  In fundamental papers from 1962 [1, 2], Heffener and Haus showed that it is
not possible to construct a linear noiseless amplifier. The implies that the
amplifier intrinsic noise sources induce random perturbations on the phase of
the incoming optical signal which translates into spectral broadening. To
achieve the minimum (quantum noise limited) induced phase fluctuation, and the
corresponding minimum spectral broadening, an optimum phase measurement method
is needed. We demonstrate that a measurement method based on the heterodyne
detection and the extended Kalman filtering approaches an optimum phase
measurement in the presence of amplifier noise. A penalty of 5 dB (numerical)
and 15 dB (experimental) compared to the quantum limited spectral broadening is
achieved. For comparison, the conventional phase measurement method's penalty
exceeds 30 dB for the measurements. Our results reveal new scientific insights
by demonstrating that the impact of amplifier noise can be significantly
reduced by using the proposed phase measurement method. An impact is envisioned
for the phase-based optical sensing system, as optical amplification could
increase sensing distance with the minimum impact on the phase.
