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Choose a route or read from the beginning. All teaching chapters stay open; the grouped index below is expandable.
Define serial transmission
Chapters 1–2. Separate watts, ratios and reference planes.
Derive correlated losses
Chapters 3–5. Compute stage losses and shared-reading covariance.
Test calibration cancellation
Chapters 6–7. Distinguish common gain from differential drift.
Full chapter index · Practice and worked checks · Photonics subject root
Optical losses in a cascade add in decibels, but their measured uncertainties do not always add as independent quantities. A power reading shared between two adjacent loss measurements creates covariance, and a common calibration factor can cancel in a power ratio. Ignoring those relationships can make a combined attenuation uncertainty larger or smaller than the evidence supports.
In a fictional two-stage optical link, input power is 10 mW, power after stage one is 8 mW and power after stage two is 4 mW. The stage transmissions are 0.80 and 0.50, so overall transmission is 0.40 and total loss is about 3.979 dB. If each of the three logarithmic power readings has independent standard uncertainty 0.10 dB, then total-loss uncertainty is about 0.141 dB, not the 0.200 dB obtained by wrongly treating adjacent stage losses as independent. This article derives the cancellation and shows what must be checked in a real fibre, sensor or photonic link. It extends the Gold Standard of Photonics.
Chapter index
Define serial transmission · Chapters 1–2
1. Begin with the actual measurement chain
2. Use the ratio and logarithmic loss definitions
Derive correlated losses · Chapters 3–5
3. Recognize a shared reading in two loss estimates
4. Calculate the independent-reading case exactly
5. Test cancellation of a common calibration factor
Test calibration cancellation · Chapters 6–7
6. Build a general covariance expression
7. Distinguish uncertainty from real optical variability
Build the error budget · Chapters 8–10
8. Design a measurement sequence that identifies common-mode errors
9. Investigate source drift and the reading sequence
10. Questions that prevent misleading uncertainty claims
Practise and continue · Chapters 11–12
11. Practise with changed power readings and uncertainty
12. Continue from the loss budget to system performance
CHAPTER 1 OF 12 · Define serial transmission
1. Begin with the actual measurement chain
Consider a source followed by two optical components and a calibrated power meter. Denote optical powers at the three planes P0, P1 and P2. If stage one transmits T1 = P1/P0 and stage two transmits T2 = P2/P1, the complete transmission is Ttot = T1 T2 = P2/P0. The intermediate power P1 cancels algebraically.
This cancellation is more than a convenient identity. It tells us that a consistent total measurement depends on the endpoints, not on the uncertainty of an intermediate reading inserted into both ratios with opposite signs. A spreadsheet that treats stage losses as independent can count the same measurement error twice.
The result assumes a clearly defined optical path, matching reference planes and measurement conditions. If the stages were characterized on different days, at different wavelengths, with different launch distributions or at different powers, multiplying their quoted transmissions may not describe a single reproducible operating state.
The numbers are fictional teaching data. This specialist article is about correlated measurement uncertainty, not about photodetector responsivity drift, background counting, the geometry of aperture clipping, or the inherent optical loss mechanisms of any one component.
CHAPTER 2 OF 12 · Define serial transmission
2. Use the ratio and logarithmic loss definitions
For transmitted power Pin and Pout under matched conditions, define linear power transmission T = Pout/Pin. When T is between zero and one, loss in decibels is LdB = 10 log10(Pin/Pout) = −10 log10 T. Decibels describe logarithmic ratios; they are not watts or absolute optical energy.
For the fictional readings 10 mW, 8 mW and 4 mW, stage one loss is L1 = 10 log10(10/8) ≈ 0.9691 dB. Stage two loss is L2 = 10 log10(8/4) ≈ 3.0103 dB. Sum is 3.9794 dB, exactly the loss obtained from Ltot = 10 log10(10/4).
If input and output are displayed in dBm, the difference of the two readings in dBm is the loss in dB, provided reference units, detector response and calibration definitions match. One milliwatt is the 0 dBm absolute-power reference. A 10 mW input corresponds to +10 dBm, while 4 mW corresponds to approximately +6.0206 dBm, giving the same 3.9794 dB loss.
Adding linear transmissions would be wrong: 0.8 + 0.5 does not represent the serial pair. Multiplying stage losses quoted in dB would also be wrong. Multiply linear transmissions or add losses in dB, then check the endpoint ratio as an independent arithmetic sanity test.
CHAPTER 3 OF 12 · Derive correlated losses
3. Recognize a shared reading in two loss estimates
Write logarithmic power readings in dBm as x0, x1 and x2. The stage losses are L1 = x0 − x1 and L2 = x1 − x2. Therefore Ltot = L1 + L2 = x0 − x2. The intermediate reading x1 appears once with a minus sign and once with a plus sign.
Suppose the measured intermediate log-power reading x1 happens to be high by +0.10 dB, while the endpoints are exact in a simplified thought experiment. Then L1 is low by 0.10 dB and L2 is high by 0.10 dB. Their sum is unchanged. The measurement errors in the two stage losses are negatively correlated through their common intermediate value.
A covariance term is therefore required when combining their uncertainties: Var(L1+L2) = Var(L1) + Var(L2) + 2 Cov(L1,L2). Leaving out the covariance discards information about how the same reading enters both quantities.
This does not say that intermediate readings are unnecessary. P1 is essential if stage-specific losses are needed for diagnosis and maintenance. It says that a shared measurement must be represented consistently in the uncertainty model rather than treated as two independent observations.
CHAPTER 4 OF 12 · Derive correlated losses
4. Calculate the independent-reading case exactly
Assume each logged power x0, x1 and x2 has zero-mean independent standard uncertainty u = 0.10 dB. Then Var(L1) = Var(x0−x1) = u²+u² = 0.020 dB², and Var(L2) = Var(x1−x2) = 0.020 dB². Each stage loss has standard uncertainty sqrt(0.020) ≈ 0.1414 dB.
Because x1 enters with opposite signs, Cov(L1,L2) = Cov(x0−x1, x1−x2) = −Var(x1) = −0.010 dB². Therefore Var(Ltot) = 0.020+0.020−0.020 = 0.020 dB² and u(Ltot) = sqrt(0.020) ≈ 0.1414 dB.
If the two stage uncertainties had been naively added in quadrature as though independent, one would get sqrt(0.020+0.020) = 0.2000 dB. That is not the correct model for three independent readings with the middle reading reused. The proper endpoint calculation u(x0−x2) also gives 0.1414 dB.
| Quantity | Formula in log-power readings | Standard uncertainty in fictional case | Shared-reading effect |
|---|---|---|---|
| Stage 1 loss | x0 − x1 | 0.1414 dB | Contains −x1 |
| Stage 2 loss | x1 − x2 | 0.1414 dB | Contains +x1 |
| Naive independent sum | sqrt(u1²+u2²) | 0.2000 dB | Double counts shared x1 |
| Correct total | x0 − x2 | 0.1414 dB | Intermediate x1 cancels |
The stated standard uncertainties are in the log-power domain, chosen only to illustrate covariance. They are not an estimate of a real instrument’s uncertainty and are not expanded 95% intervals.
CHAPTER 5 OF 12 · Derive correlated losses
5. Test cancellation of a common calibration factor
Suppose the same meter has an unknown multiplicative absolute power scale k that applies identically to all three readings in linear units. Its displayed powers are kP0, kP1 and kP2. Each transmission ratio remains unchanged because k cancels: (kP1)/(kP0) = P1/P0 and (kP2)/(kP1) = P2/P1.
In log-power units, a common multiplicative factor becomes an additive constant c = 10 log10 k on every reading. Each difference xi−xj removes c. Treating that common scale uncertainty as independent for input and output would overstate its impact on a same-meter ratio.
But this is only exact when the same factor applies to each reading under the compared conditions. Detector nonlinearity, wavelength-dependent calibration, range switching, temperature drift between measurements or different power meters can create differences that do not cancel. The distinction between common-mode calibration and differential measurement error is central.
A useful measurement model writes observed log power as xobs_i = xtrue_i + c + ei + bi, where c is shared calibration offset, ei is reading-specific random error and bi collects point-specific biases such as nonlinearity or range dependence. Differences cancel c but not necessarily ei or differences among the bi.
CHAPTER 6 OF 12 · Test calibration cancellation
6. Build a general covariance expression
For any pair of logged powers xa and xb, the variance of their difference is Var(xa−xb) = Var(xa)+Var(xb)−2Cov(xa,xb). Positive covariance between readings can reduce the uncertainty of the ratio when both measurements share a stable calibration term. Independent noise does not receive that cancellation benefit.
For multiple serial segments, define Lj = x(j−1)−xj. Their sum telescopes to x0−xn. If intermediate readings are reused consistently, adjacent segment-loss errors inherit covariance. Total uncertainty can be computed from the full covariance matrix, or, under a justified direct-endpoint measurement model, from the two endpoint readings.
Not every covariance is negative. If separate stage tests share a common input reference in the same direction, errors can be positively correlated. The sign follows how each measured quantity depends on its underlying observations, not a universal rule that correlation always improves precision.
Never enter a numerical correlation coefficient solely to make the uncertainty budget look smaller. Derive it from the measurement arrangement, repeated data, calibrated uncertainty components or a documented physical model. An unsupported covariance estimate is as misleading as assuming complete independence.
CHAPTER 7 OF 12 · Test calibration cancellation
7. Distinguish uncertainty from real optical variability
An optical component can show variable loss because of polarization, temperature, wavelength, connector alignment, launch conditions or bending. Those are physical changes in its transmission, not necessarily errors in measuring a fixed transmission. A complete test design should separate measurement uncertainty from the variability of the device under different legitimate conditions.
If two stages interact—for example, if the first changes modal distribution or polarization that changes the second stage’s response—the individual stage transmissions measured separately may not multiply as assumed. The correct cascade relation still holds for powers measured at the same three planes in the assembled system, but independent off-line component specifications might not describe that assembled state.
Stray reflections and coherent interference can create dependence on connector spacing and source coherence. Upstream and downstream readings may not be strictly comparable when back-reflections alter the source operating point. Such effects call for better experimental control, not just another decimal place in a covariance matrix.
Record wavelength, input polarization, launch conditions, reference-plane definitions, dynamic range and environmental state. A transmission estimate without those conditions may be precise for one setup yet unreliable for another.
CHAPTER 8 OF 12 · Build the error budget
8. Design a measurement sequence that identifies common-mode errors
When possible, measure input and output power with the same calibrated meter, identical wavelength setting and stable source, while recording the exact sequence and any range changes. Repeat readings in an interleaved order if source drift is plausible. Reference measurements help distinguish source variation from device attenuation.
For each reading, note which uncertainty components are shared and which vary independently. A common absolute calibration scale often cancels in a ratio; detector linearity and temporal drift may not. If several meters are used, record their separate traceability and any shared reference calibration that creates covariance.
| Error component | Typical relationship across readings | Consequence for a ratio |
|---|---|---|
| One shared multiplicative meter scale | Common to all readings | Cancels if truly identical |
| Independent repeatability noise | Reading-specific | Remains in endpoint difference |
| Reused intermediate reading | Opposite sign in adjacent losses | Produces negative stage-loss covariance |
| Power-dependent nonlinearity | Changes with power or range | Does not generally cancel |
Quality control requires more than checking that L1 + L2 equals Ltotal algebraically; the definitions guarantee that identity when measurements are consistent. A stronger independent check uses repeated sequences or changed conditions to test whether the uncertainty model predicts observed scatter and bias.
CHAPTER 9 OF 12 · Build the error budget
9. Investigate source drift and the reading sequence
The identity Ltotal = x0 − x2 assumes comparable source states at the measurement endpoints. A source may drift between sequential readings, and a meter may change range or settle thermally. Correlation cannot compensate for an unmodeled change in the actual source.
One useful sequence is input, output and input again, with times recorded. The repeated reference may reveal drift, while any interpolation needs to be supported by the observed behavior. Simultaneous reference detection may remove common source fluctuations, but differences in detector gain and spectral response still need evaluation.
A report should separate repeatability under one alignment from reproducibility after connectors, launches or settings are reset. Small meter noise does not prove low system variability. Write these contributions explicitly before combining uncertainties.
CHAPTER 10 OF 12 · Build the error budget
10. Questions that prevent misleading uncertainty claims
Does adding losses in dB always give the total?
For power transmissions along one consistent serial path, yes: linear transmissions multiply and their logarithmic losses add. Independently measured stage specifications may fail to describe an assembled optical train if conditions or interactions differ.
Should I always add dB uncertainties in quadrature?
Only if the relevant estimates are uncorrelated under the chosen uncertainty model. Shared readings and shared calibration factors create covariance that must be included or canceled explicitly through the underlying measurement equation.
Does a common meter remove every calibration concern?
No. A common stable multiplicative scale cancels in ideal ratios, but detector nonlinearity, spectral mismatch, range changes, drift and other differential effects can remain. Their influence must be evaluated under the actual measurement conditions.
Can covariance make combined uncertainty larger?
Yes. Positive covariance among quantities that enter with the same sign can increase the variance of their sum. Covariance can also reduce uncertainty when common terms subtract. Always follow the algebraic signs from the measurement model.
CHAPTER 11 OF 12 · Practise and continue
11. Practise with changed power readings and uncertainty
Take a second fictional optical chain with P0 = 12 mW, P1 = 9 mW and P2 = 6 mW. The first stage transmission is 0.75, the second is 2/3 and total is 0.50. Thus total loss is 10 log10(2) ≈ 3.0103 dB, regardless of how the stage losses are rounded individually.
Suppose logged input and output readings have independent standard uncertainties of 0.08 dB and 0.06 dB. Then u(Ltot) = sqrt(0.08²+0.06²) = 0.10 dB. If the middle reading is comparatively noisy, its uncertainty affects the individual stage losses but cancels from their algebraic sum when the exact same middle observation is used in both.
Now suppose an absolute log-power calibration offset of +0.25 dB is shared across all three values. Each logged power moves up by the same amount, but the inferred stage and total losses remain unchanged. If only the output reading instead experiences an additional +0.25 dB differential offset, the inferred total loss is biased downward by 0.25 dB. Those are different error models.
As a final exercise, write a one-line measurement equation for the actual total loss and list each common and differential uncertainty component. That answer matters more than an isolated standard deviation with no declared dependencies.
CHAPTER 12 OF 12 · Practise and continue
12. Continue from the loss budget to system performance
The gold standard of cascaded optical-loss reporting is a consistent sequence of physical powers, defined reference planes, correct logarithmic arithmetic and a covariance-aware uncertainty model. It states which calibration effects cancel and which are left to test. The best error budget is the one another laboratory can reproduce from the measurement arrangement.
Return to the Gold Standard of Photonics for source, transmission and detector foundations. Read Responsivity Drift and Why a Dark Reading Does Not Calibrate Gain for a different detector-calibration failure, and Shared Background Counts for another covariance mechanism. These specialist pages separate optical transmission from different routes by which measurements become correlated.
Sources and further reading
IEC TR 61282-14:2019, uncertainty of fibre optic attenuation measurements describes attenuation ratios and covariance treatment in the measurement model. NIST: Uncertainty of Measurement Results gives a general guide to uncertainty propagation and correlated quantities. NIST: Photonic Radiometry addresses optical power calibration. The powers, uncertainties and three-plane chain above are constructed examples, not measured equipment specifications.
How X Works and The Gold Standard of Education connect this topic with a transferable test: before combining uncertain results, ask whether they share any of the same measurements.
Sources checked 21 October 2026. Worked examples are educational models with the assumptions stated above.