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Far-field transform

The far-field transform, or the Fourier coefficient with respect to spherical harmonics, of a function $ f(\theta,\phi )$ is defined as
$\displaystyle \widetilde {f}_{n,m} = \int_{S^0} Y_{n,m}(\theta,\phi)
f(\theta,\phi)dS_{\hat{\mbox{$\mathbf k $}}}$     (11)

where $ Y_{n,m}$ is the normalised spherical harmonics. Also, the inverse transform of $ \widetilde {f}_{n,m}$ is given by
$\displaystyle f(\theta,\phi ) = \sum_{n=0}^{\infty} \sum_{m=-n}^{n}
Y_{n,m}^{*}(\theta,\phi ) \widetilde {f}_{n,m}$     (12)

where $ ^*$ indicates the complex conjugate.

The integral in Eq. (12) is evaluated numerically in the manner described in 3.2 as follows:

$\displaystyle \widetilde {f}_{n,m}={\int_{S^0} Y_{n,m}(\theta,\phi)
f(\theta,\phi)dS_{\hat{\mbox{$\mathbf k $}}}}$
  $\displaystyle \approx$ $\displaystyle \sum_{J=0}^{2p} \sum_{I=1}^{p+1} w^p_I \eta^{p}_J
c_n^m P_n^m(\cos\lambda^p_I) e^{im\phi_J} f(\lambda^p_I,\phi^p_J) $ (13)

where $ c_n^m$ is the coefficient of the normalised spherical harmonics. One can use FFT in the computation of the outer sum with respect to $ J$ in Eq. (13). Using one set $ \{f(\lambda^p_I,\phi^p_J)\vert I=1,\ldots,p+1,J=0,\ldots,2p\}$ on the unit sphere one obtains another set $ \{f(\lambda^{p'}_I,\phi^{p'}_J)\vert I=1,\ldots,p'+1,J=0,\ldots,2p'\}$ via
$\displaystyle \widetilde {f}_{n,m}= \sum_{J=0}^{2p} \sum_{I=1}^{p+1} w^p_I \eta^{p}_J
c_n^m P_n^m(\cos\lambda^p_I) e^{im\phi_J} f(\lambda^p_I,\phi^p_J)$     (14)
$\displaystyle f(\lambda^{p'}_I,\phi^{p'}_J)
= \sum_{n=0}^{p'} \sum_{m=-n}^{n}
Y_{n,m}^{*}(\lambda^{p'}_I,\phi^{p'}_J) \widetilde {f}_{n,m}$     (15)

Operations in Eqs. (15) and (16) are called `interpolation' (`anterpolation') if $ p < p'(p > p')$ . In this paper, we determine the numbers $ p$ and $ p'$ following Song et al.[8]


next up previous
Next: M2M and L2L translations Up: Translation of multipole and Previous: Translation of multipole and
2001-12-14