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Unformatted text preview: 72 CHAPTER 2. OPTICAL FIBERS rectangular array which is placed inside a polyethylene tube. The mechanical strength is provided by using steel rods in the two outermost polyethylene jackets. The outer diameter of such fiber cables is about l—l.5 cm. Connectors are needed to use optical fibers in an actual communication system. They can be divided into two categories. A permanent joint between two fibers is known as a fiber splice, and a detachable connection between them is realized by using a fiber connector. Connectors are used to link fiber cable with the transmitter (or the receiver), while splices are used to join fiber segments (usually 5—10 km long). The main issue in the use of splices and connectors is related to the loss. Some power is always lost, as the two fiber ends are never perfectly aligned in practice. Splice losses below 0.1 dB are routinely realized by using the technique of fusion splicing [93]. Connector losses are generally larger. State-of—the—art connectors provide an average loss of about 0.3 dB [94]. The technology behind the design of splices and connectors is quite sophisticated. For details, the reader is referred to Ref. [95], a book devoted entirely to this issue. Problems 2.1 A multimode fiber with a 50-;tm core diameter is designed to limit the inter- modal dispersion to 10 ns/km. What is the numerical aperture of this fiber? What is the limiting bit rate for transmission over 10 km at 0.88 [.tm? Use 1.45 for the refractive index of the cladding. 2.2 Use the ray equation in the paraxial approximation [Eq. (21.8)] to prove that intermodal dispersion is zero for a graded-index fiber with a quadratic index profile. 2.3 Use Maxwell’s equations to express the field components E p, E¢, Hp, and H1, in terms of EZ and Hz and obtain Eqs. (2.2.29)—(2.2.32). 2.4 Derive the eigenvalue equation (2.2.33) by matching the boundary conditions at the core—cladding interface of a step—index fiber. 2.5 A single-mode fiber has an index step 111 — n2 2 0.005. Calculate the core radius if the fiber has a cutoff wavelength of 1 ,um. Estimate the spot size (FWHM) of the fiber mode and the fraction of the mode power inside the core when this fiber is used at 1.3 ,um. Use 111 = 1.45 mm. 2.6 A 1.55—um unchirped Gaussian pulse of 100-ps width (FWHM) is launched into ' a single—mode fiber. Calculate its FWHM after 50 km if the fiber has a dispersion of 16 ps/(km-nm). Neglect the source spectral width. 2.7 Derive an expression for the confinement factor F of single-mode fibers defined as the fraction of the total mode power contained inside the core. Use the Gaus- sian approximation for the fundamental fiber mode. Estimate 1" for V : 2. 2.8 A single-mode fiber is measured to have 22(d2n / £122) : 0.02 at 0.8 #111. Cal- culate the dispersion parameters B2 and D. PROBLEMS 73 —m0de expressions for the minimum width and the fiber length at which the minimum occurs. 2.10 Estimate the limiting bit rate for a 60—km ‘um wavelengths assuming transform—limited, 50—ps (FWHM) input pulses. As— sume that [32 = 0 and —20 psZ/km and fi3 = 0.1 ps3/km and O at 1.3- and 1.55-um wavelengths, respectively. Also assume that Va, << 1. single—mode fiber link at 1.3- and 1.55- se of a single~mode semiconductor laser for which ate is limited by B(l[33|L)1/3 < 0.324. What is the = 100 km if m = 0.1 pS3/km? Va, << 1 and show that the bit r limiting bit rate for L ystem operating at 5 Gb/s is using Gaus- M) chirped such that C = —6. What is the ength? How much will it change if the pulses phase shift induced by SPM b neglect fiber losses. 2.19 Calculate the power launched into a 40-km-long single-mode fiber for which the SPM—induced nonlinear phase shift becomes 180°. Assume l = 1.55 pm, A63 = 40 ,umz, a = 0.2 dB/km, and r72 = 2.6 >< 10*20 m2M. 74 CHAPTER 2. OPTICAL FIBERS 2.20 Find the maximum frequency shift occurring because of the SPM—induced chirp imposed on a Gaussian pulse of 20-ps width (FWHM) and 5—mW peak power af— ter it has propagated 100 km. Use the fiber parameters of the preceding problem but assume ()6 = 0. References [1] J. Tyndall, Proc. Roy. Inst. 1, 446 (1854). [2] J. L. Baird, British Patent 285,738 (1927). [3] C. W. Hansell, US. Patent 1,751,584 (1930). [4] H. Lamm, Z. Instrumentenk. 50, 579 (1930). [5] A. C. S. van Heel, Nature 173, 39 (1954). [6] B. I. Hirschowitz, L. E. Curtiss, C. W. Peters, and H. M. Pollard, Gastro—enterology 35, 50 (1958). [7] N. S. Kapany, J. Opt. Soc. Am. 49, 779 (1959). [8] N. S. Kapany, Fiber Optics: Principles and Applications, 1967. [9] K. C. Kao and G. A. Hockha 967 (1966). [10] F. P. Kapron, D. B. Kec [11] T. Miya, Y. Temnuma, T. Academic Press, San Diego, CA, In, Proc. IEE 113, 1151 (1966); A. Werts, Onde Electr. 45, k, and R. D. Maurer, Appl. Phys. Lett. 17, 423 (1970). Hosaka, and T. Miyoshita, Electron. Lett. 15, 106 (1979). [12] M. J. Adams, An Introduction to Optical Waveguides, Wiley, New York, 1981. [13] T. Okoshi, Optical Fibers, Academic Press, San Diego, CA, 1982. [14] A. W. Snyder and J. D. Love, Optical Waveguide Theory, Chapman & Hall, London, 1983. [15] L. B. Jeunhomme, Single—Mode Fiber Optics, Marcel Dekker, New York, 1990. [16] T. Li, Ed., Optical Fiber Communications, Vol. 1, Academic Press, San Diego, CA, 1985. [17] T. Izawa and S. Sudo, Optical Fibers: Materials and Fabrication. Kluwer Academic, Boston, 1987. [18] E. G. Neumann, Single-Mode Fibers, [19] D. Marcuse, Theory of Dielectric Op Diego, CA, 1991. [20] G. Cancellieri, Single—Mode Optical Fibers, [21] J. A. Buck, Fundamentals of Optical Fibers, [22] M. Born and E. Wolf, Principles of Optics, York, 1999. [23] J. Gower, Optical [24] Y. Koike, T. lshigure, and E. Nihei, [25] T. lshigure, A. Horibe, E. Nihei, an [26] U. Fiedler, G. Reiner, P. Schnitzer, and K. J. Ebeling, (1996). ‘ [27] O. Nolen, Plastic Optical Fibers for Data C0 Boston, 1996. [28] C. DeCusatis, Opt. Eng. 37, 3082 (1998). [29] F. Mederer, R. J ager, P. Schnitzer, H. Unold, M. Kicherer, K. J. Ebeling, M. Naritomi, and R. Yoshida, IEEE Photon. Technol. Lett. 12, 201 (2000). [30] P. Diament, Wave Transmission and Fiber Optics, Macmillan, New Springer, New York, 1988. tical Waveguides, 2nd ed., Academic Press, San Pergamon Press, Elmsford, NY, 1991. Wiley. New York, 1995. 7th ed., Cambridge University Press, New Communication Systems, 2nd ed., Prentice Hall, London, 1993. J. Lightwave Technol. 13, 1475 (1995). d Y. Koike, J. Lightwave Technol. 13, 1686 (1995). IEEE Photon. Technol. Lett. 8, 746 mmunications, Information Gatekeepers, York, 1990, Chap. 3. ...
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