The electrical engineering handbook
Agbo, S.O., Cherin, A.H, Tariyal, B.K. “Lightwave” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000 â 2000 by CRC Press LLC 42 Lightwave 42.1Lightwave Waveguides Ray TheoryWave Equation for Dielectric MaterialsModes in Slab WaveguidesFields in Cylindrical FibersModes in Step-Index FibersModes in Graded-Index FibersAttenuationDispersion and Pulse Spreading 42.2Optical Fibers and Cables IntroductionClassication of OpticalFibers and Attractive FeaturesFiber Transmission CharacteristicsOptical Fiber Cable Manufacturing 42.1 Lightwave Waveguides Samuel O. Agbo Lightwave waveguides fall into two broad categories: dielectric slab waveguides and optical bers. As illustrated in Fig. 42.1, slab waveguides generally consist of a middle layer (the lm) of refractive index n 1 and lower and upper layers of refractive indices n 2 and n 3 , respectively. Optical bers are slender glass or plastic cylinders with annular cross sections. The core has a refractive index, n 1 , which is greater than the refractive index, n 2 , of the annular region (the cladding). Light propagation is conned to the core by total internal reection, even when the ber is bent into curves and loops. Optical bers fall into two main categories: step-index and graded-index (GRIN) bers. For step-index bers, the refractive index is constant within the core. For GRIN bers, the refractive index is a function of radius r given by (42.1) In Eq. (42.1), D is the relative refractive index difference , a is the core radius, and a denes the type of graded-index prole. For triangular, parabolic, and step-index proles, a is, respectively, 1, 2, and Ơ . Figure 42.2 shows the raypaths in step-index and graded-index bers and the cylindrical coordinate system used in the analysis of lightwave propagation through bers. Because rays propagating within the core in a GRIN ber undergo progressive refraction, the raypaths are curved (sinusoidal in the case of parabolic prole). Ray Theory Consider Fig 42.3, which shows possible raypaths for light coupled from air (refractive index n 0 ) into the lm of a slab waveguide or the core of a step-index ber. At each interface, the transmitted raypath is governed by Snells law. As q 0 (the acceptance angle from air into the waveguide) decreases, the angle of incidence q i increases nr n r a ra nnar () ; (); = - ổ ố ỗ ử ứ ữ ộ ở ờ ờ ự ỷ ỳ ỳ < -=< ỡ ớ ù ù ợ ù ù 1 12 1 12 2 12 12 D D a / / Samuel O. Agbo California Polytechnic State University Allen H. Cherin Lucent Technologies Basant K. Tariyal Lucent Technologies © 2000 by CRC Press LLC until it equals the critical angle, q c , making q 0 equal to the maximum acceptance angle, q a . According to ray theory, all rays with acceptance angles less than q a propagate in the waveguide by total internal reflections. Hence, the numerical aperture (NA) for the waveguide, a measure of its light-gathering ability, is given by (42.2) By Snell’s law, sin q c = n 2 / n 1 . Hence, (42.3) For step-index fibers, the preceding analysis applies to meridional rays . Skew (nonmeridional) rays have larger maximum acceptance angles, q as , given by (42.4) FIGURE 42.1 Dielectric slab waveguide: (a) the Cartesian coordinates used in analysis of slab waveguides; (b) the slab waveguide; (c) light guiding in a slab waveguide. FIGURE 42.2 The optical fiber: (a) the cylindrical coordinate system used in analysis of optical fibers; (b) some graded- index profiles; (c) raypaths in step-index fiber; (d) raypaths in graded-index fiber. NA n a n c ==- æ è ç ö ø ÷ 01 2 sin sinq p q NA n n=- [] 1 2 2 2 12/ sin cos q g as NA = © 2000 by CRC Press LLC where NA is the numerical aperture for meridional rays and g is the angle between the core radius and the projection of the ray onto a plane normal to the fiber axis. Wave Equation for Dielectric Materials Only certain discrete angles, instead of all acceptance angles less than the maximum acceptance angle, lead to guided propagation in lightwave waveguides. Hence, ray theory is inadequate, and wave theory is necessary, for analysis of light propagation in optical waveguides. For lightwave propagation in an unbounded dielectric medium, the assumption of a linear, homogeneous, charge-free, and nonconducting medium is appropriate. Assuming also sinusoidal time dependence of the fields, the applicable Maxwell’s equations are Ñ ן E = –j wm H (42.5a) Ñ ן H = j w ⑀ E (42.5b) Ñ ן E = 0 (42.5c) Ñ ן H = 0 (42.5d) The resulting wave equations are Ñ 2 E – g 2 E = 0 (42.6a) Ñ 2 H – g 2 H = 0 (42.6b) where g 2 = w 2 m ⑀ = ( j k ) 2 (42.7) and (42.8) In Eq. (42.8) k is the phase propagation constant and n is the refractive index for the medium, while k 0 is the phase propagation constant for free space. The velocity of propagation in the medium is n = 1 / . FIGURE 42.3 Possible raypaths for light coupled from air into a slab waveguide or a step-index fiber. kkwm w n == =n 0 ⑀ m⑀ © 2000 by CRC Press LLC Modes in Slab Waveguides Consider a plane wave polarized in the y direction and propagating in z direction in an unbounded dielectric medium in the Cartesian coordinates. The vector wave equations (42.6) lead to the scalar equations: (42.9a) (42.9b) The solutions are E y = Ae j ( w t – k z ) (42.10a) (42.10b) where A is a constant and h = is the intrinsic impedance of the medium. Because the film is bounded by the upper and lower layers, the rays follow the zigzag paths as shown in Fig. 42.3. The upward and downward traveling waves interfere to create a standing wave pattern. Within the film, the fields transverse to the z axis, which have even and odd symmetry about the x axis, are given, respectively, by E y = A cos( hy ) e j ( w t – b z ) (42.11a) E y = A sin( hy) e j( w t – b z) (42.11b) where b and h are the components of k parallel to and normal to the z axis, respectively. The fields in the upper and lower layers are evanescent fields decaying rapidly with attenuation factors a 3 and a 2 , respectively, and are given by (42.12a) (42.12b) Only waves with raypaths for which the total phase change for a complete (up and down) zigzag path is an integral multiple of 2p undergo constructive interference, resulting in guided modes. Waves with raypaths not satisfying this mode condition interfere destructively and die out rapidly. In terms of a raypath with an angle of incidence q i = q in Fig. 42.3, the mode conditions [Haus, 1984] for fields transverse to the z axis and with even and odd symmetry about the x axis are given, respectively, by (42.13a) (42.13b) ¶ ¶ ¶ 2 2 0 E z E y y -= ¶ ¶ ¶ 2 2 0 H z H x x -= H E A e x y jt z = - = - hh wk() m⑀ EAe e y jt z y d = - - æ è ç ö ø ÷ - 3 3 2 a wb() EAe e y jt z y d = - + æ è ç ö ø ÷ - 2 2 2 a wb() tan cos sin hd n nn 2 1 1 1 22 2 2 12 æ è ç ö ø ÷ =- [] q q / tan cos sin hd n nn 22 1 1 1 22 2 2 12 - æ è ç ö ø ÷ =- [] p q q / © 2000 by CRC Press LLC where h = k cos q = (2pn 1 /l) cos q and l is the free space wavelength. Equations (42.13a) and (42.13b) are transcendental, have multiple solutions, and are better solved graphically. Let (d/l) 0 denote the smallest value of d/l, the film thickness normalized with respect to the wavelength, satisfying Eqs. (42.13a) and (42.13b). Other solutions for both even and odd modes are given by (42.14) where m is a nonnegative integer denoting the order of the mode. Figure 42.4 [Palais, 1992] shows a mode chart for a symmetrical slab waveguide obtained by solving Eqs. (42.13a) and (42.13b). For the TE m modes, the E field is transverse to the direction (z) of propagation, while the H field lies in a plane parallel to the z axis. For the TM m modes, the reverse is the case. The highest- order mode that can propagate has a value m given by the integer part of (42.15) To obtain a single-mode waveguide, d/l should be smaller than the value required for m = 1, so that only the m = 0 mode is supported. To obtain a multimode waveguide, d/l should be large enough to support many modes. Shown in Fig. 42.5 are transverse mode patterns for the electric field in a symmetrical slab waveguide. These are graphical illustrations of the fields given by Eqs. (42.11) and (42.12). Note that, for TE m , the field has m zeros in the film, and the evanescent field penetrates more deeply into the upper and lower layers for high-order modes. For asymmetric slab waveguides, the equations and their solutions are more complex than those for symmetric slab waveguides. Shown in Fig. 42.6 [Palais, 1992] is the mode chart for the asymmetric slab waveguide. Note that the TE m and TM m modes in this case have different FIGURE 42.4Mode chart for the symmetric slab waveguide with n 1 = 3.6, n 2 = 3.55. ddm n m ll q æ è ç ö ø ÷ = æ è ç ö ø ÷ + 0 1 2 cos m d nn=- [] 2 1 2 2 2 12 l / FIGURE 42.5Transverse mode field patterns in the symmetric slab waveguide. © 2000 by CRC Press LLC propagation constants and do not overlap. By contrast, for the symmetric case, TE m and TM m modes are degenerate, having the same propagation constant and forming effectively one mode for each value of m. Figure 42.7 shows typical mode patterns in the asymmetric slab waveguide. Note that the asymmetry causes the evanes- cent fields to have unequal amplitudes at the two boundaries and to decay at different rates in the two outer layers. The preceding analysis of slab waveguides is in many ways similar to, and constitutes a good introduction to, the more complex analysis of cylindrical (optical) fibers. Unlike slab waveguides, cylindrical waveguides are bounded in two dimensions rather than one. Consequently, skew rays exist in optical fibers, in addition to the meridional rays found in slab waveguides. In addition to transverse modes similar to those found in slab waveguides, the skew rays give rise to hybrid modes in optical fibers. Fields in Cylindrical Fibers Let y represent E z or H z and b be the component of k in z direction. In the cylindrical coordinates of Fig. 42.2, with wave propagation along the z axis, the wave equations (42.6) correspond to the scalar equation (42.16) The general solution to the preceding equation is y(r) = C 1 J l (hr) + C 2 Y l (hr);k 2 > b 2 (42.17a) y(r) = C 1 I l (qr) + C 2 K l (qr);k 2 < b 2 (42.17b) In Eqs. (42.17) and (42.17b), J l and Y l are Bessel functions of the first kind and second kind, respectively, of order l; I l and K l are modified Bessel functions of the first kind and second kind, respectively, of order l; C 1 and C 2 are constants; h 2 = k 2 – b 2 and q 2 = b 2 – k 2 . E z and H z in a fiber core are given by Eq. (42.17a) or (42.17b), depending on the sign of k 2 – b 2 . For guided propagation in the core, this sign is negative to ensure that the field is evanescent in the cladding. One of the FIGURE 42.6Mode chart for the asymmetric slab waveguide with n 1 = 2.29, n 2 = 1.5, and n 3 = 1.0. FIGURE 42.7Transverse mode field patterns in the asymmetric slab waveguide. ¶y ¶ ¶y ¶ ¶y ¶ kby 2 22 2 2 22 11 0 r rr r +++-= F () © 2000 by CRC Press LLC coefficients vanishes because of asymptotic behavior of the respective Bessel functions in the core or cladding. Thus, with A 1 and A 2 as arbitrary constants, the fields in the core and cladding are given, respectively, by y(r) = A 1 J l (hr) (42.18a) y(r) = A 2 k l (hr) (42.18b) Because of the cylindrical symmetry, y(r,t) = y(r,f)e j( w t – bz) (42.19) Thus, the usual approach is to solve for E z and H z and then express E r , E f , H r , and H f in terms of E z and H z . Modes in Step-Index Fibers Derivation of the exact modal field relations for optical fibers is complex. Fortunately, fibers used in optical communication satisfy the weekly guiding approximation in which the relative index difference, Ñ, is much less than unity. In this approximation, application of the requirement for continuity of transverse and tangential electric field components at the core-cladding interface (at r = a) to Eqs. (42.18a) and (42.18b) results in the following eigenvalue equation [Snyder, 1969]: (42.20) Let the normalized frequency V be defined as (42.21) Solving Eq. (42.20) allows b to be calculated as a function of V. Guided modes propagating within the core correspond to n 2 k 0 £ b £ n 1 k. The normalized frequency V corresponding to b = n 1 k is the cut-off frequency for the mode. As with planar waveguides, TE (E z = 0) and TM (H z = 0) modes corresponding to meridional rays exist in the fiber. They are denoted by EH or HE modes, depending on which component, E or H, is stronger in the plane transverse to the direction or propagation. Because the cylindrical fiber is bounded in two dimensions rather than one, two integers, l and m, are needed to specify the modes, unlike one integer, m, required for planar waveguides. The exact modes, TE lm , TM lm , EH lm , and HE lm , may be given by two linearly polarized modes, LP lm . The subscript l is now such that LP lm corresponds to HE l + 1,m , EH l – 1,m , TE l – 1,m , and TM l – 1,m . In general, there are 2l field maxima around the fiber core circumference and m field maxima along a radius vector. Figure 42.8 illustrates the correspondence between the exact modes and the LP modes and their field configurations for the three lowest LP modes. Figure 42.9 gives the mode chart for step-index fiber on a plot of the refractive index, b/k 0 , against the normalized frequency. Note that for a single-mode (LP 01 or HE 11 ) fiber, V < 2.405. The number of modes supported as a function of V is given by (42.22) haJ ha Jha qa qa qa l l l l ± ± =± 1 1 () () () () k k Vaqh ann aNA=+= - ( ) =() () 2212 01 2 2 2 12 2 / / k p l N V = 2 2 © 2000 by CRC Press LLC Modes in Graded-Index Fibers A rigorous modal analysis for optical fibers based on the solution of Maxwell’s equations is possible only for step-index fiber. For graded-index fibers, approximate methods are used. The most widely used approximation is the WKB (Wenzel, Kramers, and Brillouin) method [Marcuse, 1982]. This method gives good modal solutions FIGURE 42.8 Transverse electric field patterns and field intensity distributions for the three lowest LP modes in a step- index fiber: (a) mode designations; (b) electric field patterns; (c) intensity distribution. (Source: J. M. Senior, Optical Fiber Communications: Principles and Practice, Englewood Cliffs, N.J.: Prentice-Hall, 1985, p. 36. With permission.) FIGURE 42.9 Mode chart for step-index fibers: b = (b /k 0 – n 2 )/(n 1 – n 2 ) is the normalized propagation constant. (Source: D. B. Keck, Fundamentals of Optical Fiber Communications, M. K. Barnoski, Ed., New York: Academic Press, 1981, p. 13. With permission.) © 2000 by CRC Press LLC for graded-index fiber with arbitrary profiles, when the refractive index does not change appreciably over distances comparable to the guided wavelength [Yariv, 1991]. In this method, the transverse components of the fields are expressed as E t = y(r)e jlf e j(wt – bz) (42.23) (42.24) In Eq. (42.23), l is an integer. Equation (42.16), the scalar wave equation in cylindrical coordinates can now be written with k = n(r) k 0 as (42.25) where HE tt = b wm d dr d dr pr r 2 2 2 1 2 0++ é ë ê ê ù û ú ú =() ()y M INIATURE R ADAR n inexpensive miniaturized radar system developed at Lawrence Livermore National Labs (LLNL) may become the most successful technology ever privatized by a federal lab, with a potential market for the product estimated at between $100 million and $150 million. The micropower impulse radar was developed by engineer Tom McEwan as part of a device designed to measure the one billion pulses of light emitted from LLNL’s Nova laser in a single second. The system he developed is the size of a cigarette box and consists of about $10 worth of parts. The same measurement had been made previously using $40,000 worth of equipment. Titan Technologies of Edmonton, AL, Canada, was the first to bring to market a product using the technology when they introduced storage-tank fluid sensors incorporating the system. The new radar allowed Titan to reduce its devices from the size of an apple crate to the size of a softball, and to sell them for one-third the cost of a comparable device. The Federal Highway Administration is preparing to use the radar for highway inspections and the Army Corps of Engineers has contracted with LLNL to use the system for search and rescue radar. Other applications include a monitoring device to check the heartbeats of infants to guard against Sudden Infant Death Syndrome (SIDS), robot guide sensors, automatic on/off switches for bathroom hand dryers, hand-held tools, automobile back-up warning systems, and home security. AERES, a San Jose-based company, has developed a new approach to ground-penetrating radar using impulse radar. The first application of the technology was an airborne system for detecting underground bunkers. The design can be altered to provide high depth capability for large targets, or high resolution for smaller targets near the surface. This supports requirements in land mine searches and explosive ordinance disposal for the military. AERAS has developed both aircraft and ground-based systems designed for civilian applications as well as military. Underground utility mapping, such as locating pipes and cables; highway and bridge under-surface inspection; and geological and archeological surveying are examples of the possible civilian applications. (Reprinted with permission from NASA Tech Briefs, 20(10), 24, 1996.) A . NA is the numerical aperture for meridional rays and g is the angle between the core radius and the projection of the ray onto a plane normal to the fiber. impedance of the medium. Because the film is bounded by the upper and lower layers, the rays follow the zigzag paths as shown in Fig. 42.3. The upward and