n1=360 87° 804° 3 FIGURE 42.4 Mode chart for the symmetric slab waveguide with n,= 3.6, n,=3.55. where h= K cos 0=(2Tn, /)cos 0 and n is the free space wavelength Equations(42 13a)and(4213b)are transcendental, have multiple solutions, and are better solved graphically Let(d/)o denote the smallest value of dn, 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) 2n, cos 0 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 TEm 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 modes, the reverse is the case. The highest rder mode that can propagate has a value m given by the integer part of (42.15) obtain a single-mode waveguide, d/ should be smaller than the value required for m= l, so that only the 0 mode is supported. To obtain a multimode waveguide, d/ should be large enough to support many modes SDI I 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, the field has m zeros in the film, and the evanescent field penetrates more deeply FIGURE 42.5 Transverse mode field patterns in the into the upper and lower layers for high-order modes symmetric slab waveguide 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 and TM modes in this case have different e 2000 by CRC Press LLC
© 2000 by CRC Press LLC where h = k cos q = (2pn1/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 TEm 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 TMm modes, the reverse is the case. The highestorder 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 TEm , 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 TEm and TMm modes in this case have different FIGURE 42.4 Mode chart for the symmetric slab waveguide with n1 = 3.6, n2 = 3.55. d d m n m l l q Ê Ë Á ˆ ¯ ˜ = Ê Ë Á ˆ ¯ ˜ + 0 1 2 cos m d = - n n [ ] 2 1 2 2 2 1 2 l / FIGURE 42.5 Transverse mode field patterns in the symmetric slab waveguide
61/A/:p FIGURE 42.6 Mode chart for the asymmetric slab waveguide with n,= 2.29, n2=1.5, and n,=1.0 propagation constants and do not overlap. By contrast, for the symmetric case, TEm and TMm 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 evans- cent fields to have unequal amplitudes at the two boundaries TE 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 FIGURE427 Transverse mode field patterns in the those found in slab waveguides, the skew rays give rise to asymmetric slab waveguide. hybrid modes in optical fibers. Fields in Cylindrical Fibers Let y represent E, or H, 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 y I dy 1 dy (K2-2)y=0 (42.16) dr r d The general solution to the preceding equation is y(r)=Cu(hr)+ CrY,(hr); K2>B2 (42.17a) y(r)=C,I(qr)+ C,k,(qr); K2< B2 (42.17b) In Eqs.(42.17)and(4217b), Je and Y, are Bessel functions of the first kind and second kind, respectively, of order l; I, and K, are modified Bessel functions of the first kind and second kind, respectively, of order e; C and C2 are constants; h2=K-B and q=B2-x. E, and H, in a fiber core are given by Eq (42 17a)or(4217b), depending on the sign of x-B. For guided ropagation in the core, this sign is negative to ensure that the field is evanescent in the cladding. One of the e 2000 by CRC Press LLC
© 2000 by CRC Press LLC propagation constants and do not overlap. By contrast, for the symmetric case, TEm and TMm 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 evanescent 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 Ez or Hz 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) = C1 Jl(hr) + C2Yl (hr); k2 > b2 (42.17a) y(r) = C1 Il(qr) + C2Kl (qr); k2 < b2 (42.17b) In Eqs. (42.17) and (42.17b), Jl and Yl are Bessel functions of the first kind and second kind, respectively, of order l; Il and Kl are modified Bessel functions of the first kind and second kind, respectively, of order l; C1 and C2 are constants; h2 = k2 – b2 and q2 = b2 – k2 . Ez and Hz in a fiber core are given by Eq. (42.17a) or (42.17b), depending on the sign of k2 – b2 . For guided propagation in the core, this sign is negative to ensure that the field is evanescent in the cladding. One of the FIGURE 42.6 Mode chart for the asymmetric slab waveguide with n1 = 2.29, n2 = 1.5, and n3 = 1.0. FIGURE 42.7 Transverse mode field patterns in the asymmetric slab waveguide. ¶ y ¶ ¶y ¶ ¶ y ¶ k b y 2 2 2 2 2 1 1 2 2 0 r r r r + + + - = F ( )
Defficients vanishes because of asymptotic behavior of the respective Bessel functions in the core or cladding Thus, with A, and A2 as arbitrary constants, the fields in the core and cladding are given, respectively, by u(r)=A,t(hr) (42.18a) y(r)=A2k,(hr) 2.1 Because of the cylindrical V(r, t)=y(r, o)ejdt-Bz (42.19) Thus, the usual approach is to solve for E, and H, and then express E, E,, H, and Ho in terms of E, and H. odes in Step-Index Fibe 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, V, 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] ha,ma)=±9a( (42.20) Je(ha) Let the normalized frequency v be defined as V=a(q+h2)2= aK(ni-n2) a Q(NA) (42.21) Solving Eq(42. 20)allows B to be calculated as a function of V Guided modes propagating within the core respond to n, Ko s Bs n,K. The normalized frequency V corresponding to B=n, k is the cut-off frequenc As with planar waveguides, TE (E,=0)and TM (H =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, e and m, are needed to specify the modes, unlike one integer, m, required for planar waveguides. The exact modes, TE m, TM,m, EH,m, and HE m, may be given by two linearly polarized modes, LP(m The subscript e is now such that LP/m corresponds to HE + Lm, EH,-lm, TE, In general, there are 2( 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/Ko, again normalized frequency. Note that for a single-mode(LPo or HE) fiber, V 2.405. The number of m b e 2000 by CRC Press LLC
© 2000 by CRC Press LLC coefficients vanishes because of asymptotic behavior of the respective Bessel functions in the core or cladding. Thus, with A1 and A2 as arbitrary constants, the fields in the core and cladding are given, respectively, by y(r) = A1 Jl(hr) (42.18a) y(r) = A2kl(hr) (42.18b) Because of the cylindrical symmetry, y(r,t) = y(r,f)ej(w t – bz) (42.19) Thus, the usual approach is to solve for Ez and Hz and then express Er, Ef , Hr , and Hf in terms of Ez and Hz . 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, —, 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 n2k0 £ b£ n1k. The normalized frequency V corresponding to b = n1k is the cut-off frequency for the mode. As with planar waveguides, TE (Ez = 0) and TM (Hz = 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, TElm , TMlm, EHlm , and HElm, may be given by two linearly polarized modes, LPlm. The subscript l is now such that LPlm corresponds to HEl + 1,m , EHl – 1,m, TEl – 1,m, and TMl – 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/k0 , against the normalized frequency. Note that for a single-mode (LP01 or HE11) fiber, V < 2.405. The number of modes supported as a function of V is given by (42.22) haJ ha J ha qa qa qa l l l l ± ± 1 = ± 1 ( ) ( ) ( ) ( ) k k V = a( ) q + h = a (n - n ) = a(NA) 2 2 1 2 0 1 2 2 2 1 2 / 2 / k p l N V = 2 2
②QD EH FIGURE 42.8 Transverse electric field patterns and field intensity distributions for the three lowest LP modes in index fiber: (a)mode designations;(b)electric field patterns; (c) intensity distribution. (Sa L Senior, Optic Communications: Principles and Practice, Englewood Cliffs, N J: Prentice-Hall, 1985, P. 36. With permission. E21 FIGURE 42.9 Mode chart for step-index fibers: b=(B/K, n,)/(m-l,)is the normalized propagation constant. Source: D. B Keck, Fundamentals of Optical Fiber Com ions, M. K. Barnoski, Ed, New York: Academic Press, 1981, P. 13 with permission Modes in graded-Index fibers A rigorous modal analysis for optical fibers based on the solution of Maxwells 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 e 2000 by CRC Press LLC
© 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 stepindex 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 /k0 – n2)/(n1 – n2) 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.)
MINIATURE RADAR A 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 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 pulse 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.) 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, =y(r) H=B (42.24) In Eq (42. 23), is an integer. Equation(42. 16), the scalar wave equation in cylindrical coordinates can no be written with x n(r)Ko as +p2(r)v(r)=0 (42.25) e 2000 by CRC Press LLC
© 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 Et = y(r)ejlf ej(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) k0 as (42.25) where H E t t = b wm d dr d dr pr r 2 2 1 2 2 + + 0 È Î Í Í ˘ ˚ ˙ ˙ () () y = MINIATURE RADAR 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