# Oblique incidence over a stratified medium - the transfer matrix is not unitary?

As you've read on the title, my problem is about an oblique incidence problem over a stratified electrical medium.

I've made a quick sketch with the incidence on the first layer. Let's suppose that I have a TE incidence, even if it is not important to my question. I tried to solve the problem using the transfer matrices. For the first interface between layer 1 and 2, for example, I have:

$$\left[\begin{matrix} E_1^+ \\ E_1^- \end{matrix}\right] = \underbrace{\frac{1}{t_{12}}\left[ \begin{matrix} 1 & r_{12} \\ r_{12} & 1 \end{matrix} \right]}_{M_{12}} \left[\begin{matrix} E_2^+ \\ E_2^- \end{matrix}\right]$$

with

$$r_{12}, t_{12} \in \mathbb{C} \quad \mathrm{and} \quad M_{12}\in \mathbb{C}^{2 \times 2}$$

The coefficients $$\t_{12} \$$ and $$\r_{12}\$$ are computed using the Fresnel equations.

Obviously before I can add a second interface matrix, I should add a propagation matrix:

$$P_2 = \left[ \begin{matrix} e^{-j \beta_2 \ell_2} & 0 \\ 0 & e^{j \beta_2 \ell_2} \end{matrix} \right]$$

Where $$\\beta_2 \$$is the propagation constant in layer 2 in the axis orthogonal to the interfaces and $$\l_2\$$ is the thickness of layer 2. For N layers I have:

$$\left[\begin{matrix} E_1^+ \\ E_1^- \end{matrix}\right] = \underbrace{M_{12} P_2 M_{23} P_3 \cdots M_{N-1, N}}_{M_\mathrm{tot}} \left[\begin{matrix} E_N^+ \\ E_N^- \end{matrix}\right]$$

I numerically computed:

$$M_\mathrm{tot} = \left[ \begin{matrix} m_{11} & m_{12} \\ m_{21} & m_{22} \end{matrix} \right] = \left[ \begin{matrix} 1/t & r/t \\ r/t & 1/r \end{matrix} \right] \qquad \mathrm{and \ I \ plotted \ how} \ |r|^2= \left|\frac{m_{21}}{m_{11}}\right|^2 \ \mathrm{ varies \ with} \ \theta$$

Where $$\|r|^2\$$ is the reflection coefficient.

This gave me nonsense results: I have for some angles a reflection coefficient greater than one.

I think that the main problem is that the propagation matrices are not unitary.

If I compute the determinant for $$\M_{12}\$$, I get:

$$\det(M_{12}) = \frac{1}{t_{12}^2}(1-r_{12}^2),$$

Using the definition $$\t = 1+r\$$, then: $$\det(M_{12}) = \frac{1-r_{12}^2}{(1+r_{12})^2} = \frac{1-r_{12}}{1+r_{12}} = \frac{1-(t_{12}-1)}{t_{12}} = \frac{2}{t_{12}}-1.$$

This means that the determinant of these matrices can be also greater than one with certain angles.

I am doing something wrong, but I can't see it.

I've also thought to use the scattering coefficient S21 instead of transmissivity t, but the transfer matrix requires the use of Fresnel coefficients.

• This is probably better suited to physics.se – user16324 Jan 2 at 12:04
• Brian why? I am sure we can answer it here. – The Force Awakens Jan 2 at 12:13
• Thank you, I will ask to physics too, I've just post here before because I'm an engineer too. However I'm sure some electrical/electronics/telecommunication engineers here are skilled on this topic. – gab____ Jan 2 at 12:29
• People here are skilled in all sorts of things that aren't valid questions. – Finbarr Jan 2 at 14:37
• @TheForceAwakens then go ahead! – user16324 Jan 2 at 21:18