Find the question you need
Choose a route or read from the beginning. All teaching chapters stay open; the grouped index below is expandable.
Define what is counted
Chapters 1–2. Keep signal, gross rate and background separate.
Build the uncertainty
Chapters 3–4. Normalize the counts and add their variance contributions.
Check the experiment
Chapters 8–10. Allow for pilots, overhead, drift and detector limits.
Practise and continue
Chapters 11–12. Use changed count rates and connected reading.
Full chapter index · Practice and worked checks · Photonics subject root
A photon counter records a bright sample and a dim background. Should the dim background receive more measurement time because it has fewer counts? Under a simple independent Poisson model, that intuition can produce a much less precise background-corrected rate.
This guide solves the time-allocation question with original numerical cases. It extends the Photonics subject root and the existing particle-counting and uncertainty guide. The calculations are paper models, not instrument settings or an optical laboratory procedure.
Chapter index
Define what is counted · Chapters 1–2
1. Define signal, gross rate and background separately
2. Subtract rates when the times differ
Build the uncertainty · Chapters 3–4
3. Add independent variance contributions
4. Choose the quantity the design should minimise
Divide the time · Chapters 5–7
5. Find the square-root allocation
6. Compare four plans with the same time budget
7. Check the minimum by another route
Check the experiment · Chapters 8–10
8. Use a pilot without pretending the rates are known exactly
9. Put overhead and background drift back into the decision
10. Check detector effects and low-count boundaries
Practise and continue · Chapters 11–12
11. Practise with changed rates and worked checks
12. Questions, connected routes and sources
CHAPTER 1 OF 12 · Define what is counted
1. Define signal, gross rate and background separately
Let s be the desired detected signal rate and b the background rate. During the sample measurement the detector records both, so the gross rate is g = s + b. A separate background measurement is intended to estimate b under conditions that preserve the relevant background.
All rates here refer to registered counts per second. They are not automatically incident photon fluxes: detection efficiency and other instrument effects would be needed for that conversion.
Assume g and b remain constant, events follow independent Poisson processes in the two measurements, and counting times are known. Initially ignore switching overhead, drift, detector dead time and calibration uncertainty. These assumptions make a focused design question possible: how should a fixed usable counting time be divided?
Let G be gross counts recorded over time tg and B be background counts recorded over time tb. Estimate the net signal rate as ŝ = G/tg − B/tb.
Raw counts cannot be subtracted directly when their observation times differ. For example, 900 gross counts over 10 s and 900 background counts over 90 s give rates of 90 and 10 counts/s. Their net rate is 80 counts/s, even though their raw counts are equal.
The error to repair is treating a count as if its exposure were attached implicitly. Write the time beside every count before subtracting. The same discipline applies to a camera’s dark frame or a scattering measurement, although the appropriate background model still needs checking in each application.
A Poisson count with mean gtg has variance gtg. Dividing that count by tg gives a rate variance g/tg. The background rate estimate similarly has variance b/tb.
Because the two measurements are independent, subtraction of the rate estimates adds their variances: Var(ŝ) = g/tg + b/tb. The corresponding counting standard uncertainty is the square root of that sum.
For observed counts, a common plug-in estimate is u² = G/tg² + B/tb². It estimates this model’s counting contribution. It is not a complete uncertainty budget or a universally adequate small-count confidence interval.
Subtracting the expected background removes its expected contribution to the rate, but does not undo fluctuations already present in either measurement. In particular, √(G − B) is not the uncertainty formula for unequal-time background subtraction.
CHAPTER 4 OF 12 · Build the uncertainty
4. Choose the quantity the design should minimise
Suppose usable counting time is T, so tg + tb = T. For fixed positive planned rates, minimise v = g/tg + b/tb. The expected net signal s = g − b does not change with the split; when s is positive, minimising v also minimises the model’s relative counting uncertainty u/s.
This objective differs from collecting equal counts, using equal times or making the background’s relative uncertainty match the gross measurement’s. Each is a possible rule, but they are not the same optimisation problem.
The research note by Pauw and Tardif, The optimal division between sample and background measurement time for photon counting experiments, develops this question. Here we test the allocation using a different set of rates and explicit competing plans.
Substitute tb = T − tg into the variance expression. Its derivative with respect to tg is −g/tg² + b/(T − tg)². Setting it to zero gives tg/tb = √(g/b).
Equivalently, tg = T√g/(√g + √b), with tb = T√b/(√g + √b). Positive rates give an interior minimum: the second derivative 2g/tg³ + 2b/tb³ is positive.
The ratio uses the gross rate g, including background, rather than the net signal s. A gross count fluctuates according to everything counted during that measurement. Replacing g with s would understate that contribution.
The rule does not say the brighter measurement needs all the time. Shortening background time eventually makes b/tb large. The optimum balances the benefit of another second in each measurement.
Take planned g = 90 counts/s, b = 10 counts/s and T = 100 s. The net rate is 80 counts/s. The optimum ratio is √9 = 3, giving tg = 75 s and tb = 25 s.
| Plan | Gross time | Background time | Variance of net rate | Counting standard uncertainty |
|---|---|---|---|---|
| Equal expected counts | 10 s | 90 s | 9.111 (counts/s)² | 3.018 counts/s |
| Equal times | 50 s | 50 s | 2.000 (counts/s)² | 1.414 counts/s |
| Square-root split | 75 s | 25 s | 1.600 (counts/s)² | 1.265 counts/s |
| Mostly gross | 90 s | 10 s | 2.000 (counts/s)² | 1.414 counts/s |
The optimum lowers standard uncertainty by about 10.6% relative to equal times in this example. Relative to the 80 counts/s net signal, the ideal counting uncertainty falls from about 1.77% to 1.58%.
The equal-count plan spends nine times longer on background and gives much poorer precision for the difference. It makes the two totals equal while leaving the much shorter gross measurement as the dominant source of rate uncertainty.
At the square-root split, the minimum variance is vmin = (√g + √b)²/T. For 90, 10 and 100, this becomes (3√10 + √10)²/100 = 160/100 = 1.6 (counts/s)².
Check nearby plans independently. At 70 s gross and 30 s background, variance is 90/70 + 10/30 ≈ 1.619. At 80 s gross and 20 s background, it is 90/80 + 10/20 = 1.625. Both exceed 1.600.
The near-optimal values are close. A small timing constraint might therefore have little practical cost, while a large shift towards the equal-count plan has a substantial cost here. Calculate the consequence before treating the mathematically exact optimum as an operational requirement.
This comparison checks a model prediction. Agreement among several ways of calculating it does not verify the underlying detector, independence assumption or stability of the background.
CHAPTER 8 OF 12 · Check the experiment
8. Use a pilot without pretending the rates are known exactly
The optimum depends on g and b, which may be uncertain before measurement. A short pilot or previous comparable measurements can provide planning estimates. The final report must retain the actual counts and actual observation times.
For a separate invented observation, suppose G = 7,200 over 80 s and B = 200 over 20 s. The estimated rates are 90 and 10 counts/s, with net 80 counts/s.
| Measurement | Observed counts | Time | Estimated rate | Contribution to rate variance |
|---|---|---|---|---|
| Gross | 7,200 | 80 s | 90 counts/s | 7,200/80² = 1.125 |
| Background | 200 | 20 s | 10 counts/s | 200/20² = 0.500 |
The plug-in standard uncertainty is √1.625 ≈ 1.275 counts/s. Do not substitute the planned 75/25 times into this observed 80/20 record to claim the smaller uncertainty.
If the pilot consumes part of the fixed total time, account for it. Decisions based on a very noisy pilot can also make the planned split unstable. More elaborate adaptive designs need an analysis matching their actual sampling rules.
CHAPTER 9 OF 12 · Check the experiment
9. Put overhead and background drift back into the decision
If a 100 s session requires 10 s for switching or settling, only 90 s remains for the two counts. Applying the three-to-one split gives 67.5 s and 22.5 s. Minimum variance is then 160/90 ≈ 1.778, larger than the no-overhead result.
Drift raises a different problem. A background measured at the start may not represent the background during a later sample run. Counting that initial background more precisely does not remove a change in its true level.
Alternating measurements or monitoring a reference may help investigate stability, but introduces timing and covariance questions outside our independent two-measurement model. Choose a design for the actual uncertainty, rather than forcing every experiment into the square-root formula.
For an evidence check, plot repeated rate estimates against time and record configuration changes. If the sequence shows a systematic shift, investigate it before reporting an uncertainty based only on Poisson counting.
CHAPTER 10 OF 12 · Check the experiment
10. Check detector effects and low-count boundaries
Detector dead time, afterpulsing, saturation and correlated events can alter count statistics. NIST’s work on photon-counting calibration, background and dead time shows why these effects require their own treatment.
If no background events were observed during a short run, that does not establish b = 0 exactly. Using zero as a known planning rate and allocating no further background time would confuse a limited observation with certainty.
Likewise, a negative net estimate can arise from fluctuations. It does not mean negative physical light. Do not automatically clip it to zero and then retain an uncertainty analysis that assumes an unchanged estimator.
For very small counts or a net rate near zero, confidence limits and detection decisions need suitable methods. Our standard-uncertainty calculation is a focused model contribution, not a claim that every result has a symmetric, physically valid interval or a particular probability of detection.
CHAPTER 11 OF 12 · Practise and continue
11. Practise with changed rates and worked checks
Check one: change the rate ratio
Let g = 40 counts/s, b = 10 counts/s and T = 120 s. The square-root ratio is two, giving 80 s gross and 40 s background. Variance is 40/80 + 10/40 = 0.75, and standard uncertainty is about 0.866 counts/s.
Check two: compare equal times
For that same case, 60 s in each measurement gives variance 40/60 + 10/60 ≈ 0.8333 and standard uncertainty about 0.913 counts/s. The gain from optimisation is smaller than in the earlier nine-to-one gross/background case.
Check three: diagnose the wrong numerator
A learner uses √(s/b) with g = 90, b = 10 and s = 80. The correct planning numerator is g: the gross measurement contains fluctuations from signal plus background. Write the measurement model before selecting a ratio.
As a practical paper exercise, list several possible splits, calculate their variance and plot it against gross time. Mark the total-time boundary and identify which assumptions would need evidence before using the curve for an actual measurement plan.
CHAPTER 12 OF 12 · Practise and continue
12. Questions, connected routes and sources
Is equal time always a bad choice?
No. It can be close to optimal when gross and background rates are similar, and other design constraints can matter. Compare its predicted variance with the alternative rather than treating one rule as universally best.
Does more time remove every source of uncertainty?
No. The model’s counting variance falls with time under stable rates. Calibration, drift, background mismatch and detector effects can remain or become dominant.
Which route develops the next question?
For optical context, use Photonics and Photonics Engineering. For count models, small counts and detector limits, use particle counters, Poisson statistics and counting uncertainty. The existing confocal microscopy guide provides a separate application context; its full measurement chain requires more than this two-rate model.
Return to The Gold Standard of Education: learning that transfers into life or the How X Works subject map. The transferable move is to allocate effort according to its contribution to the uncertainty of the final answer.
Primary references are Pauw and Tardif’s photon-counting time-allocation research note and NIST’s photon-counting calibration study. All numerical rates, plans, counts and practice cases here are original educational examples.
Sources checked 21 October 2026. Worked examples are educational models with the assumptions stated above.