60 PROPAGATION OF SIGNALS IN OPTICAL FIBER in the cladding and partly in the core, and thus their propagation constants fl satisfy kn2 < fl < knl. Instead of the propagation constant of a mode, we can consider its effective index neff = fl/k. The effective index of a mode thus lies between the refractive indices of the cladding and the core. For a monochromatic wave in a single-mode fiber, the effective index is analogous to the refractive index: the speed at which the wave propagates is C/neff. We will discuss the propagation constant further in Section 2.3. The solution of (2.10) and (2.11) is discussed in [Agr97, Jeu90]. We only state some important properties of the solution in the rest of this section. The core radius a, the core refractive index n l, and the cladding refractive index n2 must satisfy the cutoff condition def 2rr / V = ~avn~ - n~ < 2.405 (2.12) in order for a fiber to be single moded at wavelength ~. The smallest wavelength ~. for which a given fiber is single moded is called the cutoff wavelength and denoted by )~cutoff. Note that V decreases with a and A - (nl - n2)/nl. Thus single-mode fibers tend to have small radii and small core-cladding refractive index differences. Typical values are a = 4/zm and A 0.003, giving a V value close to 2 at 1.55 tzm. The calculation of the cutoff wavelength )~cutoff for these parameters is left as an exercise (Problem 2.4). Since the value of A is typically small, the refractive indices of the core and cladding are nearly equal, and the light energy is not strictly confined to the fiber core. In fact, a significant portion of the light energy can propagate in the fiber cladding. For this reason, the fiber modes are said to be weakly guided. For a given mode, for example, the fundamental mode, the proportion of light energy that propagates in the core depends on the wavelength. This gives rise to spreading of pulses through a phenomenon called waveguide dispersion, which we will discuss in Section 2.3. A fiber with a large value of the V parameter is called a multimode fiber and supports several modes. For large V, the number of modes can be approximated by V2/2. For multimode fibers, typical values are a = 25/zm and A = 0.005, giving a V value of about 28 at 0.8/zm. Thus a typical multimode fiber supports a few hundred propagation modes. The parameter V can be viewed as a normalized wave number since for a given fiber (fixed a, nl, and n2) it is proportional to the wave number. It is useful to know the propagation constant/5 of the fundamental mode Supported by a fiber as a function of wavelength. This is needed to design components such as filters whose operation depends on coupling energy from one mode to another, as will become clear in Chapter 3. For example, such an expression can be used to calculate the velocity with which pulses at different wavelengths propagate in the fiber. The exact 2.1 Light Propagation in Optical Fiber 61 determination of/3 must be done numerically. But, analogous to the normalized wave number, we can define a normalized propagation constant (sometimes called a normalized effective index), b, by neff /'/2 bde f f12 kZn 2 _ 2 2 This normalized propagation constant can be approximated with a relative error less than 0.2% by the equation b(V) ,~ (1.1428- 0.9960/V) 2 for V in the interval (1.5,2.5); see [Neu88, p. 71] or [Jeu90, p. 25], where the result is attributed to [RN76]. This is the range of V that is of interest in the design of single-mode optical fibers. Polarization We defined a fiber mode as a solution of the wave equations that satisfies the boundary conditions at the core-cladding interface. Two linearly independent solutions of the wave equations exist for all ~, however large. Both these solutions correspond to the fundamental mode and have the same propagation constant. The other solutions exist only for ~. < ~.cutoff. Assume that the electric field E(r, c0) is written as E(r, co)= Ex~x + Eyey +/~zez, where ~x, Oy, and Oz are the unit vectors along the x, y, and z directions, respectively. Note that each of Ex, Ey, and Ez can depend, in general, on x, y, and z. We take the direction of propagation (fiber axis) as z and consider the two linearly independent solutions to (2.10) and (2.11) that correspond to the fundamental mode. It can be shown (see Ueu90]) that one of these solutions has E~ = 0 but Ey, k'z -r 0, whereas the other has Ey = 0 but Ex, Ez ~ O. Since z is also the direction of propagation, Ez is called the longitudinal component. The other nonzero component, which is either Ex or Ey, is called the transverse component. Before we discuss the electric field distributions of the fundamental mode further, we need to understand the concept of polarization of an electric field. Note that this is different from the dielectric polarization P discussed above. Since the electric field is a vector, for a time-varying electric field, both the magnitude and the direction can vary with time. A time-varying electric field is said to be linearly polarized if its direction is a constant, independent of time. If the electric field associated with an electromagnetic wave has no component along the direction of propagation of the wave, the electric field is said to be transverse. For the fundamental mode of a single-mode fiber, the magnitude of the longitudinal component fEz) is much smaller than the magnitude of the transverse component (Ex or Ey). Thus the electric field 62 PROPAGATION OF SIGNALS IN OPTICAL FIBER associated with the fundamental mode can effectively be assumed to be a transverse field. With this assumption, the two linearly independent solutions of the wave equa- tions for the electric field are linearly polarized along the x and y directions. Since these two directions are perpendicular to each other, the two solutions are said to be orthogonally polarized. Since the wave equations are linear, any linear combination of these two linearly polarized fields is also a solution and thus a fundamental mode. The state of polarization (SOP) refers to the distribution of light energy among the two polarization modes. The reason the fiber is still termed single mode is that these two polarization modes are degenerate; that is, they have the same propagation con- stant, at least in an ideal, perfectly circularly symmetric fiber. Thus, though the energy of a pulse is divided between these two polarization modes, since they have the same propagation constant, it does not give rise to pulse spreading by the phenomenon of dispersion. In practice, fibers are not perfectly circularly symmetric, and the two orthogonally polarized modes have slightly different propagation constants; that is, practical fibers are slightly birefringent. Since the light energy of a pulse propagating in a fiber will usually be split between these two modes, this birefringence gives rise to pulse spreading. This phenomenon is called polarization-mode dispersion (PMD). This is similar, in principle, to pulse spreading in the case of multimode fibers, but the effect is much weaker. We will study the effects of PMD on optical communication systems in Section 5.7.4. PMD is illustrated in Figure 2.5. The assumption here is that the propagation constants of the two polarizations are constant throughout the length of the fiber. If the difference in propagation constants is denoted by Aft, then the time spread, or differential group delay (DGD), due to PMD after the pulse has propagated through a unit length of fiber is given by Ar =/x~/,o. A typical value of the DGD is A r = 0.5 ps/km, which suggests that after propagating through 100 km of fiber, the accumulated time spread will be 50 ps comparable to the bit period of 100 ps for a 10 Gb/s system. This would effectively mean that 10 Gb/s transmission would not be feasible over any reasonable distances due to the effects of PMD. However, the assumption of fixed propagation constants for each polarization mode is unrealistic for fibers of practical lengths since the fiber birefringence changes over the length of the fiber. (It also changes over time due to temperature and other environmental changes.) The net effect is that the PMD effects are not nearly as bad as indicated by this model since the time delays in different segments of the fiber vary randomly and tend to cancel each other. This results in an inverse dependence of the 2.1 Light Propagation in Optical Fiber 63 Figure 2.5 Illustration of pulse spreading due to PMD. The energy of the pulse is assumed to be split between the two orthogonally polarized modes, shown by horizontal and vertical pulses, in (a). Due to the fiber birefringence, one of these components travels slower than the other. Assuming the horizontal polarization component travels slower than the vertical one, the resulting relative positions of the horizontal and vertical pulses are shown in (b). The pulse has been broadened due to PMD since its energy is now spread over a larger time period. DGD not on the link length, but on the square root of the link length. Typical values lie in the range 0.1-1 ps/k~/-k-m. We undertake a quantitative discussion of the effects of PMD, and the system limitations imposed by it, in Section 5.7.4. Many optical materials and components constructed using them respond dif- ferently to the different polarization components in the input light. Some compo- nents where these polarization effects are used include isolators, circulators, and acousto-optic tunable filters, which we will study in Chapter 3. The two polarization modes also see slightly different losses in many of these components. This depen- dence of the loss through a component on the state of polarization of the input light is termed the polarization-dependent loss (PDL) and is an important characteristic that has to be specified for most components. Light Propagation in Dielectric Waveguides A dielectric is a material whose conductivity is very small; silica is a dielectric material. Any dielectric region of higher refractive index placed in another dielectric of lower refractive index for the purpose of guiding (optical) waves can be called a dielectric waveguide. Thus an optical fiber is also a dielectric waveguide. However, the term is more often used to refer to a device where the guiding occurs in some region of a glass or dielectric slab. Examples of such devices include semiconductor amplifiers, semiconductor lasers, dielectric switches, multiplexers, and other integrated optic devices. In many applications, the guiding region has a rectangular cross section. In 64 PROPAGATION OF SIGNALS IN OPTICAL FIBER contrast, the guiding region of an optical fiber is its core, which has a circular cross section. The propagation of light in waveguides can be analyzed in a fashion similar to that of propagation in optical fiber. In the ray theory approach, which is applicable when the dimensions of the guiding region are much larger than the wavelength, the guiding process is due to total internal reflection; light that is launched into the waveguide at one end is confined to the guiding region. When we use the wave theory approach, we again find that only certain distributions of the electromagnetic fields are supported or guided by the waveguide, and these are called the modes of the waveguide. Furthermore, the dimensions of the waveguide can be chosen so that the waveguide supports only a single mode, the fundamental mode, above a certain cutoff wavelength, just as in the case of optical fiber. However, the modes of a rectangular waveguide are quite different from the fiber modes. For most rectangular waveguides, their width is much larger than their depth. For these waveguides, the modes can be classified into two groups: one for which the electric field is approximately transverse, called the TE modes, and the other for which the magnetic field is approximately transverse, called the TM modes. (The transverse approximation holds exactly if the waveguides have infinite width; such waveguides are called slab waveguides.) If the width of the waveguide is along the x direction (and much larger than the depth), the TE modes have an electric field that is approximately linearly polarized along the x direction. The same is true for the magnetic fields of TM modes. The fundamental mode of a rectangular waveguide is a TE mode. But in some applications, for example, in the design of isolators and circulators (Section 3.2.1), the waveguide is designed to support two modes: the fundamental TE mode and the lowest-order TM mode. For most waveguides, for example, those made of silica, the propagation constants of the fundamental TE mode and lowest-order TM mode are very close to each other. The electric field vector of a light wave propagating in such a waveguide can be expressed as a linear combination of the TE and TM modes. In other words, the energy of the light wave is split between the TE and TM modes. The proportion of light energy in the two modes depends on the input excitation. This proportion also changes when gradual or abrupt discontinuities are present in the waveguide. In some applications, for example, in the design of acousto-optic tunable filters (Section 3.3.9), it is desirable for the propagation constants of the fundamental TE mode and lowest-order TM mode to have a significant difference. This can be arranged by constructing the waveguide using a birefringent material, such as lithium niobate. For such a material, the refractive indices along different axes are quite different, resulting in the effective indices of the TE and TM modes being quite different. 2.2 Loss and Bandwidth 65 2.2 Loss and Bandwidth Although we neglected the attenuation loss in the fiber in the derivation of propaga- tion modes, its effect can be modeled easily as follows" the output power Pout at the end of a fiber of length L is related to the input power Pin by Pout - Pin e-C~ L. Here the parameter c~ represents the fiber attenuation. It is customary to express the loss in units of dB/km; thus a loss of ~dB dB/km means that the ratio Pout~Pin for L - 1 km satisfies Pout 10 log10 Pin cedB or C~dB = (10 log10 e)~ ~ 4.343~. The two main loss mechanisms in an optical fiber are material absorption and Rayleigh scattering. Material absorption includes absorption by silica as well as the impurities in the fiber. The material absorption of pure silica is negligible in the entire 0.8-1.6 #m band that is used for optical communication systems. The reduction of the loss due to material absorption by the impurities in silica has been very important in making optical fiber the remarkable communication medium that it is today. The loss has now been reduced to negligible levels at the wavelengths of interest for optical communication so much so that the loss due to Rayleigh scattering is the dominant component in today's fibers in all the three wavelength bands used for optical communication: 0.8 lzm, 1.3 #m, and 1.55/zm. Figure 2.6 shows the attenuation loss in silica as a function of wavelength. We see that the loss has local minima at these three wavelength bands with typical losses of 2.5, 0.4, and 0.25 dB/km. (In a typical optical communication system, a signal can undergo a loss of about 20-30 dB before it needs to be amplified or regenerated. At 0.25 dB/km, this corresponds to a distance of 80-120 km.) The attenuation peaks separating these bands are primarily due to absorption by the residual water vapor in the silica fiber. The bandwidth can be measured either in terms of wavelength A~. or in terms of frequency Af. These ~re related by the equation C Af ~ ~-~ A)~. This equation can be derived by differentiating the relation f = c/)~ with respect to 4. Consider the long wavelength 1.3 and 1.5/zm bands, which are the primary bands used today for optical communication. The usable bandwidth of optical fiber in 66 PROPAGATION OF SIGNALS IN OPTICAL FIBER r~ 0 20 1~ f 5 2 1 0.5 0.2 9 , i i i i 9 9 9 08 1 12 14 16 Wavelength, )~ (~tm) Figure 2.6 Attenuation loss in silica as a function of wavelength. (After [Agr97].) these bands, which we can take as the bandwidth over which the loss in decibels per kilometer is within a factor of 2 of its minimum, is approximately 80 nm at 1.3/~m and 180 nm at 1.55 ~m. In terms of optical frequency, these bandwidths correspond to about 35,000 GHz! This is an enormous amount of bandwidth indeed, considering that the bit rate needed for most user applications today is no more than a few tens of megabits per second. The usable bandwidth of fiber in most of today's long-distance networks is limited by the bandwidth of the erbium-doped fiber amplifiers (see Section 3.4) that are widely deployed, rather than by the bandwidth of the silica fiber. Based on the availability of amplifiers, the low-loss band at 1.55 ~m is divided into three regions, as shown in Figure 2.7. The middle band from 1530 to 1565 nm is the conventional or C-band where WDM systems have operated using conventional erbium-doped fiber amplifiers. The band from 1565 to 1625 nm, which consists of wavelengths longer than those in the C-band, is called the L-band and is today being used in high-capacity WDM systems, with the development of gain-shifted erbium-doped amplifiers (see Section 3.4) that provide amplification in this band. The band below 1530 nm, consisting of wavelengths shorter than those in the C-band, is called the S-band. Fiber Raman amplifiers (Section 3.4.4) provide amplification in this band. Lucent introduced a new kind of single-mode optical fiber, called AllWave fiber, in 1998, which virtually eliminates the absorption peaks due to water vapor. This fiber has an even larger bandwidth and is expected to be deployed in metropolitan-area networks that do not use erbium-doped fiber amplifiers. 2.2 Loss and Bandwidth 67 0.30 0.28 0.26 r~ e 0.24 0.22 S-band C-band 0.20 ' ' ' ' ' ' ' ' ' ' ' ' 1450 1500 1550 Wavelength (nm) L-band v , l , , ~ ~ i 1600 1650 Figure 2.7 The three bands, S-band, C-band, and L-band, based on amplifier availabil- ity, within the low-loss region around 1.55 #m in silica fiber. (After [Kan99].) 2.2.1 As we saw earlier in this section, the dominant loss mechanism in optical fiber is Rayleigh scattering. Rayleigh scattering arises because of fluctuations in the density of the medium (silica) at the microscopic level. We refer to [BW99] for a detailed description of the scattering mechanism. The loss due to Rayleigh scattering is a fundamental one and decreases with increasing wavelength. The loss coefficient dR due to Rayleigh scattering at a wavelength Z can be written as dR = A/Z 4, where A is called the Rayleigh scattering coefficient. Note that the Rayleigh scattering loss decreases rapidly with increasing wavelength due to the Z -4 dependence. Glasses with substantially lower Rayleigh attenuation coefficients at 1.55 #m are not known. In order to reduce the fiber loss below the current best value of about 0.2 dB/km, one possibility is to operate at higher wavelengths, so as to reduce the loss due to Rayleigh scattering. However, at such higher wavelengths, the material absorption of silica is quite significant. It may be possible to use other materials such as fluorozirconate (ZiFr4) in order to realize the low loss that is potentially possible by operating at these wavelengths [KK97, p. 69]. Bending Loss Optical fibers need to be bent for various reasons both when deployed in the field and particularly within equipment. Bending leads to "leakage" of power out of the 68 PROPAGATION OF SIGNALS IN OPTICAL FIBER fiber core into the cladding, resulting in additional loss. A bend is characterized by the bend radius~the radius of curvature of the bend (radius of the circle whose arc approximates the bend). The "tighter" the bend, the smaller the bend radius and the larger the loss. The bend radius must be of the order of a few centimeters in order to keep the bending loss low. Also, the bending loss at 1550 nm is higher than at 1310 nm. The ITU-T standards specify that the additional loss at 1550 nm due to bending must be in the range 0.5-1 dB, depending on the fiber type, for 100 turns of fiber wound with a radius of 37.5 mm. Thus a bend with a radius of 4 cm results in a bending loss of < 0.01 dB. However, the loss increases rapidly as the bend radius is reduced, so that care must be taken to avoid sharp bends, especially within equipment. 2.3 Chromatic Dispersion Dispersion is the name given to any effect wherein different components of the transmitted signal travel at different velocities in the fiber, arriving at different times at the receiver. We already discussed the phenomenon of intermodal dispersion in Section 2.1 and polarization-mode dispersion in Section 2.1.2. Our main goal in this section will be to understand the phenomenon of chromatic dispersion and the system limitations imposed by it. Other forms of dispersion and their effect on the design of the system are discussed in Section 5.7. Chromatic dispersion is the term given to the phenomenon by which different spectral components of a pulse travel at different velocities. To understand the effect of chromatic dispersion, we must understand the significance of the propagation constant. We will restrict our discussion to single-mode fiber since in the case of multimode fiber, the effects of intermodal dispersion usually overshadow that of chromatic dispersion. So the propagation constant in our discussions will be that associated with the fundamental mode of the fiber. Chromatic dispersion arises for two reasons. The first is that the refractive in- dex of silica, the material used to make optical fiber, is frequency dependent. Thus different frequency components travel at different speeds in silica. This component of chromatic dispersion is termed material dispersion. Although this is the principal component of chromatic dispersion for most fibers, there is a second component, called waveguide dispersion. To understand the physical origin of waveguide disper- sion, recall from Section 2.1.2 that the light energy of a mode propagates partly in the core and partly in the cladding. Also recall that the effective index of a mode lies between the refractive indices of the cladding and the core. The actual value of the effective index between these two limits depends on the proportion of power that is contained in the cladding and the core. If most of the power is contained in the 2.3 Chromatic Dispersion 69 2.3.1 core, the effective index is closer to the core refractive index; if most of it propagates in the cladding, the effective index is closer to the cladding refractive index. The power distribution of a mode between the core and cladding of the fiber is itself a function of the wavelength. More accurately, the longer the wavelength, the more power in the cladding. Thus, even in the absence of material dispersion~so that the refractive indices of the core and cladding are independent of wavelength~if the wavelength changes, this power distribution changes, causing the effective index or propagation constant of the mode to change. This is the physical explanation for waveguide dispersion. A mathematical description of the propagation of pulses in the presence of chro- matic dispersion is given in Appendix E. Here we just note that the shape of pulses propagating in optical fiber is not preserved, in general, due to the presence of chro- matic dispersion. The key parameter governing the evolution of pulse shape is the second derivative f12 = d2~/do) 2 of the propagation constant ft. /32 is called the group velocity dispersion parameter, or simply the G VD parameter. The reason for this terminology is as follows. If fll = dfl/do), 1 ~ill is the velocity with which a pulse propagates in optical fiber and is called the group velocity. The concept of group velocity is discussed in greater detail in Appendix E. Since/32 is related to the rate of change of group velocity with frequency, chromatic dispersion is also called group velocity dispersion. In the absence of chromatic dispersion,/32 = 0, and in this ideal situation, all pulses would propagate without change in shape. In general, not only is f12 -~ 0, it is also a function of the optical frequency or, equivalently, the optical wavelength. For most optical fibers, there is a so-called zero-dispersion wavelength, which is the wavelength at which the GVD parameter I32 = 0. If f12 > 0, the chromatic dispersion is said to be normal. When f12 < 0, the chromatic dispersion is said to be anomalous. Chirped Gaussian Pulses We next discuss how a specific family of pulses changes shape as they propagate along a length of single-mode optical fiber. The pulses we consider are called chirped Gaussian pulses. An example is shown in Figure 2.8. The term Gaussian refers to the envelope of the launched pulse. Chirped means that the frequency of the launched pulse changes with time. Both aspects are illustrated in Figure 2.8, where the center frequency w0 has been greatly diminished for the purposes of illustration. We consider chirped pulses for three reasons. First, the pulses emitted by semicon- ductor lasers when they are directly modulated are considerably chirped, and such transmitters are widely used in practice. As we will see in Chapter 5, this chirp has a significant effect on the design of optical communication systems. The second reason is that some nonlinear effects that we will study in Section 2.4 can cause otherwise . ' ' ' ' ' ' 1450 1500 1550 Wavelength (nm) L-band v , l , , ~ ~ i 1600 1650 Figure 2.7 The three bands, S-band, C-band, and L-band, based on amplifier availabil-. called the cutoff wavelength and denoted by )~cutoff. Note that V decreases with a and A - (nl - n2)/nl. Thus single-mode fibers tend to have small radii and small core-cladding refractive. values are a = 25/zm and A = 0.005, giving a V value of about 28 at 0.8/zm. Thus a typical multimode fiber supports a few hundred propagation modes. The parameter V can be viewed as a normalized