光的世界︰矩陣光學五‧一

已知光在均質空間裡走直線,由此推導其『矩陣表現』︰

自由空間

 

自由空間一

自由空間二

 

讀來輕鬆快意,甚至可用 sympy.physics.optics 驗證一番︰

假使光在光軸 z 上的投影 d 等於 a + b ,那麼不管先走了 a ,再走了 b ,一定等於 先走了 b ,再走了 a

pi@raspberrypi:~ ipython3 Python 3.4.2 (default, Oct 19 2014, 13:31:11)  Type "copyright", "credits" or "license" for more information.  IPython 2.3.0 -- An enhanced Interactive Python. ?         -> Introduction and overview of IPython's features. %quickref -> Quick reference. help      -> Python's own help system. object?   -> Details about 'object', use 'object??' for extra details.  In [1]: from sympy import *  In [2]: from sympy.physics.optics import FreeSpace  In [3]: init_printing()  In [4]: d, a, b = symbols('d, a, b')  In [5]: Fd = FreeSpace(d)  In [6]: type(Fd) Out[6]: sympy.physics.optics.gaussopt.FreeSpace  In [7]: Fd Out[7]:  ⎡1  d⎤ ⎢    ⎥ ⎣0  1⎦  In [8]: Fa = FreeSpace(a)  In [9]: Fb = FreeSpace(b)  In [10]: Fb * Fa Out[10]:  ⎡1  a + b⎤ ⎢        ⎥ ⎣0    1  ⎦  In [11]: Fa * Fb Out[11]:  ⎡1  a + b⎤ ⎢        ⎥ ⎣0    1  ⎦  In [12]: F = Matrix([[1, d], [0,1]])  In [13]: F Out[13]:  ⎡1  d⎤ ⎢    ⎥ ⎣0  1⎦  In [14]: F.det() Out[14]: 1  In [15]: F.inv() Out[15]:  ⎡1  -d⎤ ⎢     ⎥ ⎣0  1 ⎦  In [16]:  </pre>    但思此矩陣中之『參數』為什麼不與『折射率』相關呢?再想分明是發生於z_1面、介質空間長度d、z_2面間的三件事,怎麼會不見那『兩面』的說法哩!若是講因為在那『兩面』上什麼也沒發生 ── 沒有『反射』,也沒有『折射』 ── 故無事可說,一般以為不必說也!這樣當真無說的耶??如果將之註解為︰此處假設z_1面之前是相同介質,z_2面之後還是相同介質,如此果然有說的乎!!  所以『面』之概念內含豐富,有  無事之面︰此處說法。  反射之面︰當下回頭。  折射之面︰折屈上下。  理論之面︰尚未言及。  ……  實在是『抽象』得很矣??!!  <div class="wc-shortcodes-row wc-shortcodes-item wc-shortcodes-clearfix"><div class="wc-shortcodes-column wc-shortcodes-content wc-shortcodes-one-half wc-shortcodes-column-first ">  <a href="http://www.freesandal.org/wp-content/uploads/枯籐老樹昏鴉.jpg"><img class="alignnone size-full wp-image-9763" src="http://www.freesandal.org/wp-content/uploads/枯籐老樹昏鴉.jpg" alt="枯籐老樹昏鴉" width="1685" height="786" /></a>  <a href="http://www.freesandal.org/wp-content/uploads/小橋流水人家.jpg"><img class="alignnone size-full wp-image-9766" src="http://www.freesandal.org/wp-content/uploads/小橋流水人家.jpg" alt="小橋流水人家" width="2244" height="1389" /></a>  <a href="http://www.freesandal.org/wp-content/uploads/趙亭玉古道西風瘦馬.jpg"><img class="alignnone size-full wp-image-9767" src="http://www.freesandal.org/wp-content/uploads/趙亭玉古道西風瘦馬.jpg" alt="趙亭玉古道西風瘦馬" width="660" height="260" /></a>  <a href="http://www.freesandal.org/wp-content/uploads/kandinsky-1923x.jpg"><img class="alignnone size-full wp-image-9762" src="http://www.freesandal.org/wp-content/uploads/kandinsky-1923x.jpg" alt="kandinsky-1923x" width="1280" height="886" /></a>  <a href="http://www.freesandal.org/wp-content/uploads/275px-PicassoGuernica.jpg"><img class="alignnone size-full wp-image-9761" src="http://www.freesandal.org/wp-content/uploads/275px-PicassoGuernica.jpg" alt="275px-PicassoGuernica" width="275" height="160" /></a>  </div><div class="wc-shortcodes-column wc-shortcodes-content wc-shortcodes-one-half wc-shortcodes-column-last ">  <span style="color: #ff99cc;">【<strong>越調</strong>】天淨沙‧<strong>秋思</strong></span> <span style="color: #ff99cc;"> <strong>元</strong> <strong>馬致遠</strong></span>  <span style="color: #ff99cc;">枯籐老樹昏鴉,</span> <span style="color: #ff99cc;"> 小橋流水人家,</span> <span style="color: #ff99cc;"> 古道西風瘦馬。</span> <span style="color: #ff99cc;"> 夕陽西下,</span> <span style="color: #ff99cc;"> 斷腸人在天涯。</span>  <strong>繪畫</strong>、<strong>照片</strong>與<strong>書法</strong>是哪種比較『<strong>寫實</strong>』?哪個又較為『<strong>抽象</strong>』?《<span style="color: #ff99cc;"><strong>秋思</strong></span>》中用具象《『<span style="color: #008000;"><strong>藤</strong></span>』、『<span style="color: #008000;"><strong>樹</strong></span>』、『<span style="color: #008000;"><strong>鴉</strong></span>』》,來虛寫『<span style="color: #808080;"><strong>時光變化</strong></span>』之《『<span style="color: #808080;"><strong>枯</strong></span>』、『<span style="color: #808080;"><strong>老</strong></span>』、『<span style="color: #808080;"><strong>昏</strong></span>』》;以實景《『<span style="color: #99ccff;"><strong>橋</strong></span>』、『<span style="color: #99ccff;"><strong>水</strong></span>』、『<span style="color: #99ccff;"><strong>家</strong></span>』》,將串成『<span style="color: #ff99cc;"><strong>應歸之所</strong></span>』的《『<span style="color: #ff99cc;"><strong>小</strong></span>』、『<span style="color: #ff99cc;"><strong>流</strong></span>』、『<span style="color: #ff99cc;"><strong>人</strong></span>』》;終至於『<span style="color: #ff0000;"><strong>道</strong></span>』得『<span style="color: #ff0000;"><strong>古</strong></span>』、『<span style="color: #ff0000;"><strong>風</strong></span>』是『<span style="color: #ff0000;"><strong>西</strong></span>』、『<span style="color: #ff0000;"><strong>馬</strong></span>』又怎能不『<span style="color: #ff0000;"><strong>廋</strong></span>』?此刻也許只該是『<span style="color: #808080;"><strong>夕陽西下</strong></span>』??否則哪歸結的出『<span style="color: #808080;"><strong>胡不歸去</strong></span>』之『<span style="color: #808080;"><strong>斷腸人在天涯</strong></span>』!!  <span style="color: #808080;">或許『<strong>失重</strong>』的『<strong>實物</strong>』漂浮於天之涯海之角,反而更顯得『<strong>虛幻</strong>』的了!!而『<strong>抽象</strong>』的『<strong>圖形</strong>』一旦擬似具體構物,或會因『<strong>沈重</strong>』終將失去『<span style="color: #808080;"><strong>空</strong></span><strong>靈</strong>』的嗎??</span>  </div></div>  ─── 摘自《<a href="http://www.freesandal.org/?p=9701">λ 運算︰計物數《中》</a>》     <span style="color: #003300;">要是再補足以z_1或且z_2$ 『介面』上可能發生一事

平面折射

 

class sympy.physics.optics.gaussopt.FlatRefraction

Ray Transfer Matrix for refraction.
Parameters :

n1 : refractive index of one medium

n2 : refractive index of other medium

Examples

>>> from sympy.physics.optics import FlatRefraction
>>> from sympy import symbols
>>> n1, n2 = symbols('n1 n2')
>>> FlatRefraction(n1, n2)
Matrix([
[1,     0],
[0, n1/n2]])

 

總能得之的吧!!!