 Maximum-exposure constraint in paired Raman acquisition

A 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 satisfyFor a fixed pair count and wall-clock budget, readout overhead in Raman exposure allocation leaves $T=T_{\mathrm{on}}+T_{\mathrm{off}}=W-2nd$. Using the full budget is feasible only if $0<T\le 2n{t}_{\max }$; otherwise one must change the pair count, shorten the acquisition or allow unused time.

Under the independent large-count shot-noise model, minimize $V=\alpha /T_{\mathrm{on}}+\beta /T_{\mathrm{off}}$. The feasible on-exposure interval isThe 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 $W=1\mathrm{s}$, $d=0.010\mathrm{s}$, $t_{\max }=0.097\mathrm{s}$, $\alpha =2\times 10^{-6}\mathrm{s}$ and $\beta =0.5\times 10^{-6}\mathrm{s}$. 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 $T_{\mathrm{on}}=0.582\mathrm{s}$, $T_{\mathrm{off}}=0.298\mathrm{s}$, with $V\approx 5.1143\times 10^{-6}$. Five and seven pairs give approximately $5.3285\times 10^{-6}$ and $5.2326\times 10^{-6}$ 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.^[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.