Worked Uncertainty Budget: In-Vacuum Interferometric Wavefront Error
Every term named, combined and expanded, for a wavefront measurement made at temperature under vacuum. A worked example against build-out targets.
Reference budget for the measurement chain in build-out. The terms are the terms this measurement has, evaluated and combined the way they have to be; the magnitudes are representative rather than measured. The structure is the deliverable, and it is published so the standard can be argued with before it is used to accept hardware.
A complete uncertainty budget for one measurement: the RMS wavefront error of an optical article held at temperature under vacuum and measured through a chamber window at 632.8 nm. Nine terms are named, including the residual that cannot be separated and that most published budgets quietly omit. Each carries its type, its assumed distribution, and the basis a reader needs in order to argue with it. The terms are combined by root-sum-square, expanded at a coverage factor of two, and set against the published acceptance criterion to establish what the measurement can decide. No test house in this market publishes this document. It is published here because the measurement chain is the product.
Document record
- Document number
- CUB-2026-01
- Version
- 1.0
- Published
- 5 August 2026
- Last revised
- 5 August 2026 (first publication)
- Authors
- Constanellis Aerospace
- DOI
- Not yet registered. Cite the document number, which is stable and is the handle this company guarantees. Deposit is planned so that each document carries a version DOI pinning the exact revision cited, alongside a concept DOI that always resolves to the newest one.
- Licence
- CC BY 4.0Copy, redistribute, adapt, and build on this document for any purpose, including commercially, provided you credit the source and say whether you changed anything.
- Review
- Internal review only. Read and checked by Constanellis engineering staff against the data module it rests on. Not peer reviewed, and not presented as peer reviewed.
- Named authors
- Authored by the Constanellis engineering staff. Individual contributors are named on request to reviewers and to press under embargo. When individual authorship is cleared for publication, each named author publishes with an ORCID iD so the attribution resolves to a person rather than to a string.
Constanellis Aerospace. "Worked Uncertainty Budget: In-Vacuum Interferometric Wavefront Error." Constanellis Aerospace CUB-2026-01, version 1.0. 5 August 2026. https://constanellis.com/methods/cub-2026-01
@techreport{cub202601,
author = {{Constanellis Aerospace}},
title = {Worked Uncertainty Budget: In-Vacuum Interferometric Wavefront Error},
institution = {Constanellis Aerospace, Inc.},
number = {CUB-2026-01},
type = {Technical Report},
version = {1.0},
year = {2026},
month = {August},
address = {Washington, DC, USA},
url = {https://constanellis.com/methods/cub-2026-01},
note = {Licensed CC BY 4.0}
}- 1.05 August 2026First publication.
No corrections issued. Corrections policy and log
What is being measured, stated before anything is measured.
The RMS wavefront error of an optical article, in double pass, at a wavelength of 632.8 nm, with the article held at a specified temperature under vacuum and viewed through the chamber window. The measurement is made in place. A measurement taken after the chamber returns to ambient describes the recovery, not the condition anyone asked about.
Every term below is expressed in nanometres of wavefront RMS at that wavelength, so every sensitivity coefficient is unity. Naming that rather than leaving it implicit is the point: carrying the thermal term back to its temperature source introduces a real sensitivity coefficient, and that extension is planned.
Nine terms, including the one nobody names.
| Source | Type | Estimate (nm) | Distribution | Divisor | u(x) (nm) | u² |
|---|---|---|---|---|---|---|
| Reference surface calibration | B | 6.00 | normal (k=2) | 2.000 | 3.00 | 9.000 |
| Instrument repeatability | A | 1.20 | normal | 1.000 | 1.20 | 1.440 |
| Chamber window contribution | B | 7.00 | rectangular | 1.732 | 4.04 | 16.333 |
| Fixture-induced mounting distortion | B | 4.00 | rectangular | 1.732 | 2.31 | 5.333 |
| Thermal gradient at the article | B | 3.50 | rectangular | 1.732 | 2.02 | 4.083 |
| Alignment and retrace | B | 2.60 | rectangular | 1.732 | 1.50 | 2.253 |
| Aperture definition in reduction | B | 1.70 | rectangular | 1.732 | 0.98 | 0.963 |
| Operator, setup to setup | A | 1.50 | normal | 1.000 | 1.50 | 2.250 |
| Unseparated residual | B | 3.00 | rectangular | 1.732 | 1.73 | 3.000 |
| Combined standard uncertainty, uc = √Σu² | 6.68 | 44.657 | ||||
From a standard uncertainty to a number a customer can use.
An uncertainty consumes tolerance, and this is how much.
The published build-out acceptance criterion is λ/20 RMS at 632.8 nm, which is 31.64 nm. The decision rule accepts an article when the measured value plus the expanded uncertainty falls inside the criterion. So the largest measured value this measurement can still accept is 31.64 − 13.4 = 18.27 nm (λ/35).
Read that consequence carefully, because it is the whole reason to publish a budget. An article measuring λ/25, comfortably better than a λ/20 requirement on paper, sits inside the guard band and cannot be accepted under this rule. The measurement is not good enough to defend it. An article measuring λ/40 can.
That is the result, and it is why the budget is worth publishing. A supplier who publishes a criterion without its uncertainty has not told a customer whether their part can be accepted; they have given them a number and left the risk on the customer's side of the table. Everything in the measurement chain, the window, the fixture, the calibration, exists to make that guard band smaller, and the budget is what says which term to attack first. Here it is the window, by a wide margin, and that is where the engineering goes.
What this document does not establish.
- L01
Magnitudes are representative of the chain in build-out. The structure, the term set, the distributions, and the combination are the deliverable; the numbers acquire their final values at commissioning and this document takes a new revision when they do.
- L02
The terms are treated as uncorrelated. Window deformation and thermal gradient both track the same thermal cycle, so that is the assumption most worth attacking. It gets measured at commissioning rather than assumed forward.
- L03
A coverage factor of two gives approximately 95 percent coverage for a roughly normal output distribution with adequate degrees of freedom. Effective degrees of freedom are not evaluated here, so the coverage is stated as approximate. That calculation is the next revision.
- L04
The budget covers the measurement, not the article. It establishes how well a wavefront can be known under these conditions and says nothing about whether any particular article meets any particular requirement.
- L05
Sensitivity coefficients are unity because every input is already expressed in nanometres of wavefront RMS at the measurement wavelength. Carrying the thermal term back to its temperature source would introduce a real one, and that is a planned extension.