'''Ray transfer matrix analysis''' (also known as '''ABCD matrix analysis''') is a mathematical form for performing [[Ray tracing (physics)|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{{nowrap|2 × 2}} '''ray transfer [[matrix (mathematics)|matrix]]''' which operates on a [[vector space|vector]] describing an incoming [[ray (optics)|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 ''{{mvar|θ''}} relative to the [[optical axis]] of the system, such that the approximation <{{math>\|1=sin \theta''θ'' \approx≈ \theta</math>''θ''}} remains valid. A small {{mvar|θ}} further implies that the transverse extent of the ray bundles (''{{mvar|x''}} and ''{{mvar|y''}}) is small compared to the length of the optical system (thus "paraxial"). Since a decent imaging system where this is ''{{em|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 [[Optical aberration|aberrations]] still need to be evaluated using full [[Ray tracing (physics)#Optical design|ray-tracing]] techniques.<ref>Extension of matrix methods to tracing (non-paraxial) meridional rays is included [http://spie.org/ETOP/1991/389_1.pdf here].</ref>
