因為 HyperPhysics 所使用之矩陣光學記號法和符號正負慣例與作者選取的不同，特別列於此處以免不必要之誤解及困擾︰

System Matrix

For systems of multiple thick lenses, it is sometimes useful to represent the system by a system matrix. The matrix is built up by multiplying the refraction matrices and translation matrices. The positions of the principal planes, the front and back surface powers, and the equivalent focal length of Gullstrand’s equation can be calculated from the system matrix.

This is the form of the system matrix used by Meyer-Arendt. An alternate form is used by Hecht.

Calculations

Alternate form of system matrix: Hecht form

Refraction and Translation Matrices

The system matrix for a thick lens is obtained by multiplying the translation matrix associated with the thickness of the lens times refraction matrix of the first surface and then multiplying by the refraction matrix of the back surface.

Refraction matrix development

Translation matrix development

Alternate form : Hecht form

 

如是當可解讀這個『變焦模型』之構想︰

Design of Scale-Model Eye

The scale model eye was developed by scaling an actual cross-sectional picture of an eye from Light & Vision by Mueller & Rudolph. Then a total distance of 2.4 cm from cornea to retina was adopted from Hecht, along with his values for the indices of refraction for all components. The actual measurements for radii of curvature and separation were scaled using a solver program (TK!Solver) until a parallel incoming ray fell on the retina. It is not known whether the eye shown by Mueller&Rudolph is typical.

Accommodation Model Being inherently a thick lens problem, image formation by the eye can be approached using a system matrix. By modeling the fixed portion of the eye with a matrix, assuming the front surface of the crystalline lens to be fixed, then the thickness and back surface power of the lens can be varied to study the accommodation process. By computing the system matrix for the entire imaging process, constraining the image to fall on the retina, the object distance on which the eye is focused can be calculated. (If an image is to be formed, the object distance must be negative since the Cartesian sign convention is used.) Alternatively, by choosing and object distance, the required back surface power can be calculated.

Scale model eye

Discussion of modeling accommodation

 

由於我們已經知道角膜的

【主平面表述】

主平面角膜表達式 = 後主平面 * 角膜 * 前主平面

= 後主平面 * 角膜後緣 * 角膜厚度 * 角膜前緣 * 前主平面

用前、後主平面作參考系，角膜光學矩陣表述可以簡化為

此處 = 主平面角膜表達式.C 。

 

實際等效於『曲面折射』。在此假設水晶體前緣為『平面折射』，變焦後之等效『半徑』是 以及『厚度』是 ，這兩者與水晶體的折射率 亦相關，終究是以能聚焦視網膜清晰成像為依歸。再者因房水之折射率 幾乎等於玻璃體的折射率 ，且設其相等，得︰

pi@raspberrypi:~ $ ipython3
Python 3.4.2 (default, Oct 19 2014, 13:31:11) 
Type "copyright", "credits" or "license" for more information.

IPython 2.3.0 -- An enhanced Interactive Python.
?         -> Introduction and overview of IPython's features.
%quickref -> Quick reference.
help      -> Python's own help system.
object?   -> Details about 'object', use 'object??' for extra details.

In [1]: from sympy import *

In [2]: from sympy.physics.optics import FreeSpace, FlatRefraction, ThinLens, GeometricRay, CurvedRefraction

In [3]: init_printing()

In [4]: R, d, N, n = symbols('R, d, N, n')

In [5]: 水晶體前部 = FlatRefraction(n, N)

In [6]: 水晶體前部
Out[6]: 
⎡1  0⎤
⎢    ⎥
⎢   n⎥
⎢0  ─⎥
⎣   N⎦

In [7]: 水晶體厚度 = FreeSpace(d)

In [8]: 水晶體厚度
Out[8]: 
⎡1  d⎤
⎢    ⎥
⎣0  1⎦

In [9]: 水晶體後部 = CurvedRefraction(-R, N, n)

In [10]: 水晶體後部
Out[10]: 
⎡    1      0⎤
⎢            ⎥
⎢-(N - n)   N⎥
⎢─────────  ─⎥
⎣   R⋅n     n⎦

In [11]: 水晶體 = 水晶體後部 * 水晶體厚度 * 水晶體前部

In [12]: 水晶體
Out[12]: 
⎡                  d⋅n       ⎤
⎢    1             ───       ⎥
⎢                   N        ⎥
⎢                            ⎥
⎢             ⎛N   d⋅(N - n)⎞⎥
⎢           n⋅⎜─ - ─────────⎟⎥
⎢-(N - n)     ⎝n      R⋅n   ⎠⎥
⎢─────────  ─────────────────⎥
⎣   R⋅n             N        ⎦

In [13]: 水晶體.det()
Out[13]: 1

In [14]: (水晶體.D - 水晶體.B * 水晶體.C).simplify()
Out[14]: 1

In [15]: 

 

能知其意指也！