定义：
一个2*2矩阵，用来描述光学元件对激光光束的作用。
ABCD矩阵或者光线传输矩阵是一个2*2矩阵，是描述某一光学元件在激光光束中的作用。可以用于光线光学，其中光以几何的射线传输，或者在高斯光束传输时可以用到。ABCD矩阵计算时通常需要采用傍轴近似，也就是光束角度或者发散角在计算过程中是非常小的。 

光线光学 
最初，为了计算横向偏移为 r，偏移角为θ 的几何光束的传输发展了这一概念。当角度很小时（参阅傍轴近似），在经过光学元件之前与之后，坐标r和θ 之间具有线性关系。下面的矩阵方程可以用来计算该光学元件对参数的改变情况： 

其中加引号的量（方程左侧）代表光束经过光学元件后的坐标值。ABCD矩阵是每一光学元件的特征量。 
例如，焦距为f的薄透镜的ABCD矩阵为： 

这表明偏移r 是不变的，而偏移角θ 的变化正比于r。 
在自由空间中传输d 距离，用矩阵表示为： 

表明偏移角不变，而偏移r根据角度会增大或者减小。 
下面有更多关于ABCD矩阵的例子。 
光束在电介质中传输时，可以采用一个更方便的修正后的光束矢量，即将角度变为其与折射率的乘积。这在有些情况下可以简化矩阵。 

高斯光束的传输 
ABCD矩阵可用来计算光学元件对高斯光束参数的影响。为了简化计算需引入参数q，包含了光束半径 w和波前的曲率半径R的信息： 

下面的方程表示参数q经过光学元件后的变化： 
 

重要光学元件的ABCD矩阵 
下面给出一些常用光学元件的ABCD矩阵。 
空气中传输距离d后： 

（如果在透明介质中传输，长度需要除以折射率，如果采用以上提到的修正的定义，那么下面的成分，也就是角度，需要乘以折射率。） 
焦距为f的透镜（f大于0代表会聚透镜）： 

弯曲半径为R的镜子（>0代表凹透镜），水平面上的入射角为θ： 

其中Re = R cos θ 是在切平面上（水平方向），而矢状面（竖直方向）则有Re = R / cos θ。 
管： 

其中径向变化的折射率为：

许多教科书中给出了很多其它光学元件的ABCD矩阵。 

多个光学元件的结合 
当光束通过几个光学元件后（包含其间的空气），这表明矢量 (r θ) 需要乘以多个矩阵。可以将这些单个矩阵的乘积用另一个矩阵表示。一定要注意的是，第一个透过的光学元件的矩阵是处于矩阵乘积的最右侧。 

典型应用 
ABCD矩阵算法的一些典型应用包括： 

• 很多时候需要研究激光光束通过一些光学装置后的情况。光纤的几何路径和光束半径的演化都可以通过这种算法计算出来。 
• 光束在谐振腔中循环一周后参数的变化也可以由ABCD矩阵描述。横向的谐振腔模式可以通过矩阵元素得到。 
• 一种扩展的算法采用了ABCDEF矩阵（3*3矩阵，其中包含一些常数），它可以用来计算激光器谐振腔的校准灵敏度。 

注意不要将ABCD矩阵与计算多层介质膜的反射和透射性质的矩阵相混淆。 

Definition: a 2-by-2 matrix describing the effect of an optical element on a laser beam

Alternative term: ray transfer matrix

An ABCD matrix [1] 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.

Ray Optics

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:

Figure 1: Definition of r and θ before an after an optical system.

where the primed quantities (left-hand side) refer to the beam after passing the optical component. The ABCD matrix (also called ray transfer matrix) is a characteristic of each optical element.

For example, a thin lens with focal length f has the following ABCD matrix:

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.

One can show that the determinant of an ABCD matrix (A D − B C) always must be 1 as long as the refractive index is the same on the input and output side; otherwise, it is the input refractive index divided by the output refractive index.

Modified Matrices

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. In many cases of free-space optics it makes no difference, since the beams are considered only at positions in air where n ≈ 1. However, equations for interfaces between different media, for example, are affected.

The determinant of such a modified ABCD matrix (A D − B C) is always 1.

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 R_e = R cos θ in the tangential plane (horizontal direction) and R_e = R / cos θ in the sagittal plane (vertical direction).

Gaussian duct:

where the radially varying refractive index is

and the modified definition of beam vectors – with the angle multiplied with the refractive index (see above) – is used.

Various textbooks (see e.g. Ref. [4]) 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.

Example:

• 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:

Typical Applications

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 [3].

The ABCD matrix method should not be confused with a different matrix method for calculating the reflection and transmission properties of dielectric multilayer coatings.

Bibliography