System Matrix: Two Thin Lenses

The position of the image formed by a pair of thin lenses can be found by the matrix method. The general matrix method involves multiplying a vector form of the incident vergence successively by matrices representing (1)the refraction by the first lens, (2) the translation to the second lens, and (3) the refraction by the second lens. If the system matrix is calculated, it can be used to directly multiply the incident vergence to obtain the exit vergence. From that exit vergence the image distance is calculated.

This example is for a two thin lenses surrounded by air (n=1). It involves the powers of the lens and the separation d of the lenses. 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.

A default set of values for the parameters of this calculation is provided. You can see the default calculation by entering a 0 into one of the lens parameters, which will trigger the entry of the default parameters for the lenses. Any of them may be then changed to explore the behavior of the lens system.

The power for lens 1 is P₁ = m^-1

该计算的参数提供了一组默认值。你可以通过将其中一个透镜参数输入0来查看默认计算，这将触发透镜的默认参数输入。之后可以更改任何参数以探索透镜系统的特性。透镜1的功率为P1 = m - 1。

中文译文中的待填/计算数值依次对应：1：p1。实际数值以上方原输入框为准。

The power for lens 2 is P₂ = m^-1

透镜2的功率为 P₂ = m - 1

中文译文中的待填/计算数值依次对应：1：p2。实际数值以上方原输入框为准。

这个例子是关于两个薄透镜被空气（n=1）包围的情况。它涉及透镜的焦力和透镜之间的距离d。它还涉及主平面H1和H2。如同大多数普通几何光学一样，它仅适用于小角度（近轴光线）。该发展也遵循笛卡尔符号约定。

The lens separation d = m

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。

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透镜间距 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。

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 f_v2 = 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 = f_v2 + 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 f_v1 = -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. ****************************

Location of Image with System Matrix

系统矩阵中的像的位置

Reminder: Using the Cartesian sign convention, the object distance is typically a negative number since it points opposite to the direction of light travel. 提示：使用笛卡尔符号约定时，物距通常为负数，因为其方向与光的传播方向相反。

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 gives: kV'= k=. 进行指定的矩阵乘法得到：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。

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