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Choose a route or read from the beginning. All teaching chapters stay open; the grouped index below is expandable.
Separate the two errors
Chapters 1–2. Define responsivity and the measured dark offset.
Rebuild the calibration
Chapters 3–5. Calculate a changed detector and compare corrections.
Test what the line means
Chapters 6–8. Check residuals, reference uncertainty and conditions.
Practise and continue
Chapters 11–12. Solve a changed calibration and choose further reading.
Full chapter index · Practice and worked checks · Photonics subject root
You block the light, measure a new dark reading and subtract it. The corrected detector output looks tidy. Yet the optical power can still be wrong if the detector’s response per watt has changed.
This guide separates an additive offset from a multiplicative response. An original worked case shows how two reference powers identify a line, why one dark measurement cannot determine its slope and what further evidence the calibration needs. All numerical readings are fictional and belong to a fixed-wavelength, steady, linear educational model.
Chapter index
Separate the two errors · Chapters 1–2
1. Name what the detector reports
2. Separate an offset change from a response change
Rebuild the calibration · Chapters 3–5
3. Build observations from the changed detector
4. Recover slope and offset from two references
5. Compare the corrections on the same reading
Test what the line means · Chapters 6–8
6. A line through two points is not a linearity test
7. Carry reference uncertainty through the inverse calculation
8. Keep the calibration conditions attached
Use evidence carefully · Chapters 9–10
9. Turn the distinction into practical evidence checks
10. Questions readers ask
Practise and continue · Chapters 11–12
11. Transfer checks with worked answers
12. Choose the next measurement question
For our model, total measured current is I = RP + b. Optical power P falls on a defined detector area, R is responsivity in amperes per watt and b is an additive current offset. The inverse calculation is P = (I − b)/R, provided R is nonzero and the model applies.
A microampere divided by a microwatt has the same numerical ratio as an ampere divided by a watt. Keep that unit cancellation explicit. Current, optical power and photon rate are different quantities, even when one instrument converts between them internally.
The offset b can include detector dark current and readout offset. A blocked-light reading measures the combined baseline of the stated setup; it does not automatically isolate a semiconductor property. Ambient light present only during the illuminated measurement would need its own treatment.
CHAPTER 2 OF 12 · Separate the two errors
2. Separate an offset change from a response change
An offset change shifts the current-versus-power line vertically. A responsivity change changes its slope. At P = 0, the model gives I = b regardless of the value of R. That is the precise reason a dark observation cannot identify the slope.
Imagine an earlier calibration with R_old = 0.80 A/W and b_old = 0.50 μA. Now stipulate R_now = 0.64 A/W and b_now = 2.50 μA. These are deliberately conspicuous model changes, not typical drift specifications for a particular detector.
The current offset increased by 2.00 μA while responsivity fell by 20%. Correcting one change leaves the other in place. An unchanged dark reading would likewise provide no proof that responsivity stayed unchanged.
CHAPTER 3 OF 12 · Rebuild the calibration
3. Build observations from the changed detector
At a reference power of 10.0 μW, the current model predicts 0.64 × 10.0 + 2.50 = 8.90 μA. At 30.0 μW, it predicts 21.70 μA. An unknown producing 15.30 μA has net current 12.80 μA and therefore power 20.0 μW.
| Observation | Power, μW | Current, μA |
|---|---|---|
| Blocked-light baseline | 0.0 | 2.50 |
| Lower reference | 10.0 | 8.90 |
| Unknown, stipulated for checking | 20.0 | 15.30 |
| Upper reference | 30.0 | 21.70 |
Only the references and measured current would be available in a real inverse problem. We stipulate the unknown’s true value here so that competing correction methods can be checked transparently. Agreement with an invented answer is a mathematical consistency check, not laboratory validation.
CHAPTER 4 OF 12 · Rebuild the calibration
4. Recover slope and offset from two references
With two distinct known powers, subtract the response equations: I₂ − I₁ = R(P₂ − P₁). The common offset cancels if it remains constant between observations. Thus R = (21.70 − 8.90)/(30.0 − 10.0) = 0.64 A/W.
Then b = I₁ − RP₁ = 8.90 − 0.64 × 10.0 = 2.50 μA. Using these current parameters, the unknown gives (15.30 − 2.50)/0.64 = 20.0 μW. A separately measured dark baseline provides a useful consistency comparison with the fitted intercept.
Alternatively, one known nonzero power plus an independently measured baseline identifies R in this exact model. It still needs uncertainty, stability and linearity checks. A dark observation alone supplies only one constraint for two unknown parameters.
CHAPTER 5 OF 12 · Rebuild the calibration
5. Compare the corrections on the same reading
Apply several methods to the identical 15.30 μA unknown. Holding the observation fixed makes the source of each difference visible.
| Parameters used | Estimated power, μW | Error relative to 20.0 μW |
|---|---|---|
| Current offset and current responsivity | 20.0 | 0% |
| Current offset but old responsivity | 16.0 | −20% |
| Old offset and old responsivity | 18.5 | −7.5% |
| Old offset but current responsivity | 23.125 | +15.625% |
The outdated pair happens to be closer than correcting only the offset. Its two errors partly cancel for this particular power. That accidental proximity does not make either old parameter reliable; cancellation changes with power.
When the offset is correctly subtracted but old responsivity is retained, P_est/P_true = R_now/R_old. When only the offset is wrong, the power error is (b_now − b_used)/R_now. One error scales with the signal, while the other contributes an additive power bias.
CHAPTER 6 OF 12 · Test what the line means
6. A line through two points is not a linearity test
Any two points at distinct powers define a straight line. They leave no independent residual with which to challenge that line. Add suitable reference levels, repeated observations and held-out checks across the intended range.
For example, the current line predicts 12.10 μA at 15.0 μW. A reproducible independent reading of 12.80 μA would differ by 0.70 μA. Whether that disagreement is significant depends on reference and measurement uncertainties; it could indicate nonlinearity, time drift or changed conditions.
Repeat the sequence in a different order to help distinguish power dependence from time dependence. Bracket an unknown with reference or baseline checks where justified by the method. A smooth fit cannot reveal an error that all reference observations share.
CHAPTER 7 OF 12 · Test what the line means
7. Carry reference uncertainty through the inverse calculation
Suppose, in a separate simplified uncertainty case, net current has standard uncertainty 0.128 μA and responsivity has independent standard uncertainty 0.0064 A/W. At net current 12.80 μA and R = 0.64 A/W, each contributes 1% relative standard uncertainty.
The resulting first-order relative standard uncertainty is √(0.01² + 0.01²) ≈ 1.414%, giving about 0.283 μW at 20.0 μW. This is not a confidence interval or a complete budget; it assumes the stated independence and small uncertainties.
Fitted slope and intercept are usually related statistically. If using I, b and R directly, P = (I − b)/R has sensitivities 1/R, −1/R and −P/R. Include covariance terms when inputs are correlated. Do not treat a reused baseline or fitted intercept as fresh independent information for every observation.
The NIST uncertainty-propagation guidance provides the general framework. The numerical uncertainty case above is an original example with deliberately stated inputs.
CHAPTER 8 OF 12 · Test what the line means
8. Keep the calibration conditions attached
Responsivity is spectral. A value established at one wavelength is not a universal current-to-power conversion. Broadband illumination requires a spectral response model or a justified effective calibration for the actual spectrum.
Beam position, illuminated area, bias, temperature, readout range and optical configuration may matter. NIST’s calibration FAQ specifically discusses differences in beam conditions and spatial response. Record such conditions rather than treating a calibration date as sufficient evidence of applicability.
Our fitted R describes the stipulated current-measurement chain. If an amplifier gain also changes, an effective slope can change even without an intrinsic photodiode change. The data may reveal a conversion error without uniquely identifying which component caused it.
CHAPTER 9 OF 12 · Use evidence carefully
9. Turn the distinction into practical evidence checks
First specify the measurand: optical power at which reference plane, wavelength and spatial acceptance? Then document the current measurement, baseline procedure, reference power uncertainties and intended operating range.
Inspect stability during the actual sequence. Test for saturation and range-dependent behaviour using appropriate procedures. Compare fitted intercept with independently observed blocked-light baseline, while recognising that their acquisition conditions must match.
Finally evaluate repeatability and residuals against the required decision. A dark subtraction can improve an estimate without completing the calibration. The NIST photonic-radiometry programme treats spectral response, optical-power standards and detector linearity as distinct measurement concerns.
Does subtracting dark current remove its noise?
It removes an estimated mean contribution under the model. Fluctuations and uncertainty remain. A longer baseline observation may reduce some random uncertainty while leaving drift or a mismatched baseline unresolved.
Can a ratio remove responsivity drift?
If numerator and denominator truly share the same multiplicative response under matched conditions, cancellation may occur. Different wavelengths, detector states or measurement times can prevent it. Also subtract appropriate offsets before assuming a current ratio equals a power ratio.
Is a two-reference check a complete accredited calibration?
No. It is a mathematical method for estimating two parameters under a model. Traceable references, uncertainty, procedures, operating conditions and validation determine the metrological status of a real result. Do not attach a certification claim to this classroom calculation.
A changed detector gives 5.0 μA at 5.0 μW and 13.0 μA at 25.0 μW. Find the line before estimating an unknown of 9.8 μA. The slope is 8.0/20.0 = 0.40 A/W, the intercept is 3.0 μA, and the unknown power is (9.8 − 3.0)/0.40 = 17.0 μW.
Now retain that slope but mistakenly use a 2.0 μA baseline. The estimate becomes 19.5 μW: a 1.0 μA offset error creates a 2.5 μW power error. Correcting the baseline changes an additive error; changing responsivity changes the scale.
Finally suppose both reference powers were assigned values 5% above their actual powers. The fitted slope versus assigned power would be too low, and an unknown reported on that scale would be 5% high under this ideal model. More repeated current readings cannot remove a shared reference-scale error.
Reconnect the optical chain
Use the Photonics root for physical quantities, Photonics Engineering for qualified interfaces, and Optical Engineering for system performance and calibration context.
Compare different correction problems
For counting measurements, read dead-time correction before background subtraction, shared background and correlated corrected readings, and dividing a fixed measurement time. Their count-rate models should not be substituted for the steady-current line used here. Lock-in detection adds a separate gain, phase and bandwidth question. Scintillation detectors and gain stability show how drift affects a different measurement, the pulse-height energy scale.
Sources and transferable learning
For specialist reference, use NIST spectral responsivity and spatial uniformity, its calibration FAQ and photonic radiometry. These resources support the measurement distinctions; our detector values and worked cases are invented.
Continue through How X Works and the Gold Standard of Education. The useful habit transfers to other sensors: identify whether a correction changes the zero, the scale or the model itself.
Sources checked 21 October 2026. Worked examples are educational models with the assumptions stated above.