Chapter 2 Optical Fibers The phenomenon of total internal reflection, responsible for guiding of light in opti- cal fibers, has been known since 1854 [1]. Although glass fibers were made in the 1920s [2]–[4], their use became practical only in the 1950s, when the use of a cladding layer led to considerable improvement in their guiding characteristics [5]–[7]. Before 1970, optical fibers were used mainly for medical imaging over short distances [8]. Their use for communication purposes was considered impractical because of high losses (∼ 1000 dB/km). However, the situation changed drastically in 1970 when, fol- lowing an earlier suggestion [9], the loss of optical fibers was reduced to below 20 dB/km [10]. Further progress resulted by 1979 in a loss of only 0.2 dB/km near the 1.55- µ m spectral region [11]. The availability of low-loss fibers led to a revolution in the field of lightwave technology and started the era of fiber-optic communications. Several books devoted entirely to optical fibers cover numerous advances made in their design and understanding [12]–[21]. This chapter focuses on the role of optical fibers as a communication channel in lightwave systems. In Section 2.1 we use geometrical- optics description to explain the guiding mechanism and introduce the related basic concepts. Maxwell’s equations are used in Section 2.2 to describe wave propagation in optical fibers. The origin of fiber dispersion is discussed in Section 2.3, and Section 2.4 considers limitations on the bit rate and the transmission distance imposed by fiber dispersion. The loss mechanisms in optical fibers are discussed in Section 2.5, and Section 2.6 is devoted to a discussion of the nonlinear effects. The last section covers manufacturing details and includes a discussion of the design of fiber cables. 2.1 Geometrical-Optics Description In its simplest form an optical fiber consists of a cylindrical core of silica glass sur- rounded by a cladding whose refractive index is lower than that of the core. Because of an abrupt index change at the core–cladding interface, such fibers are called step-index fibers. In a different type of fiber, known as graded-index fiber, the refractive index decreases gradually inside the core. Figure 2.1 shows schematically the index profile and the cross section for the two kinds of fibers. Considerable insight in the guiding 23 Fiber-Optic Communications Systems, Third Edition. Govind P. Agrawal Copyright  2002 John Wiley & Sons, Inc. ISBNs: 0-471-21571-6 (Hardback); 0-471-22114-7 (Electronic) 24 CHAPTER 2. OPTICAL FIBERS Figure 2.1: Cross section and refractive-index profile for step-index and graded-index fibers. properties of optical fibers can be gained by using a ray picture based on geometrical optics [22]. The geometrical-optics description, although approximate, is valid when the core radius a is much larger than the light wavelength λ . When the two become comparable, it is necessary to use the wave-propagation theory of Section 2.2. 2.1.1 Step-Index Fibers Consider the geometry of Fig. 2.2, where a ray making an angle θ i with the fiber axis is incident at the core center. Because of refraction at the fiber–air interface, the ray bends toward the normal. The angle θ r of the refracted ray is given by [22] n 0 sin θ i = n 1 sin θ r , (2.1.1) where n 1 and n 0 are the refractive indices of the fiber core and air, respectively. The re- fracted ray hits the core–cladding interface and is refracted again. However, refraction is possible only for an angle of incidence φ such that sin φ < n 2 /n 1 . For angles larger than a critical angle φ c , defined by [22] sin φ c = n 2 /n 1 , (2.1.2) where n 2 is the cladding index, the ray experiences total internal reflection at the core– cladding interface. Since such reflections occur throughout the fiber length, all rays with φ > φ c remain confined to the fiber core. This is the basic mechanism behind light confinement in optical fibers. 2.1. GEOMETRICAL-OPTICS DESCRIPTION 25 Figure 2.2: Light confinement through total internal reflection in step-index fibers. Rays for which φ < φ c are refracted out of the core. One can use Eqs. (2.1.1) and (2.1.2) to find the maximum angle that the incident ray should make with the fiber axis to remain confined inside the core. Noting that θ r = π /2− φ c for such a ray and substituting it in Eq. (2.1.1), we obtain n 0 sin θ i = n 1 cos φ c =(n 2 1 − n 2 2 ) 1/2 . (2.1.3) In analogy with lenses, n 0 sin θ i is known as the numerical aperture (NA) of the fiber. It represents the light-gathering capacity of an optical fiber. For n 1  n 2 the NA can be approximated by NA = n 1 (2∆) 1/2 , ∆ =(n 1 − n 2 )/n 1 , (2.1.4) where ∆ is the fractional index change at the core–cladding interface. Clearly, ∆ should be made as large as possible in order to couple maximum light into the fiber. How- ever, such fibers are not useful for the purpose of optical communications because of a phenomenon known as multipath dispersion or modal dispersion (the concept of fiber modes is introduced in Section 2.2). Multipath dispersion can be understood by referring to Fig. 2.2, where different rays travel along paths of different lengths. As a result, these rays disperse in time at the output end of the fiber even if they were coincident at the input end and traveled at the same speed inside the fiber. A short pulse (called an impulse) would broaden considerably as a result of different path lengths. One can estimate the extent of pulse broadening simply by considering the shortest and longest ray paths. The shortest path occurs for θ i = 0 and is just equal to the fiber length L. The longest path occurs for θ i given by Eq. (2.1.3) and has a length L/sin φ c . By taking the velocity of propagation v = c/n 1 , the time delay is given by ∆T = n 1 c  L sin φ c − L  = L c n 2 1 n 2 ∆. (2.1.5) The time delay between the two rays taking the shortest and longest paths is a measure of broadening experienced by an impulse launched at the fiber input. We can relate ∆T to the information-carrying capacity of the fiber measured through the bit rate B. Although a precise relation between B and ∆T depends on many details, 26 CHAPTER 2. OPTICAL FIBERS such as the pulse shape, it is clear intuitively that ∆T should be less than the allocated bit slot (T B = 1/B). Thus, an order-of-magnitude estimate of the bit rate is obtained from the condition B∆T < 1. By using Eq. (2.1.5) we obtain BL < n 2 n 2 1 c ∆ . (2.1.6) This condition provides a rough estimate of a fundamental limitation of step-index fibers. As an illustration, consider an unclad glass fiber with n 1 = 1.5 and n 2 = 1. The bit rate–distance product of such a fiber is limited to quite small values since BL < 0.4 (Mb/s)-km. Considerable improvement occurs for cladded fibers with a small index step. Most fibers for communication applications are designed with ∆ < 0.01. As an example, BL < 100 (Mb/s)-km for ∆ = 2× 10 −3 . Such fibers can communicate data at a bit rate of 10 Mb/s over distances up to 10 km and may be suitable for some local-area networks. Two remarks are in order concerning the validity of Eq. (2.1.6). First, it is obtained by considering only rays that pass through the fiber axis after each total internal re- flection. Such rays are called meridional rays. In general, the fiber also supports skew rays, which travel at angles oblique to the fiber axis. Skew rays scatter out of the core at bends and irregularities and are not expected to contribute significantly to Eq. (2.1.6). Second, even the oblique meridional rays suffer higher losses than paraxial meridional rays because of scattering. Equation (2.1.6) provides a conservative estimate since all rays are treated equally. The effect of intermodal dispersion can be considerably re- duced by using graded-index fibers, which are discussed in the next subsection. It can be eliminated entirely by using the single-mode fibers discussed in Section 2.2. 2.1.2 Graded-Index Fibers The refractive index of the core in graded-index fibers is not constant but decreases gradually from its maximum value n 1 at the core center to its minimum value n 2 at the core–cladding interface. Most graded-index fibers are designed to have a nearly quadratic decrease and are analyzed by using α -profile, given by n( ρ )=  n 1 [1− ∆( ρ /a) α ]; ρ < a, n 1 (1− ∆)=n 2 ; ρ ≥ a, (2.1.7) where a is the core radius. The parameter α determines the index profile. A step-index profile is approached in the limit of large α .Aparabolic-index fiber corresponds to α = 2. It is easy to understand qualitatively why intermodal or multipath dispersion is re- duced for graded-index fibers. Figure 2.3 shows schematically paths for three different rays. Similar to the case of step-index fibers, the path is longer for more oblique rays. However, the ray velocity changes along the path because of variations in the refractive index. More specifically, the ray propagating along the fiber axis takes the shortest path but travels most slowly as the index is largest along this path. Oblique rays have a large part of their path in a medium of lower refractive index, where they travel faster. It is therefore possible for all rays to arrive together at the fiber output by a suitable choice of the refractive-index profile. 2.1. GEOMETRICAL-OPTICS DESCRIPTION 27 Figure 2.3: Ray trajectories in a graded-index fiber. Geometrical optics can be used to show that a parabolic-index profile leads to nondispersive pulse propagation within the paraxial approximation. The trajectory of a paraxial ray is obtained by solving [22] d 2 ρ dz 2 = 1 n dn d ρ , (2.1.8) where ρ is the radial distance of the ray from the axis. By using Eq. (2.1.7) for ρ < a with α = 2, Eq. (2.1.8) reduces to an equation of harmonic oscillator and has the general solution ρ = ρ 0 cos(pz)+( ρ  0 /p) sin(pz), (2.1.9) where p =(2∆/a 2 ) 1/2 and ρ 0 and ρ  0 are the position and the direction of the input ray, respectively. Equation (2.1.9) shows that all rays recover their initial positions and directions at distances z = 2m π /p, where m is an integer (see Fig. 2.3). Such a complete restoration of the input implies that a parabolic-index fiber does not exhibit intermodal dispersion. The conclusion above holds only within the paraxial and the geometrical-optics ap- proximations, both of which must be relaxed for practical fibers. Intermodal dispersion in graded-index fibers has been studied extensively by using wave-propagation tech- niques [13]–[15]. The quantity ∆T /L, where ∆T is the maximum multipath delay in a fiber of length L, is found to vary considerably with α . Figure 2.4 shows this varia- tion for n 1 = 1.5 and ∆ = 0.01. The minimum dispersion occurs for α = 2(1− ∆) and depends on ∆ as [23] ∆T /L = n 1 ∆ 2 /8c. (2.1.10) The limiting bit rate–distance product is obtained by using the criterion ∆T < 1/B and is given by BL < 8c/n 1 ∆ 2 . (2.1.11) The right scale in Fig. 2.4 shows the BL product as a function of α . Graded-index fibers with a suitably optimized index profile can communicate data at a bit rate of 100 Mb/s over distances up to 100 km. The BL product of such fibers is improved by nearly three orders of magnitude over that of step-index fibers. Indeed, the first generation 28 CHAPTER 2. OPTICAL FIBERS Figure 2.4: Variation of intermodal dispersion ∆T/L with the profile parameter α for a graded- index fiber. The scale on the right shows the corresponding bit rate–distance product. of lightwave systems used graded-index fibers. Further improvement is possible only by using single-mode fibers whose core radius is comparable to the light wavelength. Geometrical optics cannot be used for such fibers. Although graded-index fibers are rarely used for long-haul links, the use of graded- index plastic optical fibers for data-link applications has attracted considerable atten- tion during the 1990s [24]–[29]. Such fibers have a relatively large core, resulting in a high numerical aperture and high coupling efficiency but they exhibit high losses (typically exceeding 50 dB/km). The BL product of plastic fibers, however, exceeds 2 (Gb/s)-km because of a graded-index profile [24]. As a result, they can be used to transmit data at bit rates > 1 Gb/s over short distances of 1 km or less. In a 1996 demonstration, a 10-Gb/s signal was transmitted over 0.5 km with a bit-error rate of less than 10 −11 [26]. Graded-index plastic optical fibers provide an ideal solution for transferring data among computers and are becoming increasingly important for Eth- ernet applications requiring bit rates in excess of 1 Gb/s. 2.2 Wave Propagation In this section we consider propagation of light in step-index fibers by using Maxwell’s equations for electromagnetic waves. These equations are introduced in Section 2.2.1. The concept of fiber modes is discussed in Section 2.2.2, where the fiber is shown to support a finite number of guided modes. Section 2.2.3 focuses on how a step-index fiber can be designed to support only a single mode and discusses the properties of single-mode fibers. 2.2. WAVE PROPAGATION 29 2.2.1 Maxwell’s Equations Like all electromagnetic phenomena, propagation of optical fields in fibers is governed by Maxwell’s equations. For a nonconducting medium without free charges, these equations take the form [30] (in SI units; see Appendix A) ∇× E = − ∂ B/ ∂ t, (2.2.1) ∇× H = ∂ D/ ∂ t, (2.2.2) ∇· D = 0, (2.2.3) ∇· B = 0, (2.2.4) where E and H are the electric and magnetic field vectors, respectively, and D and B are the corresponding flux densities. The flux densities are related to the field vectors by the constitutive relations [30] D = ε 0 E + P, (2.2.5) B = µ 0 H + M, (2.2.6) where ε 0 is the vacuum permittivity, µ 0 is the vacuum permeability, and P and M are the induced electric and magnetic polarizations, respectively. For optical fibers M = 0 because of the nonmagnetic nature of silica glass. Evaluation of the electric polarization P requires a microscopic quantum-mechanical approach. Although such an approach is essential when the optical frequency is near a medium resonance, a phenomenological relation between P and E can be used far from medium resonances. This is the case for optical fibers in the wavelength region 0.5–2 µ m, a range that covers the low-loss region of optical fibers that is of interest for fiber-optic communication systems. In general, the relation between P and E can be nonlinear. Although the nonlinear effects in optical fibers are of considerable in- terest [31] and are covered in Section 2.6, they can be ignored in a discussion of fiber modes. P is then related to E by the relation P(r,t)= ε 0  ∞ −∞ χ (r,t − t  )E(r,t  )dt  . (2.2.7) Linear susceptibility χ is, in general, a second-rank tensor but reduces to a scalar for an isotropic medium such as silica glass. Optical fibers become slightly birefringent because of unintentional variations in the core shape or in local strain; such birefrin- gent effects are considered in Section 2.2.3. Equation (2.2.7) assumes a spatially local response. However, it includes the delayed nature of the temporal response, a feature that has important implications for optical fiber communications through chromatic dispersion. Equations (2.2.1)–(2.2.7) provide a general formalism for studying wave propaga- tion in optical fibers. In practice, it is convenient to use a single field variable E.By taking the curl of Eq. (2.2.1) and using Eqs. (2.2.2), (2.2.5), and (2.2.6), we obtain the wave equation ∇× ∇× E = − 1 c 2 ∂ 2 E ∂ t 2 − µ 0 ∂ 2 P ∂ t 2 , (2.2.8) 30 CHAPTER 2. OPTICAL FIBERS where the speed of light in vacuum is defined as usual by c =( µ 0 ε 0 ) −1/2 . By introduc- ing the Fourier transform of E(r,t) through the relation ˜ E(r, ω )=  ∞ −∞ E(r,t) exp(i ω t) dt, (2.2.9) as well as a similar relation for P(r,t), and by using Eq. (2.2.7), Eq. (2.2.8) can be written in the frequency domain as ∇× ∇× ˜ E = − ε (r, ω )( ω 2 /c 2 ) ˜ E, (2.2.10) where the frequency-dependent dielectric constant is defined as ε (r, ω )=1 + ˜ χ (r, ω ), (2.2.11) and ˜ χ (r, ω ) is the Fourier transform of χ (r,t). In general, ε (r, ω ) is complex. Its real and imaginary parts are related to the refractive index n and the absorption coefficient α by the definition ε =(n + i α c/2 ω ) 2 . (2.2.12) By using Eqs. (2.2.11) and (2.2.12), n and α are related to ˜ χ as n =(1 + Re ˜ χ ) 1/2 , (2.2.13) α =( ω /nc)Im ˜ χ , (2.2.14) where Re and Im stand for the real and imaginary parts, respectively. Both n and α are frequency dependent. The frequency dependence of n is referred to as chromatic dispersion or simply as material dispersion. In Section 2.3, fiber dispersion is shown to limit the performance of fiber-optic communication systems in a fundamental way. Two further simplifications can be made before solving Eq. (2.2.10). First, ε can be taken to be real and replaced by n 2 because of low optical losses in silica fibers. Second, since n(r, ω ) is independent of the spatial coordinate r in both the core and the cladding of a step-index fiber, one can use the identity ∇× ∇× ˜ E≡ ∇(∇· ˜ E)− ∇ 2 ˜ E = −∇ 2 ˜ E, (2.2.15) where we used Eq. (2.2.3) and the relation ˜ D = ε ˜ E to set ∇· ˜ E = 0. This simplification is made even for graded-index fibers. Equation (2.2.15) then holds approximately as long as the index changes occur over a length scale much longer than the wavelength. By using Eq. (2.2.15) in Eq. (2.2.10), we obtain ∇ 2 ˜ E + n 2 ( ω )k 2 0 ˜ E = 0, (2.2.16) where the free-space wave number k 0 is defined as k 0 = ω /c = 2 π / λ , (2.2.17) and λ is the vacuum wavelength of the optical field oscillating at the frequency ω . Equation (2.2.16) is solved next to obtain the optical modes of step-index fibers. 2.2. WAVE PROPAGATION 31 2.2.2 Fiber Modes The concept of the mode is a general concept in optics occurring also, for example, in the theory of lasers. An optical mode refers to a specific solution of the wave equation (2.2.16) that satisfies the appropriate boundary conditions and has the property that its spatial distribution does not change with propagation. The fiber modes can be classified as guided modes, leaky modes, and radiation modes [14]. As one might expect, sig- nal transmission in fiber-optic communication systems takes place through the guided modes only. The following discussion focuses exclusively on the guided modes of a step-index fiber. To take advantage of the cylindrical symmetry, Eq. (2.2.16) is written in the cylin- drical coordinates ρ , φ , and z as ∂ 2 E z ∂ρ 2 + 1 ρ ∂ E z ∂ρ + 1 ρ 2 ∂ 2 E z ∂φ 2 + ∂ 2 E z ∂ z 2 + n 2 k 2 0 E z = 0, (2.2.18) where for a step-index fiber of core radius a, the refractive index n is of the form n =  n 1 ; ρ ≤ a, n 2 ; ρ > a. (2.2.19) For simplicity of notation, the tilde over ˜ E has been dropped and the frequency de- pendence of all variables is implicitly understood. Equation (2.2.18) is written for the axial component E z of the electric field vector. Similar equations can be written for the other five components of E and H. However, it is not necessary to solve all six equa- tions since only two components out of six are independent. It is customary to choose E z and H z as the independent components and obtain E ρ , E φ , H ρ , and H φ in terms of them. Equation (2.2.18) is easily solved by using the method of separation of variables and writing E z as E z ( ρ , φ ,z)=F( ρ )Φ( φ )Z(z). (2.2.20) By using Eq. (2.2.20) in Eq. (2.2.18), we obtain the three ordinary differential equa- tions: d 2 Z/dz 2 + β 2 Z = 0, (2.2.21) d 2 Φ/d φ 2 + m 2 Φ = 0, (2.2.22) d 2 F d ρ 2 + 1 ρ dF d ρ +  n 2 k 2 0 − β 2 − m 2 ρ 2  F = 0. (2.2.23) Equation (2.2.21) has a solution of the form Z = exp(i β z), where β has the physical significance of the propagation constant. Similarly, Eq. (2.2.22) has a solution Φ = exp(im φ ), but the constant m is restricted to take only integer values since the field must be periodic in φ with a period of 2 π . Equation (2.2.23) is the well-known differential equation satisfied by the Bessel functions [32]. Its general solution in the core and cladding regions can be written as F( ρ )=  AJ m (p ρ )+A  Y m (p ρ ); ρ ≤ a, CK m (q ρ )+C  I m (q ρ ); ρ > a, (2.2.24) 32 CHAPTER 2. OPTICAL FIBERS where A, A  , C, and C  are constants and J m , Y m , K m , and I m are different kinds of Bessel functions [32]. The parameters p and q are defined by p 2 = n 2 1 k 2 0 − β 2 , (2.2.25) q 2 = β 2 − n 2 2 k 2 0 . (2.2.26) Considerable simplification occurs when we use the boundary condition that the optical field for a guided mode should be finite at ρ = 0 and decay to zero at ρ = ∞. Since Y m (p ρ ) has a singularity at ρ = 0, F(0) can remain finite only if A  = 0. Similarly F( ρ ) vanishes at infinity only if C  = 0. The general solution of Eq. (2.2.18) is thus of the form E z =  AJ m (p ρ )exp(im φ )exp(i β z) ; ρ ≤ a, CK m (q ρ )exp(im φ )exp(i β z); ρ > a. (2.2.27) The same method can be used to obtain H z which also satisfies Eq. (2.2.18). Indeed, the solution is the same but with different constants B and D, that is, H z =  BJ m (p ρ )exp(im φ )exp(i β z) ; ρ ≤ a, DK m (q ρ )exp(im φ )exp(i β z); ρ > a. (2.2.28) The other four components E ρ , E φ , H ρ , and H φ can be expressed in terms of E z and H z by using Maxwell’s equations. In the core region, we obtain E ρ = i p 2  β ∂ E z ∂ρ + µ 0 ω ρ ∂ H z ∂φ  , (2.2.29) E φ = i p 2  β ρ ∂ E z ∂φ − µ 0 ω ∂ H z ∂ρ  , (2.2.30) H ρ = i p 2  β ∂ H z ∂ρ − ε 0 n 2 ω ρ ∂ E z ∂φ  , (2.2.31) H φ = i p 2  β ρ ∂ H z ∂φ + ε 0 n 2 ω ∂ E z ∂ρ  . (2.2.32) These equations can be used in the cladding region after replacing p 2 by −q 2 . Equations (2.2.27)–(2.2.32) express the electromagnetic field in the core and clad- ding regions of an optical fiber in terms of four constants A, B, C, and D. These constants are determined by applying the boundary condition that the tangential com- ponents of E and H be continuous across the core–cladding interface. By requiring the continuity of E z , H z , E φ , and H φ at ρ = a, we obtain a set of four homogeneous equations satisfied by A, B, C, and D [19]. These equations have a nontrivial solution only if the determinant of the coefficient matrix vanishes. After considerable algebraic details, this condition leads us to the following eigenvalue equation [19]–[21]:  J  m (pa) pJ m (pa) + K  m (qa) qK m (qa)  J  m (pa) pJ m (pa) + n 2 2 n 2 1 K  m (qa) qK m (qa)  = m 2 a 2  1 p 2 + 1 q 2  1 p 2 + n 2 2 n 2 1 1 q 2  , (2.2.33) [...]... of optical communication systems by broadening optical pulses as they propagate inside the fiber Fiber losses represent another limiting factor because they reduce the signal power reaching the receiver As optical receivers need a certain minimum amount of power for recovering the signal accurately, the transmission distance is inherently limited by fiber losses In fact, the use of silica fibers for optical... wavelength range 1.3–1.6 µ m that is of interest for optical communication systems Typical values of D are in the range 15–18 ps/(km-nm) near 1.55 µ m This wavelength region is of considerable interest for lightwave systems, since, as discussed in Section 2.5, the fiber loss is minimum near 1.55 µ m High values of D limit the performance of 1.55-µ m lightwave systems Since the waveguide contribution D W depends... Eq (2.3.1) was used The parameter β 2 = d 2 β /d ω 2 is known as the GVD parameter It determines how much an optical pulse would broaden on propagation inside the fiber In some optical communication systems, the frequency spread ∆ω is determined by the range of wavelengths ∆λ emitted by the optical source It is customary to use ∆λ in place of ∆ω By using ω = 2π c/λ and ∆ω = (−2π c/λ 2)∆λ , Eq (2.3.3)... during propagation In the case of optical pulses, the polarization state will also be different for different spectral components of the pulse The final polarization state is not of concern for most lightwave systems as photodetectors used inside optical receivers are insensitive to the state of polarization unless a coherent detection scheme is employed What affects such systems is not the random polarization... general conditions We use it in the next section to find the limiting bit rate of optical communication systems 2.4.3 Limitations on the Bit Rate The limitation imposed on the bit rate by fiber dispersion can be quite different depending on the source spectral width It is instructive to consider the following two cases separately Optical Sources with a Large Spectral Width 1 in Eq (2.4.23) Consider first the... by 3 dB: |H( f3 dB )/H(0)| = 1 2 (2.4.37) Note that f 3 dB is the optical bandwidth of the fiber as the optical power drops by 3 dB at this frequency compared with the zero-frequency response In the field of electrical communications, the bandwidth of a linear system is defined as the frequency at which electrical power drops by 3 dB Optical fibers cannot generally be treated as linear with respect to... drops down to 20% for V = 1 For this reason most telecommunication single-mode fibers are designed to operate in the range 2 < V < 2.4 2.3 Dispersion in Single-Mode Fibers It was seen in Section 2.1 that intermodal dispersion in multimode fibers leads to considerable broadening of short optical pulses (∼ 10 ns/km) In the geometrical-optics CHAPTER 2 OPTICAL FIBERS 38 description, such broadening was attributed... foregoing discussion assumes that the optical source used to produce the in∆ ω0 put pulses is nearly monochromatic such that its spectral width satisfies ∆ω L (under continuous-wave, or CW, operation), where ∆ω 0 is given by Eq (2.4.14) This CHAPTER 2 OPTICAL FIBERS 50 condition is not always satisfied in practice To account for the source spectral width, we must treat the optical field as a stochastic process... frequency components of a pulse acquire different polarization states, resulting in pulse broadening This phenomenon is called polarization-mode dispersion (PMD) and becomes a limiting factor for optical communication systems operating at high bit rates It is possible to make fibers for which random fluctuations in the core shape and size are not the governing factor in determining the state of polarization... ∼ 1 ps/(km-nm)] For a semiconductor laser, the spectral width ∆λ is 2–4 nm even when the laser operates in several longitudinal modes The BL product of such lightwave systems can exceed 100 (Gb/s)-km Indeed, 1.3-µ m telecommunication systems typically operate at a bit rate of 2 Gb/s with a repeater spacing of 40–50 km The BL product of single-mode fibers can exceed 1 (Tb/s)-km when singlemode semiconductor . chapter focuses on the role of optical fibers as a communication channel in lightwave systems. In Section 2.1 we use geometrical- optics description to explain. for optical fibers in the wavelength region 0.5–2 µ m, a range that covers the low-loss region of optical fibers that is of interest for fiber -optic communication