An ABCD matrix or ray transfer matrix  is a 2-by-2 matrix associated with an optical element which can be used for describing the element's effect on a laser beam. It can be used both in ray optics, where geometrical rays are propagated, and for propagating Gaussian beams. The paraxial approximation is always required for ABCD matrix calculations, i.e., the involved beam angles or divergence angles must stay small for the calculations to be accurate.
Originally, the concept was developed in geometrical optics for calculating the propagation of light rays with some transverse offset r and offset angle θ from a reference axis (Figure 1). As long as the angles involved are small enough (→ paraxial approximation), there is a linear relation between the r and θ coordinates before and after an optical element. The following equation can then be used for calculating how these parameters are modified by an optical element:
where the primed quantities (left-hand side) refer to the beam after passing the optical component. The ABCD matrix is a characteristic of each optical element.
This shows that the offset r remains unchanged, whereas the offset angle θ experiences a change in proportion to r.
Propagation through free space over a distance d is associated with the matrix
which shows that the angle remains unchanged, whereas the beam offset is increasing or decreasing according to the angle.
Further examples for ABCD matrices are given below.
For situations where beams propagate through dielectric media, it is convenient to use a modified kind of beam vectors, where the lower component (the angle) is multiplied by the refractive index. This can somewhat simplify the ABCD matrices for certain situations.
Propagation of Gaussian Beams
ABCD matrices can also be used for calculating the effect of optical elements on the parameters of a Gaussian beam. A convenient quantity for that purpose is the complex q parameter, which contains information on both the beam radius w and the radius of curvature R of the wavefronts:
The following equation shows how the q parameter is modified by an optical element:
ABCD Matrices of Important Optical Elements
The following list gives the ABCD matrices of frequently used optical elements.
Air space with length d:
(For propagation in a transparent medium, the length d has to be divided by the refractive index n, if the above mentioned modified definition is used where the lower component (the angle) is multiplied by the refractive index.)
Lens with focal length f (where positive f applies for a focusing lens):
Curved mirror with curvature radius R (>0 for concave mirror), incidence angle θ in the horizontal plane:
with Re = R cos θ in the tangential plane (horizontal direction) and Re = R / cos θ in the sagittal plane (vertical direction).
where the radially varying refractive index is:
Various textbooks (see e.g. Ref. ) specify the ABCD matrices for other types of optical components.
Combining Multiple Optical Elements
If a beam propagates through several optical elements (including any air spaces in between), this means that the (r θ) vector is subsequently multiplied by various matrices. Instead, a single matrix may be used, which is the matrix product of all the single matrices. Note that the first optical element must be on the right-hand side of that product – matrix multiplications are not commutative, and the same holds for optical elements.
- combined matrix for free-space propagation length with distance d, followed by a lens with focal length f:
- combined matrix for a lens with focal length f, followed by free-space propagation length with distance d:
Some typical applications of the ABCD matrix algorithm are:
- It is often of interest how a laser beam propagates through some optical setup. Both the geometric path of a ray and the evolution of the beam radius can be calculated.
- The changes of beam parameters within one complete round trip in a resonator can be described with an ABCD matrix. The transverse resonator modes can then be obtained from the matrix components.
- An extended algorithm, involving an ABCDEF matrix (a 3-by-3 matrix with some constant components), can be used for calculating the alignment sensitivity of a laser resonator .
The ABCD matrix method should not be confused with a different matrix method for calculating the reflection and transmission properties of dielectric multilayer coatings.
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