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An exact teaching model makes the peak suppression explicit. Assume the narrowband field correlation separates as $g_{12}^{(1)}(B,u)=\gamma (B)g_t^{(1)}(u)$ after delay compensation. Average each channel's light intensity over the same rectangular gate of duration $T$. The two time integrals have triangular relative-delay weightIntegrating the Siegert relation against this weight givesThe factor $p$ is one for a selected polarization of waves and $1/2$ for two independent equally populated polarizations. The signal fraction ${\rho }_j=s_j/(s_j+b_j)$ uses mean stellar signal $s_j$ and independent additive background $b_j$ in each detector. The photon correlation loss and background model then supplies ${\rho }_1{\rho }_2$. This assumes no correlated sky background or electronic coupling.

For the original model $|g_t^{(1)}(u){|}^2=e^{-|u|/{\tau }_0}$, elementary integration givesThe area-based optical coherence time of this intensity envelope is ${\tau }_c=\int |g_t^{(1)}(u){|}^{2}du=2{\tau }_0$. For $T\gg {\tau }_0$, $\beta \approx {\tau }_c/T$; for $T\ll {\tau }_0$, $\beta \to 1$. With the chosen ${\tau }_0=1\mathrm{ps}$, $T=50\mathrm{ps}$, $p=1$, and ${\rho }_1={\rho }_2=0.8$, $\beta \approx 0.0392$ and $g_{\mathrm{obs}}^{(2)}(0,0)=1.025088$. At $B=10\mathrm{m}$ the stipulated disk gives approximately $1.013617$. A nearly unit measured $g^{(2)}$ can therefore conceal a substantial $V^2$.

This rectangular-gate calculation specifies a counting model. An analog instrument's amplifier bandwidth and impulse response instead weight the fluctuations it actually passes; the Sirius amplifiers were bandpass devices. Both require measuring the response and calibrating the zero-baseline excess. When $p$, ${\rho }_j$, and the temporal response are unchanged, the ratio of excesses recovers $V^2$:A sufficiently unresolved calibration source can supply the denominator; dividing the full $g^{(2)}$ values would give the wrong result. Narrower optical bandwidth lengthens optical coherence time, whereas wider usable electronic bandwidth can resolve faster fluctuations; these are distinct bandwidths. Background changes, unequal spectral responses, drift, or averaging over changing interferometric baselines can invalidate a common calibration. Resolving a disk means establishing a reproducible decline of calibrated correlation, supported by those controls.

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]