64 CHAPTER 2. OPTICAL FIBERS
2.6.2 Nonlinear Phase Modulation
The refractive index of silica was assumed to be power independent in the discussion of
fiber modes in Section 2.2. In reality, all materials behave nonlinearly at high intensities
and their refractive index increases with intensity. The physical origin of this effect
lies in the anharmonic response of electrons to optical fields, resulting in a nonlinear
susceptibility [73]. To include nonlinear refraction, we modify the core and cladding
indices of a silica fiber as [31]
n′j=nj+n ̄ 2 (P/Aeff), j= 1 , 2 , (2.6.12)where ̄n 2 is thenonlinear-index coefficient,Pis the optical power, andAeffis the effec-
tive mode area introduced earlier. The numerical value of ̄n 2 is about 2. 6 × 10 −^20 m^2 /W
for silica fibers and varies somewhat with dopants used inside the core. Because of this
relatively small value, the nonlinear part of the refractive index is quite small (< 10 −^12
at a power level of 1 mW). Nevertheless, it affects modern lightwave systems consider-
ably because of long fiber lengths. In particular, it leads to the phenomena of self- and
cross-phase modulations.
Self-Phase Modulation
If we use first-order perturbation theory to see how fiber modes are affected by the
nonlinear term in Eq. (2.6.12), we find that the mode shape does not change but the
propagation constant becomes power dependent. It can be written as [31]
β′=β+k 0 n ̄ 2 P/Aeff≡β+γP, (2.6.13)whereγ= 2 πn ̄ 2 /(Aeffλ)is an important nonlinear parameter with values ranging from
1to5W−^1 /km depending on the values ofAeffand the wavelength. Noting that the
optical phase increases linearly withzas seen in Eq. (2.4.1), theγterm produces a
nonlinear phase shift given by
φNL=∫L0(β′−β)dz=∫L0γP(z)dz=γPinLeff, (2.6.14)whereP(z)=Pinexp(−αz)accounts for fiber losses andLeffis defined in Eq. (2.6.7).
In deriving Eq. (2.6.14)Pinwas assumed to be constant. In practice, time depen-
dence ofPinmakesφNLto vary with time. In fact, the optical phase changes with time
in exactly the same fashion as the optical signal. Since this nonlinear phase modula-
tion is self-induced, the nonlinear phenomenon responsible for it is calledself-phase
modulation(SPM). It should be clear from Eq. (2.4.12) that SPM leads to frequency
chirping of optical pulses. In contrast with the linear chirp considered in Section 2.4,
the frequency chirp is proportional to the derivativedPin/dtand depends on the pulse
shape. Figure 2.19 shows how chirp varies with time for Gaussian (m=1) and super-
Gaussian pulses (m=3). The SPM-induced chirp affects the pulse shape through GVD
and often leads to additional pulse broadening [31]. In general, spectral broadening of
the pulse induced by SPM [79] increases the signal bandwidth considerably and limits
the performance of lightwave systems.