## Abstract

The angular spectrum (AS) method is a popular solution to the Helmholtz equation without the use of approximations. Modified band-limited AS methods are of particular interest for the cases of high-off-axis and large distance propagation problems, because conventional AS methods are impractical due to requirements regarding memory and computational effort. However, these techniques make use of rectangular-shaped filters that introduce ringing artifacts in the calculated field that are related to the Gibbs phenomenon. This work proposes AS algorithms based on a smooth band-limiting filter for accurate field computation as well as techniques that evaluate only nonzero components of the field. This enables accurate field calculations with an acceptable level of computational effort that cannot be offered by current AS methods reported in the scientific literature.

© 2013 Optical Society of America

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### Equations (6)

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(1)
$$A[z]=\mathrm{FFT}\left\{\left[\begin{array}{ccc}{\varsigma}_{l,t}& {\varsigma}_{l,N}& {\varsigma}_{l,t}\\ {\varsigma}_{M,t}& u[z]& {\varsigma}_{M,t}\\ {\varsigma}_{l,t}& {\varsigma}_{l,N}& {\varsigma}_{l,t}\end{array}\right]\right\},$$
(2)
$${f}_{x}=\frac{1}{\lambda}\frac{{x}_{d}-{x}_{s}}{\sqrt{{({z}_{0})}^{2}+{({x}_{d}-{x}_{s})}^{2}}},$$
(3)
$$A[z]=A[(p+\frac{\alpha}{P})\mathrm{\Delta}{f}_{x},(q+\frac{\beta}{Q})\mathrm{\Delta}{f}_{y},z]\phantom{\rule{0ex}{0ex}}=\sum _{\alpha =0}^{P-1}\sum _{\beta =0}^{Q-1}{H}_{a,\beta}[p,q,z],$$
(4)
$$u[{z}_{0}]=\sum _{\alpha =0}^{P-1}\sum _{\beta =0}^{Q-1}\widehat{u}[{z}_{0}]\mathrm{exp}\left[j2\pi (\frac{m\alpha}{M}+\frac{q\beta}{N})\right].$$
(5)
$$\widehat{u}[z]=\sum _{p=p1}^{p2}\sum _{p=p1}^{p2}{H}_{a,\beta}[p,q,{z}_{0}]{e}^{j2\pi (p\mathrm{\Delta}{f}_{x}m\mathrm{\Delta}x+q\mathrm{\Delta}{f}_{y}n\mathrm{\Delta}y)={e}^{j2\pi (\frac{{p}_{1}m}{M}+\frac{{q}_{1}{n}_{y}}{N})}}\times \sum _{p=0}^{p2-p1}\sum _{q=0}^{q2-q1}{H}_{a,\beta}[p+{p}_{1},q+{q}_{1},{z}_{0}]{e}^{j2\pi (\frac{pm}{M}+\frac{qn}{N})},$$
(6)
$$\underset{\text{size}=({K}_{x})\times ({K}_{y})}{\underbrace{\widehat{u}[{m}^{\prime},{n}^{\prime},{z}_{0}]}}=\frac{\mathrm{\Phi}[{m}^{\prime},{n}^{\prime}]}{{R}_{x}{R}_{y}}\underset{\text{size}=({K}_{x})\times ({K}_{y})}{\underbrace{\mathrm{IFFT}\{A[{z}_{0}]\}}},$$