= Photon correlation loss and background model
{title2=$g^{(2)}_{\rm obs}=1+\rho_1\rho_2(g^{(2)}_{\rm sig}-1)$}
Independent loss alone leaves normalized <factorial moments> invariant: if every input <photon> is retained with <probability> $\eta$, <binomial thinning conditional on a total> gives $\langle K\rangle=\eta\mu$ and $\langle K(K-1)\rangle=\eta^2\langle N(N-1)\rangle$. Both powers cancel from $g^{(2)}(0)$. Fewer coincidences still mean poorer statistical precision.
Now let each channel contain a signal rate $s_i$ and an independent <Poisson process> background rate $b_i$, independent also of the signal. Put $\rho_i=s_i/(s_i+b_i)$. The pair density is $s_1s_2g^{(2)}_{\rm sig}+s_1b_2+b_1s_2+b_1b_2$. Dividing by $(s_1+b_1)(s_2+b_2)$ proves the displayed formula, for the same bin and timing response. For chosen equal signal fractions $\rho_1=\rho_2=4/5$, ideal signal values $0,1,2$ become $9/25,1,41/25$. Background pulls both dips and peaks toward one.
This calculation assumes additive, mutually independent background counts. Scattered coherent <light> that interferes with the signal, correlated electronic noise, or recovery effects require a different model. Correcting a dip without independently estimating the backgrounds cannot establish <photon antibunching>.
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