One-sided derivatives at an optimal Raman frame-count switch ($V_{\ast +}^{\prime }={\min }_{n\in A}{V}_n^{\prime },V_{\ast -}^{\prime }={\max }_{n\in A}{V}_n^{\prime }$)

One-sided derivatives at an optimal Raman frame-count switch describe the change in the best acquisition variance when several integer designs tie. Let $V_{\ast }(W)={\min }_n{V}_n(W)$ as in frame-count switching in Raman acquisition-budget sensitivity, and let $A$ be the active designs with $V_n(W_{\ast })=V_{\ast }(W_{\ast })$. Suppose the relevant branches are differentiable near the tie and nonactive designs have a positive value gap.

Writing their slopes as $s_n=V_n^{\prime }(W_{\ast })$ givesIndeed, $V_n(W_{\ast }+h)=V_{\ast }(W_{\ast })+s_{n}h+o(h)$. Positive $h$ selects the smallest slope; negative $h$ selects the largest because multiplication reverses their ordering. A single derivative exists only when the selected left and right slopes agree. If an active branch itself has a kink, use its appropriate one-sided slope on each side.

For an original balanced-noise example, take $\alpha =\beta =a=10^{-6}\mathrm{s}$, exposure cap $c=0.080\mathrm{s}$ and per-frame readout $d=0.015\mathrm{s}$. Near the three-pair/four-pair switch,The branches tie at $W_{\ast }=6c+8d=0.60\mathrm{s}$, with variance about $8.3333\times 10^{-6}$. The three-pair branch has slope zero; the four-pair branch has slope $-4a/(6c{)}^2\approx -17.3611\times 10^{-6}{\mathrm{s}}^{-1}$. Thus these are the left and right optimum slopes. Smaller pair counts have less capacity and larger counts have more readout near this tie, so they do not change the local optimum.

Joint time-and-readout perturbations at a Raman design tie require following the direction in both resource coordinates; the scalar slopes above hold when only $W$ changes. They also quantify a local sensitivity, not a universal prescription to change exposure count whenever a small budget fluctuation occurs.

The primary stimulated-Raman apparatus limited exposures for saturation and switched pump state during shuttered readout.^[ref] The rounded timings here are a separate mathematical example. Read noise, drift or switching costs can change the competing branches and their derivatives.

Joint time-and-readout perturbations at a Raman design tie describe the directional response of the best acquisition variance when both available wall time and per-frame readout cost change. A tie generally has different derivatives on opposite sides, so there need not be one gradient for the optimum. The active branch derivatives must be combined along the actual perturbation direction.

Use the original balanced-noise model in one-sided derivatives at an optimal Raman frame-count switch. With $a=10^{-6}\mathrm{s}$, per-exposure cap $c=0.080\mathrm{s}$ and readout $d$, the locally competing branches areThey tie along $W=6c+8d$. At $(W_{\ast },d_{\ast })=(0.60,0.015)\mathrm{s}$, define $K=4a/(6c{)}^2\approx 17.3611\times 10^{-6}{\mathrm{s}}^{-1}$. The three-pair branch is locally constant, while the four-pair gradient is $(-K,8K)$.

For a path $W=W_{\ast }+ϵu_W$, $d=d_{\ast }+ϵu_d$ with $ϵ\downarrow 0$, the optimum has directional derivativeThe sign of the net live-time change $u_W-8u_d$ selects the improving branch. This conclusion assumes the branches remain feasible and nonactive designs have a positive value gap nearby.

For original finite examples, adding8ms of budget while adding1ms to each readout leaves $W-8d$ exactly unchanged, so these two branches remain tied. Adding10ms while adding1ms per readout increases that live time by2ms: the four-pair variance becomes $4\times 10^{-6}/0.482\approx 8.2988\times 10^{-6}$, below the three-pair value $8.3333\times 10^{-6}$. The first-order improvement is $-K(0.002)\approx -3.4722\times 10^{-8}$. Adding only6ms with the same readout increase instead makes the four-pair branch worse; the three-pair optimum remains unchanged locally.

The resource distinction has a physical motivation. The primary stimulated-Raman experiment used alternating pump states and shuttered detector readout; its exposure and readout intervals were separate parts of the acquisition sequence.^[ref] The rounded timings and noise parameter above are a mathematical planning example, not that instrument's measured values. Changing exposure caps, state-switching costs or noise terms requires the corresponding branch model to be recomputed.