7.11 ABCD matrices

The transformation of the beam parameter can be performed by the ABCD matrix-formalism [3450]. When a beam passes an optical element or freely propagates though space, the initial beam parameter q1 is transformed into q2. This transformation can be described by four real coefficients as follows:
q -q2- A-n11-+-B- n = C q1 + D , (154 ) 2 n1
with the coefficient matrix
( ) A B M = C D , (155 )
n1 being the index of refraction at the beam segment defined by q1, and n2 the index of refraction at the beam segment described by q2. 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
View Image

Figure 48:
with R C being the radius of curvature of the spherical surface. The sign of the radius is defined such that RC is negative if the centre of the sphere is located in the direction of propagation. The curvature shown above (in Figure 48View Image), 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
View Image

Figure 49:
The reflection at the back surface can be described by the same type of matrix by setting C = 2n2∕RC.

Transmission through a beam splitter:

A beam splitter is understood as a single surface with an arbitrary angle of incidence α1. The matrices for transmission and reflection are different for the sagittal and tangential planes (Ms and Mt):
View Image

Figure 50:
with α2 given by Snell’s law:
n1 sin (α1) = n2 sin (α2), (156 )
and Δn by
n cos(α ) − n cos(α ) Δn = --2------2-----1------1 . (157 ) cos (α1 )cos(α2 )
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:
A −→ 1∕A, D −→ 1∕D. (158 )

Reflection at a beam splitter:

The reflection at the front surface of a beam splitter is given by:
View Image

Figure 51:
To describe a reflection at the back surface the matrices have to be changed as follows:
R −→ − R , C C n1 − → n2, (159 ) α1 − → − α2.

Transmission through a thin lens:

A thin lens transforms the beam parameter as follows:
View Image

Figure 52:
where f 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 n = 1.

Transmission through a free space:

As mentioned above, the beam in free space can be described by one base parameter q0. In some cases it is convenient to use a matrix similar to that used for the other components to describe the z-dependency of q(z) = q0 + z. On propagation through a free space of the length L and index of refraction n, the beam parameter is transformed as follows.
View Image

Figure 53:
The matrix for the opposite direction of propagation is identical.


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