Nonlinear Fiber Optics - 4 ed. Agrawal

80 Chapter 4. Self-Phase Modulation
4.1.1 Nonlinear Phase Shift
In terms of the normalized amplitude U(z,T ) defined as in Eq. (3.1.3), the pulsepropagation
equation (3.1.4) in the limit β 2 = 0 becomes
∂U
∂z = ie−αz
L NL
|U| 2 U, (4.1.1)
where α accounts for fiber losses. The nonlinear length is defined as
L NL =(γP 0 ) −1 , (4.1.2)
where P 0 is the peak power and γ is related to the nonlinear-index coefficient n 2 as in
Eq. (2.3.29). Equation (4.1.1) can be solved by using U = V exp(iφ NL ) and equating
the real and imaginary parts so that
∂V
∂z = 0;
∂φ NL
∂z
= e−αz
L NL
V 2 . (4.1.3)
As the amplitude V does not change along the fiber length L, the phase equation can be
integrated analytically to obtain the general solution
where U(0,T ) is the field amplitude at z = 0 and
U(L,T )=U(0,T )exp[iφ NL (L,T )], (4.1.4)
φ NL (L,T )=|U(0,T )| 2 (L eff /L NL ). (4.1.5)
The effective length L eff for a fiber of length L is defined as
L eff =[1 − exp(−αL)]/α. (4.1.6)
Equation (4.1.4) shows that SPM gives rise to an intensity-dependent phase shift
but the pulse shape remains unaffected. The nonlinear phase shift φ NL in Eq. (4.1.5)
increases with fiber length L. The quantity L eff plays the role of an effective length that
is smaller than L because of fiber losses. In the absence of fiber losses, α = 0, and
L eff = L. The maximum phase shift φ max occurs at the pulse center located at T = 0.
With U normalized such that |U(0,0)| = 1, it is given by
φ max = L eff /L NL = γP 0 L eff . (4.1.7)
The physical meaning of the nonlinear length L NL is clear from Eq. (4.1.7): It is the
effective propagation distance at which φ max = 1. If we use a typical value γ = 2
W −1 /km in the 1.55-μm wavelength region, L NL = 50 km at a power level P 0 = 10 mW
and it decreases inversely with an increase in P 0 .
Spectral changes induced by SPM are a direct consequence of the time dependence
of φ NL . This can be understood by recalling from Section 3.2 that a temporally varying
phase implies that the instantaneous optical frequency differs across the pulse from its
central value ω 0 . The difference δω is given by
δω(T )=− ∂φ NL
∂T
= − (
Leff
L NL
) ∂
∂T |U(0,T )|2 , (4.1.8)