Paraxial approximation: Difference between revisions

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{{Short description|Small angle approximation in geometric optics}}

In [[geometric optics]], the '''paraxial approximation''' is a [[small-angle approximation]] used in [[Ray tracing (physics)|ray tracing]] of light through an optical system (such as a [[lens (optics)|lens]]).<ref name=Greivenkamp>{{cite book | first=John E. | last=Greivenkamp | year=2004 | title=Field Guide to Geometrical Optics | publisher=SPIE | others=SPIE Field Guides vol. '''FG01''' | isbn=0-8194-5294-7 |pages=19–20 }}</ref>

[[File:Small angle compare error.svg|thumbnail|The error associated with the paraxial approximation. In this plot the cosine is approximated by {{nowrap|1 - θ²/2}}.]]

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">{{citeCite book | firstisbn =John E.0-8194-5294-7 | lasttitle =Greivenkamp Field Guide to Geometrical Optics | yearlast1 =2004 Greivenkamp | titlefirst1 =Field GuideJohn toE. Geometrical| Opticsyear = 2004 | publisher = [[SPIE]] | othersseries = SPIE Field Guides vol.| '''FG01'''volume | isbn=0-8194-5294-7 1 | pages = 19–20 }}</ref><ref>{{cite web

| last=Weisstein

| author-link=Eric W. 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}}</ref>

A '''paraxial ray''' is a [[Ray (optics)|ray]] whichthat 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, namely:<ref name=Greivenkamp/>

:<math>\begin{align}

\sin( \theta) &\approx \theta,\\quad

\tan( \theta) &\approx \theta

:<math>\quad \text{and}\quad\cos( \theta) \approx 1.</math>

\end{align}</math>

and

:<math>\cos(\theta) \approx 1</math>

The paraxial approximation is used in [[Gaussian optics]] and ''first-order'' raytracingray and [[Gaussian optics]]tracing.<ref name="Greivenkamp" /> [[Ray transfer matrix analysis]] is one method that uses the approximation.

In some cases, the second-order approximation is also called "paraxial". The approximations above for sine and tangent aredo alreadynot accuratechange tofor secondthe "second-order" in ''θ'', but theparaxial approximation for cosine needs to be expanded by including (the nextsecond term in thetheir [[Taylor series]] expansion. Theis thirdzero), while approximationfor cosine the second thenorder approximation becomesis

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

The paraxialsecond-order approximation is fairly accurate within 0.5% for angles under about 10°, but isits inaccuracy grows inaccuratesignificantly for larger angles.<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}] -->

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.

Use of the small angle approximations replaces dimensionless trigonometric functions with angles in radians. In [[dimensional analysis]] on optics equations radians are dimensionless and therefore can be ignored.

A paraxial approximation is also commonly used in [[physical optics]]. It is used in the derivation of the paraxial wave equation from the homogeneous [[Maxwell's equations]] and, consequently, [[Gaussian beam]] optics.

==References==

{{reflistReflist}}

== External links ==

* [http://demonstrations.wolfram.com/ParaxialApproximationAndTheMirror/ Paraxial Approximation and the Mirror] by David Schurig, [[The Wolfram Demonstrations Project]].

[[Category:Geometrical optics]]

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[[fr:Approximation de Gauss]]

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