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Quantitative investigation of the validity conditions for the Beckmann-Kirchhoff scattering model

Hooshmand, Helia; Liu, Mingyu; Leach, Richard; Piano, Samanta

Quantitative investigation of the validity conditions for the Beckmann-Kirchhoff scattering model Thumbnail


Research Fellow in Optical Metrology

Mingyu Liu


Approximate and rigorous methods are widely used to model light scattering from a surface. The boundary element method (BEM) is a rigorous model that accounts for polarization and multiple scattering effects. BEM is suitable to model the scattered light from surfaces with complex geometries containing overhangs and re-entrant features. The Beckmann-Kirchhoff (BK) scattering model, which is an approximate model, can be used to predict the scattering behavior of slowly varying surfaces. Although the approximate BK model cannot be applied to complex surface geometries that give rise to multiple scattering effects, it has been used to model the scattered field due to its fast and simple implementation. While many of the approximate models are restricted to surface features with relatively small height variations (typically less than half the wavelength of the incident light), the BK model can predict light scattering from surfaces with large height variations, as long as the surfaces are locally flat with small curvatures. Thus far, attempts have been made to determine the validity conditions for the BK model. The primary validity condition is that the radius of curvature of any surface irregularity should be significantly greater than the wavelength of the light. However, to have the most accurate results for the BK model, quantifying the validity conditions is critical. This work aims to quantify the validity conditions of the BK model according to different surface specifications, e.g., slope angles (SA) and curvatures. For this purpose, the scattered fields from various sinusoidal and combinations of sinusoidal profiles are simulated using the BEM and the BK models and their differences are compared. The result shows that the BK model fails when there are high SA (≳ 38 deg) and small radii of curvature (≲ 10 λ) within a sinusoidal profile. Moreover, it is shown that for a combination of sinusoidal profiles the BK model is valid for profiles with a high maximum slope angle value (≳ 38 deg) if the average of positive SA is low (≲ 5 deg).


Hooshmand, H., Liu, M., Leach, R., & Piano, S. (2022). Quantitative investigation of the validity conditions for the Beckmann-Kirchhoff scattering model. Optical Engineering, 61(12), Article 124113.

Journal Article Type Article
Acceptance Date Dec 13, 2022
Online Publication Date Dec 31, 2022
Publication Date Dec 31, 2022
Deposit Date Dec 19, 2022
Publicly Available Date Dec 31, 2022
Journal Optical Engineering
Print ISSN 0091-3286
Electronic ISSN 1560-2303
Publisher SPIE-Intl Soc Optical Eng
Peer Reviewed Peer Reviewed
Volume 61
Issue 12
Article Number 124113
Keywords light scattering, boundary element method, Beckmann-Kirchhoff, validity conditions, slope and curvature
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