322 CHAPTER 7. DISPERSION MANAGEMENT
7.12Solve Eqs. (7.6.1) and (7.6.2) and show that the transfer function of a Bragg
grating is indeed given by Eq. (7.6.4).
7.13Write a computer program to solve Eqs. (7.6.1) and (7.6.2) for chirped fiber
gratings such that bothδandκvary withz. Use it to plot the amplitude and
phase of the reflectivity of a grating in which the period varies linearly by 0.01%
over the 10-cm length. AssumeκL=4 and the Bragg wavelength of 1.55μmat
the input end of the grating.
7.14Use the dispersion relationq^2 =δ^2 −κ^2 of a Bragg grating to show that the
second- and third-order dispersion parameters of the grating are given by Eq.
(7.6.5).
7.15Explain how a chirped fiber grating compensates for GVD. Derive an expression
for the GVD parameter of such a grating when the grating period varies linearly
by∆Λover the grating lengthL.
7.16Explain how midspan OPC compensates for fiber dispersion. Show that the OPC
process inverts the signal spectrum.
7.17Prove that both SPM and GVD can be compensated through midspan OPC only
if the fiber lossα=0. Show also that simultaneous compensation of SPM and
GVD can occur whenα =0 if GVD decreases along the fiber length. What is
the optimum GVD profile of such a fiber?
7.18Prove that the phase conjugator should be located at a distance given in Eq.
(7.7.9) when the frequencyωcof the phase-conjugated field does not coincide
with the signal frequencyωs.
7.19Derive the variational equations for the pulse width and chirp using the La-
grangian density given in Eq. (7.8.6).
7.20Solve the variational equations (7.8.7) and (7.8.8) after settingγ=0 and find the
pulse width and chirp after one map period in terms of their initial values.
References
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