= Siegert relation
{c}
{title2=$g^{(2)}_{12}(\tau)=1+|g^{(1)}_{12}(\tau)|^2$}
For one selected <polarization of waves>, write the detected complex fields as $E_1(t),E_2(t+\tau)$ and their <light intensities> as $I_j=|E_j|^2$, absorbing a common conversion constant. With stationary <ensemble averages> and nonzero $\bar I_j=\langle I_j\rangle$, define
$$
g^{(1)}_{12}(B,\tau)=\frac{\langle E_1^*(t)E_2(t+\tau)\rangle}{\sqrt{\bar I_1\bar I_2}},
\qquad
g^{(2)}_{12}(B,\tau)=\frac{\langle I_1(t)I_2(t+\tau)\rangle}{\bar I_1\bar I_2}.
$$
Here <second-order optical coherence> concerns distinct detector channels, with the corresponding <normal ordering> in the quantum formulation. Independent detection has $g^{(2)}=1$. Subtracting the means gives the normalized <covariance>
$$
\frac{\langle\delta I_1(t)\delta I_2(t+\tau)\rangle}{\bar I_1\bar I_2}=g^{(2)}_{12}(B,\tau)-1.
$$
Thus multiplying raw <photocurrents> contains a large mean-product contribution; multiplying their fluctuations isolates the small excess. Independent detector <shot noise> does not supply a mean cross-channel excess, although it increases <measurement uncertainty>.
The displayed <Siegert relation> requires a zero-mean, jointly circular complex <Gaussian random field>, the usual chaotic <thermal light> model. Circular means invariance under a common arbitrary complex phase rotation; in particular $\langle E_iE_j\rangle=0$. Apply the <Gaussian moment pairing theorem> to the real and imaginary parts:
$$
\langle E_1^*E_1E_2^*E_2\rangle
=\langle|E_1|^2\rangle\langle|E_2|^2\rangle
+|\langle E_1^*E_2\rangle|^2.
$$
The remaining pairing vanishes by circularity. Division proves the <Siegert relation>. Gaussianity is an assumption about fourth moments, not a consequence of merely measuring <first-order optical coherence>; a <coherent state> or <photon antibunching> cannot be interpreted by this thermal formula.
For equal-intensity beams combined in a common optical mode with matched <wave polarizations>, <interferometric visibility> is $V=|g^{(1)}_{12}|$. Consequently ideal thermal <stellar intensity interferometry> measures $g^{(2)}-1=V^2$, while the full $g^{(2)}$ includes the independent-count reference one. Unpolarized <thermal light> with two independent equally populated <wave polarizations> has half this excess: adding their <covariances> doubles the numerator while multiplying total means quadruples the denominator. Finite detector response further suppresses the peak and must be calibrated.<Spatial intensity interferometry on three bright stars by William Guerin and others>[Guerin and others, Spatial intensity interferometry on three bright stars, arXiv:1805.06653v2 (13 August 2018), §§2.1–2.3: normalized correlation, thermal contrast and temporal-response calibration]{ref}
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