Agbo, S.O., Cherin, A H, Tariyal, B.K. Lightwave The Electrical Engineering Handbook Ed. Richard C. Dorf Boca raton crc Press llc. 2000
Agbo, S.O., Cherin, A.H, Tariyal, B.K. “Lightwave” The Electrical Engineering Handbook Ed. Richard C. Dorf Boca Raton: CRC Press LLC, 2000
42 Lightwave Samuel O Agbo 42. 1 Lightwave waveguides Lay the Wave Equation for Dielectric Materials. Modes in Slab California Polytechnic State Waveguides. Fields in Cylindrical Fibers.Modes in Step-Index Fibers· Modes in graded- Index Fibers· Attenuation· Dispersion Allen h. Cherin and Pulse Spreadin 2.2 Optical Fibers and Cables ntroduction. Classification of OpticalFibers and Attractive Basant K. Tariyal Features. Fiber Transmission Characteristics. Optical Fiber Cable Lucent Technologies 42.1 Lightwave Waveguides Samuelo. agbo Lightwave waveguides fall into two broad categories: dielectric slab waveguides and optical fibers. As illustrated Fig 42.1, slab waveguides generally consist of a middle layer(the film)of refractive index n, and lower and upper layers of refractive indices n, and ny respectively. Optical fibers are slender glass or plastic cylinders with annular cross sections. The core has a than the refractive ind is confined to the core by total internal reflection, even when the fiber is bent into curves and loops fibers fall into two main categories: step-index and graded-index(GRIN) fibers. For step-index fibers, the refractive index is constant within the core For GRIN fibers, the refractive index is a function of radius rgiven by r (42.1) (1-2△) In Eq. (42. 1), A is the relative refractive index difference, a is the core radius, and a defines the type of graded-index profile For triangular, parabolic, and step-index profiles, a is, respectively, 1, 2, and oo Figure 42.2 shows the raypaths in step-index and graded-index fibers and the cylindrical coordinate system used in the nalysis of lightwave propagation through fibers. Because rays propagating within the core in a grin fib dergo progressive refraction, the raypaths are curved (sinusoidal in the case of parabolic profile) ay theory Consider Fig 42.3, which shows possible raypaths for light coupled from air (refractive index no)into the film of a slab waveguide or the core of a step-index fiber. At each interface, the transmitted raypath is governed by Snell's law As B(the acceptance angle from air into the waveguide)decreases, the angle of incidence A increases c 2000 by CRC Press LLC
© 2000 by CRC Press LLC 42 Lightwave 42.1 Lightwave Waveguides Ray Theory • Wave Equation for Dielectric Materials • Modes in Slab Waveguides • Fields in Cylindrical Fibers • Modes in Step-Index Fibers • Modes in Graded-Index Fibers • Attenuation • Dispersion and Pulse Spreading 42.2 Optical Fibers and Cables Introduction • Classification of OpticalFibers and Attractive Features • Fiber Transmission Characteristics • Optical Fiber Cable Manufacturing 42.1 Lightwave Waveguides Samuel O. Agbo Lightwave waveguides fall into two broad categories: dielectric slab waveguides and optical fibers. As illustrated in Fig. 42.1, slab waveguides generally consist of a middle layer (the film) of refractive index n1 and lower and upper layers of refractive indices n2 and n3, respectively. Optical fibers are slender glass or plastic cylinders with annular cross sections. The core has a refractive index, n1, which is greater than the refractive index, n2, of the annular region (the cladding). Light propagation is confined to the core by total internal reflection, even when the fiber is bent into curves and loops. Optical fibers fall into two main categories: step-index and graded-index (GRIN) fibers. For step-index fibers, the refractive index is constant within the core. For GRIN fibers, 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 defines the type of graded-index profile. For triangular, parabolic, and step-index profiles, a is,respectively, 1, 2, and •. Figure 42.2 shows the raypaths in step-index and graded-index fibers and the cylindrical coordinate system used in the analysis of lightwave propagation through fibers. Because rays propagating within the core in a GRIN fiber undergo progressive refraction, the raypaths are curved (sinusoidal in the case of parabolic profile). Ray Theory Consider Fig 42.3, which shows possible raypaths for light coupled from air (refractive index n0) into the film of a slab waveguide or the core of a step-index fiber. At each interface, the transmitted raypath is governed by Snell’s law.As q0 (the acceptance angle from air into the waveguide) decreases, the angle of incidence qi increases n r n r a r a n n a r ( ) ; ( ) ; = - Ê Ë Á ˆ ¯ ˜ È Î Í Í ˘ ˚ ˙ ˙ < - = < Ï Ì Ô Ô Ó Ô Ô 1 1 2 1 1 2 2 1 2 1 2 D D a / / Samuel O. Agbo California Polytechnic State University Allen H. Cherin Lucent Technologies Basant K. Tariyal Lucent Technologies
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 fiber;(d until it equals the critical angle, A making Ao equal to the maximum acceptance angle, 8. According to ray theory, all rays with acceptance angles less than 0, 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 NA = o sin Ba= n, sin--0 (42.2) By Snell's law, sin 0= n2/n. Hence For step-index fibers, the preceding analysis applies to meridional rays. Skew(nonmeridional) rays have larger maximum acceptance angles, Aas, given by NA e 2000 by CRC Press LLC
© 2000 by CRC Press LLC until it equals the critical angle, qc , making q0 equal to the maximum acceptance angle, qa . According to ray theory, all rays with acceptance angles less than qa 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 qc = n2 /n1. Hence, (42.3) For step-index fibers, the preceding analysis applies to meridional rays. Skew (nonmeridional) rays have larger maximum acceptance angles, qas , 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 gradedindex profiles; (c) raypaths in step-index fiber; (d) raypaths in graded-index fiber. NA = n a = n - c Ê Ë Á ˆ ¯ 0 1 ˜ 2 sin q sin p q NA = - n n [ 1 ] 2 2 2 1/2 sin cos q g as NA =
FIGURE 42.3 Possible raypaths for light coupled from air into a slab waveguide or a step-index fiber. where NA is the numerical aperture for meridional rays and y 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, harge-free, and nonconducting medium is appropriate. Assuming also sinusoidal time dependence of the fields, the applicable Maxwells equations are V×E V×H=jO∈E (42.5b) V×E=0 (42.5c) V×H=0 42.5d) The resulting wave equations are V2E-Y2E=0 PH-y2H=0 where Y2=02uE=(jK)2 (42.7) (42.8) s In Eq (42.8)K is the phase propagation constant and n is the refractive index for the medium, while K, is phase propagation constant for free space. The velocity of propagation in the medium is v=l/wue e 2000 by CRC Press LLC
© 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 — 2 E = –jwmH (42.5a) — 2 H = jweE (42.5b) — 2 E = 0 (42.5c) — 2 H = 0 (42.5d) The resulting wave equations are —2E – g2E = 0 (42.6a) —2H – g 2H = 0 (42.6b) where g2 = w2me = (jk)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 k0 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. k k wm w n == = n 0 e me
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 d-e 0 (42.9a) d-H a2H=0 (42.9b) The solutions are E= Aej(t-Kz (42.10a) EA (42.10b) n where a is a constant and n= whe 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,=A cos(hy)e/t-pa) (42.11a) Ey= A sin(hy) (42.11b) where Band 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, and a2, respectively, and are b E= A,e (42.12b) Only waves with raypaths for which the total phase change for a complete(up and down) zigzag path is an egral multiple of 2T 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 8, =0 in Fig. 42.3, the mode conditions [Haus, 1984] for fields transverse to the z axis and with ven and odd symmetry about the x axis are given, respectively, by tan sin-8-n2 (42.13a) sin e (42.13b) e 2000 by CRC Press LLC
© 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 Ey = Aej(wt – kz) (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 Ey = A cos(hy)ej(wt – bz) (42.11a) Ey = A sin(hy)ej(wt – bz) (42.11b) where band 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 a3 and a2 , 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 qi = 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 = - = - h h ( ) w k me E Ae e y jt z y d = - - Ê Ë Á ˆ ¯ ˜ - 3 3 2 a ( ) w b E Ae e y jt z y d = - + Ê Ë Á ˆ ¯ ˜ - 2 2 2 a ( ) w b tan cos sin hd n n n 2 1 1 1 2 2 2 2 Ê 1 2 Ë Á ˆ ¯ ˜ = - [ ] q q / tan cos sin hd n n n 2 2 1 1 1 2 2 2 2 1 2 - Ê Ë Á ˆ ¯ ˜ = - [ ] p q q /