Ray transfer matrix analysis: Difference between revisions

Revision as of 14:53, 26 May 2025

Latest revision as of 04:47, 25 July 2025

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[[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

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<math display="block"> \begin{bmatrix}x_N \\ \theta_N \end{bmatrix} = \lambda^N \begin{bmatrix}x_1 \\ \theta_1\end{bmatrix}. </math>

If the waveguide is stable, no ray should stray arbitrarily far from the main axis, that is, {{mvar|λ{{sup|N}}}} must not grow without limit. Suppose {{nowrap|<math> g^2 > 1</math>.}} Then both eigenvalues are real. Since {{nowrap|<math> \lambda_+ \lambda_- = 1</math>,}} one of them has to be bigger than 1 (in [[absolute value]]), which implies that the ray which corresponds to this eigenvector would not converge. Therefore, in a stable waveguide, {{nowrap|<math> g^2 \leq 1</math>,}} and the eigenvalues can be represented by complex numbers:

<math display="block"> \lambda_{\pm} = g \pm i \sqrt{1 - g^2} = \cos(\phi) \pm i \sin(\phi) = e^{\pm i \phi} , </math>

with the substitution {{math|1=''g'' = cos(''ϕ'')}}.

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After {{mvar|N}} waveguide sectors, the output reads

<math display="block"> \mathbf{M}^N (c_+ r_+ + c_- r_-) = \lambda_+^N c_+ r_+ + \lambda_-^N c_- r_- = e^{i N \phi} c_+ r_+ + e^{- i N \phi} c_- r_- , </math>

which represents a [[periodic function]].

== Gaussian beams ==