Dispersion curves of a slab waveguide with a nonlinear covering medium and an exponential graded-index thin film (transverse magnetic case)
Bibliographic record
Abstract
A slab waveguide that comprises a linear substrate, an exponentially graded-index guiding layer, and a power-dependent refractive index covering medium is investigated. All layers are assumed dielectric. The guiding film index is described by a function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>n</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> , which changes exponentially from the cover-film interface to the substrate. The solution of the Helmholtz equation in the guiding layer is found as Bessel’s functions of the first and second kinds, the order of which is <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>V</mml:mi> <mml:msqrt> <mml:mi>b</mml:mi> </mml:msqrt> </mml:math> , where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>V</mml:mi> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>b</mml:mi> </mml:math> are two asymmetry parameters called normalized film thickness and normalized guide index, respectively. The dispersion relation is obtained, and the dispersion properties are investigated. We found that there is no cutoff thickness for small asymmetry coefficient ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>a</mml:mi> </mml:math> ) values due to the nonlinear covering medium. The properties of the waveguide structure are determined by the graded-index function, and the effect of the cover layer nonlinearity is highly reduced and contributes almost nothing to the dispersion properties for large values of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>a</mml:mi> </mml:math> . The <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>b</mml:mi> </mml:math> -values do not exceed unity, which means that the modes carried by the waveguide structure are guided modes.
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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 | 0.000 |
| 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".