# ABCD Matrix

Author: the photonics expert Dr. Rüdiger Paschotta

Definition: a 2-by-2 matrix describing the effect of an optical element on a laser beam

Alternative term: ray transfer matrix

An ABCD matrix [1] is a 2-by-2 matrix associated with an optical element which can be used for describing the element's effect on a laser beam. It can be used both in *ray optics*, where geometrical rays are propagated, and for propagating Gaussian beams. The paraxial approximation is always required for ABCD matrix calculations, i.e., the involved beam angles or divergence angles must stay small for the calculations to be accurate.

## Ray Optics

Originally, the concept was developed in geometrical optics for calculating the propagation of light rays with some transverse offset <$r$> and offset angle <$\theta$> from a reference axis (Figure 1). As long as the angles involved are small enough (→ *paraxial approximation*), there is a linear relation between the <$r$> and <$\theta$> coordinates before and after an optical element. The following equation can then be used for calculating how these parameters are modified by an optical element:

where the primed quantities (left-hand side) refer to the beam after passing the optical component. The ABCD matrix (also called *ray transfer matrix*) is a characteristic of each optical element.

For example, a thin lens with focal length <$f$> has the following ABCD matrix:

$$\left( {\begin{array}{*{20}{c}} 1&0\\ { - 1/f}&1 \end{array}} \right)$$This shows that the offset <$r$> remains unchanged, whereas the offset angle <$\theta$> experiences a change in proportion to <$r$>.

Propagation through free space over a distance <$d$> is associated with the matrix

$$\left( {\begin{array}{*{20}{c}} 1&d\\ 0&1 \end{array}} \right)$$which shows that the angle remains unchanged, whereas the beam offset is increasing or decreasing according to the angle.

Further examples of ABCD matrices are given below.

One can show that the determinant of an ABCD matrix (<$A D - B C$>) always must be 1 as long as the refractive index is the same on the input and output side; otherwise, it is the input refractive index divided by the output refractive index. This also implies that an ABCD matrix can never be singular; it can always be inverted.

### ABCD Matrix Calculations

The software RP Resonator uses ABCD matrices – or actually ABCDEF matrices, also treating misalignment effects – for calculating the mode properties of optical resonators. This is vital for laser development, for example. This tool is particularly flexible, allowing even rather sophisticated resonator design optimizations. For example, you can take into account alignment sensitivity and thermal lensing, and even design a resonator for operation in a specific stability zone.

### Modified Matrices

For situations where beams propagate through dielectric media, it is convenient to use a modified kind of beam vectors, where the lower component (the angle) is multiplied by the refractive index:

$$\left( {\begin{array}{*{20}{c}} {r'}\\ {n\;\theta '} \end{array}} \right) = \left( {\begin{array}{*{20}{c}} A&B\\ C&D \end{array}} \right)\left( {\begin{array}{*{20}{c}} r\\ {n\;\theta } \end{array}} \right)$$This can somewhat simplify the ABCD matrices for certain situations. In numerous instances of free-space optics it makes no difference, since the beams are considered only at positions in air where <$n \approx 1$>. However, equations for interfaces between different media, for example, are affected.

The determinant of such a modified ABCD matrix (<$A D - B C$>) is always 1.

## Propagation of Gaussian Beams

ABCD matrices can also be used for calculating the effect of optical elements on the parameters of a Gaussian beam. A convenient quantity for that purpose is the complex <$q$> parameter, which contains information on both the beam radius <$w$> and the radius of curvature <$R$> of the wavefronts:

$$\frac{1}{q} = - i\frac{\lambda }{{\pi {w^2}}} + \frac{1}{R}$$The following equation shows how the <$q$> parameter is modified by an optical element:

$$q' = \frac{{Aq + B}}{{Cq + D}}$$## ABCD Matrices of Important Optical Elements

The following list gives the ABCD matrices of frequently used optical elements.

Air space with length <$d$>:

$$\left( {\begin{array}{*{20}{c}} 1&d\\ 0&1 \end{array}} \right)$$(For propagation in a transparent medium, the length <$d$> has to be divided by the refractive index <$n$>, if the above-mentioned modified definition is used where the lower component (the angle) is multiplied by the refractive index.)

Lens with focal length <$f$> (where positive <$f$> applies for a focusing lens):

$$\left( {\begin{array}{*{20}{c}} 1&0\\ { - 1/f}&1 \end{array}} \right)$$Curved mirror with curvature radius <$R$> (>0 for concave mirror), incidence angle <$\theta$> in the horizontal plane:

$$\left( {\begin{array}{*{20}{c}} 1&0\\ { - 2/{R_{\rm{e}}}}&1 \end{array}} \right)$$with <$R_\textrm{e} = R \: \cos \theta$> in the tangential plane (horizontal direction) and <$R_\textrm{e} = R / \cos \theta$> in the sagittal plane (vertical direction).

Gaussian duct:

$$\left( {\begin{array}{*{20}{c}} {\cos \gamma z}&{{{({n_0}\gamma )}^{ - 1}}\sin \gamma z}\\ { - ({n_0}\gamma )\sin \gamma z}&{\cos \gamma z} \end{array}} \right)$$where the radially varying refractive index is

$$n(r) = {n_0} - \frac{1}{2}{n_2}{r^2},\quad {\gamma ^2} = {n_2}/{n_0}$$and the modified definition of beam vectors – with the angle multiplied with the refractive index (see above) – is used.

Various textbooks (see e.g. Ref. [4]) specify the ABCD matrices for other types of optical components.

Note that there are optical elements (e.g. axicons) which can not be described with ABCD matrices, and which convert a Gaussian beam into a non-Gaussian beam.

## Combining Multiple Optical Elements

If a beam propagates through several optical elements (including any air spaces in between), this means that the (<$r \theta$>) vector is subsequently multiplied by various matrices. Instead, a single matrix may be used, which is the matrix product of all the single matrices. Note that the first optical element must be on the right-hand side of that product – matrix multiplications are not commutative, and the same holds for optical elements.

Example:

- combined matrix for free-space propagation length with distance <$d$>, followed by a lens with focal length <$f$>:

- combined matrix for a lens with focal length <$f$>, followed by free-space propagation length with distance <$d$>:

## Typical Applications

Some typical applications of the ABCD matrix algorithm are:

- It is often of interest how a laser beam propagates through some optical setup. Both the geometric path of a ray and the evolution of the beam radius can be calculated.
- The changes of beam parameters within one complete round trip in a resonator can be described with an ABCD matrix. The transverse resonator modes can then be obtained from the matrix components.
- An extended algorithm, involving an ABCDEF matrix (a 3-by-3 matrix with some constant components), can be used for calculating the alignment sensitivity of a laser resonator [3].

The ABCD matrix method should not be confused with a different matrix method for calculating the reflection and transmission properties of dielectric multilayer coatings.

## More to Learn

Encyclopedia articles:

### Bibliography

[1] | H. Kogelnik and T. Li, “Laser beams and resonators”, Appl. Opt. 5 (10), 1550 (1966); https://doi.org/10.1364/AO.5.001550 |

[2] | P. A. Bélanger, “Beam propagation and the ABCD ray matrices”, Opt. Lett. 16 (4), 196 (1991); https://doi.org/10.1364/OL.16.000196 |

[3] | O. E. Martínez, “Matrix formalism for dispersive laser cavities”, IEEE J. Quantum Electron. 25 (3), 296 (1989); https://doi.org/10.1109/3.18543 |

[4] | A. E. Siegman, Lasers, University Science Books, Mill Valley, CA (1986) |

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2022-05-25

When calculating the propagation of a Gaussian beam, should the used beam dimension be <$1/e^2$>, <$1/e$>, or full width half maximum?

The author's answer:

For the beam radius <$w$>, use the Gaussian beam radius, as explained in the article on beam radius. This is the radius where the electric field strength drops to <$1/e$> (≈ 37%) of the maximum value, or the intensity to <$1/e^2$>.