ABCD Matrices Tutorial

Definition: a 2-by-2 matrix associated with an optical element which characterizes the element's effect on a light ray

In ABCD matrix analysis (also known as Ray transfer matrix analysis) a 2-by-2 matrix associated with an optical element is used to describe the element's effect on a light ray.
The main points one are as follows:

The input beam is described by a vector (with some transverse offset r and offset θ angle from a reference axis)
An ABCD matrix which accurately describes the optical element is formed (see below for more information on how to form the ABCD matrices)
The ABCD matrix is multiplied by the input vector
The result is a vector which describes the output beam

The primed quantites refer to the beam after passing through the optical component.

Note: this technique uses the paraxial approximation of ray optics, which means all rays are assumed to be at a smal angle and a small distance relative to the optical axis of the system. The approximation is valid as long as sin(θ) ≈ θ .

ABCD Matrix Calculator

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ABCD Matrices for simple optical components

In short an ABCD matrix allows the output of an optical element to be written in terms of its input. The hard part about ABCD matrix analysis is coming up with the appropriate ABCD matrices for the elements in the system to be analyzed. Below is the derivation of the ABCD transfer matrix of a beam traveling through free space or in a medium of constant refractive index. The derivations of the transfer matrices for other simple optical elements follow the same pattern and are relatively straightforward. The detailed derivations of the transfer matrices of other simple optical elements will not be discussed in this tutorial, but a table is included which gives the general form of the transfer matrices for several optical elements.

Table of ABCD Matrices for simple optical components

Element	Matrix	Remarks
Propagation in free space or in a medium of constant refractive index		d = distance
Refaction at a flat interface		n₁ = initial refractive index n₂ = final refractive index
Refaction at a curved interface		R = radius of curvature, R > for convex (centre of curvature after interface) n₁ = initial refractive index n₂ = final refractive index
Reflection from a flat mirror		Identity matrix
Reflection from a curved mirror		R = radius of curvature, R > 0 for convex
Thin lens		f = focal length of lens where f > 0 for convex/positive (converging) lens. Valid if and only if the focal length is much greater than the thickness of the lens

ABCD Matrix Example

Below is an example of a number of simple optical elements combined in order to form an optical system. Following the diagram there is a short discussion on how to obtain an ABCD matrix which characterizes the entire system. Or if you would like to use an ABCD calculator for your own system click here

Step by step explanation of the formulation of the ABCD transfer matrix of the above system

Note: Each element in the numbered list corresponds with a 2 by 2 transfer matrix in the expression below (they are numbered respectively)

☆ ☆ ☆ One of the properties of ABCD matrices that makes them extremely useful is that a system of several optical elements can be described by a single ABCD matrix. This ABCD matrix is the product of the ABCD matrices of each of the optical elements in the system. ☆ ☆ ☆

Transfer matrix of the medium between points 1 and 2 (with constant index of refraction n₁)
Transfer matrix of the flat interface at point 2
Transfer matrix of the medium between points 2 and 3 (with constant index of refraction n₂)
Transfer matrix of the flat interface at point 3
Transfer matrix of the medium between points 3 and 4 (with constant index of refraction n₁)
Transfer matrix of the thin lens with focal length f and index of refraction n₃
Transfer matrix of the medium between points 4 and 5 (with constant index of refraction n₁)
Transfer matrix of the curved interface at point 5
Transfer matrix of the medium between points 5 and 6 (with constant index of refraction n₄)
Transfer matrix of the flat interface at point 6
Transfer matrix of the medium between points 6 and 7 (with constant index of refraction n₁)
Transfer matrix of the flat mirror at point 7
Transfer matrix of the medium between points 7 and 8 (with constant index of refraction n₁)