 Frame-count switching in Raman acquisition-budget sensitivity

Frame-count switching in Raman acquisition-budget sensitivity occurs when a different integer number of exposure pairs becomes optimal as the available time changes. For each feasible pair count $n$, let $V_n(W)$ be the minimum predicted ratio variance after optimizing its live exposures. The complete design isThe marginal value of unused time below a Raman frame cap gives slopes for a fixed $n$. At a branch switch, those slopes need not be the derivative of the lower envelope on both sides.

A simple original example uses equal shot-noise coefficients $\alpha =\beta =a$, readout duration $d$ per frame and cap $c$ per exposure. Equal on/off totals are optimal, givingTake $a=10^{-6}\mathrm{s}$, $c=0.10\mathrm{s}$ and $d=0.010\mathrm{s}$. Two pairs saturate their $0.40\mathrm{s}$ live-time capacity once $W\ge 0.44\mathrm{s}$. Three pairs have live time $W-0.06\mathrm{s}$ near this budget. Their variances agree atJust below this value the two-pair variance is the constant $10\times 10^{-6}$. Just above it the three-pair variance is $4\times 10^{-6}/(W-0.06\mathrm{s})$ and begins decreasing. One pair has lower exposure capacity, while four or more pairs incur more readout and have less useful live time near the switch, so these two branches determine the global optimum there.

The one-sided derivatives at an optimal Raman frame-count switch are zero on the left and $-25\times 10^{-6}{\mathrm{s}}^{-1}$ on the right. The optimal variance is continuous at this switch, but has a kink. At $W=0.45\mathrm{s}$ two pairs give $10\times 10^{-6}$; at $0.47\mathrm{s}$ three pairs give about $9.7561\times 10^{-6}$. Extra time becomes useful again after the discrete design changes, despite the previous fixed-count plateau.

The primary Raman experiment's saturation-limited exposures and shuttered on/off readout motivate these competing resources.^[ref] The example uses chosen rounded timings. Read noise, drift and switching requirements can move the branch crossings and must be included when choosing an actual acquisition protocol.