= Raman pump-on pump-off gain ratio
{c}
A \b[Raman pump-on pump-off gain ratio] compares the probe spectrum with and without the narrowband Raman pump, after correcting detector background and normalizing by a reference. For positive dark-corrected probe values $S$ and reference values $R$,
$$
G=\frac{(S/R)_{\rm on}}{(S/R)_{\rm off}}
=\frac{S_{\rm on}R_{\rm off}}{R_{\rm on}S_{\rm off}}.
$$
Unity denotes no pump-induced change in this model; $G-1$ is the fractional gain. The primary FSRS experiment explicitly performs dark subtraction, reference normalization and the on/off division.<Femtosecond Broadband Stimulated Raman: A New Approach for High-Performance Vibrational Spectroscopy by David W. McCamant and others>[Data Collection, equation 4; Instrumentation/Figure 2, separate probe/reference spectra]{ref} These operations define a measured optical gain observable, not a direct count of Raman photons alone.
An original offset example has true probe readings $100000$ with pump off and $102000$ with pump on, with an unchanged reference ratio. The true gain is $1.02$. A common unremoved detector offset of $50000$ changes the raw ratio to
$$
\frac{102000+50000}{100000+50000}=1.01333\ldots.
$$
Reference normalization does not automatically remove an additive detector background. Uncertain subtraction also contributes uncertainty to the corrected ratio.
The <shot-noise precision of a Raman gain ratio> gives a separate statistical limit. If the four inputs are independent Poisson photoelectron counts with large means, first-order propagation gives
$$
\frac{u^2(G)}{G^2}\simeq
\frac1{\mu_{S,\rm on}}+\frac1{\mu_{R,\rm off}}
+\frac1{\mu_{R,\rm on}}+\frac1{\mu_{S,\rm off}}.
$$
For weak gain and all four means approximately $N$, $u(G)\simeq2/\sqrt N$. At $N=250000$, that is about $0.004$: a one-percent gain is only about2.5times this standard uncertainty. These are chosen photoelectron-count assumptions, not an uncertainty specification for arbitrary digitized detector counts.
Shared laser fluctuations, read noise, uncertain references and non-Raman nonlinear effects require their own covariance and response model. The ratio can reveal a weak band only when the acquisition and background control justify its precision. The parent <femtosecond stimulated Raman spectroscopy> article explains the pump/probe apparatus and supplies a distinct reference-fluctuation example.
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