
Transfer matrix (of a lens)
The transfer matrix of a ray is the matrix that describes the propagation of rays "close" to the ray. On this page we discuss the transfer matrix of an axially symmetric lens, which is the transfer matrix of an axial ray.
An alternative to the ABCD matrix is the transfer matrix. At any point along its trajectory a ray is described by its transverse coordinate q and transverse momentum p. By transverse we mean that these quantities describe the components of position and momentum perpendicular to the optical axis. The transfer matrix of a lens is a matrix that describes how the paraxially transforms the initial coordinate q and momentum p at a specified input plane to the final coordinate q' and momentum p' at a specified output plane.

The transformation of ray coordinates is accomplished by right-multiplication of the momentum-coordinate vector by the transfer matrix:

The form of the tansfer 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 |
The order of the basis used to represent the transfer operator in the literature is (q,p) or (p,q) depending on the author. Here we use (p,q).
The effect of the tansfer matrix for various operations can be written in a manner independent of the order of the basis, as follows:
q → q + pL/n | Propagation by distance L |
p → p - q/f | Refraction by a thin lens of focal length f |
p → p + qc(n-n') | Refraction from material of index n to material of index n' across an interface of curvature c |
Note that the definition used above for the focal length is the ratio of the image height to the initial transverse momentum.
From the above examples, you can see that mqp has units of length, that mpq has units of inverse length, and that mpp and mqq are dimensionless.
The coefficients of the transfer matrix are not independent, because the matrix must describe a canonical transformation (i.e., conserve brightness). This means that the matrix must be symplectic. For the 2x2 matrix this means that the determinant of the matrix is equal to unity, which is a simpler condition than that of the ABCD matrix.
On the Trace listing LensForge calculates and displays the transfer matrix relative to the traced ray. The paraxial transfer matrix of the lens itself is obtained by tracing an axial ray (i.e., one with Hx=Hy=Px=Py=0).