為著方便討論起見，首先引用

《光的世界︰【□○閱讀】話眼睛《二》》文本之

眼睛之古典球面近似『模型參數』

角膜→房水【前室】→虹膜【瞳孔】→晶狀體→玻璃體【後部】→視網膜

【角膜】 cornea
非球面
折射率︰1.376
前緣半徑︰ 7.8 mm
後緣半徑︰6.4 mm
厚度︰ 0.6 mm

【房水】 aqueous humor
折射率︰1.336
厚度︰3.0 mm

【瞳孔】 pupil
直徑變化︰ 1.5 mm ~ 8.0 mm

【晶狀體】 crystalline lens
折射率︰梯度， 1.386 ~ 1.406
前緣半徑︰10.1 mm
後緣半徑︰6.1 mm
厚度︰4.0 mm

【玻璃體】 vitreous humor
厚度︰16.9 mm

 

假設晶珠可用

晶珠前緣球面折射半徑→晶珠厚度→晶珠後緣球面折射半徑

之光學矩陣構件模擬︰

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, RayTransferMatrix

In [3]: init_printing()

In [4]: 房水折射率 = 1.336

In [5]: 晶珠折射率低 = 1.386

In [6]: 晶珠折射率高 = 1.406

In [7]: 晶珠前緣半徑 = 10.1

In [8]: 晶珠後緣半徑 = -6.1

In [9]: 晶珠厚度 = 4.0

In [10]: 玻璃體折射率 = 1.336

In [11]: 晶珠前部 = CurvedRefraction(晶珠前緣半徑, 房水折射率, 晶珠折射率低)

In [12]: 晶珠前部
Out[12]: 
⎡         1                    0        ⎤
⎢                                       ⎥
⎣-0.00357178575000356  0.963924963924964⎦

In [13]: 晶珠中段 = FreeSpace(晶珠厚度)

In [14]: 晶珠中段
Out[14]: 
⎡1  4.0⎤
⎢      ⎥
⎣0   1 ⎦

In [15]: 晶珠後部 = CurvedRefraction(晶珠後緣半徑, 晶珠折射率低, 玻璃體折射率)

In [16]: 晶珠後部
Out[16]: 
⎡         1                   0       ⎤
⎢                                     ⎥
⎣-0.00613527044272109  1.0374251497006⎦

In [17]: 晶珠 = 晶珠後部 * 晶珠中段 * 晶珠前部

In [18]: 晶珠
Out[18]: 
⎡ 0.985712856999986    3.85569985569986 ⎤
⎢                                       ⎥
⎣-0.00975307532295808  0.976344238639321⎦

In [19]: 晶珠前部 = CurvedRefraction(晶珠前緣半徑, 房水折射率, 晶珠折射率高)

In [20]: 晶珠後部 = CurvedRefraction(晶珠後緣半徑, 晶珠折射率高, 玻璃體折射率)

In [21]: 晶珠後部 * 晶珠中段 * 晶珠前部
Out[21]: 
⎡0.980282523273664   3.80085348506401 ⎤
⎢                                     ⎥
⎣-0.013607662259734  0.967353030338363⎦

In [22]: 晶珠中段 = FreeSpace(8.0)

In [23]: 晶珠後部 * 晶珠中段 * 晶珠前部
Out[23]: 
⎡ 0.960565046547329   7.60170697012802 ⎤
⎢                                      ⎥
⎣-0.0134383013867043  0.934706060676725⎦

In [24]: 晶珠中段 = FreeSpace(2.0)

In [25]: 晶珠後部 * 晶珠中段 * 晶珠前部
Out[25]: 
⎡ 0.990141261636832   1.90042674253201 ⎤
⎢                                      ⎥
⎣-0.0136923426962489  0.983676515169181⎦

In [26]: 

 

為何不管以最高、最低的晶珠折射率估算，它的屈光力變化也只有 3、4 dpt 呢？甚至改變它的厚度，彷彿也沒有什麼用的耶？？當真自然造物之神奇，早已走上了藉著折射率梯度屈光的道路！！

Gradient-index optics

Gradient-index (GRIN) optics is the branch of optics covering optical effects produced by a gradual variation of the refractive index of a material. Such variations can be used to produce lenses with flat surfaces, or lenses that do not have the aberrations typical of traditional spherical lenses. Gradient-index lenses may have a refraction gradient that is spherical, axial, or radial.

A gradient-index lens with a parabolic variation of refractive index (n) with radial distance (x). The lens focuses light in the same way as a conventional lens.

In nature

The lens of the eye is the most obvious example of gradient-index optics in nature. In the human eye, the refractive index of the lens varies from approximately 1.406 in the central layers down to 1.386 in less dense layers of the lens (Hecht 1987, p. 178). This allows the eye to image with good resolution and low aberration at both short and long distances (Shirk et al., 2006).

Another example of gradient index optics in nature is the common mirage of a pool of water appearing on a road on a hot day. The pool is actually an image of the sky, apparently located on the road since light rays are being refracted (bent) from their normal straight path. This is due to the variation of refractive index between the hot, less dense air at the surface of the road, and the denser cool air above it. The variation in temperature (and thus density) of the air causes a gradient in its refractive index, causing it to increase with height (Tsiboulia, 2003). This index gradient causes refraction of light rays (at a shallow angle to the road) from the sky, bending them into the eye of the viewer, with their apparent location being the road’s surface.

The Earth’s atmosphere acts as a GRIN lens, allowing observers to see the sun for a few minutes after it is actually below the horizon, and observers can also view stars that are below the horizon (Tsiboulia, 2003). This effect also allows for observation of electromagnetic signals from satellites after they have descended below the horizon, as in radio occultation measurements.

 

有興趣者可以讀讀

Abstract

A new lens model based on the gradient-index Luneburg lens and composed of two oblate half spheroids of different curvatures is presented. The spherically symmetric Luneburg lens is modified to create continuous isoindicial contours and to incorporate curvatures that are similar to those found in a human lens. The imaging capabilities of the model and the changes in the gradient index profile are tested for five object distances, for a fixed geometry and for a fixed image distance. The central refractive index decreases with decreasing object distance. This indicates that in order to focus at the same image distance as is required in the eye, a decrease in refractive power is needed for rays from closer objects that meet the lens surface at steeper angles compared to rays from more distant objects. This ensures a highly focused image with no spherical aberration.

© 2015 Optical Society of America

 

想像類似人眼之人工創作物，進入 GRIN 的天地乎☆☆

Luneburg lens

A Luneburg lens (originally Lüneburg lens, often incorrectly spelled Luneberg lens) is a spherically symmetric gradient-index lens. A typical Luneburg lens’s refractive index n decreases radially from the center to the outer surface. They can be made for use with electromagnetic radiation from visible light to radio waves.

For certain index profiles, the lens will form perfect geometrical images of two given concentric spheres onto each other. There are an infinite number of refractive-index profiles that can produce this effect. The simplest such solution was proposed by Rudolf Luneburg in 1944.^[1] Luneburg’s solution for the refractive index creates two conjugate foci outside of the lens. The solution takes a simple and explicit form if one focal point lies at infinity, and the other on the opposite surface of the lens. J. Brown and A. S. Gutman subsequently proposed solutions which generate one internal focal point and one external focal point.^[2][3] These solutions are not unique; the set of solutions are defined by a set of definite integrals which must be evaluated numerically.^[4]

Cross-section of the standard Luneburg lens, with blue shading proportional to the refractive index

Designs

Luneburg’s solution

Each point on the surface of an ideal Luneburg lens is the focal point for parallel radiation incident on the opposite side. Ideally, the dielectric constant  of the material composing the lens falls from 2 at its center to 1 at its surface (or equivalently, the refractive index  falls from  to 1), according to

where  is the radius of the lens. Because the refractive index at the surface is the same as that of the surrounding medium, no reflection occurs at the surface. Within the lens, the paths of the rays are arcs of ellipses.