= First-order optical coherence
{title2=$g^{(1)}_{12}(\tau)$}
Consider one selected <polarization of waves>, represented by complex fields $E_1(t)$ and $E_2(t)$ at two sampling positions. Suppose their second moments are finite and stationary: $J_j=\langle|E_j(t)|^2\rangle$ and the following uncentered <wave-field correlation> do not depend on the absolute time $t$. Define
$$
\Gamma_{12}(\tau)=\langle E_1^*(t)E_2(t+\tau)\rangle,
\qquad g^{(1)}_{12}(\tau)=\frac{\Gamma_{12}(\tau)}{\sqrt{J_1J_2}},
\qquad J_1,J_2>0.
$$
The star denotes the <complex conjugate>; brackets denote an <ensemble average>. A long detector average estimates this quantity only if it adequately samples the fluctuations. The superscript $1$ identifies a field correlation: it does not mean the <autocorrelation> of the measured <light intensity>. The one-position version is $g^{(1)}(\tau)=\langle E^*(t)E(t+\tau)\rangle/\langle|E|^2\rangle$, so $g^{(1)}(0)=1$ for every nonzero field covered by this definition; this alone says nothing about its correlation at nonzero delay. https://www.rp-photonics.com/coherence_time.html[Rüdiger Paschotta, RP Photonics Encyclopedia, Coherence Time: stationary complex field correlation]{ref}.
The <Cauchy-Schwarz inequality> gives $0\leq|g^{(1)}_{12}(\tau)|\leq1$. At a specified delay, magnitude one means $E_2(t+\tau)=aE_1(t)$ in mean square for a fixed nonzero complex $a$: the samples have a reproducible relation in both <wave amplitude> and <wave phase>. Magnitude zero means the averaged field cross term vanishes. It need not mean statistical <independence>. A deterministic <phase difference> changes the argument of $g^{(1)}$, not its <complex modulus>; a distorted but fixed <wavefront> can therefore remain fully coherent. https://www.rp-photonics.com/coherence.html[Paschotta, Coherence: static distortions versus fluctuating distortions]{ref}.
To see the observable, recombine the samples in the same detected spatial mode with matched <wave polarizations>. With a common positive conversion factor $C$, let $I_j=CJ_j$ be the single-arm <light intensities> reaching that detector. Adding $E_1(t)+e^{i\phi}E_2(t+\tau)$ gives
$$
I(\phi,\tau)=I_1+I_2+2\sqrt{I_1I_2}\,\operatorname{Re}\!\left[e^{i\phi}g^{(1)}_{12}(\tau)\right],
\qquad \mathcal V(\tau)=\frac{2\sqrt{I_1I_2}}{I_1+I_2}|g^{(1)}_{12}(\tau)|.
$$
This follows simply by expanding the squared <complex modulus> and averaging, then taking the maximum and minimum as $\phi$ spans a full cycle. The extra <wave phase> scan must leave $I_1,I_2$ and the magnitude of the correlation effectively unchanged. Thus <interferometric visibility> equals $|g^{(1)}|$ only for equal nonzero <light intensities> and ideal mode matching. For $I_2/I_1=1/4$, perfectly coherent beams have <interferometric visibility> $0.8$; a measured value $0.4$ would imply $|g^{(1)}|=0.5$ under these assumptions. Orthogonal <linear polarizations>, background light, unresolved <interference fringes> or vibrations can reduce measured contrast further. One absent arm gives zero contrast and makes the normalized two-sample correlation undefined.
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