Fundamentals of Photonics

CHAPTER 8 FIBER OPTICS 8.1 STEP-INDEX FIBERS A. Guided Rays B. Guided Waves C. Single-Mode Fibers 8.2 GRADED-INDEX FIBERS A. Guided Waves B. Propagation Constants and Velocities 8.3 ATTENUATION AND DISPERSION A. Attenuation B. Dispersion C. Pulse Propagation Dramatic improvements in the development of low-loss materials for optical fibers are responsible for the commercial viability of fiber-optic communications. Corning Incorpo- rated pioneered the development and manu- facture of ultra-low-loss glass fibers. C 0 R N I N G 272 Fundamentals of Photonics Bahaa E. A. Saleh, Malvin Carl Teich Copyright © 1991 John Wiley & Sons, Inc. ISBNs: 0-471-83965-5 (Hardback); 0-471-2-1374-8 (Electronic) An optical fiber is a cylindrical dielectric waveguide made of low-loss materials such as silica glass. It has a central core in which the light is guided, embedded in an outer cladding of slightly lower refractive index (Fig. 8.0-l). Light rays incident on the core-cladding boundary at angles greater than the critical angle undergo total internal reflection and are guided through the core without refraction. Rays of greater inclina- tion to the fiber axis lose part of their power into the cladding at each reflection and are not guided. As a result of recent technological advances in fabrication, light can be guided through 1 km of glass fiber with a loss as low as = 0.16 dB (= 3.6 %). Optical fibers are replacing copper coaxial cables as the preferred transmission medium for electro- magnetic waves, thereby revolutionizing terrestrial communications. Applications range from long-distance telephone and data communications to computer communications in a local area network. In this chapter we introduce the principles of light transmission in optical fibers. These principles are essentially the same as those that apply in planar dielectric waveguides (Chap. 71, except for the cylindrical geometry. In both types of waveguide light propagates in the form of modes. Each mode travels along the axis of the waveguide with a distinct propagation constant and group velocity, maintaining its transverse spatial distribution and its polarization. In planar waveguides, we found that each mode was the sum of the multiple reflections of a TEM wave bouncing within the slab in the direction of an optical ray at a certain bounce angle. This approach is approximately applicable to cylindrical waveguides as well. When the core diameter is small, only a single mode is permitted and the fiber is said to be a single-mode fiber. Fibers with large core diameters are multimode fibers. One of the difficulties associated with light propagation in multimode fibers arises from the differences among the group velocities of the modes. This results in a variety of travel times so that light pulses are broadened as they travel through the fiber. This effect, called modal dispersion, limits the speed at which adjacent pulses can be sent without overlapping and therefore the speed at which a fiber-optic communication system can operate. Modal dispersion can be reduced by grading the refractive index of the fiber core from a maximum value at its center to a minimum value at the core-cladding boundary. The fiber is then called a graded-index fiber, whereas conventional fibers Figure 8.0-I An optical fiber is a cylindrical dielectric waveguide. 273 274 FIBER OPTICS W -- (cl -- n1 I n2 --- a-- r n1 me-_ -A- -- n2 --- l} nl --- -- Figure 8.0-2 Geometry, refractive-index profile, and typical rays in: (a) a multimode step-index fiber, (b) a single-mode step-index fiber, and (c) a multimode graded-index fiber. with constant refractive indices in the core and the cladding are called step-index fibers. In a graded-index fiber the velocity increases with distance from the core axis (since the refractive index decreases). Although rays of greater inclination to the fiber axis must travel farther, they travel faster, so that the travel times of the different rays are equalized. Optical fibers are therefore classified as step-index or graded-index, and multimode or single-mode, as illustrated in Fig. 8.0-2. This chapter emphasizes the nature of optical modes and their group velocities in step-index and graded-index fibers. These topics are presented in Sets. 8.1 and 8.2, respectively. The optical properties of the fiber material (which is usually fused silica), including its attenuation and the effects of material, modal, and waveguide dispersion on the transmission of light pulses, are discussed in Sec. 8.3. Optical fibers are revisited in Chap. 22, which is devoted to their use in lightwave communication systems. 8.1 STEP-INDEX FIBERS A step-index fiber is a cylindrical dielectric waveguide specified by its core and cladding refractive indices, ~zr and n2, and the radii a and b (see Fig. 8.0-l). Examples of standard core and cladding diameters 2a/2b are S/125, 50/125, 62.5/125, 85/125, 100/140 (units of pm). The refractive indices differ only slightly, so that the fractional refractive-index change A = ‘l - n2 (8.1-1) nl is small (A < 1). Almost all fibers currently used in optical communication systems are made of fused silica glass (SiO,) of high chemical purity. Slight changes in the refractive index are STEP-INDEX FIBERS 275 made by the addition of low concentrations of doping materials (titanium, germanium, or boron, for example). The refractive index y1r is in the range from 1.44 to 1.46, depending on the wavelength, and A typically lies between 0.001 and 0.02. A. Guided Rays An optical ray is guided by total internal reflections within the fiber core if its angle of incidence on the core-cladding boundary is greater than the critical angle 8, = sin - ‘(n,/nt ), and remains so as the ray bounces. Meridional Rays The guiding condition is simple to see for meridional rays (rays in planes passing through the fiber axis), as illustrated in Fig. 8.1-l. These rays intersect the fiber axis and reflect in the same plane without changing their angle of incidence, as if they were in a planar waveguide. Meridional rays are guided if their angle 8 with the fiber axis is smaller than the complement of the critical angle GC = VT/~ - 8, = cos-l&/n,). Since rrr = n2, 8, is usually small and the guided rays are approximately paraxial. Meridional plane Figure 8.1-1 The trajectory of a meridional ray lies in a plane passing through the fiber axis. The ray is guided if 8 < aC = cos-‘(n,/n,). Skewed Rays An arbitrary ray is identified by its plane of incidence, a plane parallel to the fiber axis and passing through the ray, and by the angle with that axis, as illustrated in Fig. 8.1-2. The plane of incidence intersects the core-cladding cylindrical boundary at an angle C#I with the normal to the boundary and lies at a distance R from the fiber axis. The ray is identified by its angle 8 with the fiber axis and by the angle 4 of its plane. When 4 # 0 (R f 0) the ray is said to be skewed. For meridional rays C$ = 0 and R = 0. A skewed ray reflects repeatedly into planes that make the same angle 4 with the core-cladding boundary, and follows a helical trajectory confined within a cylindrical shell of radii R and a, as illustrated in Fig. 8.1-2. The projection of the trajectory onto the transverse (x-y) plane is a regular polygon, not necessarily closed. It can be shown that the condition for a skewed ray to always undergo total internal reflection is that its angle 0 with the z axis be smaller than aC. Numerical Aperture A ray incident from air into the fiber becomes a guided ray if upon refraction into the core it makes an angle 8 with the fiber axis smaller than gC. Applying Snell’s law at the air-core boundary, the angle 8, in air corresponding to gC in the core is given by the relation 1 - sin 0, = nr sin gC, from which (see Fig. 8.1-3 and Exercise 1.2-5) sin e a = n (1 I - cos2e )li2 c = n,[l - (n2/n1)2]‘/2 = (ny - n;)‘/2. Therefore 8, = sin-’ NA, (8.1-2) 276 FIBER OPTICS a x Figure 8.1-2 A skewed ray lies in a plane offset from the fiber axis by a distance R. The ray is identified by the angles 8 and 4. It follows a helical trajectory confined within a cylindrical shell of radii R and a. The projection of the ray on the transverse plane is a regular polygon that is not necessarily closed. where 1 NA = (4 - w2 = n1wY2 1 ,,,,,ica,~;;‘,;“,; is the numerical aperture of the fiber. Thus 8, is the acceptance angle of the fiber. It Unguided Guided Small NA Large NA Figure 8.1-3 (a) The acceptance angle 8, of a fiber. Rays within the acceptance cone are guided by total internal reflection. The numerical aperture NA = sin 8,. (b) The light-gathering capacity of a large NA fiber is greater than that of a small NA fiber. The angles 8, and gC are typically quite small; they are exaggerated here for clarity. STEP-INDEX FIBERS 277 determines the cone of external rays that are guided by the fiber. Rays incident at angles greater than 8, are refracted into the fiber but are guided only for a short distance. The numerical aperture therefore describes the light-gathering capacity of the fiber. When the guided rays arrive at the other end of the fiber, they are refracted into a cone of angle 8,. Thus the acceptance angle is a crucial parameter for the design of systems for coupling light into or out of the fiber. EXAMPLE 8.1-l. C/added and Uncladded Fibers. In a silica glass fiber with izl = 1.46 and A = (n, - n2)/n1 = 0.01, the complementary critical angle gC = cos-‘(n/n,) = 8.1”, and the acceptance angle 8, = 11.9”, corresponding to a numerical aperture NA = 0.206. By comparison, an uncladded silica glass fiber (n, = 1.46, n2 = 1) has e, = 46.8”, Ba = 90”, and NA = 1. Rays incident from all directions are guided by the uncladded fiber since they reflect within a cone of angle SC = 46.8” inside the core. Although its light-gathering capacity is high, the uncladded fiber is not a suitable optical waveguide because of the large number of modes it supports, as will be shown subsequently. B. Guided Waves In this section we examine the propagation of monochromatic light in step-index fibers using electromagnetic theory. We aim at determining the electric and magnetic fields of guided waves that satisfy Maxwell’s equations and the boundary conditions imposed by the cylindrical dielectric core and cladding. As in all waveguides, there are certain special solutions, called modes (see Appendix C), each of which has a distinct propagation constant, a characteristic field distribution in the transverse plane, and two independent polarization states. Spatial Distributions Each of the components of the electric and magnetic fields must satisfy the Helmholtz equation, V2U + n2kzU = 0, where n = ~1~ in the core (r < a) and n = n2 in the cladding (r > a) and k, = 27r/A, (see Sec. 5.3). We assume that the radius b of the cladding is sufficiently large that it can safely be assumed to be infinite when examining guided light in the core and near the core-cladding boundary. In a cylindrical coordinate system (see Fig. 8.1-4) the Helmholtz equation is a2u 1 au 1 a2u a2u p+;~+~~+~+n2k~U=0, r a4 (8.1-4) Cladding Figure 8.1-4 Cylindrical coordinate system. 278 FIBER OPTICS where the complex amplitude U = U(r, 4, z) represents any of the Cartesian compo- nents of the electric or magnetic fields or the axial components E, and Hz in cylindrical coordinates. We are interested in solutions that take the form of waves traveling in the z direction with a propagation constant /3, so that the z dependence of U is of the form e . -j@ Since U must be a periodic function of the angle 4 with period 2~, we assume that the dependence on 4 is harmonic, e-j@‘, where I is an integer. Substituting U( r, 4, z) = u( r)e-ir4e-jpz, I = 0, * 1, * 2,. . . , (8.1-5) into (8.1-4), an ordinary differential equation for U(T) is obtained: d2u 1 du z+;~+ (8.1-6) As in Sec. 7.2B, the wave is guided (or bound) if the propagation constant is smaller than the wavenumber in the core (p < n,k,) and greater than the wavenumber in the cladding (/3 > n,k,). It is therefore convenient to define and kf = n:kz - p2 (8.1-7a) y2 = p2 - nzkz, (8.1-7b) so that for guided waves k; and y2 are positive and k, and y are real. Equation (8.1-6) may then be written in the core and cladding separately: r < a (core), (8.1-8a) r > a (cladding). (8.1-8b) Equations (8.1-8) are well-known differential equations whose solutions are the family of Bessel functions. Excluding functions that approach 03 at r = 0 in the core or at r + a in the cladding, we obtain the bounded solutions: u(r) a J&r) 7 r < a (core) KkYr) 7 r > a (cladding), (8.1-9) where J[(x) is the Bessel function of the first kind and order 1, and K,(x) is the modified Bessel function of the second kind and order 1. The function J,(x) oscillates like the sine or cosine functions but with a decaying amplitude. In the limit x z+ 1, J,(X) = (h)l/lw~[~ - (I + +);I, x x=- 1. (8.1-10a) u(r) t STEP-INDEX FIBERS 279 44 t *~ o 0 a r (a) lb) Figure 8.1-5 Examples of the radial distribution U(T) given by (8.1-9) for (a) 1 = 0 and (b) 1 = 3. The shaded areas represent the fiber core and the unshaded areas the cladding. The parameters k, and y and the two proportionality constants in (8.1-9) have been selected such that u(r) is continuous and has a continuous derivative at r = a. Larger values of k, and y lead to a greater number of oscillations in U(T). In the same limit, K,(x) decays with increasing x at an exponential rate, 412 - 1 1 + 8x ew( -x>, x x=- 1. (8.1-lob) Two examples of the radial distribution U(T) are shown in Fig. 8.1-S. The parameters k, and y determine the rate of change of U(T) in the core and in the cladding, respectively. A large value of k, means faster oscillation of the radial distribution in the core. A large value of y means faster decay and smaller penetration of the wave into the cladding. As can be seen from (8.1-7), the sum of the squares of k, and y is a constant, k; + y2 = (nf - n;)k; = NA2 + k;, (8.1-11) so that as k, increases, y decreases and the field penetrates deeper into the cladding. As k, exceeds NA- k,, y becomes imaginary and the wave ceases to be bound to the core. The V Parameter It is convenient to normalize k, and y by defining X = kTa, Y= ya. (8.1-12) In view of (&l-11), x2 + Y2 = v2, (8.1-13) where V = NA * k,a, from which 1 V = 2rFNA. (8.1-14) 0 V Parameter As we shall see shortly, V is an important parameter that governs the number of modes 280 FIBER OPTICS of the fiber and their propagation constants. It is called the fiber parameter or V parameter. It is important to remember that for the wave to be guided, X must be smaller than V. Modes We now consider the boundary conditions. We begin by writing the axial components of the electric- and magnetic-field complex amplitudes E, and Hz in the form of (8.1-5). The condition that these components must be continuous at the core-cladding boundary r = a establishes a relation between the coefficients of proportionality in (8.1-9), so that we have only one unknown for E, and one unknown for Hz. With the help of Maxwell’s equations, jwe0n2E = V x H and -jopOH = V X E, the remaining four components E,, H4, E,., and Hr are determined in terms of E, and Hz. Continuity of E, and H4 at r = a yields two more equations. One equation relates the two unknown coefficients of proportionality in E, and Hz; the other equation gives a condition that the propagation constant p must satisfy. This condition, called the characteristic equation or dispersion relation, is an equation for p with the ratio a/h, and the fiber indices n 1, n2 as known parameters. For each azimuthal index I, the characteristic equation has multiple solutions yielding discrete propagation constants plm, m = 1,2,. . . , each solution representing a mode. The corresponding values of k, and y, which govern the spatial distributions in the core and in the cladding, respectively, are determined by use of (8.1-7) and are denoted kTlm and ylm. A mode is therefore described by the indices 1 and m characterizing its azimuthal and radial distributions, respectively. The function U(T) depends on both I and m; 2 = 0 corresponds to meridional rays. There are two independent configurations of the E and H vectors for each mode, corresponding to two states of polarization. The classification and labeling of these configurations are generally quite involved (see specialized books in the reading list for more details). Characteristic Equation for the Weakly Guiding Fiber Most fibers are weakly guiding (i.e., y1r = n2 or A < 1) so that the guided rays are paraxial (i.e., approximately parallel to the fiber axis). The longitudinal components of the electric and magnetic fields are then much weaker than the transverse components and the guided waves are approximately transverse electromagnetic (TEM). The linear polarization in the x and y directions then form orthogonal states of polarization. The linearly polarized (I, m) mode is usually denoted as the LP,, mode. The two polariza- tions of mode (I, m) travel with the same propagation constant and have the same spatial distribution. For weakly guiding fibers the characteristic equation obtained using the procedure outlined earlier turns out to be approximately equivalent to the conditions that the scalar function U(T) in (8.1-9) is continuous and has a continuous derivative at r = a. These two conditions are satisfied if The derivatives Ji and Ki of the Bessel functions satisfy the identities (8.1-15) J,(x) J/(x) = +Jlqzl(~) + I- X K,(x) K/(x) = -K/.,(X) T l- x * STEP-INDEX FIBERS 281 Substituting these identities into (8.1-15) and using the normalized parameters X = k,a and Y = ya, we obtain the characteristic equation (8.1-16) Characteristic Equation X2+Y2=V2 Given V and I, the characteristic equation contains a single unknown variable X (since Y2 = V2 - X2). Note that J-&x) = (- l)‘J[(x) and K-,(x) = K,(x), so that if I is replaced with -I, the equation remains unchanged. The characteristic equation may be solved graphically by plotting its right- and left-hand sides (RHS and LHS) versus X and finding the intersections. As illustrated in Fig. 8.1-6 for I = 0, the LHS has multiple branches and the RHS drops monotonically with increase of X until it vanishes at X = V (V = 0). There are therefore multiple intersections in the interval 0 < X i V. Each intersection point corresponds to a fiber mode with a distinct value of X. These values are denoted Xlm, m = 1,2,. . . , M[ in order of increasing X. Once the X,, are found, the corresponding transverse propaga- tion constants kTlm, the decay parameters yfm, the propagation constants firm, and the radial distribution functions ulm(r) may be readily determined by use of (&l-12), (8.1-7), and (8.1-9). The graph in Fig. 8.1-6 is similar to that in Fig. 7.2-2, which governs the modes of a planar dielectric waveguide. Each mode has a distinct radial distribution. The radial distributions U(T) shown in Fig. 8.1-5, for example, correspond to the LP,, mode (I = 0, m = 1) in a fiber with V = 5; and the LP,, mode (I = 3, m = 4) in a fiber with V = 25. Since the (1, m) and (-I, m) modes have the same propagation constant, it is interesting to examine the spatial distribution of their superposition (with equal weights). The complex amplitude of the sum is proportional to U/~(T) cos Z4 exp( -jplmz). The intensity, which is proportional to u;~(I-) cos2 Z& is illustrated in Fig. 8.1-7 for the LP,, and LP,, modes (the same modes for which U(T) is shown in Fi