Nonlinear Fiber Optics - 4 ed. Agrawal

322 Chapter 8. Stimulated Raman Scattering
with an intense pump beam. Assume g R I p (0) =2km −1 and α p = α s = 0.2
dB/km.
8.3 Perform the integration in Eq. (8.1.9) using the method of steepest descent and
derive Eq. (8.1.9).
8.4 Use Eq. (8.1.9) to derive the Raman-threshold condition given in Eq. (8.1.13).
8.5 Solve Eqs. (8.1.2) and (8.1.3) analytically after assuming α p = α s .
8.6 Calculate the threshold pump power of a 1.55-μm Raman laser whose cavity
includes a 1-km-long fiber with 40-μm 2 effective core area. Use α p = 0.3 dB/km
and total cavity losses of 6 dB. Use the Raman gain from Figure 8.2.
8.7 Explain the technique of time-dispersion tuning used commonly for synchronously
pumped Raman lasers. Estimate the tuning range for the laser of Problem
8.7.
8.8 Solve Eqs. (8.3.1) and (8.3.2) analytically after setting β 2p = β 2s =0.
8.9 Use the results of the preceding problem to plot the output pulse shapes for Raman
amplification in a 1-km-long fiber. Assume λ p = 1.06 μm, λ s = 1.12 μm,
γ p =10W −1 /km, g R = 1 × 10 −3 m/W, d = 5 ps/m, and A eff = 40 μm 2 . Input
pump and Stokes pulses are Gaussian with the same 100-ps width (FWHM) and
with 1 kW and 10 mW peak powers, respectively.
8.10 Solve Eqs. (8.3.16) and (8.3.17) numerically using the split-step Fourier method
and reproduce the results shown in Figures 8.11 and 8.12.
8.11 Solve Eqs. (8.3.16) and (8.3.17) in the case of anomalous dispersion and reproduce
the results shown in Figure 8.19. Explain why the Raman pulse forms
a soliton when the walk-off length and the dispersion length have comparable
magnitudes.
8.12 Design an experiment for amplifying 50-ps (FWHM) Gaussian pulses through
SRS by at least 30 dB such that they are also compressed by a factor of 10. Use
numerical simulations to verify your design.
8.13 Derive Eq. (8.5.3) by using Eq. (8.5.1) and taking into account the properties of
the χ (3) tensor discussed in Section 6.1.
8.14 Derive the set of two coupled vector equations given as Eqs. (8.5.9) and (8.5.10)
starting from the nonlinear polarization given in Eq. (8.5.1).
8.15 Convert Eqs. (8.5.9) and (8.5.10) to the Stokes space and prove that the Stokes
vectors for the pump and signal indeed satisfy Eqs. (8.5.14) and (8.5.15).
References
[1] C. V. Raman, Indian J. Phys. 2, 387 (1928).
[2] E. J. Woodbury and W. K. Ng, Proc. IRE 50, 2347 (1962).
[3] G. Eckhardt, R. W. Hellwarth, F. J. McClung, S. E. Schwarz, and D. Weiner, Phys. Rev.
Lett. 9, 455 (1962).