User:Gengkw/Parratt recursive formalism

For a system with several interfaces (a film or a multilayer) the Fresnel-coefficients rj,j+1 for all interfaces are alculated separately. According to Parratt (Phys.Rev 95(1954), 359) a recursion formalism is developed for the ratio Xj of reflected and transmitted amplitude in the   medium between interfaces j and j+1

Xj= Rj/Tj = e(-2ikz,jzj) rj.j+1 + Xj+1e(2ikz,j+1zj) --- 1+ rj,j+1Xj+1e(2ikz,j+1zj)

with the Fresnel-coefficients

rj,j+1 = kzj - kz,j+1

kzj + kz,j+1

For a N-interface system on a semi-infinite substrate the recursion starts with RN+1 = 0.

All Xj can be calculated successively.

The Parratt formalism inherently takes into account multiple scattering, that is it is "dynamic".

Roughness can be taken into account by the factor

rj,j+1 =  rj,j+1 e(-2kz,jkz,j+1 σj2 ) rough    Ideal

as long as the roughness is much smaller than the layer thicknes