Analytical solutions for beams passing apertures with sharp boundaries

Abstract

An approximation is elaborated for the paraxial propagation of diffracted beams, with both one-and twodimensional cross sections, which are released from apertures with sharp boundaries. The approximation applies to any beam under the condition that the thickness of its edges is much smaller than any other length scale in the beam's initial profile. The approximation can be easily generalized for any beam whose initial profile has several sharp features. Therefore, this method can be used as a tool to investigate the diffraction of beams on complex obstacles. The analytical results are compared to numerical solutions and experimental findings, which demonstrates high accuracy of the approximation. For an initially uniform field confined by sharp boundaries, this solution becomes exact for any propagation distance and any sharpness of the edges. Thus, it can be used as an efficient tool to represent the beams, produced by series of slits with a complex structure, by a simple but exact analytical solution.

Analytical boundary-based method for diffraction calculations

“…(36) of Ref. [15], the boundaries were smooth with a nonzero transition width). As can be seen in the figure, the agreement between the two methods is perfect.…”

“…Accordingly, the paraxial wave equation is most commonly used to model in slowly expanding beams [24,25]. It should be stressed that diffraction due to spatial confinement of sharp (hard-edged) boundaries does not violate the paraxial assumption [22,15,26,14,27].…”

“…In the present work, we develop an essential generalization of Ref. [15]. The approach is based on the possibility that, similar to the one-dimensional (1D) case, where the diffraction effect, in the near-field area, is mainly determined by the field's value at the boundaries, in the 2D case the diffracted field should be mainly determined by values of the field at the aperture's boundaries, in accordance with BDWT.…”

“…The solid black curve represents the exact numerical calculation produced by Eq. (4), and the dashed red curve corresponds to Eq (15)…”

“…As inFig. 2, the dashed red and solid black curves, in the lower-right panel, represent solutions given by Eqs (15). and(4)at ( ) ,0, 70mm xz = , respectively (i.e., along the cross-section marked by the dashed black curve in the upper right panel).…”