## Multislice Approximation

For thick samples the sample can be divided into n slices of thickness q0. At the entrance of each slice (x, y), a spherical wave with amplitude y) and wave vector k is created. The amplitude at the exit of the slice (x', y' ) is a superposition of all spherical waves:

Hx>, y) = f exp(iaVp(x, y)) ^M2^*', x, y, y)) dxdy r (8)

In the case of |x — x'|, |y — y'| ^ q0 the Fresnel approximation can be used:

Therefore one obtains exp(2wikq0 )

2q02 2q02

This is a convolution defined as f(x)* g(x) = f f(x' )g(x - x )dxx

The electron wave function at the entrance of the first slice is y, 0), and, at the exit, which serves as the entrance of the second slice, it is ft(x, y, q0). Therefore we obtain

V qo

where Vpt(x,y) is the projected potential of the nth slice. The propagation of the electron can therefore be calculated successively. With the Fourier theorem of the convolution the convolution product can be replaced by a normal product in the Fourier space. £ is the symbol for the Fourier transform. Therefore one obtains

£(x, y, nq0) = £((x, y, (n — 1)qo)) exp(iVPt(x, y))

0 0