Glossary entry

ABCD matrix (ray transfer matrix)

A 2×2 matrix that describes how an optical element or a stretch of free space changes a paraxial ray's height and angle. Matrices multiply along a system, and the same four numbers propagate a Gaussian beam's q parameter and decide whether a laser resonator is stable.

Optics & beams Optics fundamentals Lasers & gainUpdated October 2026

In the paraxial approximation a ray at a given plane is described by two numbers, its height $y$ above the axis and its angle $θ$ , and every simple optical element changes them linearly:

(\begin{array}{c} y_{2} \\ θ_{2} \end{array}) = (\begin{array}{cc} A & B \\ C & D \end{array}) (\begin{array}{c} y_{1} \\ θ_{1} \end{array}) .

Free space of length $d$ has the matrix $(\begin{array}{cc} 1 & d \\ 0 & 1 \end{array})$ , a thin lens of focal length $f$ has $(\begin{array}{cc} 1 & 0 \\ - 1 / f & 1 \end{array})$ , and a curved mirror of radius $R$ acts as a lens with $f = R / 2$ . A system is described by the product of its elements' matrices, written right to left in the order the light meets them. The determinant $A D - B C$ equals the ratio of the refractive indices at input and output, one in a system that starts and ends in the same medium. The elements have direct meanings: $C = - 1 / f$ gives the system's focal length, $B = 0$ means the output plane images the input plane, and $C = 0$ means the system is afocal, like a beam expander.

Gaussian beams

The same matrix propagates a Gaussian beam. Its complex beam parameter $q$ , defined by $1 / q = 1 / R - i λ / (π w^{2})$ with $R$ the wavefront radius and $w$ the beam radius, transforms as

q_{2} = \frac{A q_{1} + B}{C q_{1} + D} .

At a waist, $q = i z_{R}$ , where $z_{R}$ is the Rayleigh range. As an example, a collimated beam of 1 mm radius at 1064 nm, with its waist at a lens of 100 mm focal length, is focused to a waist of 33.8 µm radius located 99.9 mm behind the lens; the simple estimate $λ f / (π w)$ gives 33.9 µm because the input Rayleigh range, 2.95 m, is much longer than the focal length.

Resonator stability

A light ray in a laser cavity or a Fabry-Perot resonator makes repeated round trips, and stays near the axis only if the round-trip matrix satisfies $- 1 < (A + D) / 2 < 1$ . For two mirrors of radii $R_{1}$ and $R_{2}$ a distance $L$ apart, this becomes $0 < g_{1} g_{2} < 1$ with $g_{i} = 1 - L / R_{i}$ . The plane-parallel ( $g_{1} g_{2} = 1$ ), symmetric confocal ( $L = R$ , $g_{1} g_{2} = 0$ ) and symmetric concentric ( $L = 2 R$ , $g_{1} g_{2} = 1$ ) resonators lie on the boundary. Two mirrors of 1 m radius spaced 0.5 m apart have $g_{1} g_{2} = 0.25$ , well inside the stable region, and the self-consistent Gaussian mode found from the round-trip matrix has a waist radius of 383 µm at the center and 442 µm on the mirrors at 1064 nm.

Limits

The formalism is paraxial and first-order: it ignores aberrations, which are what ray-tracing programs add, and apertures, which clip beams that the matrices would propagate unchanged. Misaligned elements need a 3×3 extension, and systems with rotated cylindrical or otherwise non-orthogonal astigmatic elements need a 4×4 one. The 2×2 matrices used for thin-film coatings, in the characteristic matrix method, are a different formalism that acts on the fields at the layer boundaries rather than on rays.

References: A. E. Siegman, Lasers (University Science Books, 1986), Ch. 15 and 19; H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).