= Wall-clock slack in saturation-limited Raman frame allocation
{title2=$S_n=\max(0,W-2n(d+c))$}
\b[Wall-clock slack in saturation-limited Raman frame allocation] is time that remains unused when a fixed number of detector frames has already reached its permitted exposure caps. With $n$ pump-on/off pairs, per-frame cap $c$ and readout time $d$ per frame,
$$
T_n=\min(W-2nd,2nc),\qquad
S_n=W-2nd-T_n=\max(0,W-2n(d+c)),
$$
assuming $W>2nd$. The <integer frame-count choice under a Raman saturation limit> compares these possibilities over different pair counts. Slack at fixed $n$ does not imply that the continuous exposure allocation was optimized incorrectly.
If $S_n>0$, both states can use their full live totals $nc$, and the shot-noise relative <variance> is $V_n=(\alpha+\beta)/(nc)$. Increasing only the wall-clock allowance cannot improve it: the <marginal value of unused time below a Raman frame cap> is zero in this capped regime. Another frame pair changes both the overhead and the allowable division of live time.
For an original example, set $W=0.50\,\mathrm s$, $n=2$, $c=0.10\,\mathrm s$ and $d=0.020\,\mathrm s$. The live time is $0.40\,\mathrm s$, readout uses $0.08\,\mathrm s$, and slack is $0.02\,\mathrm s$. With $\alpha=2\times10^{-6}\,\mathrm s$ and $\beta=0.5\times10^{-6}\,\mathrm s$, equal capped totals $0.20\,\mathrm s$ give $V_2=12.5\times10^{-6}$.
Three pairs increase readout time to $0.12\,\mathrm s$, leaving only $0.38\,\mathrm s$ live time. Yet the larger per-state cap permits the preferred ratio two to one: $T_{\rm on}\approx0.2533\,\mathrm s$ and $T_{\rm off}\approx0.1267\,\mathrm s$. Their <variance> is $V_3=4.5\times10^{-6}/0.38\approx11.8421\times10^{-6}$, lower despite less total live time. Relieving the weak state's cap outweighs the extra readout cost in this chosen model.
The primary Raman apparatus limited exposures to avoid detector saturation and alternated pump state during shuttered readout.<Femtosecond Broadband Stimulated Raman: A New Approach for High-Performance Vibrational Spectroscopy by David W. McCamant and others>[Data Collection, exposure limit, 10 ms readout and shutter sequence preceding equation 4]{ref} The timings above are separate illustrative choices. Read noise, sample drift and switching constraints can change the comparison; unused time and useful precision are different optimization quantities.
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