Ray transfer matrix analysis: Difference between revisions

Revision as of 22:49, 11 August 2024

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== Some examples ==

=== Free space example ===

*As Forone 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{matrix}

x_2 & = & x_1 + d\theta_1 \\

\theta_2 & = & \theta_1

\end{matrix} </math>

* 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 ''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'': <math display="block">\mathbf{L}\mathbf{S} = \begin{bmatrix} 1 & 0 \\ -\frac{1}{f} & 1\end{bmatrix}

=== 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>