Nonlinear Fiber Optics - 4 ed. Agrawal

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4.2. Effect of Group-Velocity Dispersion 101 Figure 4.17: Measured (solid lines) intensity and chirp profiles (a) at the output of a Yb-doped fiber amplifier with 30-dB gain and (b) after propagation in a 2-m-long passive fiber. The circles show numerical results based on the NLS equation, dotted lines show the asymptotic solution, and dashed lines show the intensity profile of a sech pulse. (After Ref. [66]; c○2000 APS.) shape is parabolic. A somewhat surprising feature of the solution in Eq. (4.2.10) is the total absence of optical wave breaking in the normal-GVD regime. It was discovered in a 1993 study that a parabolic shape represent the only shape for which optical wave breaking can be suppressed [53]. Optical amplifiers facilitate the production of a parabolic shape since new frequency components are generated continuously through SPM in such a way that the pulse maintains a linear chirp as it is amplified and its width increases through dispersion. An important property of the self-similar solution given in Eqs. (4.2.10) through (4.2.13) is that the amplified pulse depends on the energy of the input pulse, but not on its other properties such as its shape and width. Parabolic pulses has been observed in several experiments in which picosecond or femtosecond pulses were amplified in the normal-dispersion regime of a fiber amplifier. Figure 4.17(a) shows the intensity and phase profiles (deduced from FROG traces) observed at the output end of a 3.6-m-long Yb-doped fiber amplifier with 30-dB gain when a 200-fs pulse with 12-pJ energy was launched into it [66]. Part (b) shows the two profiles after further propagation in a 2-m-long passive fiber. The experimental results agree well with both the numerical results obtained by solving the NLS equation (circles) and the asymptotic solution corresponding to a parabolic pulse (dotted lines). As shown by the dashed lines in Figure 4.17, the observed intensity profile is far from that of a “sech” pulse. Self-similar evolution toward a parabolic pulse has also been observed in fiberbased Raman amplifiers [69]. In such amplifiers the nonlinear phenomenon of stimulated Raman scattering is used to amplify the input pulse (see Section 8.3.6). Mode-
102 Chapter 4. Self-Phase Modulation locked Yb-doped fiber lasers can also emit parabolic-shape pulses under suitable conditions [70]. The self-similar nature of parabolic pulses is helpful for generating highenergy pulses from lasers and amplifiers based on Yb-doped fibers [73]–[76]. 4.3 Semianalytic Techniques The results of the preceding section are based on the numerical solutions of the NLS equation with the split-step Fourier method of Section 2.4.1. Although a numerical solution is necessary for accuracy, considerable physical insight is gained if the NLS equation can be solved approximately in a semianalytic fashion. In this section we employ two such techniques for solving the NLS equation (2.3.45). Using A = √ P 0 e −αz U, this equation can be written in the form 4.3.1 Moment Method i ∂U ∂z − β 2 ∂ 2 A 2 ∂T 2 + γP 0e −αz |A| 2 A = 0. (4.3.1) The moment method was used as early as 1971 in the context of nonlinear optics [77]. It can be used to solve Eq. (4.3.1) approximately, provided one can assume that the pulse maintains a specific shape as it propagates down a fiber even though its amplitude, width, and chirp change in a continuous fashion [78]–[80]. This assumption may hold in some limiting cases. For example, it was seen in Section 3.2 that a Gaussian pulse maintains its shape in a linear dispersive medium even though its amplitude, width, and chirp change during propagation. If the nonlinear effects are relatively weak (L NL ≫ L D ), a Gaussian shape may remain approximately valid. Similarly, it was seen in Section 4.1 that a pulse maintains its shape even when nonlinear effects are strong provided dispersive effect are negligible (L NL ≪ L D ). It will be seen in Chapter 5 in the context of solitons that, even when L NL and L D are comparable, a pulse may maintain its shape under certain conditions. The basic idea behind the moment method is to treat the optical pulse like a particle whose energy E p , RMS width σ p , and chirp C p are related to U(z,T ) as E p = C p = ∫ ∞ −∞ |U| 2 dT, i ∫ ∞ T E p −∞ σp 2 = 1 ∫ ∞ T 2 |U| 2 dT, (4.3.2) E p ( U ∗ ∂U ∂U ∗ −U ∂T ∂T −∞ ) dT. (4.3.3) As the pulse propagates inside the fiber, these three moments change. To find how they evolve with z, we differentiate Eqs. (4.3.2) and (4.3.3) with respect to z and use Eq. (4.3.1). After some algebra, we find that dE p /dz = 0butσp 2 and C p satisfy dσp 2 dz = β ∫ ∞ ( 2 T 2 Im U ∗ ∂ 2 ) U E p −∞ ∂T 2 dT, (4.3.4) dC p dz = 2β ∫ ∞ 2 ∂U 2 E p ∣ ∂T ∣ dT + γP 0 e −αz 1 ∫ ∞ |U| 4 dT. (4.3.5) E p −∞ −∞
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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
Page 56 and 57: Chapter 3 Group-Velocity Dispersion
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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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8.1. Basic Concepts 279 10 mW. Howe
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8.1. Basic Concepts 281 0.5 Raman R
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8.2. Quasi-Continuous SRS 283 four
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8.2. Quasi-Continuous SRS 285 Figur
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8.2. Quasi-Continuous SRS 291 wavel
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8.2. Quasi-Continuous SRS 293 1 0.8
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8.3. SRS with Short Pump Pulses 295
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8.4. Soliton Effects 307 Figure 8.1
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8.4. Soliton Effects 313 ton laser
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8.5. Polarization Effects 315 maxim
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8.5. Polarization Effects 317 where
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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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