Under the same conditions described in the previous problem, the fluctuations of the length of \(\mathbf{J}_{12}(T)\) are caused mainly by the fluctuations in the real part of the noise. Making this assumption, show that the rms signal-to-(self-) noise ratio associated with the measurement of \(\left|\mathbf{J}_{12}\right|\) is given by

\[ \left(\frac{S}{N}\right)_{\mathrm{rms}} \approx \frac{\left|\mathbf{J}_{12}\right|}{\sigma_{\Re}}=\sqrt{\frac{2 \mu_{12}^{2}}{1+\mu_{12}^{2}} \frac{T}{\tau_{c}}} \]

Note that this is the rms signal-to-(self-) noise ratio with which the visibility of the fringe in a Young's experiment (i.e., amplitude interferometry) can be measured, assuming that \(\overline{I_{1}}\) and \(\overline{I_{2}}\) are known perfectly and that the noise is dominated by self-noise (a condition not generally true at optical frequencies).