Matrix Method

The optical properties of thin film samples under monochromatic illumination can be calculated using a transfer matrix approach [12, 18, 20]. An electromagnetic, monochromatic plane wave can be described by its amplitude A and the k-vector. Following Vignaud et al. [20], we use a notation of upwards (+) and downwards (−) travelling waves in z-direction:

\(u^{\pm}(z) =A_j^{\pm}\exp(\pm ik_{z,j}z),\)

which fully describe the state of the system in every layer j at the vertical position z. The wave in vacuum, outside the full stack of layers, is given by the following equation:

\(\left[ \begin{matrix} u^+(z_{vac}) \\ u^-(z_{vac}) \end{matrix} \right]=\mathbf{M}\cdot \left[\begin{matrix} u^+(z_{substrate}) \\ u^-(z_{substrate}) \end{matrix}\right]\)

which contains the 2 × 2 transfer matrix M, defined as a product of reflection (R_j/k ) and transmission (T_j ) matrices [20]. For a thin film of a specific material (mat.) on a substrate, including a contamination layer (cont.), an oxide layer (ox.), and an interdiffusion layer (diff.), this transfer matrix reads:

\( \mathbf{M} = \mathbf{R}_{vac. / cont.} \cdot \mathbf{T}_{cont.} \cdot \mathbf{R}_{cont. / ox.} \cdot \mathbf{T}_{ox.} \cdot \mathbf{R}_{ox. / mat.} \cdot \mathbf{T}_{mat.} \cdot \mathbf{R}_{mat. / diff.} \cdot \mathbf{T}_{diff.} \cdot \mathbf{R}_{diff. / subs.}\)

Further layers can be incorporated by additional transmission and reflection terms. A transmission matrix T has diagonal form, accounting for absorption and accumulated spectral phase inside the material:

\(\mathbf{T}_j = \left[ \begin{array} 0\exp(ik_{z,j}h_j ) & 0 \\ 0 & \exp(-ik_{z,j}h_j )\end{array}\right]\)

with h_j being the thickness of the respective layer. The reflection matrix is calculated from the Fresnel
coefficients of the individual interfaces:

\(\mathbf{R}_{j/k} = \left[ \begin{array} 0 p_{j/j-1} & m_{j/j-1}\\ m_{j/j-1} & p_{j/j-1}\end{array} \right]\)

with the following coefficients:

\(p_{j/j-1} = \frac{k_{z,j} + k_{z,j-1}}{2k_{z,j}}\cdot\exp ( -\frac{1}{2}(k_{z,j} + k_{z,j-1})^2)\cdot s_{j/j-1}^2 )\),

\(m_{j/j-1} = \frac{k_{z,j}- k_{z,j-1}}{2k_{z,j}} \cdot \exp(-\frac{1}{2}(k_{z,j} - k_{z,j-1})^2\cdot s_{j/j-1}^2 ).\)

Imperfections at the interfaces reduce the specular signal. This effect is included here by the exponential term, introducing the interfacial roughness parameter s_j/j−1 between material j and (j −1) [19]. The optical constants of the materials themselves are encoded in the z-component of the wave vectors inside the materials via the refractive index n_j of layer j: k_z,j = n_j ·k_z,vac. To this end, the reflection coefficient R is the ratio of the incoming and outgoing intensity in vacuum R = I_out/ I_in , calculated from the upwards and  downwards traveling plane wave solutions of equation 2. It is assumed that no upwards traveling wave in the substrate exists. The radiation is either completely absorberd in the substrate (which is the case in the UV and soft X-ray spectral ranges) or any back-reflected radiation in a transparent substrate is spatially separated from the interaction volume of the incident radiation.