An $O(N^{3})$ Algorithm for the Calculation of Far Field Radiation Patterns

We present a new method for computing the near-to-far-field (NTFF) transformation in finite-difference time-domain simulations which has an overall scaling of
$O(N^{3})$
instead of the standard
$O(N^{4})$
. By mapping the far field with a cartesian coordinate system, the 2-D surface integral can be split into two successive 1-D integrals. For a near field spanned by
$N_{X}\times N_{Y}$
discrete sample points, and a far field spanned by
$N_{\theta}\times N_{\phi}$
points, the NTFF calculation can then be performed in
$O(N_{\theta}N_{X}N_{Y})+O(N_{\theta}N_{\phi}N_{Y})$
operations instead of
$O(N_{\theta}N_{\phi}N_{X}N_{Y})$
.