Ray transfer matrix analysis

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 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).

The technique is based on two reference planes perpendicular to the optical axis of the system. At the first plane, a light ray crosses the plane at a distance x₁ from the optical axis at an angle θ₁. Some distance along the optical axis, at the second plane, the ray crosses at distance x₂ and an angle θ₂.

These quantities are related by the expression:

<math> {x_2 \choose \theta_2} = \begin{pmatrix} A & B \\ C & D \end{pmatrix}{x_1 \choose \theta_1} <math>.

This relates the ray vectors at the first and second reference planes by the ray transfer matrix (RTM) which represents the arbitrary optical system between the two reference planes.

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

<math> \mathbf{S} = \begin{pmatrix} 1 & d \\ 0 & 1 \end{pmatrix} <math>,

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

<math> {x_2 \choose \theta_2} = \mathbf{S}{x_1 \choose \theta_1} <math>,

and this relates the parameters of the two rays as:

<math> \begin{matrix} x_2 & = & x_1 + d\theta_1 \\

\theta_2 & = & \theta_1 \end{matrix} <math>

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

<math> \mathbf{L} = \begin{pmatrix} 1 & 0 \\ \frac{-1}{f} & 1 \end{pmatrix} <math>,

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:

<math>\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} <math>.

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:

<math> \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} <math>.

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.

RTM analysis is particularly used when modelling 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 duct or lens waveguide. The RTM of each section of the waveguide is thus M=LS as shown above, substituting f=R/2.

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 refocused 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:

<math> {x_2 \choose \theta_2} = \lambda {x_1 \choose \theta_1} <math>.

This gives:

<math> \mathbf{M}{x_1 \choose \theta_1} = \lambda {x_1 \choose \theta_1} <math>,

which is an eigenvalue equation:

<math> \left[ \mathbf{M} - \lambda\mathbf{I} \right] {x_1 \choose \theta_1} = 0 <math>

where I is the 2x2 identity matrix.

After N passes through the system, we have:

<math> {x_N \choose \theta_N} = \lambda^N {x_1 \choose \theta_1} <math>.

If the waveguide is stable, λ^N must not grow without limit. Solving the eigenvalue equation gives us a periodic solution of the form:

<math> \lambda^N = e^{\pm i N \phi} <math>,

where cos(φ) = 1 - d/(2f) = g. The value g is known as the stability parameter, and the solution is valid as long as g²-1 is negative. The condition for stability of the waveguide is thus |g|<1.

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.

The matrix formalism is also useful to describe Gaussian optics. If we have a Gaussian beam of wavelength λ, radius of curvature R and beam radius ω, it is possible to define a "complex beam parameter" q by:

<math> \frac{1}{q} = \frac{1}{R} - \frac{i\lambda}{\pi \omega^2} <math>.

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

<math> {q_2 \choose 1} = k \begin{pmatrix} A & B \\ C & D \end{pmatrix}

{q_1 \choose 1} <math>,

where k is a normalisation constant chosen to keep the second component of the ray vector equal to 1.