查看“︁近軸近似”︁的源代码

[[File:Small angle compare error.svg|thumb|300px|近軸近似的誤差，圖中的cosθ是用{{nowrap|1 - θ<sup>2</sup>/2}}近似]]
'''近軸近似'''是[[幾何光學]]中的[[高斯光學]]及{{le|光線追蹤 (物理)|Ray tracing (physics)|光線追蹤}}用的[[小角度近似]]，可以用在光學系統（例如[[透镜]]）的分析<ref name="Greivenkamp">{{Cite book | isbn = 0-8194-5294-7 | title = Field Guide to Geometrical Optics | last1 = Greivenkamp | first1 = John E. | year = 2004 | publisher = SPIE | series = SPIE Field Guides | volume = 1  | pages = 19–20  }}</ref>
<ref>{{cite web
 | last=Weisstein
 | first=Eric W.
 | title=Paraxial Approximation
 | url=http://scienceworld.wolfram.com/physics/ParaxialApproximation.html
 | work=ScienceWorld
 | publisher=[[Wolfram Research]]
 | accessdate=15 January 2014
 | year=2007
 | archive-date=2021-09-10
 | archive-url=https://web.archive.org/web/20210910104312/https://scienceworld.wolfram.com/physics/ParaxialApproximation.html
 | dead-url=no
 }}</ref>

'''近軸光線'''是指[[光線]]和[[光轴 (光学)|光轴]]角度很小，而光線接近光學系統的軸。<ref name=Greivenkamp/>

在近軸近似及近軸光線下，在計算光的路徑時，可以使用以下的近似（''θ''為[[弧度]]）<ref name=Greivenkamp/>

:<math>
\sin \theta \approx \theta,\quad
\tan \theta \approx \theta
\quad \text{and}\quad\cos \theta \approx 1.</math>

近軸近似可用在[[高斯光学]]及一階{{le|光線追蹤 (物理)|Ray tracing (physics)|光線追蹤}}中<ref name=Greivenkamp/>。像[[光線轉換矩陣分析]]就使用了這種近似方式。

有時二階近似也稱為近軸近似，對於<math>sin</math>及<math>tan</math>函數，其[[泰勒級數]]的二階項為0，因此二階近似和一階近似相同，而<math>cos</math>函數的二階近似如下：

:<math> \cos \theta \approx 1 - { \theta^2 \over 2 } \ .</math>

二階近似在角度小於10°時，其準確度在0.5%以內，若角度變大，誤差就會顯著提昇<ref>
{{cite web
 | title=Paraxial approximation error plot
 | url=http://www.wolframalpha.com/input/?i=Plot%5B{%28x+Deg+-+Sin%5Bx+Deg%5D%29%2FSin%5Bx+Deg%5D%2C+%28Tan%5Bx+Deg%5D+-+x+Deg%29%2FTan%5Bx+Deg%5D%2C+%281+-+Cos%5Bx+Deg%5D%29%2FCos%5Bx+Deg%5D}%2C+{x%2C+0%2C+15}%5D
 | work=[[Wolfram Alpha]]
 | publisher=[[Wolfram Research]]
 | accessdate=26 August 2014}}</ref> 
<!-- This plots the error plot of the paraxial approximation, i.e. the 3 curves for small angles: Plot[{(x Deg - Sin[x Deg])/Sin[x Deg], (Tan[x Deg] - x Deg)/Tan[x Deg], (1 - Cos[x Deg])/Cos[x Deg]}, {x, 0, 15}] -->

若光線和光軸的夾角較大時，需區分和光軸共平面的{{le|子午光线|meridional ray}}及不共平面的{{le|弧矢光线|sagittal ray}}。

==相關條目==
*{{le|高频近似|High frequency approximation}}

== 参考文献 ==
{{Reflist|30em}}

==外部連結==
* [http://demonstrations.wolfram.com/ParaxialApproximationAndTheMirror/ Paraxial Approximation and the Mirror] {{Wayback|url=http://demonstrations.wolfram.com/ParaxialApproximationAndTheMirror/ |date=20210507034921 }} by David Schurig, The Wolfram Demonstrations Project.

[[Category:幾何光學]]