Nonlinear Fiber Optics - 4 ed. Agrawal

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Chapter 9 Stimulated Brillouin Scattering Stimulated Brillouin scattering (SBS) is a nonlinear process that can occur in optical fibers at input power levels much lower than those needed for stimulated Raman scattering (SRS). It manifests through the generation of a backward-propagating Stokes wave that carries most of the input power, once the Brillouin threshold is reached. For this reason, SBS limits the channel power in optical communication systems. At the same time, it can be useful for making fiber-based Brillouin lasers and amplifiers. This chapter is devoted to the SBS phenomenon in optical fibers. Section 9.1 presents the basic concepts behind SBS with emphasis on the spectrum of Brillouin gain. Section 9.2 focuses on the Brillouin threshold in the case of a continuous-wave (CW) pump and the techniques used to control it. Fiber-Brillouin amplifiers and their properties are discussed in Section 9.3. The dynamic aspects of SBS are the focus of Section 9.4, where we discuss SBS for a pulsed pump together with phenomena such as SBSinduced modulation instability and optical chaos. Section 9.5 is devoted to Brillouin lasers operating continuously or pulsed. 9.1 Basic Concepts The nonlinear phenomenon of SBS, first observed in 1964, has been studied extensively [1]–[10]. SBS is similar to SRS inasmuch as it manifests through the generation of a Stokes wave whose frequency is downshifted from that of the incident light by an amount set by the nonlinear medium. However, several major differences exist between the two. For example, the Stokes wave propagates backward when SBS occurs in a single-mode optical fiber, in contrast to SRS that can occur in both directions. The Stokes shift (∼10 GHz) is smaller by three orders of magnitude for SBS compared with that of SRS. The threshold pump power for SBS depends on the spectral width associated with the pump wave. It can be as low as 1 mW for a CW pump, or when the pumping is in the form of relatively wide pulses (width >1 μs). In contrast, SBS nearly ceases to occur for short pump pulses (width
330 Chapter 9. Stimulated Brillouin Scattering 9.1.1 Physical Process The process of SBS can be described classically as a nonlinear interaction between the pump and Stokes fields through an acoustic wave. The pump field generates an acoustic wave through the process of electrostriction [10]. The acoustic wave in turn modulates the refractive index of the medium. This pump-induced index grating scatters the pump light through Bragg diffraction. Scattered light is downshifted in frequency because of the Doppler shift associated with a grating moving at the acoustic velocity v A . The same scattering process can be viewed quantum mechanically as if annihilation of a pump photon creates a Stokes photon and an acoustic phonon simultaneously. As both the energy and the momentum must be conserved during each scattering event, the frequencies and the wave vectors of the three waves are related by Ω B = ω p − ω s , k A = k p − k s , (9.1.1) where ω p and ω s are the frequencies, and k p and k s are the wave vectors, of the pump and Stokes waves, respectively. The frequency Ω B and the wave vector k A of the acoustic wave satisfy the standard dispersion relation Ω B = v A |k A |≈2v A |k p |sin(θ/2), (9.1.2) where θ is the angle between the pump and Stokes fields, and we used |k p |≈|k s | in Eq. (9.1.1). Equation (9.1.2) shows that the frequency shift of the Stokes wave depends on the scattering angle. In particular, Ω B is maximum in the backward direction (θ = π) and vanishes in the forward direction (θ = 0). In a single-mode optical fiber, only relevant directions are the forward and backward directions. For this reason, SBS occurs only in the backward direction with the Brillouin shift given by ν B = Ω B /2π = 2n p v A /λ p , (9.1.3) where Eq. (9.1.2) was used with |k p | = 2πn p /λ p and n p is the effective mode index at the pump wavelength λ p . Ifweusev A = 5.96 km/s and n p = 1.45, the values appropriate for silica fibers, ν B ≈ 11.1 GHz at λ p = 1.55 μm. Even though Eq. (9.1.2) predicts correctly that SBS should occur only in the backward direction in single-mode fibers, spontaneous Brillouin scattering can occur in the forward direction. This happens because the guided nature of acoustic waves leads to a relaxation of the wave-vector selection rule. As a result, a small amount of Stokes light is generated in the forward direction. This phenomenon is referred to as guidedacoustic-wave Brillouin scattering [11]. In practice, the Stokes spectrum shows multiple lines with frequency shifts ranging from 10 to 1000 MHz. Because of its extremely weak character, this phenomenon is not considered further in this chapter. 9.1.2 Brillouin-Gain Spectrum Similar to the case of SRS, the growth of the Stokes wave is characterized by the Brillouin-gain spectrum g B (Ω) peaking at Ω = Ω B . However, in contrast to the SRS case, the spectral width of the gain spectrum is very small (∼10 MHz in lieu of ∼10 THz) because it is related to the damping time of acoustic waves related to the
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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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6.6. Random Birefringence 219 Figur
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References 221 6.9 Derive the dispe
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References 223 [63] P. Kockaert, M.
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References 225 [140] A. El Amari, N
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7.1. XPM-Induced Nonlinear Coupling
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7.2. XPM-Induced Modulation Instabi
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7.2. XPM-Induced Modulation Instabi
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7.3. XPM-Paired Solitons 233 This t
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7.3. XPM-Paired Solitons 235 The co
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7.3. XPM-Paired Solitons 237 It is
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7.4. Spectral and Temporal Effects
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7.5. Applications of XPM 249 Probe
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7.5. Applications of XPM 251 by hig
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7.5. Applications of XPM 253 Figure
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7.6. Polarization Effects 255 where
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7.6. Polarization Effects 257 Figur
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7.6. Polarization Effects 259 1 0.8
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7.6. Polarization Effects 261 4 Pum
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7.6. Polarization Effects 263 Figur
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7.7. XPM Effects in Birefringent Fi
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7.7. XPM Effects in Birefringent Fi
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Problems 269 7.2 Derive the dispers
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References 271 [33] M. Lisak, A. H
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References 273 [105] B. V. Vu, A. S
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8.1. Basic Concepts 275 Figure 8.1:
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8.1. Basic Concepts 277 set of two
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10.3. Phase-Matching Techniques 379
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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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Nonlinear Fiber Optics - 4 ed. Agrawal

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