Nonlinear Fiber Optics - 4 ed. Agrawal

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Problems 269 7.2 Derive the dispersion relation (7.2.7) for XPM-induced modulation instability starting from Eqs. (7.1.15) and (7.1.16). Under what conditions can modulation instability occur in the normal-GVD regime of a fiber? 7.3 Write a computer program in MATLAB (or any other programming language) and reproduce the gain curves shown in Figure 7.1. 7.4 Verify that the pair of bright and dark solitons given by Eqs. (7.3.1) and (7.3.2) indeed satisfies the coupled NLS equations. 7.5 Solve Eqs. (7.4.1) and (7.4.2) analytically after neglecting the GVD terms with second derivatives. 7.6 Starting from the solution given in Eq. (7.4.5), derive an expression for the XPMinduced phase shift imposed on a probe pulse by a copropagating pump pulse. Assume that the two pulses have “sech” shape, the same width, and are launched simultaneously. 7.7 Use the result obtained in the preceding problem to calculate the frequency chirp imposed on the probe pulse. Plot the chirp profile when the pump pulse is 10 ps wide and is launched into a 1-km-long fiber with 10-W peak power. Assume γ = 2W −1 /km for the probe and d = 0.1 ps/m. 7.8 Make a figure similar to Figure 7.3 using the same parameter values but assume that both pulses have “sech” shapes. 7.9 Explain why XPM produces a shift in the wavelength of a 0.53-μm probe pulse when a 1.06-μm pump pulse is launched with it simultaneously (no initial time delay). Can you predict the sign of a wavelength shift for standard optical fibers? 7.10 Write a computer program using the split-step Fourier method and solve Eqs. (7.4.18) and (7.4.19) numerically. Reproduce the results shown in Figure 7.6. 7.11 Use the program developed for the preceding problem to study XPM-induced pulse compression. Reproduce the results shown in Figure 7.9. 7.12 Use the third-order instantaneous nonlinear response of silica fibers and derive Eq. (7.6.3) when the total optical field is of the form given in Eq. (7.6.1). 7.13 Derive the vector form of the nonlinear coupled equations given in Eqs. (7.6.5) and (7.6.6) using Eqs. (7.6.3) and (7.6.4). 7.14 Write a computer program (using MATLAB or other software) and solve Eqs. (7.6.7) and (7.6.8) numerically. Use it to reproduce the results shown in Figures 7.11 and 7.12. 7.15 Write a computer program (using MATLAB or other software) and solve Eqs. (7.6.18) and (7.6.19) numerically. Use it to reproduce the results shown in Figures 7.14 and 7.15. 7.16 Derive the dispersion relation for XPM-induced modulation instability starting from Eqs. (7.7.15) and (7.7.16). Assume that one of the CW beams is polarized along the slow axis while the other beam is linearly polarization at an angle θ from that axis.
270 Chapter 7. Cross-Phase Modulation 7.17 Use the dispersion relation obtained in the preceding problem to plot the gain spectra of modulation instability for several values of the polarization angle (θ equals 0, 20, 45, 70, and 90 ◦ ). Discuss your results physically. References [1] S. A. Akhmanov, R. V. Khokhlov, and A. P. Sukhorukov, in Laser Handbook, Vol. 2, F. T. Arecchi and E. O. Schulz-Dubois, Eds. (North-Holland, Amsterdam, 1972), Chap. E3. [2] J. I. Gersten, R. R. Alfano, and M. Belic, Phys. Rev. A 21, 1222 (1980). [3] A. R. Chraplyvy and J. Stone, Electron. Lett. 20, 996 (1984). [4] R. R. Alfano, Q. X. Li, T. Jimbo, J. T. Manassah, and P. P. Ho, Opt. Lett. 14, 626 (1986). [5] M. N. Islam, L. F. Mollenauer, R. H. Stolen, J. R. Simpson, and H. T. Shang, Opt. Lett. 12, 625 (1987). [6] R. R. Alfano, P. L. Baldeck, F. Raccah, and P. P. Ho, Appl. Opt. 26, 3491 (1987). [7] D. Schadt and B. Jaskorzynska, Electron. Lett. 23, 1090 (1987); J. Opt. Soc. Am. B 4, 856 (1987); J. Opt. Soc. Am. B 5, 2374 (1988). [8] B. Jaskorzynska and D. Schadt, IEEE J. Quantum Electron. 24, 2117 (1988). [9] R. R. Alfano and P. P. Ho, IEEE J. Quantum Electron. 24, 351 (1988). [10] R. R. Alfano, Ed., The Supercontinuum Laser Source, 2nd ed. (Springer-Verlag, New York, 2006). [11] G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987). [12] G. P. Agrawal, P. L. Baldeck, and R. R. Alfano, Phys. Rev. A 39, 3406 (1989). [13] C. J. McKinstrie and R. Bingham, Phys. Fluids B 1, 230 (1989). [14] C. J. McKinstrie and G. G. Luther, Physica Scripta 30, 31 (1990). [15] W. Huang and J. Hong, J. Lightwave Technol. 10, 156 (1992). [16] J. E. Rothenberg, Phys. Rev. Lett. 64, 813 (1990). [17] G. P. Agrawal, Phys. Rev. Lett. 64, 814 (1990). [18] M. Yu, C. J. McKinstrie, and G. P. Agrawal, Phys. Rev. E 48, 2178 (1993). [19] I. O. Zolotovskii and D. I. Sementsov, Opt. Spectroscopy 96, 789 (2004). [20] S. M. Zhang, F. Y. Lu, W. C. Xu, and J. Wang, Opt. Fiber Technol. 11, 193 (2005). [21] A. S. Gouveia-Neto, M. E. Faldon, A. S. B. Sombra, P. G. J. Wigley, and J. R. Taylor, Opt. Lett. 13, 901 (1988). [22] E. J. Greer, D. M. Patrick, P. G. J. Wigley, and J. R. Taylor, Electron. Lett. 18, 1246 (1989); Opt. Lett. 15, 851 (1990). [23] D. M. Patrick and A. D. Ellis, Electron. Lett. 29, 227 (1993). [24] Y. Inoue, J. Phys. Soc. Jpn. 43, 243 (1977). [25] M. R. Gupta, B. K. Som, and B. Dasgupta, J. Plasma Phys. 25, 499 (1981). [26] S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, Opt. Lett. 13, 871 (1988). [27] V. V. Afanasjev, E. M. Dianov, A. M. Prokhorov, and V. N. Serkin, JETP Lett. 48, 638 (1988). [28] V. V. Afanasjev, Y. S. Kivshar, V. V. Konotop, and V. N. Serkin, Opt. Lett. 14, 805 (1989). [29] V. V. Afanasjev, E. M. Dianov, and V. N. Serkin, IEEE J. Quantum Electron. 25, 2656 (1989). [30] M. Florjanczyk and R. Tremblay, Phys. Lett. 141, 34 (1989). [31] L. Wang and C. C. Yang, Opt. Lett. 15, 474 (1990). [32] V. V. Afanasjev, L. M. Kovachev, and V. N. Serkin, Sov. Tech. Phys. Lett. 16, 524 (1990).
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Preface Since the publication of th
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Chapter 1 Introduction This introdu
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1.2. Fiber Characteristics 3 Figure
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1.2. Fiber Characteristics 7 1.49 1
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1.2. Fiber Characteristics 11 faste
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1.3. Fiber Nonlinearities 13 Figure
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1.3. Fiber Nonlinearities 15 Sectio
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1.3. Fiber Nonlinearities 17 1.3.3
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1.4. Overview 19 briefly. The last
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References 21 1.11 Equation (1.3.2)
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References 23 [63] M. Ibanescu, Y.
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Chapter 2 Pulse Propagation in Fibe
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2.2. Fiber Modes 27 where Ẽ(r,ω)
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2.2. Fiber Modes 29 across the core
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2.3. Pulse-Propagation Equation 31
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2.4. Numerical Methods 41 Equation
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2.4. Numerical Methods 43 Figure 2.
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2.4. Numerical Methods 45 the basic
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References 47 2.5 Derive an express
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References 49 [44] V. I. Karpman an
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Chapter 3 Group-Velocity Dispersion
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3.2. Dispersion-Induced Pulse Broad
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3.3. Third-Order Dispersion 63 This
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3.3. Third-Order Dispersion 65 Figu
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3.3. Third-Order Dispersion 67 6 5
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3.3. Third-Order Dispersion 69 Noti
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3.4. Dispersion Management 71 Figur
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3.4. Dispersion Management 73 10 3
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3.4. Dispersion Management 75 Figur
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References 77 3.10 An optical commu
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Chapter 4 Self-Phase Modulation An
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4.1. SPM-Induced Spectral Changes 8
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4.2. Effect of Group-Velocity Dispe
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4.3. Semianalytic Techniques 103 In
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4.3. Semianalytic Techniques 105 (1
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4.4. Higher-Order Nonlinear Effects
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Problems 115 4.2 Plot the spectrum
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References 117 [40] D. Marcuse, J.
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References 119 [109] J. Santhanam a
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5.1. Modulation Instability 121 of
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5.1. Modulation Instability 123 2 L
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5.1. Modulation Instability 125 Fig
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5.1. Modulation Instability 127 Fig
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5.2. Fiber Solitons 129 Figure 5.5:
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5.2. Fiber Solitons 131 and write i
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5.2. Fiber Solitons 133 Physically,
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5.3. Other Types of Solitons 141 1
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5.3. Other Types of Solitons 143 pr
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5.3. Other Types of Solitons 145 Th
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5.4. Perturbation of Solitons 147 w
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5.4. Perturbation of Solitons 149 w
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5.4. Perturbation of Solitons 151 t
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5.4. Perturbation of Solitons 153 b
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5.4. Perturbation of Solitons 155 F
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5.5. Higher-Order Effects 157 where
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5.5. Higher-Order Effects 159 Figur
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5.5. Higher-Order Effects 161 1 N =
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5.5. Higher-Order Effects 163 Integ
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5.5. Higher-Order Effects 165 Figur
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5.5. Higher-Order Effects 167 that
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Problems 169 Problems 5.1 Solve Eq.
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References 171 [27] E. Brainis, D.
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References 173 [93] L. F. Mollenaue
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References 175 [165] V. V. Afanasje
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Chapter 6 Polarization Effects As d
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6.1. Nonlinear Birefringence 179 wh
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6.1. Nonlinear Birefringence 181 re
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6.2. Nonlinear Phase Shift 183 dA y
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6.2. Nonlinear Phase Shift 185 wher
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6.2. Nonlinear Phase Shift 187 side
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6.3. Evolution of Polarization Stat
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6.4. Vector Modulation Instability
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6.5. Birefringence and Solitons 207
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6.6. Random Birefringence 213 where
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6.6. Random Birefringence 215 PMD,
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6.6. Random Birefringence 217 As in
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8.5. Polarization Effects 319 and s
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Problems 321 Figure 8.25: (a) Avera
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References 323 [4] W. Kaiser and M
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References 325 [75] M. Nakazawa, T.
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References 327 [144] V. I. Kruglov,
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Chapter 9 Stimulated Brillouin Scat
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9.1. Basic Concepts 331 Figure 9.1:
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9.2. Quasi-CW SBS 333 [20]. A part
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9.2. Quasi-CW SBS 335 (see Figure 8
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9.2. Quasi-CW SBS 337 pseudorandom
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9.2. Quasi-CW SBS 339 Figure 9.4: S
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9.3. Brillouin Fiber Amplifiers 341
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9.3. Brillouin Fiber Amplifiers 343
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9.4. SBS Dynamics 345 9.4.1 Coupled
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9.4. SBS Dynamics 347 Pump Power (k
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9.4. SBS Dynamics 349 Figure 9.11:
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9.4. SBS Dynamics 351 Δn SBS . If
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9.5. Brillouin Fiber Lasers 357 Fig
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9.5. Brillouin Fiber Lasers 359 Fig
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9.5. Brillouin Fiber Lasers 361 rin
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References 363 9.4 Estimate SBS thr
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References 365 [47] Y.-X. Fan, F.-Y
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References 367 [119] R. G. Harrison
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10.1. Origin of Four-Wave Mixing 36
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10.2. Theory of Four-Wave Mixing 37
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10.3. Phase-Matching Techniques 377
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10.4. Parametric Amplification 387
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10.5. Polarization Effects 401 Bril
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10.5. Polarization Effects 403 foll
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10.5. Polarization Effects 405 Jone
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10.5. Polarization Effects 407 to s
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10.5. Polarization Effects 409 Figu
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10.6. Applications of Four-Wave Mix
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Problems 417 Figure 10.24: Output s
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References 419 [5] R. W. Boyd, Nonl
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References 421 [79] A. Durécu-Legr
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References 423 [140] M. D. Levenson
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11.1. Nonlinear Parameter 425 11.1.
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11.1. Nonlinear Parameter 427 Figur
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11.1. Nonlinear Parameter 433 when
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11.2. Fibers with Silica Cladding 4
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11.3. Tapered Fibers with Air Cladd
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11.3. Tapered Fibers with Air Cladd
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11.4. Microstructured Fibers 441 he
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11.4. Microstructured Fibers 443 Ef
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11.5. Non-Silica Fibers 445 0.05 0.
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11.5. Non-Silica Fibers 447 Figure
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References 449 wavelength range of
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References 451 [57] H. Yokota, E. S
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Chapter 12 Novel Nonlinear Phenomen
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12.1. Intrapulse Raman Scattering 4
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12.2. Four-Wave Mixing 465 Figure 1
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12.2. Four-Wave Mixing 467 Figure 1
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12.3. Supercontinuum Generation 469
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12.4. Temporal and Spectral Evoluti
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12.5. Harmonic Generation 495 Figur
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12.5. Harmonic Generation 497 traps
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12.5. Harmonic Generation 501 analy
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12.5. Harmonic Generation 505 The p
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References 507 12.6 Derive an expre
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References 509 [45] J. E. Sharping,
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References 511 [111] J. Y. Y. Leong
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References 513 [180] A. L. Calvez,
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Appendix A 515 converted into decib
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Appendix B 517 % This code solves t
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Appendix C List of Acronyms Each sc
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Index acoustic response function, 4
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Index 523 distributed fiber sensors
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Index 525 four-photon, 412 gain-swi
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Index 527 Raman amplification with,
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Index 529 spectral hole burning, 36
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