= Maximum-exposure constraint in paired Raman acquisition
{title2=$T_{\rm on},T_{\rm off}\leq nt_{\max}$}
A \b[maximum-exposure constraint in paired Raman acquisition] bounds each detector frame so that a bright channel stays below saturation. For $n$ pump-on/off pairs with common per-frame cap $t_{\max}$, the total live exposures satisfy
$$
0<T_{\rm on}\leq nt_{\max},\qquad0<T_{\rm off}\leq nt_{\max}.
$$
For a fixed pair count and wall-clock budget, <readout overhead in Raman exposure allocation> leaves $T=T_{\rm on}+T_{\rm off}=W-2nd$. Using the full budget is feasible only if $0<T\leq2nt_{\max}$; otherwise one must change the pair count, shorten the acquisition or allow unused time.
Under the independent large-count shot-noise model, minimize $V=\alpha/T_{\rm on}+\beta/T_{\rm off}$. The feasible on-exposure interval is
$$
\max(0,T-nt_{\max})\leq T_{\rm on}\leq\min(T,nt_{\max}).
$$
The objective is strictly convex for positive on/off exposures. Its unconstrained optimum is $T\sqrt\alpha/(\sqrt\alpha+\sqrt\beta)$; clip that value to the feasible interval, retaining positive time in both states. This follows from the derivative changing sign once, rather than requiring a new unconstrained allocation rule.
For an original model, set $W=1\,\mathrm s$, $d=0.010\,\mathrm s$, $t_{\max}=0.097\,\mathrm s$, $\alpha=2\times10^{-6}\,\mathrm s$ and $\beta=0.5\times10^{-6}\,\mathrm s$. With six pairs, $T=0.88\,\mathrm s$ and the unconstrained on total $0.58667\,\mathrm s$ exceeds its cap $0.582\,\mathrm s$. The constrained choice is $T_{\rm on}=0.582\,\mathrm s$, $T_{\rm off}=0.298\,\mathrm s$, with $V\approx5.1143\times10^{-6}$. Five and seven pairs give approximately $5.3285\times10^{-6}$ and $5.2326\times10^{-6}$ respectively under the same model. Six is preferable among these three choices.
The <integer frame-count choice under a Raman saturation limit> must also include switching cadence, drift and read noise when relevant. The primary apparatus used 97 ms exposures to avoid detector saturation and 10 ms shuttered readout.<Femtosecond Broadband Stimulated Raman: A New Approach for High-Performance Vibrational Spectroscopy by David W. McCamant and others>[Data Collection, exposure limit, S1/S2 shutter sequence and readout preceding equation 4]{ref} Those values motivate the stipulated numerical cap above; they are not universal detector specifications. A different optical flux or detector gain changes the usable exposure limit.
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