Editing Ray transfer matrix analysis (section)

== Matrix definition ==
[[File:RayTransferMatrixDefinitions.svg|thumb|300px|In ray transfer (ABCD) matrix analysis, an optical element (here, a thick lens) gives a transformation between {{math|(''x''{{sub|1}}, ''θ''{{sub|1}})}}  at the input plane and {{math|(''x''{{sub|2}}, ''θ''{{sub|2}})}} when the ray arrives at the output plane.]]

The ray tracing technique is based on two reference planes, called the ''input'' and ''output'' planes, each perpendicular to the optical axis of the system.  At any point along the optical train an optical axis is defined corresponding to a central ray; that central ray is propagated to define the optical axis further in the optical train which need not be in the same physical direction (such as when bent by a prism or mirror). The transverse directions {{mvar|x}} and {{mvar|y}} (below we only consider the {{mvar|x}} direction) are then defined to be orthogonal to the optical axes applying. A light ray enters a component crossing its input plane at a distance {{math|''x''{{sub|1}}}} from the optical axis, traveling in a direction that makes an angle {{math|''θ''{{sub|1}}}} with the optical axis. After propagation to the output plane that ray is found at a distance {{math|''x''{{sub|2}}}} from the optical axis and at an angle {{math|''θ''{{sub|2}}}} with respect to it. {{math|''n''{{sub|1}}}} and {{math|''n''{{sub|2}}}} are the [[index of refraction|indices of refraction]] of the media in the input and output plane, respectively.

The ABCD matrix representing a component or system relates the output ray to the input according to
<math display="block"> \begin{bmatrix}x_2 \\ \theta_2\end{bmatrix} = \begin{bmatrix} A & B \\ C & D \end{bmatrix} \begin{bmatrix}x_1 \\ \theta_1\end{bmatrix}, </math>
where the values of the 4 matrix elements are thus given by
<math display="block">A = \left.\frac{x_2}{x_1} \right|_{\theta_1 = 0} \qquad B = \left.\frac{x_2}{\theta_1} \right|_{x_1 = 0},</math>
and
<math display="block">C = \left.\frac{\theta_2}{ x_1 } \right|_{\theta_1 = 0} \qquad D = \left.\frac{\theta_2}{\theta_1 } \right|_{x_1 = 0}.</math>

This relates the ''ray vectors'' at the input and output planes by the ''ray transfer matrix'' ({{dfn|RTM}}) {{math|'''M'''}}, which represents the optical component or system present between the two reference planes.  A [[thermodynamics]] argument based on the [[blackbody]] radiation {{Citation needed|date=August 2023}} can be used to show that the [[determinant]] of a RTM is the ratio of the indices of refraction:
<math display="block">\det(\mathbf{M}) = AD - BC = \frac{n_1}{n_2}.  </math>

As a result, if the input and output planes are located within the same medium, or within two different media which happen to have identical indices of refraction, then the determinant of {{math|'''M'''}} is simply equal to 1.

A different convention for the ray vectors can be employed. Instead of using {{math|1= ''θ'' ≈ sin ''θ''}}, the second element of the ray vector is {{math|1= ''n''&thinsp;sin ''θ''}},{{sfnp|Gerrard|Burch|1994|p=[https://archive.org/details/introductiontoma0000gerr_u8i1/page/27/mode/2up 27]|ps=, called the "optical direction-cosine".}} which is proportional not to the ray angle ''per se'' but to the transverse component of the [[wave vector]].
This alters the ABCD matrices given in the table below where refraction at an interface is involved.

The use of transfer matrices in this manner parallels the {{val|2|×|2}} matrices describing electronic [[two-port network#ABCD-parameters|two-port networks]], particularly various so-called ABCD matrices which can similarly be multiplied to solve for cascaded systems.