50 PROPAGATION OF SIGNALS IN OPTICAL FIBER Figure 2.1 Cross section and longitudinal section of an optical fiber showing the core and cladding regions, a denotes the radius of the fiber core. theory model based on solving Maxwell's equations. We then devote the rest of the chapter to understanding the basics of chromatic dispersion and fiber nonlinearities. Designing advanced systems optimized with respect to these parameters is treated in Chapter 5. 2.1 2.1.1 Light Propagation in Optical Fiber An optical fiber consists of a cylindrical core surrounded by a cladding. The cross section of an optical fiber is shown in Figure 2.1. Both the core and the cladding are made primarily of silica (SiO2), which has a refractive index of approximately 1.45. The refractive index of a material is the ratio of the speed of light in a vacuum to the speed of light in that material. During the manufacturing of the fiber, certain impurities (or dopants) are introduced in the core and/or the cladding so that the refractive index is slightly higher in the core than in the cladding. Materials such as germanium and phosphorous increase the refractive index of silica and are used as dopants for the core, whereas materials such as boron and fluorine that decrease the refractive index of silica are used as dopants for the cladding. As we will see, the resulting higher refractive index of the core enables light to be guided by the core, and thus propagate through the fiber. Geometrical Optics Approach We can obtain a simplified understanding of light propagation in optical fiber using the so-called ray theory or geometrical optics approach. This approach is valid when the fiber that is used has a core radius a that is much larger than the operating wavelength k. Such fibers are termed multimode, and first-generation optical com- munication links were built using such fibers with a in the range of 25-100/~m and )~ around 0.85 ~m. 2.1 Light Propagation in Optical Fiber 51 II Figure 2.2 Reflection and refraction of light rays at the interface between two media. In the geometrical optics approach, light can be thought of as consisting of a number of "rays" propagating in straight lines within a material (or medium) and getting reflected and/or refracted at the interfaces between two materials. Figure 2.2 shows the interface between two media of refractive index n l and n2. A light ray from medium 1 is incident on the interface of medium 1 with medium 2. The angle of incidence is the angle between the incident ray and the normal to the interface between the two media and is denoted by 01. Part of the energy is reflected into medium 1 as a reflected ray, and the remainder (neglecting absorption) passes into medium 2 as a refracted ray. The angle of reflection 01r is the angle between the reflected ray and the normal to the interface; similarly, the angle of refraction 02 is the angle between the refracted ray and the normal. The laws of geometrical optics state that Olr ~ O1 and n 1 sin O1 n 2 sin 02. (2.1) Equation (2.1) is known as Snell's law. As the angle of incidence 01 increases, the angle of refraction 02 also increases. If nl > n2, there comes a point when 02 = 7r/2 radians. This happens when 01 sin -1 n2/nl. For larger values of 01, there is no refracted ray, and all the energy from the incident ray is reflected. This phenomenon is called total internal reflection. The smallest angle of incidence for which we get total internal reflection is called the critical angle and equals sin -1 n2/nl. Simply stated, from the geometrical optics viewpoint, light propagates in optical fiber due to a series of total internal reflections that occur at the core-cladding interface. This is depicted in Figure 2.3. In this figure, the coupling of light from the medium outside (taken to be air with refractive index no) into the fiber is also shown. 52 PROPAGATION OF SIGNALS IN OPTICAL FIBER Figure 2.3 Propagation of light rays in optical fiber by total internal reflection. It can be shown using Snell's law (see Problem 2.1) that only those light rays that are incident at an angle 00 < 0~ nax= sin -i V/n12 - n22 (2.2) no at the air-core interface will undergo total internal reflection at the Core-cladding interface and will thus propagate. Such rays are called guided rays, and 0~ nax is called the acceptance angle. The refractive index difference n l - n2 is usually small, and it is convenient to denote the fractional refractive index difference (nl - nz)/nl by A. For small A, 0~ nax ~ sin -1 nl,/~ As an example, if A = 0.01 which is a typical n0 " value for (multimode) fiber, and nl = 1.5, a typical value for silica, assuming we are coupling from air, so that no = 1, we obtain 0~ nax ~ 12 ~ Owing to the different lengths of the paths taken by different guided rays, the energy in a narrow (in time) pulse at the input of the fiber will be spread out over a larger time interval at the output of the fiber. A measure of this time spread, which is called intermoclal dispersion, is obtained by taking the difference in time, ~T, between the fastest and the slowest guided rays. We will see later that by suitably designing the fiber, intermodal dispersion can be significantly reduced (graded-index fiber) and even eliminated (single-mode fiber). We now derive an approximate measure of the time spread due to intermodal dispersion. Consider a fiber of length L. The fastest guided ray is the one that travels along the center of the core and takes a time Tf = Lnl/c to traverse the fiber, c being the speed of light in a vacuum. The slowest guided ray is incident at the critical angle on the core-cladding interface, and it can be shown that it takes a time T~ = Ln 2/cn2 to propagate through the fiber. Thus L n21A. ,~T=Ts-rf= c n2 2.1 Light Propagation in Optical Fiber 53 How large can ~T be before it begins to matter? That depends on the bit rate used. A rough measure of the delay variation 8T that can be tolerated at a bit rate of B b/s is half the bit period 1/2B s. Thus intermodal dispersion sets the following limit: Ln 2 1 8T - A < ~. (2.3) c n2 2B The capacity of an optical communication system is frequently measured in terms of the bit rate-distance product. If a system is capable of transmitting x_Mb/s over a distance of y km, it is said to have a bit rate-distance product of xy (Mb/s)-km. The reason for doing this is that usually the same system is capable of transmitting x' Mb/s over y~ km providing x~y ~ < xy; thus only the product of the bit rate and the distance is constrained. (This is true for simple systems that are limited by loss and/or intermodal dispersion, but is no longer true for systems that are limited by chromatic dispersion and nonlinear effects in the fiber.) From (2.3), the intermodal dispersion constrains the bit rate-distance product of an optical communication link to ln2 c BL<-~ 2n 2 A For example, if A = 0.01 and nl = 1.5(~ n2), we get BL < 10 (Mb/s)-km. This limit is plotted in Figure 2.4. Note that 0~ ax increases with increasing A, which causes the limit on the bit rate-distance product to decrease with increasing A. The value of A is typically chosen to be less than 1% so as to minimize the effects of intermodal dispersion, and since 0~ ~x is consequently small, lenses or other suitable mechanisms are used to couple light into the fiber. The fiber we have described is a step-index fiber since the variation of the refrac- tive index along the fiber cross section can be represented as a function with a step at the core-cladding interface. However, almost all multimode fibers used today are graded-index fibers, and the refractive index decreases gradually, or continuously, from its maximum value at the center of the core to the value in the cladding at the core-cladding interface. This has the effect of reducing 8T because the rays traversing the shortest path through the center of the core encounter the highest refractive index and travel slowest, whereas rays traversing longer paths encounter regions of lower refractive index and travel faster. For the optimum graded-index profile (which is very nearly a quadratic decrease of the refractive index in the core from its maximum value at the center to its value in the cladding), it can be shown that ST, the time 54 PROPAGATION OF SIGNALS IN OPTICAL FIBER +.a ~ 10 0.1 fiber 0.01 ~ Step-index 0.001 ~ 2 5 10 20 ' 50' '' i00 Distance, L (km) Figure 2.4 Limit on the bit rate-distance product due to intermodal dispersion in a step-index and a graded-index fiber. In both cases, A 0.01 and n l = 1.5. difference between the fastest and slowest rays to travel a length L of the fiber, is given by ST LnlA 2 c 8 Assuming that the condition aT < 1/2B, where B is the bit rate, must be satisfied, we get the following limit on the bit rate-distance product of a communication system employing graded-index fiber" BL< 4c nlA2" For example, if A = 0.01 and nl = 1.5, we get BL < 8 (Gb/s)-km. This limit is also plotted in Figure 2.4 along with the limit for step-index fiber. For instance, there are commercial systems operating at 200 Mb/s over a few kilometers using multimode fibers today. Graded-index fibers significantly reduce the effects of intermodal dispersion. But in order to overcome intermodal dispersion completely, you must use fibers whose core radius is appreciably smaller and of the order of the operating wavelength. Such fibers are called single-mode fibers (the precise reason for this term will become clear later). Roughly speaking, the different paths that light rays can take in a multimode 2.1 Light Propagation in Optical Fiber 55 2.1.2 fiber can be termed as different modes in which light can propagate. In a single-mode fiber, there is only one mode in which light can propagate. The physical reason for the confinement of the light within the fiber core can no longer be attributed to total internal reflection since this picture is invalid when the fiber core radius is comparable to the light wavelength, as is the case for single-mode fiber. The following physical explanation for the propagation of light in single-mode fiber is based on [Neu88]. In any medium with a constant refractive index, a narrow light beam tends to spread due to a phenomenon called diffraction. Thus in such a medium, the beam width will increase as light propagates. This effect can be counteracted by using an inhomogeneous medium in which the refractive index near the beam center is appropriately larger than the refractive index at the beam periphery. In such a medium, the beam center travels slightly slower than the beam periphery so that the medium effectively provides continuous focusing of the light to offset the spreading effect of diffraction. This allows the beam to be guided in the medium and go long distances with low loss, which would not be the case if the beam were allowed to spread out. A step-index optical fiber is an example of such an inhomogeneous medium since the refractive index in the core (beam center) is larger than that in the cladding (beam periphery). In the following sections, we will provide a more quantitative description of the propagation of light in single-mode fibers using the wave theory approach. The wave theory is more general and is applicable for all values of the fiber radius. The ray theory is an approximation that holds when the optical wavelength is much smaller than the radius of the fiber core. Our objective is to gain a quantitative understanding of two phenomena that are important in the design of fiber optic communication systems: chromatic dispersion and fiber nonlinearities. Wave Theory Approach Light is an electromagnetic wave, and its propagation in any medium is governed by Maxwell's equations. These equations are stated in Appendix D. The propagation of light can be described by specifying the evolution of the associated electric and magnetic field vectors in space and time, denoted by E(r, t) and H(r, t), respectively. Here r denotes the position vector and t denotes time. Sometimes it will be more con- venient to deal with the Fourier transforms of these vectors. The Fourier transform of E is defined as E(r, o)) - E(r, t) exp(iwt) dt. (2.4) The Fourier transform of H and other vectors that we will encounter later are defined similarly. Note that even when E(r, t) is real, l~(r, co) can be complex. It turns out to 56 PROPAGATION OF SIGNALS IN OPTICAL FIBER be quite convenient, in many cases, to allow E(r, t) to be complex valued as well. But it is understood that we should consider only the real part of the solutions obtained. The electrons in an atom are negatively charged, and the nucleus carries a positive charge. Thus when an electric field is applied to a material such as silica, the forces experienced by the nuclei and the electrons are in opposite directions. These forces result in the atoms being polarized or distorted. The induced electric polarization of the material, or dielectric polarization, can be described by a vector P, which depends both on the material properties and the applied field. The dielectric polarization can be viewed as the response of the medium to the applied electric field. We will shortly discuss the relationship between P and E in detail. It is convenient to define another vector D called the electric flux density, which is simply related to the electric field E and dielectric polarization P by D = ~oE + P, (2.5) where 6o is a constant called the permittivity of vacuum. The flux density in a vacuum is simply 60E. The magnetic polarization M and the magnetic flux density B can be defined in an analogous fashion as B = tzo(H + M). (2.6) However, since silica is a nonmagnetic material, B = ~0H, where ~0 is a constant called the permeability of vacuum. Maxwell's equations take into account the effect of material properties on the propagation of electromagnetic waves, since they not only involve E and H but also the flux densities D and the magnetic flux density B. The relationship between P and E in optical fiber due to the nature of silica is the origin of two important effects related to the propagation of light in fiber, namely, dispersion and nonlinearities. These two effects set limits on the performance of optical communication systems today. We will understand the origin of these effects in this chapter. Methods of dealing with these effects in optical communication systems will be discussed in Chapter 5. The relationship between the vectors P and E depends on the nature of the medium. Next, we discuss five characteristics of a medium and their effect on the relationship between the dielectric polarization P in the medium and the applied electric field E. Locality of Response. In a medium whose response to the applied electric field is local, P(r) at r = rl depends only on E(rl). The values of E(r) for r -~ rl have no effect on P(rl). This property holds to a good degree of approximation for silica fibers in the 0.5-2 ~m wavelength range that is of interest in optical communication systems. 2.1 Light Propagation in Optical Fiber 57 Isotropy. An isotropic medium is one whose electromagnetic properties such as the refractive index are the same in all directions. In an isotropic medium, E and P are vectors with the same orientation. Silica is an isotropic medium, and a perfectly cylindrical optical fiber is isotropic in the transverse plane. However, this is not exactly true if the cylindrical symmetry of fiber is destroyed. A medium whose re- fractive indices along two different directions, for example, the x and y axes in an appropriate coordinate system, are different is said to birefringent. Birefringence can arise due to the geometry of the medium or due to the intrinsic property of the material. An optical fiber that does not possess cylindrical symmetry is therefore said to be geometrically birefringent. Birefringence of materials such as lithium niobate is exploited in designing certain components such as modulators, isolators, and tunable filters. We will discuss these components in Chapter 3. A bent fiber is also not an isotropic medium. Bending leads to additional loss, and we discuss this in Section 2.2. Linearity. In a linear, isotropic medium, P(r, t) = eo g (r, t - t')E(r, t') dt', (X) (2.7) where x is called the susceptibility, or more accurately, linear susceptibility, of the medium (silica). Thus the induced dielectric polarization is obtained by con- volving the applied electric field with (E0 times) the susceptibility of the medium. If P and ~ denote the Fourier transforms of P and X, respectively, (2.7) can be written in terms of Fourier transforms as P(r, co) = E0)~ (r, co)l'~(r, co). (2.8) Electrical engineers will note that in this linear case, the dielectric polarization can be viewed as the output of a linear system with impulse response Eox(r, t), or transfer function E0)~(r, co), and input E(r, t) (or l~(r, co)). It is important to note that the value of P at time t depends not only on the value of E at time t but also on the values of E before time t. Thus the response of the medium to the applied electric field is not instantaneous. (In other words, )~ (r, co) is not independent of co.) This is the origin of an important type of dispersion known as chromatic dispersion, which sets a fundamental limit on the performance of optical communication systems. If the medium response is instantaneous so that the susceptibility (impulse response) is a Dirac delta function, its Fourier transform would be a constant, independent of co, and chromatic dispersion would vanish. Thus the origin of chromatic dispersion lies in the delayed response of the dielectric polarization in the silica medium to the applied electric field. 58 PROPAGATION OF SIGNALS IN OPTICAL FIBER This linear relationship between P and E does not hold exactly for silica but is a good approximation at moderate signal powers and bit rates. The effects of nonlinearities on the propagation of light will be discussed in Section 2.4. Homogeneity. A homogeneous medium has the same electromagnetic properties at all points within it. In such a medium, X, and hence )~, are independent of the position vector r, and we can write X(t) for x(r, t). Whereas silica is a homogeneous medium, optical fiber is not, since the refractive indices in the core and cladding are different. However, individually, the core and cladding regions in a step-index fiber are homogeneous. The core of a graded-index fiber is inhomogeneous. A discussion of the propagation of light in graded-index fiber is beyond the scope of this book. Losslessness. Although silica fiber is certainly not lossless, the loss is negligible and can be assumed to be zero in the discussion of propagation modes. These modes would not change significantly if the nonzero loss of silica fiber were included in their derivation. In this section, we assume that the core and the cladding regions of the silica fiber are locally responsive, isotropic, linear, homogeneous, and lossless. These as- sumptions are equivalent to assuming the appropriate properties for P, E, and X in the fiber according to the preceding discussion. Recall that the refractive index of a material n is the ratio of the speed of light in a vacuum to the speed of light in that material. It is related to the susceptibility as n 2 (co) 1 + )~ (co). (2.9) Since the susceptibility )~ is a function of the angular frequency co, so is the refractive index. Hence we have written n(co) for n in (2.9). This dependence of the refractive index on frequency is the origin of chromatic dispersion in optical fibers as we noted. For optical fibers, the value of )~ ~ 1.25, and the refractive index n ~ 1.5. With these assumptions, starting from Maxwell's equations, it can be shown that the following wave equations hold for 1~ and ISI. These equations are derived in Appendix D. o)2n2 (o)) V2]~ -~ - 1~ 0 (2.10) r co2n2 (o9) ~ V2I-7-I q - H-0. (2.11) c 2 Here V 2 denotes the Laplacian operator, which is given in Cartesian coordinates by 0 2 0 2 0 2 ax 2 + ~ + ~-72. Thus the wave equations are second-order, linear, partial differen- tial equations for the Fourier transforms of the electric and magnetic field vectors. 2.1 Light Propagation in Optical Fiber 59 Note that each wave equation actually represents three equationsnone for each component of the corresponding field vector. Fiber Modes The electric and magnetic field vectors in the core, ]~core and I~core , and the electric and magnetic field vectors in the cladding, ]~cladding and I~Icladding, must satisfy the wave equations, (2.10) and (2.11), respectively. However, the solutions in the core and the cladding are not independent; they are related by boundary conditions on 1~ and H at the core-cladding interface. Quite simply, every pair of solutions of these wave equations that satisfies these boundary conditions is a fiber mode. Assume the direction of propagation of the electromagnetic wave (light) is z. Also assume that the fiber properties such as the core diameter and the core and cladding refractive indices are independent of z. Then it turns out that the z-dependence of the electric and magnetic fields of each fiber mode is of the form eigz. The quantity fl is called the propagation constant of the mode. Each fiber mode has a different propagation constant g associated with it. (This is true for nondegenerate modes. We discuss degenerate modes in the context of polarization below.) The propagation constant is measured in units of radians per unit length. It determines the speed at which pulse energy in a mode propagates in the fiber. (Note that this concept of different propagation speeds for different modes has an analog in the geometrical optics approach. We can think of a "mode" as one possible path that a guided ray can take. Since the path lengths are different, the propagation speeds of the modes are different.) We will discuss this further in Section 2.3. The light energy propagating in the fiber will be divided among the modes supported by the fiber, and since the modes travel at different speeds in the fiber, the energy in a narrow pulse at the input of a length of fiber will be spread out at the output. Thus it is desirable to design the fiber such that it supports only a single mode. Such a fiber is called a single-mode fiber, and the mode that it supports is termed the fundamental mode. We had already come to a similar conclusion at the end of Section 2.1.1, but the wave theory approach enables us to get a clearer understanding of the concept of modes. To better understand the notion of a propagation constant of a mode, consider the propagation of an electromagnetic wave in a homogeneous medium with refractive index n. Further assume that the wave is monochromatic; that is, all its energy is concentrated at a single angular frequency ~0 or free-space wavelength )~. In this case, the propagation constant is con/c - 2Jrn/)~. The wave number, k, is defined by k - 22r/~ and is simply the spatial frequency (in cycles per unit length). In terms of the wave number, the propagation constant is kn. Thus for a wave propagating purely in the core, the propagation constant is knl, and for a wave propagating only in the cladding, the propagation constant is kn2. The fiber modes propagate partly . using the so-called ray theory or geometrical optics approach. This approach is valid when the fiber that is used has a core radius a that is much larger than the operating wavelength k 2.1) that only those light rays that are incident at an angle 00 < 0~ nax= sin -i V/n12 - n22 (2.2) no at the air-core interface will undergo total internal reflection at the Core-cladding. Core-cladding interface and will thus propagate. Such rays are called guided rays, and 0~ nax is called the acceptance angle. The refractive index difference n l - n2 is usually small, and it is