 Marginal value of unused time below a Raman frame cap

The marginal value of unused time below a Raman frame cap measures how the optimum predicted Raman ratio variance changes with the allowed acquisition budget. Hold the pair count $n$, readout time $d$ and per-frame exposure cap $c$ fixed. Put $H=nc$ and $T=W-2nd>0$. The wall-clock slack in saturation-limited Raman frame allocation becomes positive once $T>2H$; additional budget then cannot increase exposure at that pair count.

For shot-noise coefficients $\alpha >\beta >0$, define $K=(\sqrt{\alpha }+\sqrt{\beta }{)}^2$. The unconstrained on fraction is $p=\sqrt{\alpha }/(\sqrt{\alpha }+\sqrt{\beta })$. It reaches its cap when $T=H/p$. The minimized variance is thereforeThis follows by using the unconstrained ratio, then holding the weaker on state at its cap while extending the off state, and finally holding both at their caps.

Away from the transition points, the positive marginal improvement per extra wall-clock second isThe slope agrees on both sides of the first transition. At the final cap it changes to zero, because new time is unused.

For an original example, choose $n=2$, $c=0.10\mathrm{s}$, $d=0.010\mathrm{s}$, $\alpha =4\times 10^{-6}\mathrm{s}$ and $\beta =10^{-6}\mathrm{s}$. The wall-clock transitions are $0.34$ and $0.44\mathrm{s}$:At the last setting, $0.05\mathrm{s}$ is slack. Optimizing the pair count as well introduces frame-count switching in Raman acquisition-budget sensitivity; the fixed-count slopes above need not describe that full integer optimization.

The primary Raman experiment capped exposures to avoid saturation and used shuttered readout between pump states.^[ref] The numerical timings here are illustrative. These marginal values concern the stated noise objective, without additional drift, read-noise or switching-cadence costs.