ABCD matrix (ray transfer matrix)

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

Free space of length $d$ has the matrix $\begin{pmatrix} 1 & d \\ 0 & 1 \end{pmatrix}$, a thin lens of focal length $f$ has $\begin{pmatrix} 1 & 0 \\ -1/f & 1 \end{pmatrix}$, 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 $AD - BC$ 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 $1/q = 1/R - i\lambda/(\pi w^2)$ with $R$ the wavefront radius and $w$ the beam radius, transforms as

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 $\lambda f/(\pi w)$ 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 ($L = 2R$, $g_1 g_2 = 1$) 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 centre 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.

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