= Exposure-time allocation for a Raman gain ratio
{title2=$T_{\rm on}/T_{\rm off}=\sqrt{\alpha/\beta}$}
\b[Exposure-time allocation for a Raman gain ratio] must respect channels that are recorded together. The ideal <photon-budget allocation for a Raman gain ratio> permits four independently adjustable exposures. A probe/reference detector pair instead has a common pump-on exposure $T_{\rm on}$ and a common pump-off exposure $T_{\rm off}$.
Let the four positive photoelectron rates be $r_A,r_B$ during pump-on and $r_C,r_D$ during pump-off. Form the gain ratio from exposure-normalized rates. For independent large counts limited by <shot noise>, its approximate relative <variance> is
$$
V=\frac{\alpha}{T_{\rm on}}+\frac{\beta}{T_{\rm off}},\qquad
\alpha=\frac1{r_A}+\frac1{r_B},\quad
\beta=\frac1{r_C}+\frac1{r_D}.
$$
With a fixed live-time budget $T_{\rm on}+T_{\rm off}=T$, differentiation or the <Cauchy-Schwarz inequality> gives
$$
T_{\rm on}=\frac{T\sqrt\alpha}{\sqrt\alpha+\sqrt\beta},\quad
T_{\rm off}=\frac{T\sqrt\beta}{\sqrt\alpha+\sqrt\beta},\quad
V_{\min}=\frac{(\sqrt\alpha+\sqrt\beta)^2}{T}.
$$
For an original example, let both pump-on rates be $10^6\,\mathrm{s^{-1}}$ and both pump-off rates be $4\times10^6\,\mathrm{s^{-1}}$. Then $\alpha=2\times10^{-6}\,\mathrm s$, $\beta=0.5\times10^{-6}\,\mathrm s$. A one-second live-time budget is best split into $2/3$ second on and $1/3$ second off. Its predicted relative <standard deviation> is $\sqrt{4.5\times10^{-6}}\approx0.002121$, compared with $\sqrt{5\times10^{-6}}\approx0.002236$ for equal times.
The 2003 primary stimulated-Raman apparatus recorded probe and reference spectra together, using 97 ms exposures followed by 10 ms readout with the detector-exposure shutter closed. The Raman-pump shutter changed state during readout.<Femtosecond Broadband Stimulated Raman: A New Approach for High-Performance Vibrational Spectroscopy by David W. McCamant and others>[Data Collection, exposure/readout and shutter sequence preceding equation 4]{ref} These acquisition details motivate grouped exposure constraints and <readout overhead in Raman exposure allocation>. The optimization above uses live time, stationary rates and negligible read noise; a wall-clock budget, detector saturation or sample drift can change which exposures are feasible.
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