Suppose that we wish to know the standard deviation of the phase of \(\mathbf{J}_{12}(T)\) under the condition that the measurement time \(T\) is sufficiently long compared with the correlation time \(\tau_{c}\) of the wave incident on the points \(P_{1}\) and \(P_{2}\) that the noise cloud surrounding \(\left|\mathbf{J}_{12}\right|\) is small in both the real and imaginary directions, compared with the length of \(\left|\mathbf{J}_{12}\right|\) (cf. Fig. 6.8). Specifically, we assume that \(T / \tau_{c} \gg 1 / \mu_{12}^{2}\) for this to be true. Under such a condition, the phase fluctuations are quite small. If \(\theta=\arg \left\{\mathbf{J}_{12}\right\}\), show that

\[ \sigma_{\theta} \approx \sqrt{\frac{1-\mu_{12}^{2}}{2 \mu_{12}^{2}} \frac{\tau_{c}}{T}} \]