Nonlinear Fiber Optics - 4 ed. Agrawal

12.4. Temporal and Spectral Evolution 493
From a practical perspective, coherence properties of a supercontinuum are important,
if such a device is employed as a broadband source of light for medical, metrological,
and other applications. For this reason, the coherence issue has attracted considerable
attention [134]–[147]. The first question we should ask is what we mean by
the coherence in the context of a supercontinuum. If the entire spectrum is generated
with pulses propagating in a single-mode fiber, the output would be spatially coherent.
However, its temporal coherence is affected by fluctuations in the energy, width, and
arrival time of individual input pulses. As a result, spectral phase is likely to fluctuate
from pulse to pulse across the bandwidth of the supercontinuum.
A suitable measure of coherence for a supercontinuum is the degree of coherence
associated with each spectral component. In the scalar case, it is defined as [99]
〈Ã ∗ 1
g 12 (ω)=
(L,ω)Ã 2 (L,ω)〉
[〈|Ã 1 (L,ω)| 2 〉〈|Ã 2 (L,ω)| 2 , (12.4.9)
〉] 1/2
where à 1 and à 2 are the Fourier transforms of two neighboring pulses, and the angle
brackets now denote an average over the entire ensemble of pulses. Experimentally, the
degree of coherence can be measured by interfering two successive pulses in a pulse
train incident on the fiber using a Michelson interferometer, and recording the contrast
of resulting spectral fringes as a function of wavelength [140]. The ensemble average
is performed automatically by the integration time associated with the optical spectrum
analyzer.
One can calculate g 12 (ω) numerically by adding quantum noise to the input pulse
(one photon per mode) and solving repeatedly Eq. (12.4.1), or Eq. (12.4.6) if polarization
effects are important [99]. This approach shows that both the shape and spectrum
of output pulses change from pulse to pulse. Moreover, g 12 (ω) varies considerably
across the supercontinuum depending on the average value of input pulse parameters.
In particular, the degradation of coherence is quite sensitive to the input pulse width.
Figure 12.34 shows the averaged output spectra and the degree of coherence g 12 (ω)
for three different pulse widths, assuming that the peak power of input pulse is 10 kW
in all cases. Fiber parameters are identical to those used for Figure 12.21. Clearly,
the supercontinuum is much less coherent for a 150-fs pulse compared with the 100-fs
pulse. More importantly, the coherence is preserved across the entire supercontinuum
for 50-fs pulses.
To understand the features seen in Figure 12.34, we note that the soliton order N
scales with pulse width T 0 linearly. Thus, N is relatively small for the 50-fs pulse.
Also, since the dispersion length scales as T0 2 , it is 9 times shorter for the 50-fs pulse
(about 63 mm) compared with the 150-fs pulse. The nonlinear length L NL is about
1 mm in all cases, as the peak power is kept fixed. The main conclusion one can
draw is that the soliton-fission process is inherently noisy and thus sensitive to both the
amplitude and phase of the input pulse. If N is small, the pulse splits into a smaller
number of solitons and suffers less from noise. In contrast, noise is enhanced if a
large number of fundamental solitons are created through the fission. Some support for
this observation comes from the fact that, even if Raman effects are ignored by setting
f R = 0 artificially, the coherence is degraded more for wider pulses [99]. Numerical
simulations also show that coherence is preserved even for 150-fs pulses when they are
propagated in the normal-GVD regime of the fiber where soliton fission does not occur.