Computationally efficient scalar nonparaxial modeling of optical wave propagation in the far-field

Abstract

We present a scalar model to overcome the computation time and sampling interval limitations of the traditional Rayleigh-Sommerfeld (RS) formula and angular spectrum method in computing wide-angle diffraction in the far-field. Numerical and experimental results show that our proposed method based on an accurate nonparaxial diffraction step onto a hemisphere and a projection onto a plane accurately predicts the observed nonparaxial far-field diffraction pattern, while its calculation time is much lower than the more rigorous RS integral. The results enable a fast and efficient way to compute far-field nonparaxial diffraction when the conventional Fraunhofer pattern fails to predict correctly.

Iterative scalar nonparaxial algorithm for the design of Fourier phase elements

“…It has been reported [6,7] that scalar nonparaxial theory can be used, within the limit of the thin element approximation, for the design of some Fresnel DOEs having subwavelength features. In this work, we design 2D Fourier DOEs based on our efficient scalar nonparaxial propagator, which has been shown to be valid for far-field diffraction by structures having features on the order of the wavelength [8]. Figure 1(a) illustrates our method, where the diffracted field on the far-field output plane can be calculated accurately by using the Harvey propagator from the DOE plane to the tangential hemisphere [9], followed by a spherical wave projection from the hemisphere to the output plane [8]:…”

“…2(a), and the DOE was designed at δ 1 400 nm and λ 633 nm, where the maximal diffraction angle is about 29°at the corners of the grid. Our scalar nonparaxial propagator, which is equivalent to but much faster than the Rayleigh-Sommerfeld diffraction integral [8], was then used to calculate the reconstruction, where two distortions are visible. The first distortion is in the diffraction position, which is similar to the pincushion distortion in lens systems [13,14].…”

“…The other distortion is in the intensity distribution, where the power decreases from the center toward the grid corners or equivalently with increasing diffraction angle. These distortions are due to the approximations in calculating the diffraction position, diffraction amplitude, and diffraction power in the scalar paraxial theory [8].…”

“…1(c) describes the iterative projection algorithm, where the diffracted field is projected back and forth between the hemisphere and the output plane. In both algorithms, an iterative Fourier transform between the DOE plane and the hemisphere is used, and the spot intensity is corrected according to the diffraction angle (P ∝ γjUj 2 ) [8]. Other than this, the same usual IFTA constraints are used in the DOE plane for quantization and in the image plane for optimizing the diffraction efficiency and uniformity of the diffraction pattern.…”

“…The validity of our design algorithms therefore primarily depends on the thin element approximation, which is used to relate the DOE phase profile and the optical field at the DOE plane [3]. However, the accuracy of the approximation does not yet appear to be fully understood in the literature: some publications indicated that DOE feature sizes should be at least 10 times the illuminating wavelength [15,16]; recent work showed that this approximation can be used for some DOEs having features on the order of or even smaller than the illuminating wavelength [6][7][8]. Figure 3(a) shows the projected pattern in angular coordinates on the hemisphere, where a nearest neighbor interpolation [17] is used for mapping the field from spatial to angular coordinates.…”