Math 2.0

Optics, electromagnetic theory Definition

Ray transfer matrix analysis

Builds on Paraxial approximationMatrixMatrix multiplicationDeterminant

Definition

Let an optical system have an optical axis, an input plane and an output plane perpendicular to the axis. Describe a ray at a plane by the column vector (y,θ)T(y, \theta)^T. Here yy is its height above the axis and θ\theta its angle with the axis. In the paraxial approximation, the output ray depends linearly on the input ray: (y2θ2)=(ABCD)(y1θ1).\begin{pmatrix} y_2 \\ \theta_2 \end{pmatrix} = \begin{pmatrix} A & B \\ C & D \end{pmatrix} \begin{pmatrix} y_1 \\ \theta_1 \end{pmatrix}. The real matrix M=(ABCD)M = \begin{pmatrix} A & B \\ C & D \end{pmatrix} is the ray transfer matrix (ABCD matrix) of the system. Ray transfer matrix analysis computes MM as the product M=Mk⋯M2M1M = M_k \cdots M_2 M_1 of the matrices of the elements. Here M1M_1 belongs to the element that the light meets first, and MkM_k to the last one.

Remarks

Basic matrices: free propagation over a distance dd has (1d01)\begin{pmatrix} 1 & d \\ 0 & 1 \end{pmatrix}, and a thin lens of focal length ff has (10−1/f1)\begin{pmatrix} 1 & 0 \\ -1/f & 1 \end{pmatrix}. The determinant is det⁡M=n1/n2\det M = n_1 / n_2, where n1n_1 and n2n_2 are the refractive indices at the input and output planes. With the reduced angle nθn\theta in place of θ\theta, every ray transfer matrix has determinant 11.

Example: free space dd followed by a thin lens ff. Check: (10−1/f1)(1d01)=(1d−1/f1−d/f)\begin{pmatrix} 1 & 0 \\ -1/f & 1 \end{pmatrix} \begin{pmatrix} 1 & d \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & d \\ -1/f & 1 - d/f \end{pmatrix}, of determinant 11. For d=fd = f the output angle is −y1/f-y_1/f, independent of θ1\theta_1: rays from one point of the front focal plane leave parallel.

Non-example: the product written left to right in the order of travel, M1M2⋯MkM_1 M_2 \cdots M_k, is in general a different matrix, since matrix multiplication does not commute. It does not describe the system.

Reading path

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  1. Determinant concept
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Further reading

  1. Web https://en.wikipedia.org/wiki/Ray_transfer_matrix_analysis
About this page
TagEM1BMZ6P6D
URIm2://EM1BMZ6P6D
Kinddefinition
WritingW0 stub
MSC78A05
DefinesRay transfer matrix analysis
UsesParaxial approximation, Matrix, Matrix multiplication, Determinant
Changed3 November 2026

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