308 CHAPTER 7. DISPERSION MANAGEMENT
whereais the amplitude,Tis the width,Cis the chirp, andφis the phase. All four
parameters vary withz. The variational method is useful to find thezdependence of
these parameters. It makes use of the fact that Eq. (7.8.4) can be derived from the
Euler–Lagrange equation using the following Lagrangian density:
Lden=i
2(
B
∂B∗
∂z−B∗
∂B
∂z)
+
1
2
[
γ ̄(z)|B|^4 −β 2 (z)∣
∣
∣
∣
∂B
∂t∣
∣
∣
∣
2 ]
. (7.8.6)
Following the variational method, we can find the evolution equations for the four
parametersa,T,C, andφ. The phase equation can be ignored as it is not coupled to
the other three equations. The amplitude equation can be integrated to find that the
combinationa^2 Tdoes not vary withzand is related to the input pulse energyE 0 as
a^2 T=
√
πE 0. Thus, one only needs to solve the following two coupled equations:dT
dz=
β 2 C
T, (7.8.7)
dC
dz=
γ ̄E 0
√
2 πT+( 1 +C^2 )
β 2
T^2. (7.8.8)
Consider first the linear case by settingγ=0. Noting that the ratio( 1 +C^2 )/T^2 is
related to the spectral width of the pulse that remains constant in a linear medium, we
can replace it by its initial value( 1 +C^20 )/T 02 , whereT 0 andC 0 are the width and the
chirp of input pulses before they are launched into the dispersion-managed fiber link.
Equations (7.8.7) and (7.8.8) can now be solved analytically and have the following
general solution:
T^2 (z)=T 02 + 2∫z0β 2 (z)C(z)dz, C(z)=C 0 +1 +C^20
T 02
∫z0β 2 (z)dz. (7.8.9)This solution looks complicated but is is easy to perform the integrations for a two-
section dispersion map. In fact, the values ofTandCat the end of the first map period
(z=Lm) are given by
T 1 =T 0 [( 1 +C 0 d)^2 +d^2 ]^1 /^2 , C 1 =C 0 +( 1 +C 02 )d, (7.8.10)whered=β ̄ 2 Lm/T 02 andβ ̄ 2 is the average GVD value. This is exactly what one would
expect from the theory of Section 2.4. It is easy to see that whenβ ̄ 2 =0, bothTand
Creturn to their input values at the end of each map period, as they should for a linear
medium. When the average GVD of the dispersion-managed link is not zero,TandC
change after each map period, and pulse evolution is not periodic.
When the nonlinear term is not negligible, the pulse parameters do not return to
their input values for perfect GVD compensation (d=0). It was noted in several
experiments that the nonlinear system performs best when GVD compensation is only
90–95% so that some residual dispersion remains after each map period. In fact, if
the input pulse is initially chirped such thatβ ̄ 2 C<0, the pulse at the end of the fiber
link may be shorter than the input pulse. This behavior is expected for a linear system
(see Section 2.4) and follows from Eq. (7.8.10) forC 0 d<0. It also persists for weakly