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 beam optics.n2 = final refractive index.n1 = initial refractive indexn2 = final refractive index. R e = R / cos ⁡ θ {displaystyle R_{e}=R/cos heta } effective radius of curvature in the sagittal plane (vertical direction)R = radius of curvature, R > 0 for concave, valid in the paraxial approximation θ {displaystyle heta } is the mirror angle of incidence in the horizontal plane.Only valid if the focal length is much greater than the thickness 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 = center thickness of lens. 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 beam 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 sin ⁡ θ ≈ θ {displaystyle sin heta approx heta } 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 abberations still need to be evaluated using full ray-tracing techniques. 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 x1 from the optical axis, traveling in a direction that makes an angle θ1 with the optical axis. After propagation to the output plane that ray is found at a distance x2 from the optical axis and at an angle θ2 with respect to it. n1 and n2 are the indices of refraction of the media in the input and output plane, respectively.