Ray Transfer Matrix Analysis - Some Examples

Some Examples

  • For example, if there is free space between the two planes, the ray transfer matrix is given by:
,

where d is the separation distance (measured along the optical axis) between the two reference planes. The ray transfer equation thus becomes:

,

and this relates the parameters of the two rays as:

\begin{matrix} x_2 & = & x_1 + d\theta_1 \\ \theta_2 & = & \theta_1 \end{matrix}
  • Another simple example is that of a thin lens. Its RTM is given by:
,

where 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 d followed by a lens of focal length f:

\mathbf{L}\mathbf{S} = \begin{pmatrix} 1 & 0 \\ \frac{-1}{f} & 1\end{pmatrix} \begin{pmatrix} 1 & d \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & d \\ \frac{-1}{f} & 1-\frac{d}{f} \end{pmatrix} .

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:

\mathbf{SL} = \begin{pmatrix} 1 & d \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ \frac{-1}{f} & 1\end{pmatrix} = \begin{pmatrix} 1-\frac{d}{f} & d \\ \frac{-1}{f} & 1 \end{pmatrix} .

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 indices, reflection from mirrors, etc.

Read more about this topic:  Ray Transfer Matrix Analysis

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