樹莓派、樹莓派之學習、樹莓派之教育

光的世界︰幾何光學三

就算只擷取維基百科什麼是『高斯光學』文本的一小段︰

Gaussian optics is a technique in geometrical optics that describes the behaviour of light rays in optical systems by using the paraxial approximation, in which only rays which make small angles with the optical axis of the system are considered.[1] In this approximation, trigonometric functions can be expressed as linear functions of the angles. Gaussian optics applies to systems in which all the optical surfaces are either flat or are portions of a sphere. In this case, simple explicit formulae can be given for parameters of an imaging system such as focal distance, magnification and brightness, in terms of the geometrical shapes and material properties of the constituent elements.

 

也得面對許多不同的概念 ── 『技術』technique 、『近軸近似』  paraxial approximation 、『平面』 flat 、『部份球面』 portions of a sphere ……。尚且此處『近軸近似』並未言明所謂之『軸』到底指什麼？它也叫作『光軸』，就是這個光學系統所有『部份球面』之『球心』所形成的『線』！

一般或許僅以一圖簡單示意︰

GO

 

既已清楚了『線』，此『面』也就是『光軸』之『旋轉』所產生的那個『部份球面』！

既然『圓錐曲線』數理早已知道

300px-Conic_Sections.svg

 

，亦且能形成『旋轉面』！甚或有更佳的『光學聚焦』性質，奈何卻要研究『球面』的耶！！

更何況人們本曉會發生問題的乎？？

球面像差

在光學中，球面像差是發生在經過透鏡折射或面鏡反射的光線，接近中心與靠近邊緣的光線不能將影像聚集在一個點上的現象。這在望遠鏡和其他的光學儀器上都是一個缺點。這是因為透鏡和面鏡必須滿足所需的形狀，否則不能聚焦在一個點上造成的。 球面像差與鏡面直徑的四次方成正比，與焦長的三次方成反比，所以他在低焦比的鏡子，也就是所謂的「快鏡」上就比較明顯。

對使用球面鏡的小望遠鏡，當焦比低於f/10時，來自遠處的點光源（例如恆星）就不能聚集在一個點上。特別是來自鏡面邊緣的光線比來自鏡面中心的光線更不易聚焦，這造成影像因為球面像差的存在而不能很尖銳的成象。所以焦比低於f/10的望遠鏡通常都使用非球面鏡或加上修正鏡。

在透鏡系統中，可以使用凸透鏡和凹透鏡的組合來減少球面像差，就如同使用非球面透鏡一樣。

Circle_caustic

來自球面鏡的球面像差

 

800px-Spherical-aberration-slice

平行光束通過透鏡後聚焦像的縱切面，上：負球面像差，中：無球面像差，下：正球面像差。鏡子位於圖的左側

 

371px-Spherical_aberration_2.svg

球面像差。一個理想的鏡面(頂端)，能經所有入射的光線匯聚在光軸上的一個點，但一個真實的鏡面(底端)會有球面像差：靠近光軸的光線會比離光軸較遠的光線較為緊密的匯聚在一個點上，因此光線不能匯聚在一個理想的焦點上(圖較為誇張)

 

Spherical-aberration-disk

一個 點光源 在負球面像差(上) 、無球面像差(中)、和正球面像差(下)的系統中的成像情形。左面的影相是在焦點內成像，右邊是在焦點外的成像

 

祇是一代之『技術』往往影響一代之『論述』。今日『精密機械』以及『射出成型』之進步，或許人世間已經是

非球面鏡

非球面鏡是指表面不是球面或者柱面的透鏡。在攝影里，包含非球面光學元件的透鏡被稱作「非球面鏡」。

相比簡單透鏡，非球面鏡的複雜表面可以減少或者消除球差或者其他像差。單一非球面鏡可以替代很多的複雜球面鏡系統。這樣的系統設計會更小，更輕，甚至有時候會更便宜。非球面鏡元件被用來設計多光學元件的廣角鏡或者標準鏡頭，以此來減少像差。它們也可以和反射光學元件相結合，如施密特修正板，這種非球面鏡就被用在施密特攝星儀、施密特-卡塞格林望遠鏡中。小的模鑄非球面鏡通常可以用來準直雷射二極體。

非球面鏡也通常被用來製造眼鏡。這種設計可以使眼鏡更薄，同時觀察者會感到戴眼鏡的人眼睛變形更小。非球面眼鏡並不會比「最優形式」球面眼鏡有更好的視覺效果，不過在沒有降低光學性能的條件下，使眼鏡厚度更薄，表面更平。

170px-Pfeilhöhe.svg

雙凸非球面鏡

 

的天下。也許不久的未來，即將邁入

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]

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  \epsilon_r of the material composing the lens falls from 2 at its center to 1 at its surface (or equivalently, the refractive index  n falls from  {\sqrt {2}} to 1), according to

n={\sqrt {\epsilon _{r}}}={\sqrt {2-\left({\frac {r}{R}}\right)^{2}}}

where R 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.

Maxwell’s fish-eye lens

Maxwell’s fish-eye lens is also an example of the generalized Luneburg lens. The fish-eye, which was first fully described by Maxwell in 1854[5] (and therefore pre-dates Luneburg’s solution), has a refractive index varying according to

n={\sqrt {\epsilon _{r}}}={\frac {n_{0}}{1+\left({\frac {r}{R}}\right)^{{2}}}}.

It focuses each point on the spherical surface of radius R to the opposite point on the same surface. Within the lens, the paths of the rays are arcs of circles.

Publication and attribution

The properties of this lens are described in one of a number of set problems or puzzles in the 1853 Cambridge and Dublin Mathematical Journal.[6] The challenge is to find the refractive index as a function of radius, given that a ray describes a circular path, and further to prove the focusing properties of the lens. The solution is given in the 1854 edition of the same journal.[5] The problems and solutions were originally published anonymously, but the solution of this problem (and one other) were included in Niven’s The Scientific Papers of James Clerk Maxwell,[7] which was published eleven years after Maxwell’s death.

220px-Luneburg_lens.svg

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

 

220px-Maxwells_fish-eye_lens.svg

Cross-section of Maxwell’s fish-eye lens, with blue shading representing increasing refractive index

 

之際元。然而所謂之『近軸』與『角度近似』之義，『幾何光學』的實用之理，依舊持續也！！？？

Paraxial approximation

In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system (such as a lens).[1] [2]

A paraxial ray is a ray which makes a small angle (θ) to the optical axis of the system, and lies close to the axis throughout the system.[1] Generally, this allows three important approximations (for θ in radians) for calculation of the ray’s path, namely:[1]

\sin \theta \approx \theta ,\quad \tan \theta \approx \theta \quad {\text{and}}\quad \cos \theta \approx 1.

The paraxial approximation is used in Gaussian optics and first-order ray tracing.[1] Ray transfer matrix analysis is one method that uses the approximation.

In some cases, the second-order approximation is also called “paraxial”. The approximations above for sine and tangent do not change for the “second-order” paraxial approximation (the second term in their Taylor series expansion is zero), while for cosine the second order approximation is

{\displaystyle \cos \theta \approx 1 - { \theta^2 \over 2 } \ .}

The second-order approximation is accurate within 0.5% for angles under about 10°, but its inaccuracy grows significantly for larger angles.[3]

For larger angles it is often necessary to distinguish between meridional rays, which lie in a plane containing the optical axis, and sagittal rays, which do not.

600px-Small_angle_compare_error.svg

The error associated with the paraxial approximation. In this plot the cosine is approximated by 1 – θ2/2.