Ray transfer matrix analysis
Ray transfer matrix analysis (also known as ABCD matrix analysis) is a mathematical form for performing ray tracing calculations in sufficiently simple problems which can be solved considering only paraxial rays. Each optical element (surface, interface, mirror, or beam travel) is described by a 2×2 "ray transfer matrix" which operates on a vector describing an incoming light ray to calculate the outgoing ray. Multiplication of the successive matrices thus yields a concise ray transfer matrix describing the entire optical system. The same mathematics is also used in accelerator physics to track particles through the magnet installations  of a particle accelerator, see electron optics. This technique, as described below, is derived using the "paraxial approximation", which requires that all ray directions (directions normal to the wavefronts) are at small angles "θ" relative to the optical axis of the system, such that the approximation formula_1 remains valid. A small θ further implies that the transverse extent of the ray bundles ("x" and "y") is small compared to the length of the optical system (thus "paraxial"). Since a decent imaging system where this is "not" the case for all rays must still focus the paraxial rays correctly, this matrix  method will properly describe the positions of focal planes and magnifications, however aberrations still need to be evaluated using full ray-tracing techniques. 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. At any point along the optical train an optical axis is defined corresponding to a central ray; that central ray is propagated to define the optical axis further in the optical train which need not be in the same physical direction (such as when bent by  a prism or mirror). The transverse directions "x" and "y" (below we only consider the "x" direction) are then defined to be orthogonal to the optical axes applying. A light ray enters a component crossing its input plane at a distance "x" from the optical axis, traveling in a direction that makes an angle "θ" with the optical axis. After propagation to the output plane that ray is found at a distance "x" from the optical axis and at an angle "θ" with respect to it. "n" and "n" are the indices of refraction of the media in the input  and output plane, respectively. The ABCD matrix representing a component or system relates the output ray to the input according to where the values of the 4 matrix elements are thus given by and This relates the "ray vectors" at the input and output planes by the "ray transfer matrix" (RTM) M, which represents the optical component or system present 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: 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 different convention for the ray vectors can be employed. Instead of using "θ"≈sin "θ", the second element of the ray vector is "n" sin "θ", which is proportional not to the ray angle "per se" but to the transverse component of the wave vector. This alters the ABCD matrices given in the table below where refraction at an interface is involved. The use of transfer matrices in this manner  parallels the 2×2 matrices describing electronic two-port networks, particularly various so-called ABCD matrices which can similarly be multiplied to solve for cascaded systems. Some examples. 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: Thus the matrices must be ordered appropriately, with the last matrix premultiplying the second last, and so on until the first matrix is premultiplied by the second. Other matrices can be constructed to represent interfaces with media of different refractive indices, reflection from mirrors, etc. Eigenvalues of Ray  Transfer Matrix. A ray transfer matrix can be regarded as a linear canonical transformation. According to the eigenvalues of the optical system, the system can be classified into several classes. Assume the ABCD matrix representing a system relates the output ray to the input according to formula_12. We compute the eigenvalues of the matrix formula_13 that satisfy eigenequation formula_14, by calculating the determinant formula_15. Let formula_16, and we have eigenvalues formula_17. According to the values of formula_18 and formula_19, there are several possible cases. For example: Table of ray transfer matrices. for simple optical components Relation between geometrical ray optics  and wave optics. The theory of Linear canonical transformation implies the relation between ray transfermatrix (geometrical optics) and wave optics. Common Decomposition of Ray Transfer Matrix. There exist infinite ways to decompose a ray transfer matrix formula_30 into a concatenation of multiple transfer matrix. For example: Resonator stability. RTM analysis is particularly useful when modeling the behavior 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, 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 traveling down the waveguide will be periodically refocused and stay within the waveguide. To do so, we can find all the "eigenrays" of the system: the input ray  vector at each of the mentioned sections of the waveguide times a real or complex factor "λ" is equal to the output one. This gives: which is an eigenvalue equation: where I is the 2×2 identity matrix. We proceed to calculate the eigenvalues of the transfer matrix: leading to the characteristic equation where is the trace of the RTM, and is the determinant of the RTM. After one common substitution we have: where is the "stability parameter". The eigenvalues are the solutions of the characteristic equation. From the quadratic formula we find Now, consider a ray after "N" passes through  the system: If the waveguide is stable, no ray should stray arbitrarily far from the main axis, that is, "λ" must not grow without limit. Suppose formula_46. Then both eigenvalues are real. Since formula_47, one of them has to be bigger than 1 (in absolute value), which implies that the ray which corresponds to this eigenvector would not converge. Therefore, in a stable waveguide, formula_48 ≤ 1, and the eigenvalues can be represented by complex numbers: with the substitution "g" = cos("ϕ"). For formula_50 let formula_51 and formula_52 be the eigenvectors with respect to the eigenvalues formula_53 and formula_54 respectively , which span all the vector space because they are orthogonal, the latter due to formula_55. The input vector can therefore be written as for some constants formula_57 and formula_58. After "N" waveguide sectors, the output reads which represents a periodic function. Ray transfer matrices for Gaussian beams. The same matrices can also be used to calculate the evolution of Gaussian beams. propagating through optical components described by the same transmission matrices. If we have a Gaussian beam of wavelength formula_60, radius of curvature "R" (positive for diverging, negative for converging), beam spot size "w" and refractive index "n", it is  possible to define a complex beam parameter "q" by: ("R", "w", and "q" are functions of position.) If the beam axis is in the "z" direction, with waist at formula_62 and Rayleigh range formula_63, this can be equivalently written as This beam can be propagated through an optical system with a given ray transfer matrix by using the equation: where "k" is a normalization constant chosen to keep the second component of the ray vector equal to 1. Using matrix multiplication, this equation expands as and Dividing the first equation by the second eliminates the normalization constant: It is often  convenient to express this last equation in reciprocal form: Example: Free space. Consider a beam traveling a distance "d" through free space, the ray transfer matrix is and so consistent with the expression above for ordinary Gaussian beam propagation, i.e. formula_72. As the beam propagates, both the radius and waist change. Example: Thin lens. Consider a beam traveling through a thin lens with focal length "f". The ray transfer matrix is and so Only the real part of 1/"q" is affected: the wavefront curvature 1/"R" is reduced by the power of the lens 1/"f", while the lateral beam size "w  remains unchanged upon exiting the thin lens. Higher rank matrices. Methods using transfer matrices of higher dimensionality, that is 3×3, 4×4, and 6×6, are also used in optical analysis. In particular, 4×4 propagation matrices are used in the design and analysis of prism sequences for pulse compression in femtosecond lasers.