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 [[Gaussian optics]] and [[Ray tracing (physics)|ray tracing]] of light through an optical system (such as a [[lens (optics)|lens]]).<ref name=Greivenkamp>

[[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}}.]]

{{cite book

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 | pages = 19–20 }}</ref><ref>{{cite web

| first=John E.

| last=GreivenkampWeisstein

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

| 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><ref>

{{cite web

| last=[[Eric W. Weisstein|Weisstein]]

| first=Eric W.

| title=Paraxial Approximation

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

The paraxialsecond-order approximation is accurate within 0.5% for angles under about 10°, but its inaccuracy grows significantly for larger angles.<ref>

{{cite web

| title=Paraxial approximation error plot

| url=http://www.wolframalpha.com/input/?i=Plot%5B%7B{%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%7D}%2C+%7Bx{x%2C+0%2C+15%7D}%5D

| work=[[Wolfram Alpha]]

| publisher=[[Wolfram Research]]

| accessdate=1526 JanuaryAugust 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 ==