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 same analysis is also used in accelerator physics to track particles through the magnet installations of a particle accelerator, see Beam optics.
The technique that is described below uses the paraxial approximation of ray optics, which means that all rays are assumed to be at a small angle (θ in radians) and a small distance (x) relative to the optical axis of the system.
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
where
and
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:
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