Many layers

Since this translational operator is now in terms of parameters v and $\sigma$ which are constant across layer boundaries, we can form the n-layer transformation by forming the product of the n individual layer transformations. In the case of the first three terms of our truncated power series approximation of Equation 8, this leads to

$$\pmatrix{ I & 0 \cr 0 & I }$$

$$- i \omega z \pmatrix{ 0 & \sum_{j=1}^{n} s_{j} \cr \sum_{j=1}^{n} \rho_{j} & 0 }$$

 

$$- \frac{\omega^{2} z^{2}}{2} \pmatrix{ \sum_{j=1}^{n} s_{... ...j} s_{j} + 2\sum_{j=2}^{n} \rho_{j} \sum_{k=1}^{j-1} s_{k} }$$ (9)