S – differential of the vector of a surface S
• d V – differential of the vector of a volume V
• D _m  = 0.122 · R ^0.21 mm – diameter of rain drops, R – rainfall rate (in mm/h)
• D _p – polarization mode dispersion factor (in fiber optic cables)
• e – charge of electron
• E _b – energy of one transmitted bit
• E _x , E _y , E _z – components of the electric field of the wave in the Cartesian coordinate system
• d _H (x, y) – Hamming distance
• G – generator matrix
• G – photo‐conductive gain of the light detector
• g(t) – a signal's envelope as a function of time
• I – light intensity
• I _k – unit ( k × k )‐matrix
• h = 6.625 · 10⁻³⁴ J · s – Planck's constant
• hν _ji – energy of photon; j and i are steady states of atoms and electrons, j > i
• H _W – parity matrix,
• H(τ) =  − τ ln τ − (1 − τ) ln(1 − τ) – entropy of the binary ensemble with parameter τ = t/n , τ > p
• – the unit imaginary number
• J _m (qr) – Bessel function of the first kind and of order m
• J _ph – photocurrent intensity
• K – Ricean fading parameter
• k _f – frequency deviation constant of frequency modulation
• k _m  = (A _m /A _c ) – modulation index of amplitude modulation
• k = 1.38 · 10⁻²³ J/K – Boltzmann's constant
• – spatial wave number for outer turbulence scale
• – spatial wave number for inner turbulence scale
• k _θ – phase deviation constant of phase modulation
• K _m (qr) – modified Hankel function or Bessel function of the second kind and of order m
• K _{α − β} [·] – modified Bessel function of the second kind of order (α − β)
• L – path loss or attenuation of optical signal
• l ₀ – inner scale of atmospheric turbulence
• L ₀ – outer scale of atmospheric turbulence
• l ₁ ≡ l _co∼1/ρ ₀ – coherence length between two coherent points of turbulence
• – first Fresnel zone scale, L – range, k = 2π/λ – wave number
• l ₃∼R/ρ ₀ k – scattering disk (turbulence) scale
• LP₀₁ (m = 0) – linear polarized mode with m = 0 in fiber optic cable
• LP₁₁ (m = 1) – linear polarized mode with m = 1 in fiber optic cable
• M – material dispersion factor
• m(x) – codeword
• m(t) – modulated message signal
• 〈m(t)〉 – average value of the modulated message signal
• M ₀ ≈  − 0.095 ps/(nm · km) – material dispersion factor at wavelength of
• np – mean optical power
• n(r) – aerosol particle distribution in the atmosphere
• n = n′ − jn″ – complex refractive index, – real part, – imaginary part
• n _eff ≡ n ₁ sin θ _i – effective refractive index
• N ₀ = 8 · 10³ m⁻² mm⁻¹ – constant number of rain drops
• N ₀ – white noise power spectral density
• N _add = N ₀ B _ω – additive (Gaussian) noise power
• N _mult – spectral density of multiplicative noise
• N _mult = N _mult B _Ω – multiplicative noise power
• N(D) – distribution of rain drops as function of their diameter D
• dN(r) – number of aerosols having radius between r and r + dr
• p – pressure in millibars, or pascals or mm Hg
• P _r – optical power incident on the detector surface
• P _oШ – error probability
• P _m – mean optical power received by the detector
• P _g ( f) – PSD of the envelope g(t)
• P _r – optical signal power
• P _r (e) – evaluated probability of the error
• P(h) – atmospheric pressure as function of altitude h
• P _m  = 〈m ²(t)〉 – power of the modulated message signal
• P(φ _i ) = (2π)⁻¹ – ray phase distribution probability function
• R – coefficient of reflection from boundary of two media
• R – data rate
• R – detector responsivity
• Re = V · l/ν – Reynolds number
• r _R – optical ray path length
• R _H – coefficient of reflection of the ray with the horizontal polarization
• R _S – bulk resistance of the photodetector
• R _V – coefficient of reflection of the ray with the vertical polarization
• – root mean square
• s(t) – bandpass signal
• T – temperature in kelvin
• T – coefficient of refraction (transfer of the wave into the medium)
• T(h) – atmospheric temperature as function of altitude h
• t _T – detector transit time
• T _b – bit period
• T _c – coherence time
• T _p (τ) =  − τ ln p − (1 − τ) ln(1 − p)
• T _S – symbol period
• x(t) – bandpass signal
• X( f) – Fourier transform of x(t)
• x _T (t) – truncated version of the signal x(t)
• X _T ( f) – Fourier transform of x _T (t)
• V _p – peak‐to‐zero value of the modulating signal m(t)
• v _ph – wave phase velocity
• y(t) – baseband signal
• Y( f) – Fourier transform of y(t)
• V – rate of data transmission (bits per seconds, bps)
• W – energy of arbitrary field
• W ₀( f) – complex baseband power spectrum in the frequency domain
• two media: n _eff ≡ n ₁ and when θ _i  = θ _c , n _eff ≡ n ₂
• z = a + ib – complex number, a – its real part, b – its imaginary part
• {x, y, z} – Cartesian coordinate system
• {ρ, φ, z} – cylindrical coordinate system
• {r, φ, θ} – spherical coordinate system
• ∇ – the nabla or del operator for an arbitrary scalar field
• Δ = ∇² – Laplacian of the vector or scalar field
• Δ – related (fractional) refractive index
• div =  ∇ · – divergence of an arbitrary vector (or “del dot the field”)
• grad Φ =  ∇ Φ – gradient of arbitrary scalar field or effect of nabla operator on the scalar field
• curl ≡ rot =  ∇ × – rotor of arbitrary vector field or the vector product of the operator nabla and the field
• Δf – peak frequency deviation of the transmitter
• α – parameter of wave attenuation in arbitrary medium
• α(λ) – light scattering coefficient
• β – parameter of phase velocity deviation in arbitrary medium
• β _f  = Δf/f _m – frequency modulation index
• β _θ  = k _θ A _m  = Δθ – phase modulation index, k _θ – phase deviation constant of phase modulation
• – attenuation factor in dB, α – attenuation factor
• γ = α + iβ – parameter of propagation in arbitrary material medium
• ε – average energy dissipation rate
• ε = ε′ + iε″ – complex permittivity of arbitrary medium
• – relative permittivity of arbitrary medium, – its real part, – its imaginary part
• – dielectric parameter of free space
• Δθ – peak phase deviation of the transmitter
• Δ(τ/l) – time delay dispersion of pulses along fiber with the length l
• μ _r – mean value of the signal envelope r
• μ – permeability of an arbitrary medium
• μ ₀ = 4π · 10⁻⁹ (H/m) – permeability of free space
• λ – wavelength
• λ ₀ ≈ 1300 nm – wavelength for material dispersion factor M = 0
• η – wave impedance in arbitrary medium
• η ₀ = 120π Ω = 377 Ω – wave impedance of free space
• η _B – bandwidth efficiency
• η _p – power efficiency
• η(p) =  − p log₂ p − (1 − p) log₂ (1 − p) – binary entropy function
• η ∈ N(0, σ ²) – normally distributed random value with zero mean and variance σ ²
• – skin layer in arbitrary material medium
• Φ – photon intensity
• λ – wavelength in arbitrary medium
• λ _g – wavelength in arbitrary waveguide structure
• K(t, t + τ) = 〈E(t) · E(t + τ)〉 – time domain autocorrelation function
• ρ – charge density in medium
• ρ – density in kg/m³
• ρ _N (h) – density of nitrogen molecules in atmosphere
• τ _R – mean electron–hole recombination time
• τ _c – correlation time
• σ – conductivity of arbitrary medium or material
• σ _τ – rms delay spread
• – variation of the signal envelope r
• – signal intensity scintillation factor
• θ _i – angle of incidence of the ray at the boundary of two media
• – critical angle of the inner total reflection from boundary of two media
• θ _full = 2 · θ _a – angle of minimum light energy spread outside the cladding of fiber
• ϑ – complexity of arrangement of coder/decoder
• σ ² = 〈(τ −  < τ>)²〉 – variance or the mean signal power in the time domain
• ω = 2πf – angular frequency
• ω _dn – Doppler shift of the n‐ray