
ABCD matrix
The ABCD matrix is the paraxial transfer operator, describing how an optical system transforms the initial ray height h and angle (tangent) u at a specified input plane to the final ray height h' and angle u' at a specified output plane.

The transformation is accomplished by right-multiplication of the height-angle vector by the ABCD matrix:

The form of the ABCD matrix for various operations is given below:
![]() | Propagation by distance L |
![]() | Refraction by a thin lens of focal length f |
![]() | Refraction from material of index n to material of index n' across an interface of curvature c |
From the above examples, you can see that B has units of length, that C has units of inverse length, and that A and D are dimensionless.
The coefficients of the ABCD matrix are not independent, because the matrix must describe a canonical transformation (i.e., conserve brightness). The means that the determinant of the matrix must be equal to n/n', where n is the refractive index of the material in which the rays are initially and n' is the refractive index of the material in which the rays are finally.