 Shot-noise precision of a Raman gain ratio

The shot-noise precision of a Raman gain ratio is the statistical precision expected from finite detected photoelectron numbers under a stated counting model. Write the Raman pump-on pump-off gain ratio as $G=AD/(BC)$, where $A,B$ are pump-on probe/reference counts and $C,D$ pump-off probe/reference counts. For large independent Poisson counts, applying first-order propagation to $\log G$ givesThe parent article derives the equal-count limit. Independence, large counts and a calibrated conversion from readout units to photoelectrons are assumptions, not consequences of taking a ratio.

An original imbalanced example has ${\mu }_A={\mu }_B={\mu }_C=10^6$ and ${\mu }_D=10^4$. The relative standard uncertainty is about $\sqrt{0.000103}=1.015\%$. Increasing the three bright channels tenfold only reduces it to about $1.0015\%$: the weakest count channel dominates. Multiplicative detector-gain corrections can make the mean ratio appropriate to the optical experiment, but they do not create additional photoelectrons.

A simple photon-budget allocation for a Raman gain ratio makes the trade-off explicit. If the four mean counts sum to $N_{\mathrm{tot}}$, the Cauchy-Schwarz inequality givesEquality in this ideal unconstrained budget requires equal counts. Near unity gain, four approximately equal means totalling $4\times 10^6$ give about $0.2\%$ relative precision. Actual optical ratios, reference splitting and exposure constraints may prevent exact equality.

The primary FSRS experiment explicitly compared its computed photoelectron shot-noise limit with observed baseline noise and found additional detector structure; it averaged repeated gain spectra and discussed pixel-dependent corrections.^[ref] A counting limit is therefore a useful benchmark, not a guarantee of achieved experimental sensitivity.