  The phase of an optical field inside a linear amplifier is widely known to
diffuse with a diffusion coefficient that is inversely proportional to the
photon number. The same process occurs in lasers which limits its intrinsic
linewidth and makes the phase uncertainty difficult to calculate. The most
commonly used simplification is to assume a narrow photon-number distribution
for the optical field (which we call the small-noise approximation). For
coherent light, this condition is determined by the average photon number. The
small-noise approximation relies on (i) the input to have a good
signal-to-noise ratio, and (ii) that such a signal-to-noise ratio can be
maintained throughout the amplification process. Here we ask: For a coherent
input, how many photons must be present in the input to a quantum linear
amplifier for the phase noise at the output to be amenable to a small-noise
analysis? We address these questions by showing how the phase uncertainty can
be obtained without recourse to the small-noise approximation. It is shown that
for an ideal linear amplifier (i.e. an amplifier most favourable to the
small-noise approximation), the small-noise approximation breaks down with only
a few photons on average. Interestingly, when the input strength is increased
to tens of photons, the small-noise approximation can be seen to perform much
better and the process of phase diffusion permits a small-noise analysis. This
demarcates the limit of the small-noise assumption in linear amplifiers as such
an assumption is less true for a nonideal amplifier.
