Ray transfer matrix analysis
Multiplication of the successive matrices thus yields a concise ray transfer matrix describing the entire optical system.
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.
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.
The ABCD matrix representing a component or system relates the output ray to the input according to
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 [citation needed] 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.
This alters the ABCD matrices given in the table below where refraction at an interface is involved.
where d is the separation distance (measured along the optical axis) between the two reference planes.
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:
Other matrices can be constructed to represent interfaces with media of different refractive indices, reflection from mirrors, etc.
A ray transfer matrix can be regarded as a linear canonical transformation.
[3] Assume the ABCD matrix representing a system relates the output ray to the input according to
effective radius of curvature in the sagittal plane (vertical direction) R = radius of curvature, R > 0 for concave, valid in the paraxial approximation θ is the mirror angle of incidence in the horizontal plane.
: coordinate of the goal There exist infinite ways to decompose a ray transfer matrix
: 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.
RTM analysis can now be used to determine the stability of the waveguide (and equivalently, the resonator).
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.
If the waveguide is stable, no ray should stray arbitrarily far from the main axis, that is, λN must not grow without limit.
, 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.
If the beam axis is in the z direction, with waist at z0 and Rayleigh range zR, this can be equivalently written as[8]
This beam can be propagated through an optical system with a given ray transfer matrix by using the equation[further explanation needed]:
where k is a normalization constant chosen to keep the second component of the ray vector equal to 1.
Consider a beam traveling a distance d through free space, the ray transfer matrix is
Consider a beam traveling through a thin lens with focal length f. The ray transfer matrix is
Methods using transfer matrices of higher dimensionality, that is 3 × 3, 4 × 4, and 6 × 6, are also used in optical analysis.
[9] In particular, 4 × 4 propagation matrices are used in the design and analysis of prism sequences for pulse compression in femtosecond lasers.
