Spectral separation of the efficiencies of the inside and outside orders of soft-x-ray-extreme-ultraviolet gratings at near normal incidence
Bibliographic record
Abstract
It is shown from both a phenomenological study and exact modeling that the reason for the experimentally observed substantial (a few angstroms or nanometers) separation in wavelength between the maxima of the inside (negative numbered) and outside (positive numbered) diffraction orders of a multilayer-coated grating, operating at near normal incidence and close to the Bragg condition in the soft-x-ray and extreme-ultraviolet (EUV) regions, is related to the different angles of deviation of the orders. This wavelength separation is also a feature of uncoated diffraction gratings, although not clearly noticeable. The widely used approximate approach for calculating the absolute efficiency, the product of the relative grating efficiency and the reflectance of its multilayer coating, has until recently been considered accurate enough for the analysis of soft-x-ray and EUV near-normal-incidence multilayer-coated gratings. The inapplicability of this approximation for the analysis of the precise positions and shapes of the efficiency curves for the inside and outside orders, despite the small ratios of wavelength and groove depth to period and the small angles of incidence, is demonstrated using gratings with realistic groove profiles and operating in the EUV region. The rigorous modified integral method (MIM), which is a variant of boundary integral equation methods and is designed for the calculation of the efficiency of multilayer gratings with arbitrary layer thicknesses and boundary shapes (including microroughness) and over a wide wavelength range, is proposed in a general operator formalism. An analysis of a derived simple phenomenological expression and the exact numerical study indicates that the spectral separation between the inside and the outside orders grows with increasing either wavelength, angle of incidence, groove frequency, or diffraction order number ∣m∣. The efficiency modeling carried out with the commercial program PCGRATE-SX, based on the MIM, gave not only the exact values of the spectral separation between the inside and outside orders of Mo4Ru6∕Be, Mo∕Si, and Mo∕Y multilayer-coated gratings with various real groove profiles measured using atomic force microscopy (AFM) but also good agreement with synchrotron radiation measurements, including high orders as well. To determine the shapes and positions of efficiency curves in the soft-x-ray-EUV range of close to normal-incidence bulk and multilayer-coated gratings with real groove profiles (measured by AFM), one should use codes based on rigorous electromagnetic theory such as the MIM. The modeling is important for developing high efficiency and dispersion gratings for high-resolution spectroscopic studies of laboratory, solar, and astrophysical radiation sources.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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 teacher head, 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".