Implementing non-scalar diffraction in Fourier optics via the Braunbek method

Abstract

Fourier optics is a powerful and efficient tool for solving many diffraction problems, but relies on the assumption of scalar diffraction theory and ignores the three-dimensional structure and material properties of the diffracting element. Recent experiments of sub-scale starshade external occulters revealed that the inclusion of these physical properties is necessary to explain the observed diffraction at 10 −10 of the incident light intensity. Here, we present a methodology for implementing non-scalar diffraction while maintaining the efficiency and ease of standard Fourier optics techniques. Our methodology is based on that of Braunbek, in which the Kirchhoff boundary values are replaced with the exact field in a narrow seam surrounding the edge of the diffracting element. In this paper, we derive the diffraction equations used to implement non-scalar diffraction and outline the computational implementation used to solve those equations. We also provide experimental results that demonstrate our model can replicate the observational signatures of non-scalar diffraction in sub-scale starshades, in effect validating our model to better than 10 −10 in relative intensity. We believe this method to be an efficient tool for including additional physics to the models of coronagraphs and other optical systems in which a full electromagnetic solution is intractable.

Optical verification experiments of sub-scale starshades

“…In Sec. 3.1.1 we posited, and the models confirm, 36 that the presence of the thick screen induces a change in the electric field in a narrow (∼λ) boundary layer around the edge. For now, we restrict ourselves to the case where the presence of the screen induces a change in amplitude only, resulting in a ∼λ wide boundary layer around the starshade edge with zero transmission.…”

“…Our theory of the edge effect is supported by a newly developed optical model 36 and its comparison to experimental results is shown in Sec. 4.3.…”

“…In this work, we use the model of Ref. 36, which uses a two-dimensional Fresnel propagator (shown to be in agreement with the boundary methods 12 ) to include non-scalar diffraction. We provide a brief summary of the optical model in Appendix C and we show it in Sec.…”

“…The optical model summarized here is presented in full in Ref. 36. The scalar diffraction problem we are solving uses the standard Fresnel and Kirchhoff assumptions that allow us to express the diffraction propagation as a Fourier transform.…”

Special Section on Starshades: Overview and a Dialogue

“…Urania: There are a number of approximations we can employ to simplify the math into a manageable problem, for example, assuming that the starshade is large enough and far enough away that we can use scalar diffraction theory. We have developed a number of methods to solve the diffraction equation for the starshade, 2,3,58,59 most recently in this section with the work of Barnett.…”