488
12 Nonlinear Processes in Optical Fibers
the pulse. The leading edge will see a positive dn/dt, whereas the trailing edge will
see a negative dn/dt.
This temporally varying index change results in a temporally varying phase
change, shown by dϕ/dt in Fig. 12.8. The consequence is that the instantaneous
optical frequency differs from its initial value ν 0 across the pulse. That is, because
the phase fluctuations are intensity-dependent, different parts of the pulse undergo
different phase shifts. This leads to what is known as frequency chirping, in that the
rising edge of the pulse experiences a red shift in frequency (toward lower frequencies
or longer wavelengths), whereas the trailing edge of the pulse experiences a blue shift
in frequency (toward higher frequencies or shorter wavelengths). Because the degree
of chirping depends on the transmitted power, SPM effects are more pronounced for
higher-intensity pulses.
For some types of fibers, the time-varying phase may result in a power penalty
owing to a GVD-induced spectral broadening of the pulse as it travels along the fiber.
In the normal dispersion region the chromatic dispersion is negative [that is, from
Eq. (3.32) the GVD parameter β 2 > 0] and the group delay decreases with wavelength.
This means that because red light has a longer wavelength than blue, the red light
travels faster in silica because n red < n blue (see Fig. 3.9). Therefore, in the normal
dispersion region the red-shifted leading edge of the pulse travels faster and thus
moves away from the center of the pulse. At the same time the blue-shifted trailing
edge travels slower, and thus also moves away from the center of the pulse. In this
case chirping worsens the effects of GVD-induced pulse broadening. On the other
hand, in the anomalous dispersion region where chromatic dispersion is positive
so that the group delay increases with wavelength, the red-shifted leading edge of
the pulse travels slower and thus moves toward the center of the pulse. Similarly,
the blue-shifted trailing edge travels faster, and also moves toward the center of the
pulse. In this case, SPM causes the pulse to narrow, thereby partly compensating for
chromatic dispersion.
12.6 Cross-Phase Modulation in WDM Systems
Cross-phase modulation (XPM) appears in WDM systems and has a similar origin
as SPM. In this case the name derives from the fact that the refractive index nonlinearity converts optical intensity fluctuations in a particular wavelength channel to
phase fluctuations in adjacent copropagating channels [2–5] In addition, because the
refractive index seen by a particular wavelength is influenced by both the optical
intensity of that wave itself and also by the optical power fluctuations of neighboring
wavelengths, SPM is always present when XPM occurs. Analogous to SPM, for two
interacting wavelengths the XPM-induced frequency shift ϕ is given by
ϕ =
dϕ
dt
= 2γ L e f f
d P
dt
(12.9)
12 Nonlinear Processes in Optical Fibers
the pulse. The leading edge will see a positive dn/dt, whereas the trailing edge will
see a negative dn/dt.
This temporally varying index change results in a temporally varying phase
change, shown by dϕ/dt in Fig. 12.8. The consequence is that the instantaneous
optical frequency differs from its initial value ν 0 across the pulse. That is, because
the phase fluctuations are intensity-dependent, different parts of the pulse undergo
different phase shifts. This leads to what is known as frequency chirping, in that the
rising edge of the pulse experiences a red shift in frequency (toward lower frequencies
or longer wavelengths), whereas the trailing edge of the pulse experiences a blue shift
in frequency (toward higher frequencies or shorter wavelengths). Because the degree
of chirping depends on the transmitted power, SPM effects are more pronounced for
higher-intensity pulses.
For some types of fibers, the time-varying phase may result in a power penalty
owing to a GVD-induced spectral broadening of the pulse as it travels along the fiber.
In the normal dispersion region the chromatic dispersion is negative [that is, from
Eq. (3.32) the GVD parameter β 2 > 0] and the group delay decreases with wavelength.
This means that because red light has a longer wavelength than blue, the red light
travels faster in silica because n red < n blue (see Fig. 3.9). Therefore, in the normal
dispersion region the red-shifted leading edge of the pulse travels faster and thus
moves away from the center of the pulse. At the same time the blue-shifted trailing
edge travels slower, and thus also moves away from the center of the pulse. In this
case chirping worsens the effects of GVD-induced pulse broadening. On the other
hand, in the anomalous dispersion region where chromatic dispersion is positive
so that the group delay increases with wavelength, the red-shifted leading edge of
the pulse travels slower and thus moves toward the center of the pulse. Similarly,
the blue-shifted trailing edge travels faster, and also moves toward the center of the
pulse. In this case, SPM causes the pulse to narrow, thereby partly compensating for
chromatic dispersion.
12.6 Cross-Phase Modulation in WDM Systems
Cross-phase modulation (XPM) appears in WDM systems and has a similar origin
as SPM. In this case the name derives from the fact that the refractive index nonlinearity converts optical intensity fluctuations in a particular wavelength channel to
phase fluctuations in adjacent copropagating channels [2–5] In addition, because the
refractive index seen by a particular wavelength is influenced by both the optical
intensity of that wave itself and also by the optical power fluctuations of neighboring
wavelengths, SPM is always present when XPM occurs. Analogous to SPM, for two
interacting wavelengths the XPM-induced frequency shift ϕ is given by
ϕ =
dϕ
dt
= 2γ L e f f
d P
dt
(12.9)