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Robert Hanbury Brown and Richard Q. Twiss sought angular sizes through correlations of detected light intensities. Their method multiplies detector outputs after detection, whereas a Michelson interferometer combines fields before detection.^[ref]

A monochromatic discharge source illuminated a half-silvered beam splitter and two photodetectors. The averaged current product showed an excess near coincidence. Moving one detector introduced a delay that removed it, supplying a control. Slow response weakened the peak. This photon bunching is compatible with fluctuating classical light intensity.^[ref]

For a source whose counting correlations are stationary, define the dimensionless correlationAssume finite moments and nonzero denominators. The numerator is in normal ordering: two detections remove two photons. The denominator is the product of the two single-count factors, so independent counts have reference value one. This is a diagonal restriction of the four-point second-order field correlation, rather than its complete definition.^[ref]

For one selected mode with number operator $N=a^{†}a$ and $\mu =⟨N⟩>0$, evaluate both detections at the same time. The canonical commutation relation $a{a}^{†}=a^{†}a+1$ implies $N^2=(a^{†}{)}^2{a}^2+N$, henceThe factorial moment counts ordered pairs of distinct photons. Using $⟨N^2⟩$ would include pairing each photon with itself and would give the wrong coincidence prediction. At $\mu =0$ this normalized expression is undefined, rather than zero.

For a single mode in thermal equilibrium, its quantum harmonic oscillator energy ladder gives a geometric distributionIts probability generating function is $F(z)=(1-q)/(1-qz)$. Differentiating twice gives $F^{\prime \prime }(1)=2q^2/(1-q{)}^2=2{\mu }^2$, hence $g^{(2)}(0)=2$. At the chosen mean $\mu =2$, $q=2/3$, $p_0=1/3$, $p_1=2/9$, $p_2=4/27$, and $\mathrm{Var}N=6$. Rare high occupations increase the pair count even though the mean matches the other preparations. The geometric weights also arise from a Gaussian mixture of coherent states.^[ref]

Single-mode selection is essential to this factor of two. If $M$ independent equally populated thermal modes are collected together, let $N={\sum }_{j=1}^M{N}_j$ with $⟨N_j⟩=\mu /M$. Adding their variances gives $\mathrm{Var}N=\mu +{\mu }^2/M$, hence $g^{(2)}(0)=1+1/M$. Averaging over many unresolved modes can make thermal pair counts approach the independent-count reference.