C.16.RM:5.1 - Separate a beam from background and detector offset
The wanted quantity is the optical power I from a beam at a detector. A supplied linear response model is:
r = g*(I + A) + b.
Here A is ambient optical power, g = 2 mV/mW is a known response coefficient and b is an electronic offset. The detector is unsaturated. During the following three observations, g, A and b remain unchanged:
| Condition | Reading |
|---|---|
| Beam on, receiver exposed | 12 mV |
| Beam off, receiver exposed to the same ambient light | 6 mV |
| Opaque cap over the receiver, excluding beam and ambient light | 2 mV |
The capped reading gives b = 2 mV. Subtracting only this offset from the beam-on reading and dividing by g gives 5 mW. That is I + A; it leaves the wanted beam contribution mixed with ambient light.
Use the exposed beam-off reading to cancel both unchanged contributions:
I = (r_on - r_off)/g = (12 - 6) mV / (2 mV/mW) = 3 mW.
The same observations give A = (6 - 2)/2 = 2 mW. Substitution reconstructs all three readings. The repaired interpretation supplies I using observations already available.
A possible arrangement repair is to shield the receiver from ambient light while preserving the beam at the detector. Under A = 0 and the same g and b, a new beam-on reading of 8 mV would also give I = 3 mW. That value is conditional until the new measurement is performed.
Suppose instead that the shield removes ambient light but transmits a known fraction alpha = 0.9 of the original beam. With the same g and b, a new reading of 7.4 mV gives (7.4 - 2)/2 = 2.7 mW for the transmitted beam. Recover the original beam through the changed relation:
I = (r - b)/(g*alpha) = (7.4 - 2)/(2*0.9) = 3 mW.
If the transmission fraction is unknown, this new reading alone leaves the original beam unresolved. The earlier valid on/off observations still support their result of 3 mW. The shield’s changed interaction must be included in any conclusion drawn from the new measurement.
For a question I ≤ 3.2 mW, suppose each on/off reading has an arbitrary additive error between -0.1 and 0.1 mV, while g and the shared background remain fixed. The difference gives 2.9 ≤ I ≤ 3.1 mW, so the bound settles the question. Drift in A between on and off would add a further term. Alternating readings reduces that uncertainty only with a supplied account of the drift; the alternation itself is insufficient.