Editing Ray transfer matrix analysis (section)

== Some examples ==
=== Free space example ===
As one example, if there is free space between the two planes, the ray transfer matrix is given by: <math display="block"> \mathbf{S} = \begin{bmatrix} 1 & d \\ 0 & 1 \end{bmatrix} , </math> where {{mvar|d}} is the separation distance (measured along the optical axis) between the two reference planes. The ray transfer equation thus becomes: <math display="block"> \begin{bmatrix} x_2 \\ \theta_2 \end{bmatrix} = \mathbf{S} \begin{bmatrix} x_1 \\ \theta_1\end{bmatrix} , </math> and this relates the parameters of the two rays as: <math display="block"> \begin{aligned}
x_2 &= x_1 + d\theta_1 \\
\theta_2 &= \hphantom{x_1 + d}\theta_1
\end{aligned} </math>

=== Thin lens example ===
Another simple example is that of a [[thin lens]]. Its RTM is given by: <math display="block"> \mathbf{L} = \begin{bmatrix} 1 & 0 \\ -\frac{1}{f} & 1 \end{bmatrix} , </math> where {{mvar|f}} is the [[focal length]] of the lens. To describe combinations of optical components, ray transfer matrices may be multiplied together to obtain an overall RTM for the compound optical system. For the example of free space of length {{mvar|d}} followed by a lens of focal length {{mvar|f}}: <math display="block">\mathbf{L}\mathbf{S} = \begin{bmatrix} 1 & 0 \\ -\frac{1}{f} & 1\end{bmatrix}
\begin{bmatrix} 1 & d \\ 0 & 1 \end{bmatrix}
= \begin{bmatrix} 1 & d \\ -\frac{1}{f} & 1-\frac{d}{f} \end{bmatrix} . </math>

Note that, since the multiplication of matrices is non-[[commutative]], this is not the same RTM as that for a lens followed by free space:
<math display="block"> \mathbf{SL} =
\begin{bmatrix} 1 & d \\ 0 & 1 \end{bmatrix}
\begin{bmatrix} 1 & 0 \\ -\frac{1}{f} & 1 \end{bmatrix}
= \begin{bmatrix} 1-\frac{d}{f} & d \\ -\frac{1}{f} & 1 \end{bmatrix} . </math>

Thus the matrices must be ordered appropriately, with the last matrix premultiplying the second last, and so on until the first matrix is premultiplied by the second. Other matrices can be constructed to represent interfaces with media of different [[refractive index|refractive indices]], reflection from [[mirror]]s, etc.