Nonlinear Fiber Optics - 4 ed. Agrawal

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5.1. Modulation Instability 125 Figure 5.3: Autocorrelation traces showing induced modulation instability at two different probe wavelengths. The modulation period can be adjusted by tuning the semiconductor laser acting as a probe. (After Ref. [13]; c○1986 AIP.) Moreover, the period could be adjusted by tuning the wavelength of the probe laser. Figure 5.3 shows the autocorrelation traces for two different probe wavelengths. As the observed pulse width is
126 Chapter 5. Optical Solitons NLS equation (5.1.1). A simple approach solves this equation in the frequency domain as a four-wave mixing problem [23]; it is discussed in detail in Chapter 10. The main disadvantage of this approach is that it cannot treat generation of higher-order sidebands located at ω 0 ±mΩ (m = 2,3,...) that are invariably created when the first-order sidebands (m = 1) become strong. The time-domain approach solves the NLS equation directly. Numerical solutions of Eq. (5.1.1) obtained with the input corresponding to a CW beam with weak sinusoidal modulation imposed on it show that the nearly CW beam evolves into a train of narrow pulses, separated by the period of initial modulation [8]. The fiber length required to realize such a train of narrow pulses depends on the initial modulation depth and is typically ∼5 L D . With further propagation, the multipeak structure deforms and eventually returns to the initial input form. This behavior is found to be generic when Eq. (5.1.1) is solved by considering arbitrary periodic modulation of the steady state [32]. The foregoing scenario suggests that the NLS equation should have periodic solutions whose forms change with propagation. Indeed, it turns out that the NLS equation has a multiparameter family of periodic solutions [32]–[39]. In their most general form, these solutions are expressed in the form of Jacobian elliptic functions. In some specific cases, the solution can be written in terms of trigonometric and hyperbolic functions [36]. From a practical standpoint, modulation instability is useful for generating a train of short optical pulses whose repetition rate can be externally controlled. As early as 1989, 130-fs pulses at a 2-THz repetition rate were generated through induced modulation instability [40]. Since then, this technique has been used to create optical sources capable of producing periodic trains of ultrashort pulses at repetition rates higher than those attainable from mode-locked lasers. Several experiments have used dispersiondecreasing fibers for this purpose [41]–[43]. Initial sinusoidal modulation in these experiments was imposed by beating two optical signals. In a 1992 experiment [42], the outputs of two distributed feedback (DFB) semiconductor lasers, operating continuously at slightly different wavelengths near 1.55 μm, were combined in a fiber coupler to produce a sinusoidally modulated signal at a beat frequency that could be varied in the 70–90 GHz range by controlling the laser temperature. In a later experiment [43], 250-fs pulses at a tunable repetition rate of 80 to 120 GHz were generated. The beat signal from two DFB lasers was amplified to power levels ∼0.8 W by using a fiber amplifier and then propagated through a 1.6-km-long dispersion-decreasing fiber, whose GVD decreased from 10 to 0.5 ps/(km-nm) over its length. Figure 5.4 shows the output pulse train (width 250 fs) at a 114-GHz repetition rate and the corresponding optical spectrum. The spectrum was shifted toward red because of intrapulse Raman scattering that becomes important for such short pulses (see Section 5.5.4). The use of a dispersion-decreasing fiber is not essential for producing pulse trains through modulation instability. In an interesting experiment, a comb-like dispersion profile was produced by splicing pieces of low- and high-dispersion fibers [44]. A dual-frequency fiber laser was used to generate the high-power signal modulated at a frequency equal to the longitudinal-mode spacing (59 GHz). When such a modulated signal was launched into the fiber, the output consisted of a 2.2-ps pulse train at the 59- GHz repetition rate. In another experiment [45], a periodic train of 1.3-ps pulses at the 123-GHz repetition rate was generated by launching the high-power beat signal into a
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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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Page 174 and 175: Problems 169 Problems 5.1 Solve Eq.
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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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Nonlinear Fiber Optics - 4 ed. Agrawal

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