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Ray transfer matrix analysis (also known as ABCD matrix analysis) is a type of ray tracing technique used in the design of some optical systems, particularly lasers. It involves the construction of a ray transfer matrix which describes the optical system; tracing of a light path through the system can then be performed by multiplying this matrix with a vector representing the light ray.

The technique, described below, uses the paraxial approximation of ray optics, i.e., all rays are assumed to be at a small angle (θ) and a small distance (x) relative to the optical axis of the system. The approximation is valid as long as sin(θ)≈θ (where θ is measured in radians). [An exact method for tracing meridional rays is available, inter alia, at <http://spie.org/etop/1991/389_1.pdf>.

Definition of the ray transfer matrix

The ray tracing technique is based on two reference planes, called the input and output planes, each perpendicular to the optical axis of the system. Without loss of generality, we will define the optical axis so that it coincides with the z-axis of a fixed coordinate system. A light ray enters the system when the ray crosses the input plane at a distance x₁ from the optical axis while traveling in a direction that makes an angle θ₁ with the optical axis. Some distance further along, the ray crosses the output plane, this time at a distance x₂ from the optical axis and making an angle θ₂. n₁ and n₂ are the indices of refraction of the medium in the input and output plane, respectively.

These quantities are related by the expression

\( {x_2 \choose \theta_2} = \begin{pmatrix} A & B \\ C & D \end{pmatrix}{x_1 \choose \theta_1}, \)

where

\( A = {x_2 \over x_1 } \bigg|_{\theta_1 = 0} \qquad B = {x_2 \over \theta_1 } \bigg|_{x_1 = 0},\)

and

\( C = {\theta_2 \over x_1 } \bigg|_{\theta_1 = 0} \qquad D = {\theta_2 \over \theta_1 } \bigg|_{x_1 = 0}.\)

This relates the ray vectors at the input and output planes by the ray transfer matrix (RTM) M, which represents the optical system between the two reference planes. A thermodynamics argument based on the blackbody radiation can be used to show that the determinant of a RTM is the ratio of the indices of refraction:

\( \det(\mathbf{M}) = AD - BC = { n_1 \over n_2 }. \)

As a result, if the input and output planes are located within the same medium, or within two different media which happen to have identical indices of refraction, then the determinant of M is simply equal to 1.

A similar technique can be used to analyze electrical circuits. See Two-port networks.
Some examples

For example, if there is free space between the two planes, the ray transfer matrix is given by:

\( \mathbf{S} = \begin{pmatrix} 1 & d \\ 0 & 1 \end{pmatrix} , \)

where d is the separation distance (measured along the optical axis) between the two reference planes. The ray transfer equation thus becomes:

\( {x_2 \choose \theta_2} = \mathbf{S}{x_1 \choose \theta_1} , \)

and this relates the parameters of the two rays as:

\( \begin{matrix} x_2 & = & x_1 + d\theta_1 \\ \theta_2 & = & \theta_1 \end{matrix} \)

Another simple example is that of a thin lens. Its RTM is given by:

\( \mathbf{L} = \begin{pmatrix} 1 & 0 \\ \frac{-1}{f} & 1 \end{pmatrix} , \)

where f is the focal length of the lens. To describe combinations of optical components, ray transfer matrices may be multiplied together to obtain an overall RTM for the compound optical system. For the example of free space of length d followed by a lens of focal length f:

\( \mathbf{L}\mathbf{S} = \begin{pmatrix} 1 & 0 \\ \frac{-1}{f} & 1\end{pmatrix} \begin{pmatrix} 1 & d \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & d \\ \frac{-1}{f} & 1-\frac{d}{f} \end{pmatrix} . \)

Note that, since the multiplication of matrices is non-commutative, this is not the same RTM as that for a lens followed by free space:

\( \mathbf{SL} = \begin{pmatrix} 1 & d \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ \frac{-1}{f} & 1\end{pmatrix} = \begin{pmatrix} 1-\frac{d}{f} & d \\ \frac{-1}{f} & 1 \end{pmatrix} . \)

Thus the matrices must be ordered appropriately. Other matrices can be constructed to represent interfaces with media of different refractive indices, reflection from mirrors, etc.
Table of ray transfer matrices

for simple optical components

Element	Matrix	Remarks
Propagation in free space or in a medium of constant refractive index	\( \begin{pmatrix} 1 & d\\ 0 & 1 \end{pmatrix} \)	d = distance
Refraction at a flat interface	\( \begin{pmatrix} 1 & 0 \\ 0 & \frac{n_1}{n_2} \end{pmatrix} \)	n₁ = initial refractive index n₂ = final refractive index.
Refraction at a curved interface	\( \begin{pmatrix} 1 & 0 \\ \frac{n_1-n_2}{R \cdot n_2} & \frac{n_1}{n_2} \end{pmatrix} \)	R = radius of curvature, R > 0 for convex (centre of curvature after interface) n₁ = initial refractive index n₂ = final refractive index.
Reflection from a flat mirror	\( \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \)
Reflection from a curved mirror	\( \begin{pmatrix} 1 & 0 \\ -\frac{2}{R} & 1 \end{pmatrix} \)	R = radius of curvature, R > 0 for concave
Thin lens	\( \begin{pmatrix} 1 & 0 \\ -\frac{1}{f} & 1 \end{pmatrix} \)	f = focal length of lens where f > 0 for convex/positive (converging) lens. Only valid if the focal length is much greater than the thickness of the lens.
Thick lens	\( \begin{pmatrix} 1 & 0 \\ \frac{n_2-n_1}{R_2} & 1 \end{pmatrix} \begin{pmatrix} 1 & \frac{t}{n_2} \\ 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 \\ \frac{n_1-n_2}{R_1} & 1 \end{pmatrix} \)	n₁ = refractive index outside of the lens. n₂ = refractive index of the lens itself (inside the lens). R₁ = Radius of curvature of First surface. R₂ = Radius of curvature of Second surface. t = thickness of lens (not taking into account thickness of curved parts. If total thickness is desired, total thickness would equal t + both curved thickness parts.
Single right angle prism	\( \begin{pmatrix} k & \frac{d}{nk} \\ 0 & \frac{1}{k} \end{pmatrix} \)	\( k = (cos\psi/cos\phi) \) is the beam expansion factor, where \( \phi \) is the angle of incidence, \psi is the angle of refraction, d = prism path length, n = refractive index of the prism material. This matrix applies for orthogonal beam exit.

Resonator stability

RTM analysis is particularly useful when modeling the behaviour of light in optical resonators, such as those used in lasers. At its simplest, an optical resonator consists of two identical facing mirrors of 100% reflectivity and radius of curvature R, separated by some distance d. For the purposes of ray tracing, this is equivalent to a series of identical thin lenses of focal length f=R/2, each separated from the next by length d. This construction is known as a lens equivalent duct or lens equivalent waveguide. The RTM of each section of the waveguide is, as above,

\( \mathbf{M} =\mathbf{L}\mathbf{S} = \begin{pmatrix} 1 & d \\ \frac{-1}{f} & 1-\frac{d}{f} \end{pmatrix} .\)

RTM analysis can now be used to determine the stability of the waveguide (and equivalently, the resonator). That is, it can be determined under what conditions light travelling down the waveguide will be periodically refocussed and stay within the waveguide. To do so, we can find all ray vectors where the output of each section of the waveguide is equal to the input vector multiplied by some real or complex constant λ:

\( {x_2 \choose \theta_2} = \lambda {x_1 \choose \theta_1} . \)

This gives:

\( \mathbf{M}{x_1 \choose \theta_1} = \lambda {x_1 \choose \theta_1} , \)

which is an eigenvalue equation:

\( \left[ \mathbf{M} - \lambda\mathbf{I} \right] {x_1 \choose \theta_1} = 0 \)

where I is the 2x2 identity matrix.

Simplifying, we have

\( \operatorname{det} \left[ \mathbf{M} - \lambda\mathbf{I} \right] = 0 \)

which leads to the characteristic equation

\( \lambda^2 - \operatorname{tr}(\mathbf{M}) \lambda + \operatorname{det}( \mathbf{M}) = 0 \)

where

\( \operatorname{tr} ( \mathbf{M} ) = A + D = 2 - { d \over f } \)

is the trace of the RTM, and

\( \operatorname{det}(\mathbf{M}) = AD - BC = 1 \)

is the determinant of the RTM. Simplifying, we have

\( \lambda^2 - 2g \lambda + 1 = 0 \, \)

where

\( g \ \stackrel{\mathrm{def}}{=}\ { \operatorname{tr}(\mathbf{M}) \over 2 } = 1 - { d \over 2 f } \)

is the stability parameter. The eigenvalues are the solutions of the characteristic equation. From the quadratic formula, we find

\( \lambda = g \pm \sqrt{g^2 - 1} \, \)

After N passes through the system, we have:

\( {x_N \choose \theta_N} = \lambda^N {x_1 \choose \theta_1} .\)

If the waveguide is stable, λN must not grow without limit. This observation implies that λ cannot take on purely real values, but must have a non-zero imaginary part. Thus,

\( \operatorname{Im} \{ \lambda \} \ne 0 \)

As a result,

\( g^2 - 1 < 0 \, \)

\( |g| < 1 \, \)

Solving the eigenvalue equation gives us a periodic solution of the form:

\( \lambda^N = e^{\pm i N \phi} , \)

\( \lambda = e^{ \pm i \phi } , \)

where

\( \cos(\phi) = \operatorname{Re} \{ \lambda \} = g = { \operatorname{tr}(\mathbf{M}) \over 2 } = 1 - { d \over 2 f } \)

The technique may be generalised for more complex resonators by constructing a suitable matrix M for the cavity from the matrices of the components present.
Ray transfer matrices for Gaussian beams

The matrix formalism is also useful to describe Gaussian beams. If we have a Gaussian beam of wavelength \( \lambda_0 \), radius of curvature R, beam spot size w and refractive index n, it is possible to define a complex beam parameter q by:

\( \frac{1}{q} = \frac{1}{R} - \frac{i\lambda_0}{\pi n w^2} . \)

This beam can be propagated through an optical system with a given ray transfer matrix by using the equation:

\( {q_2 \choose 1} = k \begin{pmatrix} A & B \\ C & D \end{pmatrix} {q_1 \choose 1} , \)

where k is a normalisation constant chosen to keep the second component of the ray vector equal to 1. Using matrix multiplication, this equation expands as

\( q_2 = k(Aq_1 + B) \, \)

and

\( 1 = k(Cq_1 + D) \, \)

Dividing the first equation by the second eliminates the normalisation constant:

\( q_2 =\frac{Aq_1+B}{Cq_1+D}, \)

It is often convenient to express this last equation in reciprocal form:

{ 1 \over q_2 } = { C + D/q_1 \over A + B/q_1 } . \)