Helmholtz equation and Multislice approach

1. Sep 29, 2014

eoghan

Hi there!
I have a problem about the proof of an equation in microscopy. I think this is the right section because it is about solving the Helmholtz equation.

http://en.wikipedia.org/wiki/Multislice
where they try to solve the Schrödinger equation for an electron passing through a medium.
In the theory section they write
$$\phi(\vec r) = 1-i\frac{\pi}{E\lambda}\int\int_{z'=-\infty}^{z'=z}V(\vec X', z')\phi(\vec X', z')\frac{1}{i\lambda(z-z')}\exp\left(ik\frac{|\vec X-\vec X'|^2}{2(z-z')}\right)d\vec X' dz'$$

At the end, removing the convolution with the Fresnel propagator (i.e. discarding the Fresnel scattering) they say that in a multislice approach (which I think that mathematically is a sort of successive approximations method) where in every slice I consider V to be constant (the slices are on the z axis), the solution is
$$\phi(\vec X, z_{n+1})=\phi(\vec X, z_n)\exp\left(-i\sigma\int _{z_n}^{z_{n+1}}V(\vec X, z')\right)dz'$$

I don't understand how this solution can be derived from the first equation. I think is something like the Dyson series, but I cannot decouple the potential V from the function phi.