Bilingual edition: English is preserved and Chinese follows each unit. Terminology uses the confirmed v20260916 glossary; automated semantic review remains traceable.
中英双语版：英文原文完整保留，中文紧随对应单元；术语采用 v20260916 确认表，自动语义审校结果可追溯。

Image from System Matrix

系统矩阵成像

The position of the image formed by a thick lens can be found by using the system matrix. This involves multiplying a vector form of the incident vergence successively by three matrices representing (1)the refraction by the first surface, (2)the translation to the second surface, and (3) the refraction by the second surface. Those three operations can be combined to form the system matrix. The entrance vergence can then be just multiplied times the system matrix to determine the exit vergence and from that vergence, the image distance is calculated.

厚透镜所成像的位置可通过系统矩阵来确定。这涉及将入射的矢量形式的发散光依次乘以三个矩阵，分别代表（1）第一表面的折射，（2）到第二表面的平移，以及（3）第二表面的折射。这三个操作可以组合成系统矩阵。然后，入射发散光可以简单地乘以系统矩阵以确定出射发散光，从而计算出像距。

The example is for a lens with index of refraction n surrounded by air (n=1). It involves the surface powers of the lens and the thickness d between the vertexes. It also involves the principal planes H1 and H2. As with most ordinary geometrical optics, it is applicable only for small angles (paraxial rays). The development also follows the cartesian sign convention.

For glass with index of refraction
nlens = n =

对于折射率为n的透镜，有n =
中文译文中的待填/计算数值依次对应：1：n。实际数值以上方原输入框为准。

the front surface power for lens radius
R1 = m is P1 = m-1

透镜半径 R₁ 的前表面功率为 m 时，P₁ = m - 1
中文译文中的待填/计算数值依次对应：1：r1 2：p1。实际数值以上方原输入框为准。

and the back surface power for lens radius
R2 = m is P2 = m-1

以及透镜半径R₂ = m时的后表面功率P₂ = m - 1
中文译文中的待填/计算数值依次对应：1：r2 2：p2。实际数值以上方原输入框为准。

Lens thickness d = m

(Note that for a double convex lens, the front surface radius R1 is positive and the back surface radius R2 is negative according to the Cartesian sign convention.)

（注意，对于双凸透镜，根据笛卡尔符号约定，前表面半径R1为正，后表面半径R2为负。）
透镜的厚度 d = m
中文译文中的待填/计算数值依次对应：1：d。实际数值以上方原输入框为准。

The symbols used by Meyer-Arendt for the elements of the system matrix will be used.

b = a =
d = c =
梅耶-阿伦特用于系统矩阵元素的符号将被使用。
中文译文中的待填/计算数值依次对应：1：mb 2：ma 3：md 4：mc。实际数值以上方原输入框为准。
**********

A number of other characteristics of the thick lens may be calculated from the system matrix.

厚透镜的许多其他特性可以从系统矩阵中计算出来。

The right-hand, or equivalent focal length of the lens is just the reciprocal of matrix element a: f2 = 1/a = m.

透镜的右手法则，或等效焦距，就是矩阵元素a的倒数：f₂ = 1/a = m。
中文译文中的待填/计算数值依次对应：1：f2。实际数值以上方原输入框为准。

The distance V2H2 from the right vertex to the associated principal plane is given by V2H2= (c-1)/a = m.

从右顶点到相关主平面的距离V₂H₂由V₂H₂ = (c-1)/a = m给出。
中文译文中的待填/计算数值依次对应：1：v2h2。实际数值以上方原输入框为准。

The right vertex focal length (or back focal length) is given by fv2 = c/a = m.

右顶点焦距（或后焦距）由 f v2 = c/a = m 给出。
中文译文中的待填/计算数值依次对应：1：fv2。实际数值以上方原输入框为准。

Note that the equivalent focal length f2 is the sum of the back vertex focal length and the distance from the vertex to the principal plane: f2 = fv2 + V2H2. Similar relationships exist for the left-hand or front vertex.

注意，等效焦距 f₂ 是后顶点焦距与顶点到主平面距离之和：f₂ = f_v2 + V₂ H₂。类似的关系也适用于左顶点或前顶点。

The left vertex focal length (or front focal length) is given by fv1 = -b/a = m.

左顶点焦距（或前焦距）由 f_v1 = -b/a = m 给出。
中文译文中的待填/计算数值依次对应：1：fv1。实际数值以上方原输入框为准。

The distance V1H1 from the left vertex to the associated principal plane is given by V1H1= (1-b)/a = m.

从左顶点到相关主平面的距离V₁H₁由V₁H₁ = (1 - b)/a = m给出。
中文译文中的待填/计算数值依次对应：1：v1h1。实际数值以上方原输入框为准。

The equivalent front focal length f1 is then given by f1 = -(b/a + V1H1) = m. **********

For object* distance o = m, the value of the input vergence V = .

对于物距 o = m 的物体，输入的 vergence 值 V = 。
中文译文中的待填/计算数值依次对应：1：od 2：vi。实际数值以上方原输入框为准。

Performing the indicated matrix multiplication at left gives:
kV'=
k=.

进行左边所示的矩阵乘法得到：kV'= k= 。
中文译文中的待填/计算数值依次对应：1：kv 2：k。实际数值以上方原输入框为准。

Dividing both by k gives the exit vergence V' = m-1

将两边都除以k得到出射 vergence V' = m - 1
中文译文中的待填/计算数值依次对应：1：kp。实际数值以上方原输入框为准。

so the image distance i = m

因此，像距 i = m
中文译文中的待填/计算数值依次对应：1：id。实际数值以上方原输入框为准。

则等效前焦点焦距 f₁ 为 f₁ = -(b/a + V₁ H₁ ) = m。
中文译文中的待填/计算数值依次对应：1：f1。实际数值以上方原输入框为准。

*Using the Cartesian sign convention, the object distance is typically a negative number since it points opposite to the direction of light travel. The input vergence is therefore negative (divergent).

根据笛卡尔符号约定，物体距离通常是一个负数，因为它方向与光的传播方向相反。因此，入射发散度为负数。

The equivalent power for the thick lens can be calculated from Gullstrand's equation:

厚透镜的等效功率可通过Gullstrand方程计算：

Note that the calculation does not take into account the change in lens thickness with the angle of the incoming ray. It is typical to do the calculation only for the paraxial rays where the departure from full thickness is negligible.

需要注意的是，该计算未考虑入射光线角度变化时透镜厚度的变化。通常只对近轴光线进行计算，因为偏离完整厚度的情况可以忽略不计。
Vergence Example
汇聚示例
Matrix definitions
矩阵定义
Example for two thin lenses
两个薄透镜示例
此例研究空气中（空气的折射率为1）折射率为n的透镜，涉及透镜两个表面的屈光力、两顶点之间的厚度d以及主平面H₁和H₂。与通常的几何光学分析一样，这里的结论只适用于小角度的近轴光线，并采用笛卡尔符号约定。
 Index

Lens concepts

Thick lens concepts
索引透镜概念 厚透镜概念
 
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