Nonlinear Fiber Optics - 4 ed. Agrawal

122 Chapter 5. Optical Solitons
where sgn(β 2 )=±1 depending on the sign of β 2 ,
Ω 2 c = 4γP 0
|β 2 | = 4
|β 2 |L NL
, (5.1.7)
and the nonlinear length L NL is defined in Eq. (3.1.5). Because of the factor exp[i(β 0 z−
ω 0 t)] that has been factored out in Eq. (2.3.21), the actual wave number and the frequency
of perturbation are β 0 ± K and ω 0 ± Ω, respectively. With this factor in mind,
the two terms in Eq. (5.1.5) represent two different frequency components, ω 0 +Ω and
ω 0 − Ω, that are present simultaneously. It will be seen later that these frequency components
correspond to the two spectral sidebands that are generated when modulation
instability occurs.
The dispersion relation (5.1.6) shows that steady-state stability depends critically
on whether light experiences normal or anomalous GVD inside the fiber. In the case of
normal GVD (β 2 > 0), the wave number K is real for all Ω, and the steady state is stable
against small perturbations. By contrast, in the case of anomalous GVD (β 2 < 0), K
becomes imaginary for |Ω| < Ω c , and the perturbation a(z,T ) grows exponentially with
z as seen from Eq. (5.1.5). As a result, the CW solution (5.1.2) is inherently unstable
for β 2 < 0. This instability is referred to as modulation instability because it leads
to a spontaneous temporal modulation of the CW beam and transforms it into a pulse
train. Similar instabilities occur in many other nonlinear systems and are often called
self-pulsing instabilities [28]–[31].
5.1.2 Gain Spectrum
The gain spectrum of modulation instability is obtained from Eq. (5.1.6) by setting
sgn(β 2 )=−1 and g(Ω) =2Im(K), where the factor of 2 converts g to power gain.
The gain exists only if |Ω| < Ω c and is given by
g(Ω)=|β 2 Ω|(Ω 2 c − Ω 2 ) 1/2 . (5.1.8)
Figure 5.1 shows the gain spectra for three values of the nonlinear length (L NL = 1, 2,
and 5 km) for an optical fiber with β 2 = −5 ps 2 /km. As an example, L NL = 5kmata
power level of 100 mW if we use γ = 2W −1 /km in the wavelength region near 1.55 μm.
The gain spectrum is symmetric with respect to Ω = 0 such that g(Ω) vanishes at Ω = 0.
The gain becomes maximum at two frequencies given by
with a peak value
Ω max = ± Ω ( ) 1/2
c 2γP0
√ = ± , (5.1.9)
2 |β 2 |
g max ≡ g(Ω max )= 1 2 |β 2|Ω 2 c = 2γP 0 , (5.1.10)
where Eq. (5.1.7) was used to relate Ω c to P 0 . The peak gain does not depend on β 2 ,
but it increases linearly with the incident power such that g max L NL = 2.
The modulation-instability gain is affected by the loss parameter α that has been
neglected in the derivation of Eq. (5.1.8). The main effect of fiber losses is to decrease
the gain along fiber length because of reduced power [9]–[11]. In effect, Ω c in Eq.