= Shot-noise precision of a Raman gain ratio
The \b[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 $\log G$ gives
$$
\frac{u^2(G)}{G^2}\simeq
\frac1{\mu_A}+\frac1{\mu_B}+\frac1{\mu_C}+\frac1{\mu_D}.
$$
The 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 $N_{\rm tot}$, the <Cauchy-Schwarz inequality> gives
$$
\left(\sum_i\mu_i\right)\left(\sum_i\mu_i^{-1}\right)\geq16,
\qquad \frac{u(G)}G\gtrsim\frac4{\sqrt{N_{\rm tot}}}.
$$
Equality in this ideal unconstrained budget requires equal counts. Near unity gain, four approximately equal means totalling $4\times10^6$ 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.<Femtosecond Broadband Stimulated Raman: A New Approach for High-Performance Vibrational Spectroscopy by David W. McCamant and others>[Data Collection; Results and Discussion, solute spectra and shot-noise-limit discussion for Figure 3 spectra]{ref} A counting limit is therefore a useful benchmark, not a guarantee of achieved experimental sensitivity.
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