樹莓派、樹莓派之學習、樹莓派之教育

光的世界︰幾何光學四

馬克士威曾斷言『光』是一種『電磁波』，後由海因里希‧赫茲作實驗證明。因此『光』之行為必得滿足馬克士威方程組，如是知道『光』在不同介質的界面上會同時發生『反射』與『折射』現象。那麼『高斯光學』為什麼不會因產生『萬花筒』效應以致無法清晰成像耶？？

Kaleidoscope

A kaleidoscope is an optical instrument, typically a cylinder with mirrors containing loose, colored objects such as beads or pebbles and bits of glass. As the viewer looks into one end, light entering the other end creates a colorful pattern, due to repeated reflection in the mirrors.

Etymology

Coined in 1817 by Scottish inventor Sir David Brewster,[1] “kaleidoscope” is derived from the Ancient Greek καλός (kalos), “beautiful, beauty”,[2] εἶδος (eidos), “that which is seen: form, shape”[3] and σκοπέω (skopeō), “to look to, to examine”,[4] hence “observation of beautiful forms.”[5]

Description

A kaleidoscope is an optical instrument in which bits of glass, held loosely at the end of a rotating tube, are shown in continually changing symmetrical forms by reflection in two or more mirrors set at angles to each other. A kaleidoscope operates on the principle of multiple reflection, where several mirrors are placed at an angle to one another. Typically there are three rectangular mirrors set at 60° to each other so that they form an equilateral triangle, but other angles and configurations are possible. The 60° angle generates an infinite regular grid of duplicate images of the original, with each image having six possible angles and being a mirror image or an unreversed image.

As the tube is rotated, the tumbling of the colored objects presents varying colors and patterns. Arbitrary patterns show up as a beautiful symmetrical pattern created by the reflections. A two-mirror kaleidoscope yields a pattern or patterns isolated against a solid black background, while the three-mirror (closed triangle) type yields a pattern that fills the entire visual field. For a deeper discussion, see the article about reflection symmetry.

Modern kaleidoscopes are made of brass tubes, stained glass, wood, steel, gourds or almost any material an artist can use. The section containing objects to be viewed is called the “object chamber” or “object cell”, and may contain almost any material. Sometimes the object cell is filled with a viscous liquid so the items float and move gracefully through the object cell in response to slight movements from the viewer.

An alternative version of the instrument adapts a telescope, omits the object cell, and allows the observer to view current surroundings; the teleidoscope can transform a portion of any scene into an abstract repeating mosaic.

 

1280px-Kaleidoscope_San_Diego

A woman looks into a large kaleidoscope

 

800px-Kaleidoscopes

Patterns when seen through a kaleidoscope tube

 

與其談論推導複雜電磁學方程式之繁瑣無趣，在此就直接援引

《Reflection at a dielectric boundary》

之文本的結果來配飯下酒︰

 

img2490

 

img2563

Figure 57 shows the coefficients of reflection (solid curves) and transmission (dashed curves) for oblique incidence from air (n_1 = 1.0) to glass (n_2 = 1.5). The left-hand panel shows the wave polarization for which the electric field is parallel to the boundary, whereas the right-hand panel shows the wave polarization for which the magnetic field is parallel to the boundary. In general, it can be seen that the coefficient of reflection rises, and the coefficient of transmission falls, as the angle of incidence increases. Note, however, that for the second wave polarization there is a particular angle of incidence, know as the Brewster angle, at which the reflected intensity is zero. There is no similar behaviour for the first wave polarization.

 

R = { \left( \frac{\alpha - \beta}{\alpha + \beta} \right) }^2

T = \alpha \beta { \left( \frac{2}{\alpha + \beta} \right) }^2

R + T =1

\alpha = \frac{\cos ( \theta_t )}{\cos ( \theta_i )} \ , \ \beta = \frac{n_2}{n_1}

 

這裡假借 57 圖中『空氣、玻璃』界面為例，僅以『近軸角度』 \theta_i = 10^{\circ} = \frac{\pi}{18} \approx 0.174 為界。依據『司乃耳折射定律』 \sin ( \theta_t ) = \frac{n_1}{n_2} \sin ( \theta_i) ，可得『折射角』 \theta_t \approx 0.116 。所以

\alpha \approx 1.009 \ , \ \beta = 1.5 ，可知『反射比率』約莫

R \approx 0.038 也，故而難得多次反射影響成像乎！！

雖然這個系列文章主要探討『屈光成像裝置』── 顯微鏡、望遠鏡 、照相機… ─── 之『矩陣光學』。無礙喜愛體驗者以樹莓派相機為『眼』，藉著參考文本為範

‧ Plane Mirror Reflection Experiment
‧ Reflection and the Ray Model of Light – Lesson 2 – Image Formation in Plane Mirrors

‧ Kaleidoscope Mirror Systems

，歡度靜思快樂時光。尚且可以研究如果兩鏡交角為 \theta ，那麼

Question from Peter, a student:

The reflecting surfaces of two intersecting flat mirrors are at an angle θ (0° < θ < 90°). For a light ray that strikes the horizontal mirror, show that the emerging ray will intersect the incident ray at an angle β = 180° – 2θ.
Hi Peter.

I drew a diagram and labelled those and other angles with variables from the Greek alphabet, using rules of reflection to indicate which angles have the same measure as other angles:

 

，此理又將如何進入『成像數』 n = \frac{{360}^{\circ}}{\theta} -1 之世界哩？？！！