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

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.