Paraxial approximation: Difference between revisions

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

[[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">{{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 | page = <!-- can't have both --> | pages = 19–20 <!-- --> }}</ref><ref>{{cite web

| last=Weisstein

<ref>{{cite web

| lastauthor-link=[[Eric W. Weisstein|Weisstein]]

| first=Eric W.

| title=Paraxial Approximation

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| 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'' ray 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 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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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 ==