### 7.11 ABCD matrices

The transformation of the beam parameter can be performed by the ABCD matrix-formalism [34, 50].
When a beam passes an optical element or freely propagates though space, the initial beam parameter
is transformed into . This transformation can be described by four real coefficients as follows:
with the coefficient matrix
being the index of refraction at the beam segment defined by , and the index of refraction at
the beam segment described by . ABCD matrices for some common optical components are given
below, for the sagittal and tangential plane.

#### Transmission through a mirror:

A mirror in this context is a single, partly-reflecting surface with an
angle of incidence of 90°. The transmission is described by
with being the radius of curvature of the spherical surface. The sign of the radius is defined such
that is negative if the centre of the sphere is located in the direction of propagation. The curvature
shown above (in Figure 48), for example, is described by a positive radius. The matrix for the transmission
in the opposite direction of propagation is identical.

#### Reflection at a mirror:

The matrix for reflection is given by
The reflection at the back surface can be described by the same type of matrix by setting
.
#### Transmission through a beam splitter:

A beam splitter is understood as a single surface with an
arbitrary angle of incidence . The matrices for transmission and reflection are different for the sagittal
and tangential planes ( and ):
with given by Snell’s law:
and by
If the direction of propagation is reversed, the matrix for the sagittal plane is identical and the
matrix for the tangential plane can be obtained by changing the coefficients A and D as follows:

#### Reflection at a beam splitter:

The reflection at the front surface of a beam splitter is given
by:
To describe a reflection at the back surface the matrices have to be changed as follows:
#### Transmission through a thin lens:

A thin lens transforms the beam parameter as follows:
where is the focal length. The matrix for the opposite direction of propagation is identical. Here it
is assumed that the thin lens is surrounded by ‘spaces’ with index of refraction .
#### Transmission through a free space:

As mentioned above, the beam in free space can be described by one
base parameter . In some cases it is convenient to use a matrix similar to that used for the other
components to describe the -dependency of . On propagation through a free
space of the length and index of refraction , the beam parameter is transformed as
follows.
The matrix for the opposite direction of propagation is identical.