## Abstract

The recent introduction of a fast Fourier transform– (FFT–)
based method for calculating the Rayleigh–Sommerfeld full diffraction
integral for tilted and offset planes permits high-speed evaluation of
integrated optical systems. An important part of introducing a new
calculational tool is its validation and an assessment of its
limitations. The validity of the new FFT-based method was
determined by comparison of that method with direct integration
(DI) of the Rayleigh–Sommerfeld integral, a well-established
method. Points of comparison were accuracy, computational speed,
memory requirements of the host computer, and applicability to various
optical modeling situations. The new FFT-based method is 228 times
faster, yet requires 14 times more memory, than the DI method for a 500
µm by 500 µm real computational window.

© 2001 Optical Society of America

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### Equations (10)

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(1)
$$s:=s+{a}_{\mathit{ik}}\times {b}_{\mathit{kj}},$$
(2)
$$\mathrm{NOF}\left(\text{new FFT}\right)=2\times {log}_{2}{N}_{\mathrm{FFT}},\mathrm{NOF}\left(\mathrm{DI}\right)=N_{\mathrm{DI}}{}^{2}.$$
(3)
$$\mathrm{NOF}\left(\text{new FFT}\right)=2\times \left(2\times {N}_{\mathrm{FFT}}\frac{{N}_{\mathrm{FFT}}\times {log}_{2}{N}_{\mathrm{FFT}}}{2}\right),\mathrm{NOF}\left(\mathrm{DI}\right)={{N}_{\mathrm{DI}}}^{4}.$$
(4)
$$\mathrm{cr}=\frac{N_{\mathrm{DI}}{}^{4}}{2\times (2\times {N}_{\mathrm{FFT}}\left[{N}_{\mathrm{FFT}}\times {log}_{2}{N}_{\mathrm{FFT}}/2\right]}=\frac{N_{\mathrm{DI}}{}^{4}}{2\times N_{\mathrm{FFT}}{}^{2}\times {log}_{2}{N}_{\mathrm{FFT}}}.$$
(5)
$$\mathrm{cr}=\frac{N_{\mathrm{DI}}{}^{2}}{32\times \left(2+{log}_{2}{N}_{\mathrm{DI}}\right)}.$$
(6)
$${\mathrm{MR}}_{\mathrm{FFT}}=16\times {N}_{\mathrm{FFT}}\times {N}_{\mathrm{FFT}}\mathrm{bytes}.$$
(7)
$${\mathrm{MR}}_{\mathrm{DI}}=8\times {N}_{\mathrm{DI}}\times {N}_{\mathrm{DI}}\mathrm{bytes}.$$
(8)
$${N}_{\mathrm{FFT}}=4\times {N}_{\mathrm{DI}}.$$
(9)
$${\mathrm{MR}}_{\mathrm{FFT}}=256\times {N}_{\mathrm{DI}}\times {N}_{\mathrm{DI}}\mathrm{bytes},{\mathrm{MR}}_{\mathrm{DI}}=8\times {N}_{\mathrm{DI}}\times {N}_{\mathrm{DI}}\mathrm{bytes},$$
(10)
$${\mathrm{MR}}_{\mathrm{FFT}}=32\times {\mathrm{MR}}_{\mathrm{DI}}.$$