RSim  Theory


Simulation of the Reflectivity $ R(\theta )$

A suitable theoretical method to describe the reflectivity and within the leaky modes of a thin layer or a multilayer system (see Figure below) is a matrix formalism after Yeh ["Optical Waves in layered media",John Wiley and Sons 1988]. The electromagnetic wave is decomposed into a left and a right propagating component.

$\displaystyle E(x) = A_i e^{j k_i^x x} + B_i e^{-j k_i^x x}$ (1)

The relation between the different Amplitudes $ A_i$ and $ B_i$ is given by a matrix multiplication over all $ N$ layers.

$\displaystyle \begin{pmatrix}A_0 \\ B_0 \end{pmatrix} = \frac{1}{t_{01}} \begin... ...{P_i} \right] \begin{pmatrix}A_{N+1}^{\prime} \\ B_{N+1}^{\prime} \end{pmatrix}$ (2)

Figure: Nomenclature of a multilayer system
\includegraphics [width=\textwidth]{figures/multilayersystem.eps}



From which the reflected intensity

$\displaystyle R=\left\vert\frac{B_0}{A_0}\right\vert^2$ (3)

can be calulated. The elements of the so called transfer Matrix $ T_{ii+1}$ are the fresnel coefficients for transmission and reflection at the interface between two different media. The optical phase difference $ \phi_i=2d_ik_i^x = 2dk_0\sqrt{\tilde{n}_i^2-n_0^2\sin(\theta)^2}$ is introduced by a phase matrix $ P_i$. The complex refractive index leads to a real exponentional factor in the elements of the phase matrix which influences strongly the intensity of the reflected light.
This matrix calculation is a very general tool and it can easily be implemented on a computer calculating a large number (practically continuous) of layers N, simulating static or dynamic refractive index profiles.