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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 ${\bar{I}}_j=⟨I_j⟩$, defineHere 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 covarianceThus 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 $⟨E_i{E}_j⟩=0$. Apply the Gaussian moment pairing theorem to the real and imaginary parts: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 $V=|g_{12}^{(1)}|$. 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.^[ref]

Stellar intensity interferometry measures a star's angular diameter by comparing fluctuations of detected light intensity at separated collectors. Each collector has its own photodetector; an analog multiplier or a digital correlator compares the electrical outputs. Changing the interferometric baseline changes the sampled spatial optical coherence. An unresolved star produces correlated fluctuations; a resolved extended source supplies different mixtures of emission directions, reducing their correlation.

This makes a useful telescope out of a surprising observable: the fluctuations discarded by an ordinary average-brightness measurement. Optical beams need not travel to a common combiner. A relative wave phase perturbation $E_2\mapsto e^{i\varphi }{E}_2$ leaves $|E_2{|}^2$ unchanged, so pure phase errors do not directly change the local light intensity. Electronic delay alignment remains necessary; intensity changes from atmospheric scintillation, collection losses, and instrumental noise still require controls.

Let $\mathbf{b}$ join the collectors and let ${\mathbf{s}}_0$ be the unit line of sight toward the star. The relevant interferometric baseline is its projection onto the plane of the sky:The reference arrival-time difference has magnitude $|\mathbf{b}\cdot {\mathbf{s}}_0|/c$, with speed of light $c$. Correct that delay, with a sign fixed by channel order, before averaging the residual correlation at $\tau =0$. The physical collector separation equals $B$ only when it is perpendicular to ${\mathbf{s}}_0$; tracking a star can change the projection even for fixed collectors.