若問一個『厚透鏡』
，此處
， 
， 
。是否也如『薄透鏡』 
一般，具有『焦距』的呢？
pi@raspberrypi:~h \cdot \left( - \alpha + 1 -\frac{z}{f} \right) + \theta \cdot \left( z \cdot ( \gamma + 1 ) + \beta \right)
z = (1 - \alpha) \cdot f
\theta \cdot \left( \gamma + 1 - \frac{z}{f} \right)
z = (1 + \gamma) \cdot f
\alpha = \frac{(n - 1) d}{n R_1}
\gamma = \frac{(n - 1) d}{n R_2}
f
*** QuickLaTeX cannot compile formula: 之值取『正號』而已。若是實際值『為負』，意味著『發散』的也。</span> <img class="alignnone size-full wp-image-57888" src="http://www.freesandal.org/wp-content/uploads/522px-Lens1b.svg.png" alt="522px-Lens1b.svg" width="522" height="345" /> <img class="alignnone size-full wp-image-57887" src="http://www.freesandal.org/wp-content/uploads/Concave_lens.jpg" alt="Concave_lens" width="800" height="378" /> <img class="alignnone size-full wp-image-57892" src="http://www.freesandal.org/wp-content/uploads/Lens_shapes.svg.png" alt="Lens_shapes.svg" width="553" height="406" /> <span style="color: #808080;">1-4為<a style="color: #808080;" title="凸透鏡" href="https://zh.wikipedia.org/wiki/%E5%87%B8%E9%80%8F%E9%95%9C">凸透鏡</a>，5-8為<a style="color: #808080;" title="凹透鏡" href="https://zh.wikipedia.org/wiki/%E5%87%B9%E9%80%8F%E9%95%9C">凹透鏡</a>，其中：</span> <span style="color: #808080;"> <b>1</b> - 對稱雙凸透鏡，<b>2</b> - 非對稱雙凸透鏡，<b>3</b> - 平凸透鏡</span> <span style="color: #808080;"> <b>4</b> - 凹凸透鏡（凸度大於凹度），<b>5</b> - 對稱雙凹透鏡</span> <span style="color: #808080;"> <b>6</b> - 非對稱雙凹透鏡， <b>7</b> - 平凹透鏡，<b>8</b> - 凸凹透鏡（凹度大於凸度） </span> <span style="color: #003300;">所謂</span> <h1 id="firstHeading" class="firstHeading" lang="zh-TW"><span style="color: #003300;"><a style="color: #003300;" href="https://zh.wikipedia.org/zh-tw/%E7%84%A6%E9%BB%9E">焦點</a></span></h1> <span style="color: #808080;"><b>焦點</b>，在<a style="color: #808080;" title="幾何光學" href="https://zh.wikipedia.org/wiki/%E5%87%A0%E4%BD%95%E5%85%89%E5%AD%A6">幾何光學</a>中有時也稱為<b>像點</b>，是源頭的<a class="mw-redirect" style="color: #808080;" title="光線" href="https://zh.wikipedia.org/wiki/%E5%85%89%E7%B7%9A">光線</a>經過物鏡後匯聚的點。<sup id="cite_ref-1" class="reference"><a style="color: #808080;" href="https://zh.wikipedia.org/wiki/%E7%84%A6%E9%BB%9E#cite_note-1">[1]</a></sup>. 然而，焦點只是概念上的點，實際上在空間上有一個範圍，稱為<a class="mw-redirect" style="color: #808080;" title="朦朧圈" href="https://zh.wikipedia.org/wiki/%E6%9C%A6%E6%9C%A7%E5%9C%88">朦朧圈</a>。這種非理想的焦點也許會導致光學影像的<a style="color: #808080;" title="像差" href="https://zh.wikipedia.org/wiki/%E5%83%8F%E5%B7%AE">像差</a>，在沒有明顯的像差下，最小的朦朧圈是<a class="mw-redirect" style="color: #808080;" title="艾里盤" href="https://zh.wikipedia.org/wiki/%E8%89%BE%E9%87%8C%E7%9B%A4">艾里盤</a>，是因為光學系統的<a style="color: #808080;" title="口徑" href="https://zh.wikipedia.org/wiki/%E5%8F%A3%E5%BE%91">開口</a>產生<a class="mw-redirect" style="color: #808080;" title="繞射" href="https://zh.wikipedia.org/wiki/%E7%B9%9E%E5%B0%84">繞射</a>造成的。當口徑加大時，像差也會變得更為嚴重，而艾里圈是在大口徑下最小的。</span> <span style="color: #808080;">一個影像，點像或區域如果能很好的被收歛就是<i><b>對焦</b></i>，如果未能良好的匯聚就是<i><b>失焦</b></i>。兩者之間的邊界有時被用來作為<a style="color: #808080;" title="模糊圈" href="https://zh.wikipedia.org/wiki/%E6%A8%A1%E7%B3%8A%E5%9C%88">模糊圈</a>的定義 。</span> <span style="color: #808080;"><b>主焦點</b>或<b>焦點</b>是球面的焦點：</span> <ul> <li><span style="color: #808080;">對一個<a class="mw-redirect" style="color: #808080;" title="透鏡" href="https://zh.wikipedia.org/wiki/%E9%80%8F%E9%8F%A1">透鏡</a>、<a style="color: #808080;" title="曲面鏡" href="https://zh.wikipedia.org/wiki/%E6%9B%B2%E9%9D%A2%E9%8F%A1">球面鏡</a>、或<a class="new" style="color: #808080;" title="拋物面鏡（頁面不存在）" href="https://zh.wikipedia.org/w/index.php?title=%E6%8B%8B%E7%89%A9%E9%9D%A2%E9%8F%A1&action=edit&redlink=1">拋物面鏡</a>，是被<a class="new" style="color: #808080;" title="瞄準（頁面不存在）" href="https://zh.wikipedia.org/w/index.php?title=%E7%9E%84%E6%BA%96&action=edit&redlink=1">瞄準</a>平行於光軸的光被聚集在光軸上的點。由於光可以從任何一個面穿越透鏡，因此透鏡在倆側各有一個焦點。在空氣中，透鏡或面鏡的<a class="new" style="color: #808080;" title="主平面（頁面不存在）" href="https://zh.wikipedia.org/w/index.php?title=%E4%B8%BB%E5%B9%B3%E9%9D%A2&action=edit&redlink=1">主平面</a>至焦點的距離稱為<i><a style="color: #808080;" title="焦距" href="https://zh.wikipedia.org/wiki/%E7%84%A6%E8%B7%9D">焦距</a></i>。</span></li> <li><span style="color: #808080;"><a class="mw-redirect" style="color: #808080;" title="橢圓" href="https://zh.wikipedia.org/wiki/%E6%A9%A2%E5%9C%93">橢圓的</a>面鏡有兩個焦點，穿過其中一個焦點後抵達鏡面上的光線 ，反射後會通過另一個焦點。</span></li> <li><span style="color: #808080;"><a class="mw-redirect" style="color: #808080;" title="雙曲線" href="https://zh.wikipedia.org/wiki/%E9%9B%99%E6%9B%B2%E7%B7%9A">雙曲線的</a>面鏡也有兩個焦點，他的特性是經過一個焦點後被反射的光，像是來自另一個焦點。</span></li> </ul> <span style="color: #808080;">一個發散（負）透鏡或凸面鏡不能將瞄準的光線匯聚在一個點上，換言之，焦點是被面鏡反射或穿透透鏡的光線看起來的發射點。凹的拋物面鏡可以將平行瞄準射入 的光線反射成看似由焦點發射出來的光；反之，經過焦點抵達拋物面鏡的光會被反射成平行的光，可以作為瞄準用的射線。凹的橢圓面鏡會將經過其中一個焦點的光 線反射至另一個焦點，好像是從那個焦點射出的，而兩個焦點都在鏡後。一個凹面的雙曲面鏡會將經過鏡前焦點的光反射，而看似是由鏡後的焦點直接發射出來的。 相反的，在<a style="color: #808080;" title="卡塞格林反射鏡" href="https://zh.wikipedia.org/wiki/%E5%8D%A1%E5%A1%9E%E6%A0%BC%E6%9E%97%E5%8F%8D%E5%B0%84%E9%8F%A1">卡塞格林反射望遠鏡</a>，會將光線匯聚在鏡後的焦點上，而這個焦點接近另一面鏡子的鏡前焦點。</span> <img class="alignnone size-full wp-image-58071" src="http://www.freesandal.org/wp-content/uploads/250px-DOF-ShallowDepthofField.jpg" alt="250px-DOF-ShallowDepthofField" width="250" height="180" /> <span style="color: #808080;">部份對焦的影像，除了中間區域以外，其它大部份區域都落在焦點外（失焦）。</span> <img class="alignnone size-full wp-image-58073" src="http://www.freesandal.org/wp-content/uploads/300px-Glasses_800_edit.png" alt="300px-Glasses_800_edit" width="300" height="225" /> <span style="color: #808080;">在玻璃上的朦朧圈是由<a class="new" style="color: #808080;" title="計算機產生影像（頁面不存在）" href="https://zh.wikipedia.org/w/index.php?title=%E8%A8%88%E7%AE%97%E6%A9%9F%E7%94%A2%E7%94%9F%E5%BD%B1%E5%83%8F&action=edit&redlink=1">計算機產生影像</a>模擬的，使用的軟體是<a style="color: #808080;" title="POV-Ray" href="https://zh.wikipedia.org/wiki/POV-Ray">POV-Ray</a>。</span> <span style="color: #003300;">不單是『幾何光學』系統之重要概念，關乎『成像法則』矣！尚且可以通達『傅立葉光學』的哩？？</span> <h3><span id="Fourier_transforming_property_of_lenses" class="mw-headline" style="color: #003300;"><a style="color: #003300;" href="https://en.wikipedia.org/wiki/Fourier_optics">Fourier transforming property of lenses</a></span></h3> <span style="color: #808080;">If a transmissive object is placed one focal length in front of a <a style="color: #808080;" title="Lens (optics)" href="https://en.wikipedia.org/wiki/Lens_%28optics%29">lens</a>, then its <a style="color: #808080;" title="Fourier transform" href="https://en.wikipedia.org/wiki/Fourier_transform">Fourier transform</a> will be formed one focal length behind the lens. Consider the figure to the right (click to enlarge)</span> <span style="color: #808080;"><img class="alignnone size-full wp-image-58079" src="http://www.freesandal.org/wp-content/uploads/Lens_FT.jpg" alt="Lens_FT" width="934" height="384" /></span> <span style="color: #999999;">On the Fourier transforming property of lenses</span> <span style="color: #808080;">In this figure, a plane wave incident from the left is assumed. The transmittance function in the front focal plane (i.e., Plane 1) <i>spatially modulates the incident plane wave</i> in magnitude and phase, <i>like on the left-hand side of eqn. (2.1)</i> (specified to <i>z</i>=0), and <i>in so doing, produces a spectrum of plane waves</i> corresponding to the FT of the transmittance function, <i>like on the right-hand side of eqn. (2.1)</i> (for <i>z</i>>0). The various plane wave components propagate at different tilt angles with respect to the optic axis of the lens (i.e., the horizontal axis). The finer the features in the transparency, the broader the angular bandwidth of the plane wave spectrum. We'll consider one such plane wave component, propagating at angle θ with respect to the optic axis. It is assumed that θ is small (<a style="color: #808080;" title="Paraxial approximation" href="https://en.wikipedia.org/wiki/Paraxial_approximation">paraxial approximation</a>), so that</span> <dl><dd><span style="color: #808080;"><img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/660113056ed3c434620c1f1b279c75288373fe8b" alt="{\frac {k_{x}}{k}}=\sin \theta \simeq \theta " /></span></dd></dl><span style="color: #808080;">and</span> <dl><dd><span style="color: #808080;"><img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6ee18b2e3c3fb795c57a0fc55ad82b491a10ff8" alt="{\frac {k_{z}}{k}}=\cos \theta \simeq 1-{\frac {\theta ^{2}}{2}}" /></span></dd></dl><span style="color: #808080;">and</span> <dl><dd><span style="color: #808080;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y"> </span><img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/26ddbbaa9787d0eb59443e5a56f26ce9bbf5b7f9" alt="{\frac {1}{\cos \theta }}\simeq {\frac {1}{1-{\frac {\theta ^{2}}{2}}}}\simeq 1+{\frac {\theta ^{2}}{2}}" /></span></dd></dl><span style="color: #808080;">In the figure, the <i>plane wave</i> phase, moving horizontally from the front focal plane to the lens plane, is</span> <dl><dd><span style="color: #808080;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y"> </span><img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac612917b2bf37d17c372cfa288877c8ec0ea3d9" alt="e^{{jkf\cos \theta }}\," /></span></dd></dl><span style="color: #808080;">and the <i>spherical wave</i> phase from the lens to the spot in the back focal plane is:</span> <dl><dd><span style="color: #808080;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y"> </span><img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1c4f4929e2694c32a82c3f5cb0e68ec62a902f32" alt="e^{{jkf/\cos \theta }}\," /></span></dd></dl><span style="color: #808080;">and the sum of the two path lengths is <i>f</i> (1 + θ<sup>2</sup>/2 + 1 - θ<sup>2</sup>/2) = 2<i>f</i> i.e., it is a constant value, independent of tilt angle, θ, for paraxial plane waves. Each paraxial plane wave component of the field in the front focal plane appears as a <a style="color: #808080;" title="Point spread function" href="https://en.wikipedia.org/wiki/Point_spread_function">point spread function</a> spot in the back focal plane, with an intensity and phase equal to the intensity and phase of the original plane wave component in the front focal plane. In other words, the field in the back focal plane is the <a style="color: #808080;" title="Fourier transform" href="https://en.wikipedia.org/wiki/Fourier_transform">Fourier transform</a> of the field in the front focal plane.</span> <span style="color: #808080;">All FT components are computed simultaneously - in parallel - at the speed of light. As an example, light travels at a speed of roughly 1 ft (0.30 m). / ns, so if a lens has a 1 ft (0.30 m). focal length, an entire 2D FT can be computed in about 2 ns (2 x 10<sup>−9</sup> seconds). If the focal length is 1 in., then the time is under 200 ps. No electronic computer can compete with these kinds of numbers or perhaps ever hope to, although new supercomputers such as the petaflop <a style="color: #808080;" title="IBM Roadrunner" href="https://en.wikipedia.org/wiki/IBM_Roadrunner">IBM Roadrunner</a> may actually prove faster than optics, as improbable as that may seem. However, their speed is obtained by combining numerous computers which, individually, are still slower than optics. The disadvantage of the optical FT is that, as the derivation shows, the FT relationship only holds for paraxial plane waves, so this FT "computer" is inherently bandlimited. On the other hand, since the wavelength of visible light is so minute in relation to even the smallest visible feature dimensions in the image i.e.,</span> <dl><dd><span style="color: #808080;"><img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e16467ca1209514d7502ac3f415baf4dc6fc8cb" alt="k^{2}\gg k_{x}^{2}+k_{y}^{2}" /></span></dd></dl><span style="color: #808080;">(for all <i>k<sub>x</sub></i>, <i>k<sub>y</sub></i> within the spatial bandwidth of the image, so that <i>k<sub>z</sub></i> is nearly equal to <i>k</i>), the paraxial approximation is not terribly limiting in practice. And, of course, this is an analog - not a digital - computer, so precision is limited. Also, phase can be challenging to extract; often it is inferred interferometrically.</span> <span style="color: #808080;">Optical processing is especially useful in real time applications where rapid processing of massive amounts of 2D data is required, particularly in relation to pattern recognition.</span> <h3><span id="The_complete_solution:_the_superposition_integral" class="mw-headline" style="color: #808080;">The complete solution: the superposition integral</span></h3> <span style="color: #808080;">A general solution to the homogeneous electromagnetic wave equation in rectangular coordinates may be formed as a weighted superposition of all possible elementary plane wave solutions as:</span> <dl><dd><span style="color: #808080;"><img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4c54785f5ea0b4c5681f8f8477ae3bfbedabe97" alt="\psi (x,y,z)=\int _{{-\infty }}^{{+\infty }}\int _{{-\infty }}^{{+\infty }}\Psi _{0}(k_{x},k_{y})~e^{{j(k_{x}x+k_{y}y)}}~e^{{\pm jz{\sqrt {k^{2}-k_{x}^{2}-k_{y}^{2}}}}}~dk_{x}dk_{y}~~~~~~~~~~~~~~~~~~(2.1)" /></span></dd></dl><span style="color: #808080;">Next, let</span> <dl><dd><span style="color: #808080;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y"> </span><img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a2367e6d7c5e7a2897f7804f67342987943a4804" alt="\psi _{0}(x,y)=\psi (x,y,z)|_{{z=0}}" />.</span></dd></dl><span style="color: #808080;">Then:</span> <dl><dd><span style="color: #808080;"><img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98c78685f09c423303a3d47d5a30d718f895e14c" alt="\psi _{0}(x,y)=\int _{{-\infty }}^{{+\infty }}\int _{{-\infty }}^{{+\infty }}\Psi _{0}(k_{x},k_{y})~e^{{j(k_{x}x+k_{y}y)}}~dk_{x}dk_{y}" /></span></dd></dl><span style="color: #808080;"><b>This plane wave spectrum representation of the electromagnetic field is the basic foundation of Fourier optics</b> (this point cannot be emphasized strongly enough), because when <i>z</i>=0, the equation above simply becomes a <b>Fourier transform (FT) relationship between the field and its plane wave content</b> (hence the name, "Fourier optics").</span> <span style="color: #808080;">Thus:</span> <dl><dd><span style="color: #808080;"><img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2ec51b08a45661d3119a2fd55be1e36838b431b" alt="\Psi _{0}(k_{x},k_{y})={\mathcal {F}}\{\psi _{0}(x,y)\}" /></span></dd></dl><span style="color: #808080;">and</span> <dl><dd><span style="color: #808080;"><img class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/587fec06dfef480e3c09639c1d1236bd016eea44" alt="\psi _{0}(x,y)={\mathcal {F}}^{{-1}}\{\Psi _{0}(k_{x},k_{y})\}" /></span></dd></dl><span style="color: #808080;">All spatial dependence of the individual plane wave components is described explicitly via the exponential functions. The coefficients of the exponentials are only functions of spatial wavenumber <i>k<sub>x</sub></i>, <i>k<sub>y</sub></i>, just as in ordinary <a style="color: #808080;" title="Fourier analysis" href="https://en.wikipedia.org/wiki/Fourier_analysis">Fourier analysis</a> and <a style="color: #808080;" title="Fourier transform" href="https://en.wikipedia.org/wiki/Fourier_transform">Fourier transforms</a>.</span> <span style="color: #003300;">這事提倡『<a style="color: #003300;" href="http://www.freesandal.org/?p=11232">光粒子說</a>』之牛頓固不知曉，他可是早就用『焦平面』</span> <img class="alignnone size-full wp-image-58019" src="http://www.freesandal.org/wp-content/uploads/Opticks.jpg" alt="Opticks" width="738" height="1000" /> <span style="color: #808080;">Cover of the first edition of Newton's <i>Opticks</i></span> <span style="color: #003300;">論述『成像法則』的人！！</span> 這裡且與一圖︰ <img class="alignnone size-full wp-image-58035" src="http://www.freesandal.org/wp-content/uploads/牛頓成像公式.png" alt="牛頓成像公式" width="1310" height="581" /> 讀者自可證明 *** Error message: Unicode character 之 (U+4E4B) leading text: $之 Unicode character 值 (U+503C) leading text: $之值 Unicode character 取 (U+53D6) leading text: $之值取 Unicode character 『 (U+300E) leading text: $之值取『 Unicode character 正 (U+6B63) leading text: $之值取『正 Unicode character 號 (U+865F) leading text: $之值取『正號 Unicode character 』 (U+300F) leading text: $之值取『正號』 Unicode character 而 (U+800C) leading text: $之值取『正號』而 Unicode character 已 (U+5DF2) leading text: $之值取『正號』而已 Unicode character 。 (U+3002) leading text: $之值取『正號』而已。 Unicode character 若 (U+82E5) leading text: $之值取『正號』而已。若 Unicode character 是 (U+662F)x \cdot x' = F \cdot F' $
的吧！！！