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）是一種沒有鏡頭的相機^[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]。

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，以毫米或英吋標示^[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]。紅光的波長是700nm，綠光的波長是546nm，藍光的波長是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：

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

經四捨五入，可得出針孔的最佳直徑是0.25mm^[9]。

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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:

Light from a distant star seen through a small telescope
Light passing through a pinhole or other small aperture, viewed from a distance much greater than the size of the hole
Light from a street light in a large-scale study of light pollution or street illumination

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。

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，故而藝術魅力也長存？？

幾何光學是門古老的學問，工藝成熟的產業，許多術語淵源流長，解釋往往言簡意賅，告誡讀者小心對待，以免望文生義，徒惹困惑不斷！！為著樹莓派上攝像頭『實用目的』，我們侷限於幾何光學的『近軸近似』

高斯光學

高斯光學是幾何光學中用近軸近似（小角近似）描述在光學系統中光線行為的技術，在近軸近似中，光線和光軸的夾角很小.^[1]，因此，夾角的一些三角函數可以用角度的線性函數來表示。高斯光學用在光學系統的表面平坦或者是為球面一部份的情形。此時可以用一些簡單的公式，配合一些像焦距、放大率及明度等參數描述影像系統，而這些參數是以組成元素的幾何形狀及材料性質來定義的。

高斯光學得名自卡爾·弗里德里希·高斯，他證明光學系統可以用很多的基本點來找出其特徵，因此可以計算其光學性質等^[2]。

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並以『矩陣分析』為方法︰

Ray transfer matrix analysis

Ray transfer matrix analysis (also known as ABCD matrix analysis) is a type of ray tracing technique used in the design of some optical systems, particularly lasers. It involves the construction of a ray transfer matrix which describes the optical system; tracing of a light path through the system can then be performed by multiplying this matrix with a vector representing the light ray. The same analysis is also used in accelerator physics to track particles through the magnet installations of a particle accelerator, see Beam optics.

The technique that is described below uses the paraxial approximation of ray optics, which means that all rays are assumed to be at a small angle (θ in radians) and a small distance (x) relative to the optical axis of the system.^[1]

Definition of the ray transfer matrix

The ray tracing technique is based on two reference planes, called the input and output planes, each perpendicular to the optical axis of the system. Without loss of generality, we will define the optical axis so that it coincides with the z-axis of a fixed coordinate system. A light ray enters the system when the ray crosses the input plane at a distance x₁ from the optical axis while traveling in a direction that makes an angle θ₁ with the optical axis. Some distance further along, the ray crosses the output plane, this time at a distance x₂ from the optical axis and making an angle θ₂. n₁ and n₂ are the indices of refraction of the medium in the input and output plane, respectively.

These quantities are related by the expression

{x_2 \choose \theta_2} = \begin{pmatrix} A & B \\ C & D \end{pmatrix}{x_1 \choose \theta_1},

where

A = {x_2 \over x_1 } \bigg|_{\theta_1 = 0} \qquad B = {x_2 \over \theta_1 } \bigg|_{x_1 = 0},

and

C = {\theta_2 \over x_1 } \bigg|_{\theta_1 = 0} \qquad D = {\theta_2 \over \theta_1 } \bigg|_{x_1 = 0}.

This relates the ray vectors at the input and output planes by the ray transfer matrix (RTM) M, which represents the optical system between the two reference planes. A thermodynamics argument based on the blackbody radiation can be used to show that the determinant of a RTM is the ratio of the indices of refraction:

\det(\mathbf{M}) = AD - BC = { n_1 \over n_2 }.

As a result, if the input and output planes are located within the same medium, or within two different media which happen to have identical indices of refraction, then the determinant of M is simply equal to 1.

Note that at least one source^[2] uses a different convention for the ray vectors. The optical direction cosine, n sin θ, is used instead of θ. This would alter some of the ABCD matrices, especially for refraction.

A similar technique can be used to analyze electrical circuits. See Two-port networks.

In ray transfer (ABCD) matrix analysis, an optical element (here, a thick lens) gives a transformation between $(x_{1},\theta _{1})$ at the input plane and $(x_{2},\theta _{2})$ when the ray arrives at the output plane.

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借力『SymPy』之工具，

Gaussian Optics

Gaussian optics.

The module implements:

Ray transfer matrices for geometrical and gaussian optics.

See RayTransferMatrix, GeometricRay and BeamParameter
Conjugation relations for geometrical and gaussian optics.

See geometric_conj*, gauss_conj and conjugate_gauss_beams

The conventions for the distances are as follows:

focal distance: positive for convergent lenses
object distance: positive for real objects
image distance: positive for real images

class sympy.physics.optics.gaussopt.RayTransferMatrix

Base class for a Ray Transfer Matrix.

It should be used if there isn’t already a more specific subclass mentioned in See Also.

Parameters :	parameters : A, B, C and D or 2×2 matrix (Matrix(2, 2, [A, B, C, D]))

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光的世界︰幾何光學二

原理