= Photon-budget allocation for a Raman gain ratio
{title2=$t_i\propto r_i^{-1/2}$}
\b[Photon-budget allocation for a Raman gain ratio] distributes measurement effort among the channels entering a ratio. For four large independent Poisson photoelectron counts with means $\mu_i$, the <shot-noise precision of a Raman gain ratio> has approximate relative variance $V=\sum_i1/\mu_i$. With a freely allocatable fixed total expected count $N$, the minimum is $16/N$ at equal counts. The constraint matters: fixed total exposure time is a different optimization problem.
Suppose channel $i$ has known photoelectron rate $r_i>0$ and an independently adjustable exposure $t_i>0$. Estimate the optical ratios from exposure-normalized rates, so the known exposure factors are included. Then $\mu_i=r_it_i$ and a total time budget $\sum_i t_i=T$ gives
$$
V\simeq\sum_i\frac1{r_it_i}.
$$
The <Cauchy-Schwarz inequality>, or differentiation with a constraint multiplier, gives
$$
t_i=\frac{T\,r_i^{-1/2}}{\sum_jr_j^{-1/2}},\qquad
V_{\min}=\frac{\left(\sum_jr_j^{-1/2}\right)^2}{T}.
$$
This <exposure-time allocation for a Raman gain ratio> spends longer on weak channels without necessarily equalizing their final counts.
For an original one-second budget, let two rates be $4\times10^6$ and two be $10^6$ photoelectrons per second. The optimal exposures are $1/6$ second for each bright channel and $1/3$ second for each weak channel. Expected counts are about $666667$ and $333333$ respectively, with relative standard uncertainty $0.0030$. Four equal exposures would give about $0.003162$ instead. The gain comes from allocating the stated resource more efficiently, not creating additional information by normalizing counts.
The primary FSRS detector records probe and reference spectra together and alternates pump-on/off exposures.<Femtosecond Broadband Stimulated Raman: A New Approach for High-Performance Vibrational Spectroscopy by David W. McCamant and others>[Data Collection, shutters, paired spectra, equation 4 and averaging]{ref} Such linked channels cannot have four independently chosen exposure times; the appropriate grouped constraints must replace the ideal model above. Read noise, dark uncertainty, saturation and sample change can alter the optimum as well.
Back to article page
