With the MLFMM formulation for inverse equivalent current problems, it became obvious that the representation with propagating plane waves on the Ewald sphere, opens up completely new possibilities for performing near-field far-field (NFFF) transformations.

The central part of this section are, however, propagating plane wave expansions on the Ewald sphere. In Section 4, multilevel hierarchical field decompositions are reviewed and discussed, which are the basis of FMM and MLFMM as well as of FIAFTA.

Propagating Plane Wave Expansion on the Ewald Sphere

An observation, which can be concluded from the existence of different exact translation operators, is that many incident wave spectra composed of propagating plane waves on the Ewald sphere can produce the correct incident field in the observation volume.

In the spherical mode expansion and in the Ewald sphere based plane wave expansion, the minimum spheres around the sources and around the considered weighting functions at the observation locations must not overlap.

Hierarchical multilevel schemes utilizing Ewald sphere based plane wave spectral representations have become famous in the so-called Multilevel Fast Multipole Method (MLFMM) or Multilevel Fast Multipole Algorithm (MLFMA) for the solution of integral equations [9].

In the case of 3D systems non-zero odd Fourier components can be also present when scattering to high angles is considered, due to Ewald sphere curvature effects.

From the performed analysis we can see that, due to the curvature of the Ewald sphere (non-zero [q.sup.z] component), we obtain non-zero odd Fourier components of the CCF when scattering from a 3D system.

Calculations were performed for a regular pentamer for the case of a flat (a) and curved (b) Ewald sphere. Calculations for the case of a flat Ewald sphere (such conditions can be achieved, for example, using high photon energy) were performed by setting the z-component of the momentum transfer vector to [q.sup.z] = 0 in (33).

In Figures 7(a)-7(d) the amplitudes [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] calculated as a function of [q.sub.2] = 1.71 [[Angstrom].sup.-1]) are presented for a flat (a), (c) and curved (b), (d) Ewald sphere. The amplitudes [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] shown in Figures 7(a) and 7(b) were calculated in dilute limit from N=10clusters, using (33) in (5), and were averaged (see (10a)) over M = [10.sup.4] realizations of the system.

Here scattering to high angles can be especially interesting due to the presence of a significant Ewald sphere curvature.