XEFI

A package for calculations of X-ray Electric Field Intensities (XEFI) using the Parratt recursive algorithm, and built to the feature-rich standards of xraysoftmat.

This package calculates discrete models of multi-layer structures, including the ability to slice simplistic models into arbitrary layers. Supports the use of the KKCalc package to calculate the index of refraction within layers.

The Model

To make this model representative of the code, we count $N+1$ layers from $i=0$ to $i=N$ inclusive, as python indexes.

Here, layers $i=0$ and $i=N$ are semi-infinite layers, typically modelling air/vacuum and a substrate respectively. Boundary conditions allow us to set the incident amplitude $T_0 = 1$, and the reflected amplitude $R_{N}=0$. We define the following quantities:

Variable	Description
$N$	The number of interfaces between the top and bottom layers, corresponding to $N+1$ layers
$i$	The layer number, indexed from 0 (i.e. 0 to $N$)
$z_i$	The depth of the $i^{th}$ interface ($z_i < 0$).
$d_i$	The thickness of the $i^{th}$ layer ($d_0 = d_N = ∞$)
$θ^t_i$	The transmitted angle of incidence in layer $i$. Same as the angle of reflection $θ^r_i$ in layer $i$.
$k_i$	The z-component of the wavevector in the $i^{th}$ layer.
$T_i$	The complex amplitude of the downward propogating electric field at interface $i$.
$R_i$	The complex amplitude of the upward propogating electric field at interface $i$.
$X_i$	The ratio of the downward and upward propogating electric field intensities at interface $i$.
$E^{Total}_i$	The total electric field in layer $i$.
$E_{beam}$	The X-ray beam energy in eV.

After recursively computing the ratio $X_i$, then solving the amplitudes $T_i$, $R_i$ at each interface, then the total electric field at depth $z$ in the film can then be calculated as the sum of downward and upward propogating waves:

$$E^{Total}_i(E_{beam}, θ^t_0, z) = T_i(E_{beam}, θ^t _0) exp(-i k_i (z-z_i)) + R_i (E_{beam}, θ^t_0) exp(i k_i (z-z_i))$$