Matrix optics

Matrix optics is a paradigm in geometric optics where we treat the behaviour of optical imaging systems as the multiplication of many matrices.

For concatenated systems, we multiply the matrices in reverse (opposite from the direction the rays are travelling). This makes sense intuitively because our rays first encounter the rightmost matrix.

$$\left[\begin{array}{c}{y}_{\text{out}}\\ {\theta }_{\text{out}}\end{array}\right]=\left[\begin{array}{cc}{A}_n& B_n\\ C_n& D_n\end{array}\right]\left[\begin{array}{cc}{A}_{n-1}& B_{n-1}\\ C_{n-1}& D_{n-1}\end{array}\right]\cdots \left[\begin{array}{cc}{A}_1& B_1\\ C_1& D_1\end{array}\right]\left[\begin{array}{c}{y}_{\text{in}}\\ {\theta }_{\text{in}}\end{array}\right]$$

Basics

Note what $y$ and $\theta$ are with respect to. In this case, $\theta$ is not with respect to the optical axis, but instead a different axis. This means that for a real image (by small-angle approximation):

$${\theta }_{\text{in}}=\frac{y_{\text{in}}}{s_o}$$

The determinant of a refraction across two different media gives us the ratio of their refractive indices:

$$\det M_i=\frac{n_{\text{in}}}{n_{\text{out}}}$$

In fact, the determinant of the final ray-transfer matrix of the entire system will give us the input to output ratio of the refractive indices.

Tips

A short list:

• For object at $s_o=\infty$, we approximate with parallel rays. Note two things:
  • The input vector has $y_{\text{in}}$ unknown, and ${\theta }_{\text{in}}=0$.
  • We omit the initial propagation matrix from $s_o$ in our computation of $y_{\text{out}}$ and ${\theta }_{\text{out}}$.

Common matrices