= Frame-count switching in Raman acquisition-budget sensitivity
{title2=$V_*(W)=\min_nV_n(W)$}
\b[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 is
$$
V_*(W)=\min_nV_n(W).
$$
The <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, giving
$$
V_n(W)=\frac{4a}{\min(W-2nd,2nc)}\qquad(W>2nd).
$$
Take $a=10^{-6}\,\mathrm s$, $c=0.10\,\mathrm s$ and $d=0.010\,\mathrm s$. Two pairs saturate their $0.40\,\mathrm s$ live-time capacity once $W\geq0.44\,\mathrm s$. Three pairs have live time $W-0.06\,\mathrm s$ near this budget. Their <variances> agree at
$$
W_*=4c+6d=0.46\,\mathrm s.
$$
Just below this value the two-pair <variance> is the constant $10\times10^{-6}$. Just above it the three-pair <variance> is $4\times10^{-6}/(W-0.06\,\mathrm s)$ 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 $-25\times10^{-6}\,\mathrm{s^{-1}}$ on the right. The optimal <variance> is continuous at this switch, but has a kink. At $W=0.45\,\mathrm s$ two pairs give $10\times10^{-6}$; at $0.47\,\mathrm s$ three pairs give about $9.7561\times10^{-6}$. 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.<Femtosecond Broadband Stimulated Raman: A New Approach for High-Performance Vibrational Spectroscopy by David W. McCamant and others>[Data Collection, exposure limit, readout and pump-shutter sequence preceding equation 4]{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.
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