Transfer-matrix method in optics

Definition

Let layers $j = 1, \dots, N$ fill the slabs $z_{j-1} < z < z_j$, with $z_0 = 0$, thickness $d_j = z_j - z_{j-1}$ and real refractive index $n_j > 0$. Let a medium of index $n_0$ fill $z < 0$ and a medium of index $n_s$ fill $z > z_N$; all media are non-magnetic. A plane wave of vacuum wave number $k_0$, polarized along $y$ ($s$-polarization), arrives from $z < 0$ at angle $\theta_0$. Let $\kappa = k_0 n_0 \sin \theta_0$, and let $q_j = \sqrt{k_0^2 n_j^2 - \kappa^2}$, assumed real and positive, with $q_0$ and $q_s$ defined in the same way. The field is $E_y = U(z) e^{i \kappa x}$, where $U'' + q_j^2 U = 0$ in layer $j$, and $U$ and $U'$ are continuous at each interface. The transfer matrix of layer $j$ is $$M_j = \begin{pmatrix} \cos(q_j d_j) & \sin(q_j d_j)/q_j \\ -q_j \sin(q_j d_j) & \cos(q_j d_j) \end{pmatrix},$$ which maps $(U, U')$ at $z_{j-1}$ to $(U, U')$ at $z_j$. The transfer-matrix method computes the matrix $M = M_N \cdots M_1$ and then the reflection and transmission coefficients $r$ and $t$ from the linear system $$\begin{pmatrix} t \\ i q_s t \end{pmatrix} = M \begin{pmatrix} 1 + r \\ i q_0 (1 - r) \end{pmatrix}.$$