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| [[File:Small angle compare error.svg|thumbnail|The error associated with the paraxial approximation. In this plot the cosine is approximated by {{nowrap|1 - θ<sup>2</sup>/2}}.]]
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| In [[geometric optics]], the '''paraxial approximation''' is a [[small-angle approximation]] used in [[Gaussian optics]] and [[Ray tracing (physics)|ray tracing]] of light through an optical system (such as a [[lens (optics)|lens]]).<ref name=Greivenkamp>
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| {{cite isbn | 0819452947 | pages=19–20}}</ref>
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| <ref>{{cite web
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| | last=[[Eric W. Weisstein|Weisstein]]
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| | first=Eric W.
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| | title=Paraxial Approximation
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| | url=http://scienceworld.wolfram.com/physics/ParaxialApproximation.html
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| | work=[[ScienceWorld]]
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| | publisher=[[Wolfram Research]]
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| | accessdate=15 January 2014|year=2007}}</ref>
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| A '''paraxial ray''' is a [[Ray (optics)|ray]] which makes a small angle (''θ'') to the [[optical axis]] of the system, and lies close to the axis throughout the system.<ref name=Greivenkamp/> Generally, this allows three important approximations (for ''θ'' in [[radian]]s) for calculation of the ray's path:<ref name=Greivenkamp/>
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| :<math>\begin{align}
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| \sin \theta &\approx \theta\\
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| \tan \theta &\approx \theta
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| \end{align}</math>
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| and
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| :<math>\cos \theta \approx 1</math> | |
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| The paraxial approximation is used in [[Gaussian optics]] and ''first-order'' raytracing.<ref name=Greivenkamp/> [[Ray transfer matrix analysis]] is one method that uses the approximation.
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| In some cases, the second-order approximation is also called "paraxial". The approximations above for sine and tangent do not change for the "second-order" paraxial approximation (the second term in their [[Taylor series]] expansion is zero), while for cosine the second order approximation is
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| :<math> \cos \theta \approx 1 - { \theta^2 \over 2 } \ .</math>
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| The second-order approximation is accurate within 0.5% for angles under about 10°, but its inaccuracy grows significantly for larger angles.<ref>
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| {{cite web
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| | title=Paraxial approximation error plot
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| | url=http://www.wolframalpha.com/input/?i=Plot[{%28x+Deg+-+Sin[x+Deg]%29%2FSin[x+Deg]%2C+%28Tan[x+Deg]+-+x+Deg%29%2FTan[x+Deg]%2C+%281+-+Cos[x+Deg]%29%2FCos[x+Deg]%2C%281-%28x+Deg%29^2%2F2-cos[x+Deg]%29%2FCos[x+Deg]}%2C+{x%2C+0%2C+15}]
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| | work=[[Wolfram Alpha]]
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| | publisher=[[Wolfram Research]]
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| | accessdate=15 January 2014}}</ref>
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| <!-- This plots 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}] -->
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| For larger angles it is often necessary to distinguish between [[meridional ray]]s, which lie in a plane containing the [[optical axis]], and [[sagittal ray]]s, which do not.
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| ==References==
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| {{reflist}}
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| == External links ==
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| * [http://demonstrations.wolfram.com/ParaxialApproximationAndTheMirror/ Paraxial Approximation and the Mirror] by David Schurig, [[The Wolfram Demonstrations Project]].
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| [[Category:Geometrical optics]]
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