Paraxial approximation: Difference between revisions

Revision as of 02:47, 18 February 2017

Latest revision as of 18:33, 13 April 2025

(10 intermediate revisions by 9 users not shown)

Line 1:

{{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, mirror(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

| last=Weisstein

<ref>{{cite web

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

| first=Eric W.

| title=Paraxial Approximation

Line 10 ⟶ 11:

| accessdate=15 January 2014|year=2007}}</ref>

A '''paraxial ray''' is a [[Ray (optics)|ray]] whichthat makes a small angle (''θ=1,2,3,4,5)'') 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>

Line 17 ⟶ 18:

\quad \text{and}\quad\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

Line 23 ⟶ 24:

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

The second-order approximation is accurate within 0.5% for angles under about 510°, but its inaccuracy grows significantly for larger angles.<ref>

{{cite web

| title=Paraxial approximation error plot

Line 30 ⟶ 31:

| 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 ==