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11 | 6 月 | 2017 | FreeSandal

GoPiGo 小汽車︰格點圖像算術《什麼是影像》一

物體無論自己發光,或是反射、漫射周遭光源,都會依據『惠更斯原理』而行︰

200px-Huygens_principle

EnvelopeAnim

220px-Plane_wave_wavefronts_3D.svg

Lens_and_wavefronts

200px-Refraction_-_Huygens-Fresnel_principle.svg

220px-Huygens_Refracted_Waves

200px-Refraction_on_an_aperture_-_Huygens-Fresnel_principle.svg

250px-FresnelDiff9_3_PEM

Circular_Aperture_Fresnel_Diffraction_high_res

惠更斯原理
在『波前』wavefront 上的每一個點都可以將它看成是產生『球面次波』Spherical secondary waves 的『點波源』,而在這『之後』任何時刻的『波前』則可看作是這一些『相同相位子波』的『包絡』Envlope 『』或者『』。

那麼什麼是『包絡』的呢?從幾何學上講,一個『曲線族』的『包絡線』與該曲線族中的每一條曲線都『相切』tangent to 於某一點。一七三四年法國數學家亞歷克西斯‧克勞德‧克萊羅 Alexis Claude de Clairault 提出 y(x)=x\frac{dy}{dx}+f\left(\frac{dy}{dx}\right) 方程式。如果將該方程式對變數 x 再次作『微分』 得到 0=\left(x+f'\left(\frac{dy}{dx}\right)\right)\frac{d^2 y}{dx^2},因此 0=\frac{d^2 y}{dx^2}  或者 0=x+f'\left(\frac{dy}{dx}\right)。假使 0=\frac{d^2 y}{dx^2},那麼 \frac{dy}{dx} = C 是一個『常數』,將之代入原方程式得到『曲線族y(x)=Cx+f(C) 的一般解。如果 0=x+f'\left(\frac{dy}{dx}\right),它的解是上述曲線族的『包絡線』。舉例而言,下圖是 f(p) = p^2 的圖示

120px-Solutions_to_Clairaut's_equation_where_f(t)=t^2

其次『波前』的形狀可以被經過的『光學系統』所改變;而『相位相同』是講在 t 時間的波前『次波』都經過了『相同』的『時距\Delta t,形成了 t^{'} = t + \Delta t 新的波前。

藉著這原理,惠更斯給出了波的『直線傳播』與『球面傳播』的『定性』解釋,並且推導出了『反射定律』與『折射定律』。但是他卻不能解釋,為什麼當光波遇到『銳邊』、『小孔』或『狹縫』時,會偏離了直線傳播,也就是說會發生『繞射』現象。除此之外『惠更斯原理』假設了『次波』只會朝著『行進方向』傳播;然而他並沒有解釋為什麼它們不可以朝反方向傳播的呢?

法國物理學家奧古斯丁‧菲涅耳 Augustin Fresnel ,是『波動光學理論』的主要創建者之一,在惠更斯原理的基礎上假設這些『次波會彼此發生干涉』 ,這就是現今所稱的『惠更斯‧菲涅耳原理』,是『惠更斯原理』與『干涉原理』的開花結果。一八一八年菲涅耳將他的論文提交給法蘭西學術院的評委會。評委會的會員西莫恩‧德尼‧帕松 Siméon Denis Poisson 認為假使菲涅耳的理論成立,那麼將光波照射於一小塊圓形擋板時,所形成的陰影之中央必定會有一個亮斑,因此他推斷這理論不正確。同時與會的弗朗索瓦‧讓‧多米尼克‧阿拉戈 François Jean Dominique Arago 親自動手做了這個實驗,結果與預測相符,證實了菲涅耳原理的正確無誤。這實驗是支持光波動說的強有力的證據,與楊氏的雙縫實驗共同反駁了牛頓主導的光粒子說。

惠更斯‧菲涅耳原理 Huygens–Fresnel principle 是研究『波傳播』問題的一種『分析方法』。它能夠正確地『解釋』與『計算』波的傳播。其後德國物理學家古斯塔夫‧羅伯特‧克希荷夫 Gustav Robert Kirchhoff 的『繞射公式』給繞射提供了一個嚴謹的數學基礎。聲波的繞射現象可能使得『聽音辨位』失了準頭,聲音來源處的『』未必是經過繞射後感覺的『其所來處』,彷彿門外樓梯間角落邊的低語聲,誤以為來於自家的大門口!!

─── 摘自《【Sonic π】聲波之傳播原理︰原理篇《二》

 

由於可見光之波長夠短,故可近似的看成光線之能量流乎︰

閱讀 Justin Peatross  和 Michael Ware 先生們寫的書是種享受,不急不徐娓娓道來,短短篇章竟能將『光』從馬克士威電磁波方程式

\nabla^2 \textbf E ( \textbf r, t) + \frac{{n(\textbf r)}^2 {\omega}^2}{c^2} E ( \textbf r, t) = 0

帶到『波前』 R(\textbf r) 之『圖像』 eikonal 方程式

\nabla R(\textbf r)  \cdot  \nabla R(\textbf r) = {n(\textbf r)}^2

wave_front

不僅解釋了德文『eikonal』命名之歷史淵源,還用摩登字『icon』描述它與『成像』的本根聯繫!!這個 \nabla R(\textbf r) 變化最快之方向,是『波前』的梯度場,也就是『能量流』之方向也︰

坡印廷向量

坡印廷向量英語:Poynting vector),亦稱能流密度向量,其方向為電磁能傳遞方向,大小為能流密度(單位面積的能量傳輸速率 )。坡印廷向量的SI單位是瓦特每平方米(W/m2)。它是以其發現者約翰·坡印廷John Henry Poynting)來命名。奧利弗·黑維塞[1]尼科萊·烏諾夫[2]:147亦獨立發現所謂的坡印廷向量。

Poynting vector

In physics, the Poynting vector represents the directional energy flux density (the rate of energy transfer per unit area) of an electromagnetic field. The SI unit of the Poynting vector is the watt per square metre (W/m2). It is named after its discoverer John Henry Poynting who first derived it in 1884.[1]:132 Oliver Heaviside[1]:132 and Nikolay Umov[2]:147 also independently discovered the Poynting vector.

DipoleRadiation

Dipole radiation of a dipole vertically in the page showing electric field strength (colour) and Poynting vector (arrows) in the plane of the page.

直 想叫『光線』現形的乎??!!

圖像方程式反思

─── 摘自《光的世界︰矩陣光學三‧上

 

所以『針孔』能成像也︰

立竿可以見影、暗箱能夠呈像︰

暗箱(英語:Camera obscura)[1],又稱暗盒,是一種光學儀器,可以把影像投在螢幕上。暗箱的概念早在公元前已經出現。自15世紀開始,被藝術家用作繪畫的輔助工具。至18世紀未,一些攝影先驅用暗箱進行攝影實驗。例如出身顯赫的湯瑪斯·威治伍德,他在1790年代開始研究硝酸銀對光線的反應,並嘗試以暗箱拍攝照片,不過以失敗告終[2]

暗箱是相機的前身[1]

250px-Camera_Obscura_box18thCentury

暗箱的工作原理。光線通過鏡頭,經過反光鏡的反射,到達磨沙玻璃,並產生一個影像。把半透明的紙張放在玻璃上,即可勾畫出景物的輪廓。

光線之直行道理,古來早已知之。依此來解釋針孔相機成像的原理卻也並不容易︰

針孔相機(英語:Pinhole camera)是一種沒有鏡頭相機[1],取代鏡頭的是一個小孔,稱為針孔。利用針孔成像原理,產生倒立的影像。

針孔相機的結構相對簡單,由不透光的容器、感光材料和針孔片組成。其中,感光材料可以是底片,也可以是相紙[2]。為了控制曝光 ,還要有快門結構[3],通常是簡單的活門。

另外,由於進光量少,用針孔相機拍照,需要較長的曝光時間[4]。曝光時間由數秒至數十分鐘不等[4],通常把相機安裝在三腳架上,或把相機放在穩固的地方[3]

一些藝術家利用針孔相機進行創作。例如,芬蘭藝術家Tarja Trygg以針孔相機,拍攝日照軌跡(Solargraphy),曝光時間長達6個月[5]

250px-Pinhole-camera.svg

針孔相機的原理。

原理

光線沿直線傳播。物體反射的光線,通過針孔,在成像面形成倒立的影像。針孔與成像面的距離,稱為焦距,以毫米英吋標示[3]。針孔接近成像面,可拍攝廣角照片[3][7]。針孔遠離成像面,可拍攝遠攝遠攝照片[3][7]

焦距越長,影像越大[3]。例如,焦距為75mm時,影像剛好覆蓋4×5英吋的底片[3]。焦距為150mm時,影像剛好覆蓋8×10英吋的底片[3]。另外,焦距越短,照片的暗角越明顯[8]

一般而言,針孔越小,影像越清晰,但針孔太小,會導致衍射,反而令影像模糊[3]

針孔的最佳直徑

據說,用來計算針孔的最佳直徑的公式,至少有50條[3]。以下是其中一條用來計算針孔的最佳直徑  \phi 的公式:

\phi ={\sqrt {2f\lambda }}

其中, f是焦距,  \lambda 是光的波長[9]光的波長是700nm光的波長是546nm,光的波長是436nm[9]。計算的時候,通常取紅光與綠光的波長的平均值,即623nm[9]。計算的時候,請把波長由nm轉換成mm。因為1nm等於10^-6mm,所以623nm等於623×10^-6mm。

以下是焦距為50mm的例子:

\phi ={\sqrt {2\times 50\times 623\times 10^{-}6}}=0.249599679

四捨五入,可得出針孔的最佳直徑是0.25mm[9]

───

物體之反射光形成『點光源』 point light source 之聚積。

Point source

A point source is a single identifiable localised source of something. A point source has negligible extent, distinguishing it from other source geometries. Sources are called point sources because in mathematical modeling, these sources can usually be approximated as a mathematical point to simplify analysis.

The actual source need not be physically small, if its size is negligible relative to other length scales in the problem. For example, in astronomy, stars are routinely treated as point sources, even though they are in actuality much larger than the Earth.

In three dimensions, the density of something leaving a point source decreases in proportion to the inverse square of the distance from the source, if the distribution is isotropic, and there is no absorption or other loss.

Mathematics

In mathematics, a point source is a singularity from which flux or flow is emanating. Although singularities such as this do not exist in the observable universe, mathematical point sources are often used as approximations to reality in physics and other fields.

Light

Generally a source of light can be considered a point source if the resolution of the imaging instrument is too low to resolve its apparent size.

Mathematically an object may be considered a point source if its angular size,  \theta , is much smaller than the resolving power of the telescope:
\theta <<\lambda /D,
where \lambda is the wavelength of light and  D is the telescope diameter.

Examples:

每 個『點光源』球狀各向發射光芒,獨有與『針孔』『成一線』者 ,方得入此間,因此光量小,故需『暗箱』護,否則難賭物,只因背景光線強 。為何那『針孔』和『像面』之距離稱『焦距』?雖說『成一線』,實乃一『光錐』,匯聚在此處,術語不虛生,因襲稱『焦距』。有人還說『針孔相機』景深無限,深得廣漠無窮三昧︰

Depth_of_field_diagram

景深(英語:Depth of field, DOF)景深是指相機對焦點前後相對清晰的成像範圍。在光學中,尤其是錄影或是攝影,是一個描述在空間中,可以清楚成像的距離範圍。雖然透鏡只能夠將光聚到某一固定的距離,遠離此點則會逐漸模糊,但是在某一段特定的距離內,影像模糊的程度是肉眼無法察覺的,這段距離稱之為景深。當焦點設在超焦距處時,景深會從超焦距的一半延伸到無限遠,對一個固定的光圈值來說,這是最大的景深。

景深通常由物距、鏡頭焦距,以及鏡頭的光圈值所決定(相對於焦距的光圈大小)。除了在近距離時,一般來說景深是由物體的放大率以及透鏡的光圈值決定。固定光圈值時,增加放大率,不論是更靠近拍攝物或是使用長焦距的鏡頭,都會減少景深的距離;減少放大率時,則會增加景深。如果固定放大率時,增加光圈值(縮小光圈)則會增加景深;減小光圈值(增大光圈)則會減少景深。

對於某些影像,例如風景照,比較適合用較大的景深,然而在人像攝影時,則經常使用小景深來構圖,造成所謂背景虛化的效果。因為數位影像的進步,影像的銳利度可以由電腦後製而改變,因此也可以由後製的方式來改變景深。

───

,故而藝術魅力也長存??

─── 摘自《光的世界︰幾何光學二

 

『等光程』亦可成像矣︰

假使從『量子光學』 Quantum optics 觀點看『圖像方程式』 eikonal equation 推導之『假設』︰

波長 → 0

,此乃『高能光子』,粒子效應十分顯著。若非被侷限在極小空間中、特殊環境裡,可忽略其波動性。因此在巨觀近軸近似下,也不特別顧慮不同折射率界面間的『鏡面反射』 。由是應能知古代製作『好鏡子』亦很難耶!

Specular reflection

Specular reflection is the mirror-like reflection of light (or of other kinds of wave) from a surface, in which light from a single incoming direction (a ray) is reflected into a single outgoing direction. Such behavior is described by the law of reflection, which states that the direction of incoming light (the incident ray), and the direction of outgoing light reflected (the reflected ray) make the same angle with respect to the surface normal, thus the angle of incidence equals the angle of reflection \theta _{i}=\theta _{r} in the figure), and that the incident, normal, and reflected directions are coplanar. This behavior was first discovered through careful observation and measurement by Hero of Alexandria (AD c. 10–70).[1]

1280px-Tso_Kiagar_Lake_Ladakh

Reflections on still water are an example of specular reflection.

……

Background

When light hits a surface, there are three possible outcomes. Light may be absorbed by the material, light may be transmitted through to the other side, or light may be reflected back. Materials often show some mix of these behaviors, with the proportion of light that goes to each depending on the properties of the material, the wavelength of the light, and the angle of incidence. For most interfaces between materials, the fraction of the light that is reflected increases with increasing angle of incidence \theta _{i}.

Reflected light can be divided into two sub-types, specular reflection and diffuse reflection. Specular reflection reflects all light at the same angle, whereas diffuse reflection reflects in a broad range of directions. An example of the distinction between specular and diffuse reflection would be glossy and matte paints. Matte paints have almost exclusively diffuse reflection, while glossy paints have both specular and diffuse reflection. A surface built from a non-absorbing powder, such as plaster, can be a nearly perfect diffuser, whereas polished metallic objects can specularly reflect light very efficiently. The reflecting material of mirrors is usually aluminum or silver.

───

依此再推導出『費馬原理』︰

費馬原理一

或當知『光程』 OPL optical path length 概念的重要性︰

In optics, optical path length (OPL) or optical distance is the product of the geometric length of the path light follows through the system, and the index of refraction of the medium through which it propagates. A difference in optical path length between two paths is often called the optical path difference (OPD). Optical path length is important because it determines the phase of the light and governs interference and diffraction of light as it propagates.

費馬原理二

梯度方向

可以得『等光程』成像法則乎???

成像一

成像二

等光程

─── 摘自《光的世界︰矩陣光學三‧中

 

苟日日親賭目見,或不必藉著『立體實像』︰

之前我們談過『海市蜃樓』現象,在此特別介紹一個也叫做『海市蜃樓』的光學玩具︰

659_g_1468431245517

這個玩具只用兩個凹面鏡,人們就可在頂部開口處,見到放在底部中央之小物的立體、全彩、週邊三百六十度之『實像』!!此處並不多說明那個有趣的玩具,因為隨著上面的 LINK ,讀者自能知道的哩。反倒是推薦一篇論文︰

A complete ray-trace analysis of the Mirage toy

Proc. SPIE 9665, Tenth International Topical Meeting on Education and Training in Optics and Photonics, 966518 (August 3, 2015); doi:10.1117/12.2207520
Text Size: A A A

Open Access

From Conference Volume 9665
  • Tenth International Topical Meeting on Education and Training in Optics and Photonics
  • Marc Nantel
  • Ottawa, Ontario, Canada | June 03, 2007

abstract

 The ‘Mirage’ (Opti-Gone International) is a well-known optics demonstration (PIRA index number 6A20.35) that uses two opposed concave mirrors to project a real image of a small object into space. We studied image formation in the Mirage by standard 2×2 matrix methods and by exact ray tracing, with particular attention to additional real images that can be observed when the mirror separation is increased beyond one focal length. We find that the three readily observed secondary images correspond to 4, 6, or 8 reflections, respectively, contrary to previous reports. © (2015) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE). Downloading of the abstract is permitted for personal use only.
 

Full text of this article:

Citation

Sriya Adhya and John W. Noé
” A complete ray-trace analysis of the Mirage toy “, Proc. SPIE 9665, Tenth International Topical Meeting on Education and Training in Optics and Photonics, 966518 (August 3, 2015); doi:10.1117/12.2207520; http://dx.doi.org/10.1117/12.2207520

引述文中所言這個現象之神奇發現史︰

mirage-history

─── 摘自《光的世界︰【□○閱讀】反射式望遠鏡《五》

 

來解釋『什麼是影像』的吧!!??

 

 

 

 

 

 

 

 

輕。鬆。學。部落客