 Exposure-time allocation for a Raman gain ratio

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 $T_{\mathrm{on}}$ and a common pump-off exposure $T_{\mathrm{off}}$.

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 isWith a fixed live-time budget $T_{\mathrm{on}}+T_{\mathrm{off}}=T$, differentiation or the Cauchy-Schwarz inequality gives

For an original example, let both pump-on rates be $10^6{\mathrm{s}}^{-1}$ and both pump-off rates be $4\times 10^6{\mathrm{s}}^{-1}$. Then $\alpha =2\times 10^{-6}\mathrm{s}$, $\beta =0.5\times 10^{-6}\mathrm{s}$. 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 $\sqrt{4.5\times 10^{-6}}\approx 0.002121$, compared with $\sqrt{5\times 10^{-6}}\approx 0.002236$ 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.^[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.