  Optical fiber signals with high power exhibit spectral broadening that seems
to limit capacity. To study spectral broadening, the autocorrelation function
of the output signal given the input signal is derived for a simplified fiber
model that has zero dispersion, distributed optical amplification (OA), and
idealized spatial noise processes. The autocorrelation function is used to
upper bound the output power of bandlimited or time-resolution limited
receivers, and thereby to bound spectral broadening and the capacity of
receivers with thermal noise. The output power scales at most as the
square-root of the launch power, and thus capacity scales at most as one-half
the logarithm of the launch power. The propagating signal bandwidth scales at
least as the square-root of the launch power. However, in practice the OA
bandwidth should exceed the signal bandwidth to compensate attenuation. Hence,
there is a launch power threshold beyond which the fiber model loses practical
relevance. Nevertheless, for the mathematical model an upper bound on capacity
is developed when the OA bandwidth scales as the square-root of the launch
power, in which case capacity scales at most as the inverse fourth root of the
launch power.
