ABCD Matrix | OptoWiki Knowledge Base

ABCD Matrixes
$\begin{pmatrix} A & B \\ C & D \end{pmatrix}$
are used in paraxial optical design.

It’s only allowed to use ABCD Matrices in the paraxial range, with $\sin{\theta} \approx \theta$
The angles are measured in radians!

A beam is described by a distance r from the optical axis and a offset angle $\theta$ from the optical axis
An ABCD Matrix that describes the optical element is formed
The ABCD Matrix is multiplied by the input vector
The result is an output vector that describes the output beam with a new distance from the optical axis and a new angle off the optical axis

$\left( \begin{array}{c}r'\\\theta'\end{array} \right) = \begin{pmatrix} A & B \\ C & D \end{pmatrix} \left( \begin{array}{c}r\\\theta\end{array} \right)$

$\left( \begin{array}{c}r'\\\theta'\end{array} \right) = \begin{pmatrix} A_1 & B_1 \\ C_1 & D_1 \end{pmatrix} \left( \begin{array}{c}r\\\theta\end{array} \right)$

is a short for for the equation system

$r' = A_1 r + B_1 \theta$

$\theta' = C_1 r + D_1 \theta$

$\left( \begin{array}{c}r''\\\theta''\end{array} \right) = \begin{pmatrix} A_2 & B_2 \\ C_2 & D_2 \end{pmatrix} \begin{pmatrix} A_1 & B_1 \\ C_1 & D_1 \end{pmatrix} \left( \begin{array}{c}r\\ \theta \end{array} \right)$

$\iff$

$\left( \begin{array}{c}r''\\\theta''\end{array} \right) = \begin{pmatrix} A_2 & B_2 \\ C_2 & D_2 \end{pmatrix} \left( \begin{array}{c}A_1 r + B_1 \theta\\\ C_1 r + D_1 \theta \end{array} \right)$

$\iff$

$\left( \begin{array}{c}r''\\\theta''\end{array} \right) = \begin{pmatrix} A_2 & B_2 \\ C_2 & D_2 \end{pmatrix} \left( \begin{array}{c}r'\\\theta'\end{array} \right)$

with

$\left( \begin{array}{c}r'\\\theta'\end{array} \right) = \begin{pmatrix} A_1 & B_1 \\ C_1 & D_1 \end{pmatrix} \left( \begin{array}{c}r\\\theta\end{array} \right)$

$\iff$

$r'' = A_2 r' + B_2 \theta'$

$\theta'' = C_2 r' + D_2 \theta'$

with

$r' = A_1 r + B_1 \theta$

$\theta' = C_1 r + D_1 \theta$

$\iff$

$r'' = A_2 (A_1 r + B_1 \theta) + B_2 (C_1 r + D_1 \theta)$

$\theta'' = C_2 (A_1 r + B_1 \theta) + D_2 (C_1 r + D_1 \theta)$

$\iff$

$r'' = A_1 A_2 r + B_1 A_2 \theta + C_1 B_2 r + D_1 B_2 \theta$

$\theta'' = A_1 C_2 r + B_1 C_2 \theta + C_1 D_2 r + D_1 D_2 \theta$

$\iff$

$r'' = (A_1 A_2 + C_1 B_2 )r + (B_1 A_2 + D_1 B_2) \theta$

$\theta'' = (A_1 C_2 + C_1 D_2) r + (B_1 C_2 + D_1 D_2) \theta$

$\iff$

$\left( \begin{array}{c}r''\\\theta''\end{array} \right) = \begin{pmatrix} A_1 A_2 + C_1 B_2 & B_1 A_2 + D_1 B_2 \\ A_1 C_2 + C_1 D_2 & B_1 C_2 + D_1 D_2 \end{pmatrix} \left( \begin{array}{c}r\\\theta\end{array} \right)$

Examples of ABCD matrices for simple optical elements :

Propagation in free space or in a medium of constant refractive index :
$\begin{pmatrix} 1 & d \\ 0 & 1 \end{pmatrix}$
Where d = reduced distance= thickness / refraction index

Propagation through a series i=1..k of planparallel media with constant refraction indices $n_i$ :

$\begin{pmatrix} 1 & d_1 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & d_2 \\ 0 & 1 \end{pmatrix} ... \begin{pmatrix} 1 & d_k \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & \sum\limits_{i=1}^k d_i\\ 0 & 1 \end{pmatrix}$

Where d_i = reduced distance_i= thickness_i / refraction index_i

Refraction at a flat surface:
$\begin{pmatrix} 1 & 0 \\ 0 & \frac{n_1}{n_2} \end{pmatrix}$
Where:
$n_1$ = initial refractive index
$n_2$ = final refractive index

Refraction at a curved surface:
$\begin{pmatrix} 1 & 0 \\ \frac{n_1 - n_2}{R n_2} & \frac{n_1}{n_2} \end{pmatrix}$
Where:
R = radius of curvature, $R > 0$ for a convex surface (centre of curvature after interface)
$n_1$ = initial refractive index
$n_2$ = final refractive index

Reflection vrom a flat mirror:
$\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$
Identity matrix

Reflection from a curved mirror:
$\begin{pmatrix} 1 & 0 \\ -\frac{2}{R_e} & 1 \end{pmatrix}$
Where:
$R_e = R\cos \theta$ effective radius of curvature in tangential plane (horizontal direction)
$R_e = \frac{R}{\cos(\theta)}$ effective radius of curvature in the sagittal plane (vertical direction)
With $R_e > 0$ for convex mirrors (centre of curvature after interface)

Refraction at a thin lens
$\begin{pmatrix} 1 & 0 \\ -\frac{1}{f} & 1 \end{pmatrix}$
Where:
f = focal length of the lens, where $f > 0$ for convex/positive/converging lenses. Valid if if and only if the focal length is much bigger than the thickness of the lens

Refraction at a thick lens

$\begin{pmatrix} 1 & 0 \\ \frac{n_2 - n_1}{R_2 n_1} & \frac{n_2}{n_1} \end{pmatrix} \begin{pmatrix} 1 & t \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ \frac{n_1 - n_2}{R_1 n_2} & \frac{n_1}{n_2} \end{pmatrix}$

Where:
$n_1$ = refractive index outside of the lens.
$n_2$ = refractive index of the lens itself (inside the lens).
$R_1$ = Radius of curvature of First surface.
$R_2$ = Radius of curvature of Second surface.
t = center thickness of lens.