 Photon-budget allocation for a Raman gain ratio

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 $V={\sum }_{i}1/{\mu }_i$. 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 ${\mu }_i=r_i{t}_i$ and a total time budget ${\sum }_i{t}_i=T$ givesThe Cauchy-Schwarz inequality, or differentiation with a constraint multiplier, givesThis 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 $4\times 10^6$ 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.^[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.