Optical filter wavefront distortion: out-of-band to in-band predictions and the effect of the illumination source bandwidth
Bibliographic record
Abstract
The wavefront distortion (WFD) of a surface with an optical filter coating is ideally measured at the operating wavelength ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>λ</mml:mi> </mml:math> ) and angle of incidence ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>θ</mml:mi> </mml:mrow> </mml:math> ) of the filter. However, this is not always possible, requiring that the filter be measured at an out-of-band wavelength and angle (typically <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>λ</mml:mi> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>633</mml:mn> </mml:mrow> <mml:mspace width="thickmathspace"/> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">n</mml:mi> <mml:mi mathvariant="normal">m</mml:mi> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>θ</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:msup> <mml:mn>0</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>∘</mml:mo> </mml:mrow> </mml:msup> </mml:math> ). Since the transmitted wavefront error (TWE) and reflected wavefront error (RWE) can depend on the measurement wavelength and angle, an out-of-band measurement may not give an accurate characterization of the WFD. In this paper, we will show how to predict the wavefront error (WFE) of an optical filter at the in-band wavelength and angle from a WFE measurement at an out-of-band wavelength and different angle. This method uses (i) the theoretical phase properties of the optical coating, (ii) the measured filter thickness uniformity, and (iii) the substrate’s WFE dependence versus the angle of incidence. Reasonably good agreement was achieved between the RWE measured directly at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>λ</mml:mi> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1050</mml:mn> </mml:mrow> <mml:mspace width="thickmathspace"/> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">n</mml:mi> <mml:mi mathvariant="normal">m</mml:mi> </mml:mrow> </mml:math> ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>θ</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:mspace width="thickmathspace"/> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>45</mml:mn> </mml:mrow> <mml:mo>∘</mml:mo> </mml:msup> </mml:math> ) and the predicted RWE based on an RWE measurement at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>λ</mml:mi> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>660</mml:mn> </mml:mrow> <mml:mspace width="thickmathspace"/> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">n</mml:mi> <mml:mi mathvariant="normal">m</mml:mi> </mml:mrow> </mml:math> ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>θ</mml:mi> </mml:mrow> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>0</mml:mn> </mml:mrow> <mml:mo>∘</mml:mo> </mml:msup> </mml:math> ). It is also shown through a series of TWE measurements using a light emitting diode (LED) and laser light sources that, if the TWE of a narrow bandpass filter (e.g., an 11 nm bandwidth centered at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>λ</mml:mi> <mml:mo>=</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1050</mml:mn> </mml:mrow> <mml:mspace width="thickmathspace"/> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">n</mml:mi> <mml:mi mathvariant="normal">m</mml:mi> </mml:mrow> </mml:math> ) is measured with a broadband LED source, the WFD can be dominated by the chromatic aberration of the wavefront measuring system—hence, a light source that has a bandwidth narrower than the optical filter bandwidth should be used.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".