The Gold Standard of Waveguides | Why Equal Modal Weights Can Produce Different Aperture Readings

Suppose a transverse coordinate x runs from zero to one across a normalised guide width. A scalar complex field E(x) describes the selected teaching model at one longitudinal position. Define a whole-region reading N as the integral of |E(x)|² from zero to one.

Now define a left-aperture reading RL by integrating the same squared field only from zero to one half. These are different observables. The first collects the complete stipulated transverse domain; the second collects only its left half. Neither should be silently substituted for the other in an attenuation account.

In an actual electromagnetic problem, fields are vector quantities and power flux has a corresponding electromagnetic definition. Our scalar model isolates one mathematical issue: orthogonality over a complete domain need not remove interference over a restricted domain. The learner should carry that distinction into a real measurement by asking what the instrument collects and how its response relates to the physical quantity being reported.

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Let u1(x) = √2 sin(πx) and u2(x) = √2 sin(2πx). Each vanishes at the endpoints of the normalised interval. Their squared integrals over zero through one are both one. We will treat them as two allowed scalar mode shapes with a fixed, known basis.

Their whole-domain overlap integral is zero. That is the orthogonality relationship we need: the integral of u1u2 over the complete interval vanishes. It does not mean that the two functions never overlap locally. At many interior positions both are nonzero.

This is a useful first-principles distinction. Orthogonality is defined using an inner product, which includes a domain and a weighting rule. It is not a visual claim that the profiles occupy different pieces of space. Before relying on cancellation, identify the particular integral for which the cancellation has been established. A change in the measurement region can change that inner product even while the underlying mode shapes stay the same.

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Using sin²θ = [1 − cos(2θ)]/2, the squared first mode is 2 sin²(πx) = 1 − cos(2πx). Its integral from zero to one is one because the sine term from the cosine integral vanishes at both endpoints.

The second mode similarly gives 2 sin²(2πx) = 1 − cos(4πx), whose full integral is one. A coefficient of magnitude A multiplying either normalised function therefore contributes A² to the whole-domain squared-field norm when the orthogonality assumptions are used.

The choice of normalisation matters. If another author uses unnormalised sine functions, coefficient magnitudes will have a different scale. Comparing their coefficients with ours without converting the basis would create a numerical disagreement that is really a convention difference. Keep function definitions and coefficient units together. Normalisation makes the bookkeeping simple, but only after the convention has been stated and verified rather than inferred from a familiar mode label.

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The product identity gives u1u2 = 2 sin(πx) sin(2πx) = cos(πx) − cos(3πx). Integrating over zero through one produces zero from both sine boundary terms. This confirms the whole-domain orthogonality.

At x = one quarter, however, u1 is one and u2 is √2. Both are present at the same point. Their local cross product is not zero. Orthogonality therefore cannot be interpreted as a point-by-point absence of overlap.

The cancellation occurs because u1 is symmetric about the midpoint while u2 changes sign there. Positive contributions from one side of the domain cancel negative contributions from the other side in the complete overlap integral. A detector that collects only one side omits the partner contributions needed for that cancellation. This explains the mechanism before we calculate any detector reading: the measurement boundary, rather than a failure of orthogonality, changes which cross terms survive.

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Write E(x) = A u1(x) + B e^(iφ)u2(x), with A and B nonnegative real coefficient magnitudes and φ the relative phase. A common overall phase has no effect on |E|², so it does not need a separate parameter in this reading model.

Expanding the squared magnitude gives A²u1² + B²u2² + 2ABu1u2 cosφ. The first two terms depend on the modal weights A² and B². The third also depends on their relative phase and on the spatial overlap relevant to the detector.

Holding the weights fixed does not hold the local reading fixed. For example, φ = zero gives a different cross term from φ = π, although A² and B² are unchanged. This is coherent superposition. Later we will distinguish a fixed unknown phase from phase that varies during an averaging measurement; those descriptions can produce different inferences even when their possible instantaneous values look similar.

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Integrate the expanded expression over the complete interval. Each squared function contributes one, and the cross-term overlap contributes zero. Hence N = A² + B², independent of φ.

For the original example choose A = B = one. The whole-region norm is two for every relative phase. The local pattern can change substantially while this defined total stays fixed. That is the mathematical reason a restricted-aperture decrease need not establish loss of the whole field.

The conclusion is conditional on our basis, domain and response definition. It is not a universal statement that all real optical detector totals are phase independent. A detector can have an uneven response, omit part of the field, select polarisation or measure another observable. A real guide may also have loss or coupling. Start with the exact model result, then identify which additional assumptions would connect it to a particular physical power measurement.

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For each individual squared mode, the integral from zero to one half is one half. For the overlap, integrate cos(πx) − cos(3πx) across that half interval. The result is 1/π − (−1)/(3π) = 4/(3π).

Therefore RL = (A² + B²)/2 + [8AB/(3π)]cosφ. Unlike the complete norm, this restricted reading retains the cross term. The aperture boundary has changed the observable while leaving the coherent field model intact.

With A = B = one, RL = 1 + [8/(3π)]cosφ. The coefficient is approximately 0.84883. Its size is large enough to make the redistribution visible, yet the reading remains nonnegative for every phase. Check the integral rather than assuming that a half-width aperture must always collect half the total. That conclusion holds for either individual squared mode here, but coherent interference changes their combined reading.

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Take the first field with φ = zero and the second with φ = π. Both have modal weights one and one, and both have whole-region norm two. Their left readings are nevertheless 1 + 8/(3π) and 1 − 8/(3π).

Numerically those are approximately 1.84883 and 0.15117 in the stipulated reading units. Nothing in this comparison changed the coefficient magnitudes or introduced a lossy segment. The relative phase changed which side of the guide receives constructive or destructive contribution in this scalar account.

This is an exact counterexample to the claim that equal modal weights necessarily give equal partial-aperture readings. It does not require an imperfect instrument. A correct detector for the defined left integral should distinguish these two fields. The lesson is to ask which information a modal-weight summary discards before using it to predict a spatially restricted observation.

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Define RR as the reading over the right half. Its overlap integral is −4/(3π), so RR = (A² + B²)/2 − [8AB/(3π)]cosφ. The interference contribution has the opposite sign from the left half.

Adding the two readings gives A² + B², exactly the complete norm. For the equal-weight examples, a large left reading is accompanied by a small right reading, and vice versa. The pair preserves the whole account even when one local number looks dramatically different.

This supplies an independent checking route. Instead of trusting one formula, verify that the complementary readings add to the full integral. If they do not, inspect the integration limits, signs and normalisation. In a real measurement, two complementary detector signals would also need compatible calibration and complete collection before their sum represented the intended total. The mathematical identity helps define that requirement; it does not certify that an unidentified detector arrangement fulfils it.

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In the present model, changing φ alone redistributes the squared field between the two halves. Changing A or B changes the whole norm as well. A single left reading cannot reliably tell which kind of change occurred without further information.

For instance, a smaller RL might come from a different phase at fixed weights, from smaller coefficient magnitudes, or from a changed collection response. Those possibilities concern different parts of the account. Labelling the observation as propagation loss selects one explanation before the evidence has distinguished it.

A useful report first states the observed reading and reference region. It then identifies which model parameters are independently known and which remain possible explanations. If a whole-region norm has also been established on a consistent basis, that additional observation can constrain the interpretation. The purpose is not to deny real attenuation, but to avoid treating every local reduction as proof of it. The next measurement should address the missing distinction.

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For two uncoupled modes at one frequency in a uniform scalar propagation model, their phase factors can accumulate at different longitudinal propagation constants β1 and β2. The relative phase becomes φ(z) = φ0 + (β2 − β1)z, up to the chosen sign convention.

If the magnitudes A and B remain fixed, the complete norm stays constant while the left-aperture reading oscillates with that relative phase. When β1 and β2 differ, the phase repeats over beat length 2π/|β2 − β1|.

This supplies one explanation for a longitudinally varying partial reading without changing modal weights. It is not evidence that every measured oscillation comes from precisely two uncoupled modes. Real guide perturbations can change coefficients, and additional modes can introduce more phase differences. Keep the two-mode example as a bounded explanation whose distinctive prediction can be tested against the actual observation and model contract.

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Suppose φ is fixed during one measurement but unknown. Then RL can take any value between its phase extrema. That is a set of possible readings, not automatically a mean or a probability distribution.

Now suppose phase varies during an averaging measurement so that its cosine averages to zero under the specified averaging rule. The averaged left reading becomes (A² + B²)/2. This is an additional temporal or ensemble assumption. It does not follow merely from our ignorance of a phase that remains stable.

For A = B = one, the averaged result would be one, although an individual coherent fixed-phase reading can differ greatly. State which observation is being modelled: a single stable phase, a sequence of resolved phase changes or an averaged response. Confusing those cases can make a theoretically correct interference calculation appear inconsistent with a detector that is actually reporting a different average.

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A compact way to describe the averaged cross term is to define γ as the average of e^(iφ) under a stated distribution or measurement average. Its magnitude is at most one. In our model, the averaged left reading is (A² + B²)/2 + [8AB/(3π)]Reγ.

A stable phase corresponds to γ = e^(iφ). Complete cancellation of the mean cross term corresponds to Reγ = zero, although that condition alone does not identify every aspect of the phase distribution. Many different distributions can have the same γ and the same mean reading.

This notation preserves the distinction between weights and phase relationships. It should not be used to invent a coherence value from a single total signal. If a later analysis needs fluctuations rather than only the average, the mean γ may also be insufficient. A summary is useful when its scope is known; it becomes misleading when information it omitted is assumed to have been determined by the act of averaging.

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Observe N = two and RL = one in our model. One explanation is A = B = one with φ = π/2, so the cross term vanishes. Another is A = √2 and B = zero, which also gives N = two and RL = one.

The two observations therefore do not by themselves establish that both modes are present, or determine their individual weights and relative phase. The first explanation is a coherent two-mode field; the second contains only one of the stipulated basis functions. They agree on the recorded summaries.

This is an information limit, rather than necessarily a problem with precision. Repeating the same two exact readings more accurately does not distinguish these constructed candidates. To resolve the ambiguity, choose another response that acts differently on the candidate fields, such as a suitable additional spatial measurement. The example does not prescribe an experimental method. It teaches the habit of constructing rival states before claiming that a measurement uniquely identifies the field.

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Even when A and B are known and nonzero, one left-aperture reading determines cosφ rather than a unique phase. For the equal-weight example, cosφ = (RL − 1)/C, where C = 8/(3π). A compatible reading must lie inside the phase range already derived.

If the reading is one, both φ = π/2 and φ = 3π/2 are compatible. More generally, opposite phase signs give the same cosine. Adding integer multiples of 2π leaves the complete field reading unchanged as well. A single scalar intensity measurement should not be described as a unique recovery of an unrestricted phase.

This does not make the observation useless. It constrains a specific phase relationship under the known-weight model. The next question is which additional information distinguishes the remaining alternatives. Preserve the distinction between a compatible set and one selected representative, especially if a later propagation calculation uses the sign of the phase rather than only its cosine.

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For another paper model, suppose an additional reading is supplied for the same equal weights and response, but with the relative phase advanced by a known π/2. This is a stipulated mathematical comparison, not a proposed method for modifying an actual guide. Denote the original and shifted readings R0 and R90.

The equations become R0 = 1 + C cosφ and R90 = 1 − C sinφ. Therefore the normalised quantities (R0 − 1)/C and (1 − R90)/C specify cosine and sine together. They must lie on the unit circle for an exact fixed-phase model. For φ = π/3, they are one half and √3/2 respectively, resolving the sign ambiguity left by cosine alone.

An incompatible pair would challenge the assumed weights, known shift, response or exact data. With an averaging coherence model, the corresponding pair can lie inside a disk instead of on the fixed-phase circle. The lesson is to connect each additional measurement with the relationship it actually supplies. Two numbers do not guarantee identification unless their observation models contribute the missing information.

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Let the detector collect x from zero to h, with h between zero and one. The individual squared-mode integrals are J11 = h − sin(2πh)/(2π) and J22 = h − sin(4πh)/(4π).

The overlap is J12 = sin(πh)/π − sin(3πh)/(3π). Therefore R(h) = A²J11 + B²J22 + 2ABJ12 cosφ. This expression shows explicitly how a moving aperture changes both the individual collection fractions and the interference term.

Check its boundaries. At h = zero, all three integrals vanish, so the reading is zero. At h = one, J11 and J22 are one while J12 is zero, recovering the whole norm. At h = one half, it returns the earlier result. Those checks examine the collection geometry as well as the superposition algebra, making them useful protections against a sign error that might otherwise produce a plausible value at only one chosen aperture.

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Consider a detector collecting the centred interval from one quarter to three quarters. The overlap integral of u1u2 over this region vanishes because its antisymmetric contributions cancel around the midpoint. Its reading is therefore independent of φ for our two modes.

The squared first-mode integral across that centred region is one half plus 1/π, while the squared second-mode integral is one half. The reading is A²(1/2 + 1/π) + B²/2. For equal weights it is 1 + 1/π, approximately 1.31831, regardless of the relative phase.

This is a counterexample to the idea that collecting less than the full field must always reveal interference between a particular pair of modes. The answer depends on the functions and region together. A centred aperture can suppress the cross term for this pair while still collecting different fractions of their separate squared profiles. State the actual overlap integral instead of relying on the general label partial detector.

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If the whole norm and centred-aperture reading are both known under this two-mode model, they can help identify the two modal weights even though the centred reading contains no phase information. Subtracting half the whole norm from the centred reading gives A²/π.

Thus A² = π[RC − N/2], and B² = N − A². The inferred values must be nonnegative. A reading outside the compatible range would challenge the assumed two-mode response model, calibration or exact data rather than justify a negative modal weight.

This identification is conditional on the specified basis and detector. It still does not recover φ, because both observables cancel the relevant cross term. Combining one more measurement does not automatically identify every unknown. A useful explanation states precisely which parameter combination the new observation determines and which information remains absent. That is more helpful than describing a detector arrangement as complete simply because it produces several numbers.

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A real collection model can be represented abstractly by a nonnegative response weight w(x). Define Rw as the integral of w(x)|E(x)|². A sharp aperture is one special case, with weight one inside the collected region and zero outside.

The weighted mode overlaps form quantities Gij = integral of w ui uj for our real scalar basis. The reading is A²G11 + B²G22 + 2ABG12 cosφ. Whole-domain orthogonality corresponds to the special uniform weight one across the whole interval, not an arbitrary response function.

This formulation makes a practical question visible: does the detector treat all collected positions equally? A spatial sensitivity variation can preserve a cross term even when the nominal aperture spans the whole guide. A calibration or response map must be connected to the quantity being reported. The mathematical weight is a model of that response, not evidence that a particular instrument has the assumed sensitivity.

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Because w is nonnegative, the weighted overlap satisfies |G12|² ≤ G11G22 by the Cauchy–Schwarz inequality. Consequently the interference contribution cannot be chosen independently of the individual collection terms.

For fixed magnitudes, phase extrema give Rmin = A²G11 + B²G22 − 2AB|G12| and Rmax with the plus sign. The lower result is nonnegative because it is at least (A√G11 − B√G22)² when the overlap bound is used.

This is a useful check on a proposed response table. If invented overlaps violate the inequality, the table cannot arise from the stated nonnegative weighted integral. A negative minimum would then reveal an inconsistent set of inputs rather than a detector producing negative squared-field energy. Preserve the common model behind the diagonal and cross terms instead of assigning each term a convenient independent interval that permits combinations no real response weight can produce.

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Suppose the detector reading changes after its position or collection boundary changes. Even with an unchanged field, the integrals G11, G22 and G12 may change. The signal variation therefore cannot automatically be attributed to changed modal coefficients.

Conversely, a fixed detector can report a changed signal when the phase or magnitudes change. The observation alone does not say which layer moved. A useful comparison records the field model, longitudinal position, aperture and response weighting together.

For learners, make one change at a time in supplied paper models. First vary φ with a fixed half aperture. Then vary h while keeping the field fixed. Finally compare equal and unequal coefficients. Predict the direction or range before calculating. That sequence helps separate three effects that can otherwise become one vague story about brightness: the generated field, its propagation state and the measurement operator applied to it.

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Our original counterexample held coefficient magnitudes fixed. If a model instead includes attenuation, A and B can vary with distance. Different modal attenuation can also change their ratio. If a perturbation couples modes, their coefficients need a corresponding evolution model rather than two independent constant magnitudes.

A whole-norm decrease under a justified complete response can provide information about those changes, while a partial reading can mix them with coherent redistribution. The two observables should be analysed together with the physical assumptions instead of treating one local trace as a complete propagation account.

The lesson is not that loss is unreal or that phase explains every observation. It is that one explanation should not be selected before competing states have been tested. A model with fixed weights supplies a useful null example: it shows how much apparent variation can occur without changing those weights. Additional physical claims require additional relationships and evidence, not merely a more elaborate verbal description of the same single reading.

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Shapira and colleagues’ 2005 paper on complete modal decomposition for optical waveguides investigates amplitudes and phases of guided vectorial modes using a specific research method and experimental system. Its emphasis on phase information illustrates why mode magnitudes alone need not describe every field observation.

Our sine functions, aperture integrals and numerical counterexamples are original scalar teaching calculations. They do not reproduce that paper’s experiment, establish a measurement protocol or claim that its vectorial modes have the profiles used here. The research connection is the importance of specifying the complex modal field, rather than replacing it with an incomplete magnitude summary.

Read the primary modal-decomposition paper with its publication date and experimental scope attached. Source route checked on 24 October 2026. Historical statements about what was newly achieved in that work should not be repeated as claims about the current state of every waveguide measurement technique.

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Consider the statement: the left detector dropped from about 1.85 to about 0.15, so almost all the guided signal was lost. Under our original fixed-weight model, those two values can arise from relative phases zero and π while the whole norm remains exactly two.

A repaired statement first identifies the partial observation. It then says that the data are consistent with a substantial redistribution of the squared field within the specified coherent model. Additional information would be needed to distinguish redistribution from attenuation or a changed detector response in an actual system.

The correction should preserve the measured fact while limiting the explanation. It should not replace one overclaim with the opposite claim that no loss occurred in a real device. Our counterexample establishes that the inference from one partial reading to almost complete loss is invalid without further assumptions. A strong learner can explain the missing assumption and propose which additional observable would address it.

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Keep the original u1 and u2 functions and the left-half aperture, but change the coefficient magnitudes to A = two and B = one. Determine the complete norm, the phase-independent part of the left reading and the coefficient multiplying cosφ.

Find the minimum and maximum left readings across all relative phases. Check whether their sum has a useful relation to the complete norm. Then find the right-half expression and verify that the two halves add to the same total at every phase.

Do not simply double the earlier equal-weight readings. The separate squared terms scale as A² and B², while the cross term scales as AB. Those different scalings are the purpose of the changed-context exercise. Explain why increasing a field coefficient by a factor of two changes its individual squared contribution by a factor of four, then check the combined expression before evaluating any decimal approximation.

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The complete norm is 2² + 1² = five. The left reading is 5/2 + [16/(3π)]cosφ. Its phase range is 2.5 minus or plus approximately 1.69765, giving limits about 0.80235 and 4.19765.

The right reading is 5/2 − [16/(3π)]cosφ. Adding left and right cancels the cross terms and gives five. The minimum and maximum possible left readings also sum to five because they are symmetric about the phase-independent part 2.5; that identity concerns this specific fixed-weight phase sweep.

As another check, apply the weighted-overlap positivity bound to the half interval. Its diagonal terms are both one half and its overlap is 4/(3π), smaller than one half. The computed minimum is therefore nonnegative, as required by its integral definition. A transfer solution should preserve the full account, phase dependence and positivity, rather than accept plausible decimal values without explaining the structure that constrains them.

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Now replace u2 with u3 = √2 sin(3πx), while retaining the same normalisation and a left-half aperture. The overlap of u1 and u3 over zero through one half is zero. Their individual squared integrals over that region are both one half.

For this pair, the left-half reading is therefore (A² + B²)/2 for every relative phase. The earlier strong phase dependence does not survive the changed functions. The result follows from a different overlap integral, not from a failure of coherent superposition.

A detector can thus be sensitive to one modal phase relationship and insensitive to another. This is a useful reason to identify the basis before interpreting a reading. A theorem about a complete orthogonal set and a calculation for one selected aperture do different jobs. Generalise the method—define the functions, response and overlaps—rather than generalising one numerical cross-term coefficient to every pair of guided patterns.

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For repair, distinguish a local squared field from its integral and verify the mode normalisation. For stabilisation, derive the half-aperture expression and check its complement. For extension, use a different aperture, a third mode or a specified weighted response and test which parameters the observations identify.

Return to the Waveguides root for geometry, modes and the difference between a model and a complete link. Read the optical-interferometry lesson for coherent phase, fringe visibility and path interpretation. That guide owns the interferometric measurement question; this article isolates restricted transverse collection of known scalar mode profiles.

For uncertainty in comparable power measurements, the cascaded optical-loss lesson develops calibration and covariance. It answers a different question from the scalar collection integral here and does not turn that integral into a calibrated physical power measurement.

For the propagation-timing question, the near-cutoff delay lesson bounds a specified mode’s group delay and checks its allowed input domain. A scalar collection integral does not determine that timing quantity.

Continue through Learning That Transfers into Life and How X Works. A useful final explanation identifies the field basis, coefficient information, phase description, response region and actual observable. The learner should be able to name which part of that account must change before a different detector reading becomes evidence of changed total propagation.

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Does orthogonality mean the modes never overlap locally?

No. It means a particular complete inner product vanishes. Both mode functions can be nonzero at the same location. Changing the integration region or response weight can leave a nonzero cross term.

Does a constant whole norm make every detector reading constant?

No. A restricted or weighted reading can vary as phase redistributes the squared field. The observation model must specify what the detector collects and how that relates to the quantity being claimed.

Does an unknown phase automatically average away?

No. A fixed unknown phase is an uncertainty about one state. Averaging the cross term away requires an appropriate phase distribution or measurement average. Ignorance alone does not supply that model.

Can the scalar norm be labelled real waveguide power?

Only with an additional justified connection to the actual electromagnetic field and measurement. This article deliberately defines a scalar squared-field observable. Vector fields, power flux, calibration and collection require their own account in a real system.

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