Optical Networks: A Practical Perspective - Part 14 pps

100 PROPAGATION OF SIGNALS IN OPTICAL FIBER called higher-order solitons. A brief quantitative discussion of soliton propagation in optical fiber appears in Section E.3. The significance of solitons for optical communication is that they overcome the detrimental effects of chromatic dispersion completely. Optical amplifiers can be used at periodic intervals along the fiber so that the attenuation undergone by the pulses is not significant, and the higher powers and the consequent soliton properties of the pulses are maintained. Solitons and optical amplifiers, when used together, offer the promise of very high-bit-rate, repeaterless data transmission over very large distances. By the combined use of solitons and erbium-doped fiber amplifiers (Section 3.4.3), repeaterless data transmission at a bit rate of 80 Gb/s over a distance of 10,000 km has been demonstrated in the laboratory [NSK99]. The use of soliton pulses is key to the realization of the very high bit rates required in OTDM systems. These aspects of solitons will be explored in Chapter 12. The main advantage of soliton systems is their relative immunity to fiber disper- sion, which in turn allows transmission at high speeds of a few tens of gigabits per second. On the other hand, in conventional on-off-keyed systems, dispersion can be managed in a much simpler manner by alternating fibers with positive and negative dispersion. We encountered this in Section 2.4.9 and we will study this further in Chapter 5. Such systems, when using special pulses called chirped RZ pulses, can also be viewed as soliton systems, albeit of a different kind, and we discuss this in the next section. 2.5.1 Dispersion-Managed Solitons Solitons can also be used in conjunction with WDM, but significant impairments arise when two pulses at different wavelengths overlap in time and position in the fiber. Such collisions, which occur frequently in the fiber, add timing jitter to the pulses. Although methods to overcome this timing jitter have been devised, commercial deployment of soliton-based systems has not been widespread for two main reasons. First, solitons require new disperson-shifted fiber with a small value of anomalous dispersion (0 < D < 1 ps/nm-km). Thus soliton-based systems cannot be used on existing fiber plants, whether based on SMF or on the popular NZ-DSF fibers. Second, solitons require amplification about every 20 km or so, which is an impracticably small spacing compared to today's WDM systems, which work with amplifier hut spacings of the order of 60-80 kin. Larger values of dispersion lead to higher levels of timing jitter, higher peak pulse powers, and even closer amplifier spacings. Summary 101 High-bit-rate transmission on widely deployed fiber plants, with reasonable am- plifier spacings, has been achieved through a combination of (1) using pulses nar- rower than a bit period but much wider than solitons, and (2) dispersion compensa- tion of the fiber plant at periodic intervals to keep the average dispersion low. The pulses used in such systems are called chirped return-to-zero (RZ) pulses and will be discussed in Section 4.1. When the characteristics of such a dispersion-managed system are mathematically analyzed, it can be shown that such a system is indeed "soliton-like" in the sense that a specific chirped Gaussian pulse shape will be trans- mitted through such a system with only periodic changes in shape, that is, with no net broadening due to dispersion, in the absence of loss. Such pulses are also called dispersion-managed (DM) solitons. We will discuss the performance of systems em- ploying such pulses in Chapter 5. By the use of chirped RZ pulses, repeaterless data transmission in a 25-channel WDM system at a bit rate of 40 Gb/s per channel, over a distance of 1500 kin, has been demonstrated in the laboratory [SKN01]. Summary The understanding of light propagation in optical fiber is key to the appreciation of not only the significant advantages of using optical fiber as a propagation medium but also of the problems that we must tackle in designing high-bit-rate WDM systems. We started by understanding how light propagates in multimode fibers using a simple ray theory approach. This introduced the concept of pulse broadening due to multimode dispersion and motivated the use of single-mode fibers. After describing the elements of light propagation in single-mode fibers, we studied the limitations imposed on optical communication systems due to the pulse-broadening effects of chromatic dispersion. Although dispersion is the most important phenomenon limiting the performance of systems at bit rates of 2.5 Gb/s and below, nonlinear effects become important at higher bit rates. The main nonlinear effects that impair high-speed WDM transmis- sion are self-phase modulation and four-wave mixing. We studied the origin of these, as well as other nonlinear effects, and briefly outlined the constraints on optical com- munication systems imposed by them. We will return to the system limitations of both dispersion and nonlinearities when we discuss the design of optical transmission systems in Chapter 5. We also studied the new types of fibers that have been introduced to mitigate the effects of dispersion and nonlinearities. Finally, we discussed solitons, which are special pulses designed to play off dispersion and nonlinearities against each other to achieve high-bit-rate, ultra-long-haul transmission. 102 PROPAGATION OF SIGNALS IN OPTICAL FIBER Further Reading The propagation of light in optical fiber is treated in several books at varying levels of detail. One of the earliest books on this subject is by Marcuse [Mar74]. The book by Green [Gre93] starts with the fundamentals of both geometrical optics and electromagnetics and describes the propagation of light using both the ray and wave theory approaches. The concepts of polarization and birefringence are also treated in some detail. However, the effects of dispersion and nonlinearities are described only qualitatively. The book on fiber optic communication by Agrawal [Agr97] focuses on the wave theory approach and treats the evolution of chirped Gaussian pulses in optical fiber and the pulse-broadening effects of chromatic dispersion in detail. Chromatic dispersion and intermodal dispersion are also treated at length in the books edited by Miller and Kaminow [MK88] and Lin [Lin89]. We recommend the book by Ramo, Whinnery, and van Duzer [RWv93] for an in-depth study of electromagnetic theory leading up to the description of light propagation in fiber. The books by Jeunhomme Ueu90] and Neumann [Neu88] are devoted to the propagation of light in single-mode fibers. Jeunhomme treats fiber modes in detail and has a more mathematical treatment. We recommend Neumann's book for its physical explanations of the phenomena involved. The paper by Gloge [Glo71] on fiber modes is a classic. In all these books, nonlinear effects are only briefly mentioned. The book by Agrawal [Agr95] is devoted to nonlinear fiber optics and contains a very detailed description of light propagation in optical fiber, including all the nonlinear effects we have discussed. Soliton propagation is also discussed. One of the earliest papers on four-wave mixing is [HJKM78]. Note that cgs units are used in this paper. The units used in the description of nonlinear effects are a source of confusion. The relationships between the various units and terminologies used in the description of nonlinear effects are described in the book by Butcher and Cotter [BC90]. This book also contains a particularly clear exposition of the fundamentals of nonlinear effects. The system impact of dispersion and nonlinearities and their interplay is discussed in detail in [KK97, Chapter 8]. Information on the new types of fibers that have been introduced to combat dispersion and nonlinearities can be found on the Web pages of the manufacturers: Corning and Lucent. Much of the data on the new fiber types for this chapter was gathered from these Web pages. The ITU has standardized three fiber types. ITU-T recommendation (standard) G.652 specifies the characteristics of standard single-mode fiber, G.653 that of DSF, and G.655 that of NZ-DSE A nice treatment of the basics of solitons appears in [KBW96]. Issues in the design of WDM soliton communication systems are discussed at length in [KK97, Chapter Problems 103 12]. A summary of soliton field trials appears in [And00]. DM solitons are discussed in [Nak00]. 2.1 2.2 2.3 2.4 2.5 2.6 2.7 Problems Note that some of these problems require an understanding of the material in the appendices referred to in this chapter. Derive (2.2). A step-index multimode glass fiber has a core diameter of 50 /,m and cladding refractive index of 1.45. If it is to have a limiting intermodal dispersion aT of 10 ns/km, find its acceptance angle. Also calculate the maximum bit rate for transmission over a distance of 20 km. Derive equation (2.11) for the evolution of the magnetic field vector I2I. Derive an expression for the cutoff wavelength ~-cutoff of a step-index fiber with core radius a, core refractive index n l, and cladding refractive index n2. Calculate the cutoff wavelength of a fiber with core radius a = 4 #m and A = 0.003. Consider a step-index fiber with a core radius of 4/,m and a cladding refractive index of 1.45. (a) For what range of values of the core refractive index will the fiber be single moded for all wavelengths in the 1.2-1.6 #m range? (b) What is the value of the core refractive index for which the V parameter is 2.0 at k = 1.55/,m? What is the propagation constant of the single mode supported by the fiber for this value of the core refractive index? Assume that, in the manufacture of single-mode fiber, the tolerance in the core radius a is +5% and the tolerance in the normalized refractive index difference A is +10%, from their respective nominal values. If the nominal value of A is specified to be 0.005, what is the largest nominal value that you can specify for a while ensuring that the resulting fiber will be single moded for )~ > 1.2/,m even in the presence of the worst-case (but within the specified tolerances) deviations of a and A from their nominal values? Assume that the refractive index of the core is 1.5. In a reference frame moving with the pulse, the basic propagation equation that governs pulse evolution inside a dispersive fiber is 8A i 82A + - o, 104 PROPAGATION OF SIGNALS IN OPTICAL FIBER 2.8 2.9 2.10 2.11 2.12 2.13 where A(z, t) is the pulse envelope. If A(0, t) = Aoexp(-t2/2T~) for some constants A0 and To, solve this propagation equation to find an expression for A(z, t). Note: You may use the following result without proof: _~exp(-(x m)2/2~) dx for all complex m and ~ provided ~t(~) > 0. Hint: Consider the Fourier transform A (z, co) of A (z, t). Starting from (E.8), derive the expression (2.13) for the width Tz of a chirped Gaus- sian pulse with initial width To after it has propagated a distance z. Show that an unchirped Gaussian pulse launched at z - 0 remains Gaussian for all z but acquires a distance-dependent chirp factor sgn (fl2 ) z / L o K(z) = 1 + (Z/LD) e" Show that the rms width of a Gaussian pulse whose half-width at the 1/e-intensity point is To is given by To/~/~. Consider a chirped Gaussian pulse for which the product xfi2 is negative that is launched at z = 0. Let x = 5. (a) For what value of z (as a multiple of LD) does the launched pulse attain its minimum width? (b) For what value of z is the width of the pulse equal to that of an unchirped pulse, for the same value of z? (Assume the chirped and unchirped pulses have the same initial pulse width.) Show that in the case of four-wave mixing, the nonlinear polarization is given by terms (2.28)through (2.32). You want to design a soliton communication system at 1.55 ~m, at which wavelength the fiber has f12 = -2 psZ/km and y = 1/W-km. The peak power of the pulses you can generate is limited to 50 mW. If you must use fundamental solitons and the bit period must be at least 10 times the full width at half-maximum (T~VHM) of the soliton pulses, what is the largest bit rate you can use? (This problem requires familiarity with the material in Appendix E.) References [Agr95] G.P. Agrawal. Nonlinear Fiber Optics, 2nd edition. Academic Press, San Diego, CA, 1995. References 105 [Agr97] G.P. Agrawal. Fiber-Optic Communication Systems. John Wiley, New York, 1997. [And00] P.A. Andrekson. High speed soliton transmission on installed fibers. In OFC 2000 Technical Digest, pages TuP2-1/229-231, 2000. [BC90] P.N. Butcher and D. Cotter. The Elements of Nonlinear Optics, volume 9 of Cambridge Studies in Modern Optics. Cambridge University Press, Cambridge, 1990. [Buc95] J.A. Buck. Fundamentals of Optical Fibers. John Wiley, New York, 1995. [BW99] M. Born and E. Wolf. Principles of Optics: Electromagnetic Theory of Propagation, Diffraction and Interference of Light. Cambridge University Press, 1999. [Glo71] D. Gloge. Weakly guiding fibers. Applied Optics, 10:2252-2258, 1971. [Gre93] P.E. Green. Fiber-Optic Networks. Prentice Hall, Englewood Cliffs, NJ, 1993. [HJKM78] K.O. Hill, D. C. Johnson, B. S. Kawasaki, and R. I. MacDonald. CW three-wave mixing in single-mode optical fibers. Journal of Applied Physics, 49(10):5098-5106, Oct. 1978. [Jeu90] L.B. Jeunhomme. Single-Mode Fiber Optics. Marcel Dekker, New York, 1990. [Kan99] J. Kani et al. Interwavelength-band nonlinear interactions and their suppression in multiwavelength-band WDM transmission systems. IEEE/OSA Journal on Lightwave Technology, 17:2249-2260, 1999. [KBW96] L.G. Kazovsky, S. Benedetto, and A. E. Willner. Optical Fiber Communication Systems. Artech House, Boston, 1996. [KK97] I.P. Kaminow and T. L. Koch, editors. Optical Fiber Telecommunications IIIA. Academic Press, San Diego, CA, 1997. [Lin89] C. Lin, editor. Optoelectronic Technology and Lightwave Communications Systems. Van Nostrand Reinhold, New York, 1989. [Liu98] Y. Liu et al. Advanced fiber designs for high capacity DWDM systems. In Proceedings of National Fiber Optic Engineers Conference, 1998. [Mar74] D. Marcuse. Theory of Dielectric Optical Waveguides. Academic Press, New York, 1974. [MK88] S.D. Miller and I. P. Kaminow, editors. Optical Fiber Telecommunications II. Academic Press, San Diego, CA, 1988. [Nak00] M. Nakazawa et al. Ultrahigh-speed long-distance TDM and WDM soliton transmission technologies. IEEE Journal of Selected Topics in Quantum Electronics, 6:363-396, 2000. [Neu88] E G. Neumann. Single-Mode Fibers. Springer-Verlag, Berlin, 1988. 106 PROPAGATION OF SIGNALS IN OPTICAL FIBER [NSK99] M. Nakazawa, K. Suzuki, and H. Kubota. Single-channel 80 Gbit/s soliton transmission over 10000 km using in-line synchronous modulation. Electronics Letters, 35:1358-1359, 1999. [RN76] H D. Rudolph and E G. Neumann. Approximations for the eigenvalues of the fundamental mode of a step-index glass fiber waveguide. Nachrichtentechnische Zeitschrift, 29(14):328-329, 1976. [RWv93] S. Ramo, J. R. Whinnery, and T. van Duzer. Fields and Waves in Communication Electronics. John Wiley, New York, 1993. [SKN01] K. Suzuki, H. Kubota, and M. Nakazawa. 1 Tb/s (40 Gb/s x 25 channel) DWDM quasi-DM soliton transmission over 1,500 km using dispersion-managed single-mode fiber and conventional C-band EDFAs. In OFC 2001 Technical Digest, pages TUN7/1-3, 2001. Components N THIS CHAPTER, we will discuss the physical principles behind the operation I of the most important components of optical communication systems. For each component, we will give a simple descriptive treatment followed by a more detailed mathematical treatment. The components used in modern optical networks include couplers, lasers, pho- todetectors, optical amplifiers, optical switches, and filters and multiplexers. Cou- plers are simple components used to combine or split optical signals. After describing couplers, we will cover filters and multiplexers, which are used to multiplex and de- multiplex signals at different wavelengths in WDM systems. We then describe various types of optical amplifiers, which are key elements used to overcome fiber and other component losses and, in many cases, can be used to amplify signals at multiple wavelengths. Understanding filters and optical amplifiers is essential to understand- ing the operation of lasers, which comes next. Semiconductor lasers are the main transmitters used in optical communication systems. Then we discuss photodetec- tors, which convert the optical signal back into the electrical domain. This is followed by optical switches, which play an important role as optical networks become more agile. Finally, we cover wavelength converters, which are used to convert signals from one wavelength to another, at the edges of the optical network, as well as inside the network. 107 108 COMPONENTS Figure 3.1 A directional coupler. The coupler is typically built by fusing two fibers together. It can also be built using waveguides in integrated optics. 3.1 Couplers A directional coupler is used to combine and split signals in an optical network. A 2 x 2 coupler consists of two input ports and two output ports, as is shown in Figure 3.1. The most commonly used couplers are made by fusing two fibers together in the middle these are called fused fiber couplers. Couplers can also be fabricated using waveguides in integrated optics. A 2 x 2 coupler, shown in Figure 3.1, takes a fraction o~ of the power from input 1 and places it on output 1 and the remaining fraction 1 - ot on output 2. Likewise, a fraction 1 - ot of the power from input 2 is distributed to output 1 and the remaining power to output 2. We call ot the coupling ratio. The coupler can be designed to be either wavelength selective or wave- length independent (sometimes called wavelength flat) over a usefully wide range. In a wavelength-independent device, ot is independent of the wavelength; in a wavelength-selective device, ot depends on the wavelength. A coupler is a versatile device and has many applications in an optical network. The simplest application is to combine or split signals in the network. For example, a coupler can be used to distribute an input signal equally among two output ports if the coupling length, l in Figure 3.1, is adjusted such that half the power from each input appears at each output. Such a coupler is called a 3 dB coupler. An n x n star coupler is a natural generalization of the 3 dB 2 x 2 coupler. It is an n-input, n-output device with the property that the power from each input is divided equally among all the outputs. An n x n star coupler can be constructed by suitably interconnecting a number of 3 dB couplers, as shown in Figure 3.2. A star coupler is useful when multiple signals need to be combined and broadcast to many outputs. However, other constructions of an n x n coupler in integrated optics are also possible (see, for example, [Dra89]). 3.1 Couplers 109 Figure 3.2 A star coupler with eight inputs and eight outputs made by combining 3 dB couplers. The power from each input is split equally among all the outputs. Couplers are also used to tap off a small portion of the power from a light stream for monitoring purposes or other reasons. Such couplers are also called taps and are designed with values of ~ close to 1, typically 0.90-0.95. Couplers are the building blocks for several other optical devices. We will explore the use of directional couplers in modulators and switches in Sections 3.5.4 and 3.7. Couplers are also the principal components used to construct Mach-Zehnder interferometers, which can be used as optical filters, multiplexers/demultiplexers, or as building blocks for optical modulators, switches, and wavelength converters. We will study these devices in Section 3.3.7. So far, we have looked at wavelength-independent couplers. A coupler can be made wavelength selective, meaning that its coupling coefficient will then depend on the wavelength of the signal. Such couplers are widely used to combine signals at 1310 nm and 1550 nm into a single fiber without loss. In this case, the 1310 nm signal on input 1 is passed through to output 1, whereas the 1550 nm signal on input 2 is passed through also to output 1. The same coupler can also be used to separate the two signals coming in on a common fiber. Wavelength-dependent couplers are . of a chirped Gaus- sian pulse with initial width To after it has propagated a distance z. Show that an unchirped Gaussian pulse launched at z - 0 remains Gaussian for all z but acquires a distance-dependent. which are special pulses designed to play off dispersion and nonlinearities against each other to achieve high-bit-rate, ultra-long-haul transmission. 102 PROPAGATION OF SIGNALS IN OPTICAL FIBER. important at higher bit rates. The main nonlinear effects that impair high-speed WDM transmis- sion are self-phase modulation and four-wave mixing. We studied the origin of these, as well as