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Integer frame-count choice under a Raman saturation limit chooses the number of pump-on/off exposure pairs as well as their durations. More pairs incur more readout overhead; too few pairs can saturate before using the available wall-clock budget. Under the model of maximum-exposure constraint in paired Raman acquisition, for positive readout time $d$ and frame cap $c$, each candidate integer $n$ with $2nd<W$ has usable live timeIts unused budget is $W-2nd-T_n\ge 0$. This wall-clock slack in saturation-limited Raman frame allocation is allowed: failure to consume the entire budget does not make a shorter acquisition physically impossible.

For each candidate, allocate $T_{\mathrm{on}}+T_{\mathrm{off}}=T_n$ with each total at most $nc$. Clip the unconstrained on total $T_n\sqrt{\alpha }/(\sqrt{\alpha }+\sqrt{\beta })$ to $[\max (0,T_n-nc),\min (T_n,nc)]$, then evaluate $V_n=\alpha /T_{\mathrm{on}}+\beta /T_{\mathrm{off}}$. This solves the continuous subproblem; comparing the finitely many integer candidates solves the stated frame-count problem. Positive counts and no other costs make additional feasible live time beneficial.

In an original example, use $W=0.60\mathrm{s}$, $d=0.010\mathrm{s}$, $c=0.097\mathrm{s}$, $\alpha =2\times 10^{-6}\mathrm{s}$ and $\beta =0.5\times 10^{-6}\mathrm{s}$. The relevant candidates areFour pairs are globally optimal in this model: one or two pairs use both states at their caps and improve as $n$ increases; three is evaluated explicitly; for every $n\ge 4$ the unconstrained ratio is feasible and $V_n=4.5\times 10^{-6}/(0.60-0.02n)$ increases with $n$. At four pairs the on/off totals are about $0.3467$ and $0.1733\mathrm{s}$.

The primary Raman apparatus used 97 ms exposures to avoid saturation and 10 ms shuttered readout, motivating these stipulated example timings.^[ref] Actual frame choice can also be constrained by read noise, drift and minimum switching cadence. Such costs must be included before treating this shot-noise-only optimum as an experimental operating prescription.

Readout overhead in Raman exposure allocation is the detector time that consumes the acquisition budget without collecting useful probe/reference photons. For $n$ pump-on/off exposure pairs, let each frame require a fixed closed-shutter readout duration $d$. A wall-clock budget $W$ leaves total live timeUnder the stationary-rate, independent shot noise model of exposure-time allocation for a Raman gain ratio, its minimum approximate relative variance at fixed $n$ isReadout changes the available live time, but not this optimum ratio when the overhead is fixed and independent of the on/off allocation. The live-time formula itself ceases to describe all constraints when exposures hit saturation limits or the number of pairs is variable.

The 2003 primary stimulated-Raman experiment used 97 ms exposures and 10 ms detector readout with the exposure shutter closed. The pump shutter switched state during readout, giving successive on/off measurements.^[ref] These reported settings give an idealized two-frame cycle of $2(97+10)=214\mathrm{ms}$, containing $194\mathrm{ms}$ live time. The paper describes the practical full cycle as approximately 220 ms, so the two-duration sum need not include every system delay.

For an original comparison at fixed wall time, the 214 ms idealization has live fraction $194/214\approx 0.9065$. Relative to a hypothetical zero-overhead detector at the same rates and on/off allocation, the shot-noise variance is multiplied by $214/194\approx 1.1031$, and its standard deviation by about $1.0503$. Using 10 ms rather than 97 ms exposures with the same 10 ms readout would give a live fraction $1/2$ and a variance penalty of two.

Longer frames reduce the fractional overhead but cannot be extended without limit. A maximum-exposure constraint in paired Raman acquisition can arise from saturation, while drift rejection can require frequent on/off alternation. Thus minimizing readout count alone is not a complete acquisition design; the applicable frame limits and switching cadence belong in the optimization.