Nonlinear Fiber Optics - 4 ed. Agrawal

Problems 115
4.2 Plot the spectrum of the chirped output pulse obtained in the preceding by using
Eq. (4.1.13). Does the number of spectral peaks agree with the prediction of Eq.
(4.1.14)?
4.3 Repeat Problem 4.1 for a hyperbolic-secant pulse. Plot the spectrum of the
chirped output pulse using Eq. (4.1.13). Comment on the impact of the pulse
shape on SPM-induced spectral broadening.
4.4 Determine the shape, width, and peak power of the optical pulse that will produce
a linear chirp at the rate of 1 GHz/ps over a 100-ps region when transmitted
through a 1-km-long fiber with a loss of 1 dB/km and an effective mode area of
50 μm 2 .
4.5 Calculate numerically the SPM-broadened spectra of a super-Gaussian pulse
(m = 3) for C = −15, 0, and 15. Assume a peak power such that φ max = 4.5π.
Compare your spectra with those shown in Figure 4.5 and comment on the main
qualitative differences.
4.6 Perform the ensemble average in Eq. (4.1.19) for a thermal field with Gaussian
statistics and prove that the coherence function is indeed given by Eq. (4.1.20).
4.7 Solve Eq. (4.2.1) numerically using the split-step Fourier method of Section 2.4.
Generate curves similar to those shown in Figures 4.8 and 4.9 for an input pulse
with (U(0,τ)=sech(τ) using N = 1 and α = 0. Compare your results with the
Gaussian-pulse case and discuss the differences qualitatively.
4.8 Use the computer program developed for the preceding problem to study numerically
optical wave breaking for an unchirped super-Gaussian pulse with m = 3
by using N = 30 and α = 0. Compare your results with those shown in Figures
4.11 and 4.12.
4.9 Perform the integrals appearing in Eqs. (4.3.4) and (4.3.5) using U(z,T ) from
Eq. (4.3.6) and reproduce Eqs. (4.3.7) and (4.3.8).
4.10 Perform the integrals appearing in Eqs. (4.3.4) and (4.3.5) using the field amplitude
in the form U(z,T )=a p sech(T /T p )exp(−iC p T 2 /2T 2 p ) and derive equations
for T p and C p .
4.11 Prove that the Euler–Lagrange equation (4.3.11) with L d given in Eq. (4.3.12)
reproduces the NLS equation (4.3.1).
4.12 Perform the integral appearing in Eqs. (4.3.10) using L d from Eq. (4.3.12) and
U(z,T ) from Eq. (4.3.6) and reproduce Eq. (4.3.13).
4.13 Show that the solution given in Eq. (4.4.9) is indeed the solution of Eq. (4.4.4)
for a Gaussian pulse. Calculate the phase profile φ(Z,τ) at sZ = 0.2 analytically
(if possible) or numerically.
4.14 Use the moment method and Ref. [109] to derive equations for the derivatives
dq/dz and dΩ p /dz for a Gaussian pulse. Use them to find an approximate expression
for the Raman-induced frequency shift Ω p .