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For stationary continuous detection, let $C_j$ count all cross-channel pairs with $t_2-t_1$ in a bin of full width $\Delta \tau$, and let $r_1,r_2$ be observed single-count rates. Let $T_j$ be their joint live observation time at that delay, approximately the common duration $T_{\mathrm{obs}}$ when delays are short. Independent streams have expected pair count $r_1{r}_2{T}_j\Delta \tau$, giving the displayed estimator. It estimates a bin average of second-order optical coherence, further blurred by detector timing response. For a finite uninterrupted run the exact overlap weighting is $T_{\mathrm{obs}}-|\tau |$ inside the delay integral, rather than a constant at long delays.

For an original numerical example, choose $r_1=2000{\mathrm{s}}^{-1}$, $r_2=3000{\mathrm{s}}^{-1}$, $T_j=100\mathrm{s}$, and $\Delta \tau =10\mathrm{ns}$. The independent reference is six pairs per bin. Chosen counts $C_j=3,6,12$ give estimates $0.5,1,2$; they are teaching inputs with large counting fluctuations, not historical measurements. For pulsed sources, compare properly normalized same-pulse and separate-pulse coincidences instead of assuming this continuous-source baseline.

Detector dead time suppresses nearby same-detector counts and can manufacture an apparent dip in a single-detector record. Two independent detector channels avoid that particular same-channel exclusion, but shared electronics can still reject pairs. Timing spread and wide bins wash out narrow features. A device recording only the first stop after each start measures a waiting-time distribution; interpreting it as all-pair correlations requires a sufficiently low stop probability over the relevant delay range.

For stationary photodetection, the normalized correlation in second-order optical coherence is $g^{(2)}(\tau )=⟨E^{-}(t)E^{-}(t+\tau )E^{+}(t+\tau )E^{+}(t)⟩/⟨E^{-}{E}^{+}{⟩}^2$. Normal ordering expresses the absorption of two photons. In a single mode,A coherent state has $g^{(2)}(0)=1$. A single-mode thermal field has $g^{(2)}(0)=2$ and bunches, while a number state with $n>0$ gives $1-1/n$; a single-photon state has zero ideal coincidence probability at zero delay. Antibunching, a zero-delay suppression relative to nearby delays, is distinct from simply reporting an intensity variance. Multimode averaging and finite detector response reduce observed contrast.

The Hanbury Brown and Twiss effect is measured by splitting the light onto two detectors and histogramming coincidences versus delay. Background counts, detector timing, dead time and afterpulsing need calibration. Two detectors avoid confusing the single detector's inability to register arbitrarily close counts with real antibunching. First-order interference and second-order correlations contain different information: weak first-order coherence does not imply an absence of intensity bunching.^[ref]

Photon antibunching means suppression of close pairs compared with pairs at some later delay: $g^{(2)}(0)<g^{(2)}(\tau )$ for some $\tau$. If correlations decay to the independent-count reference, $g^{(2)}(0)<1$ is a sufficient signature. It violates the classical intensity bounds in photon bunching. For the specified single-mode number measurement, second-order optical coherence also shows that $g^{(2)}(0)<1$ is equivalent to $\mathrm{Var}N<\mu$. This variance criterion concerns that mode or gate; it must not be applied unchanged to arbitrarily long integrated counting intervals.

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]