Paraxial approximation: Difference between revisions

Revision as of 05:49, 13 November 2010

Revision as of 08:14, 3 December 2010

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:<math>\begin{align}

\sin( \theta) &\approx \theta\\

\tan( \theta) &\approx \theta

\end{align}</math>

and

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

The paraxial approximation is used in ''first-order'' raytracing and [[Gaussian optics]].<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". To second order, the approximations above for sine and tangent do not change (the next term in their [[Taylor series]] expansion is zero), while for cosine the second order approximation is

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

The paraxial approximation is fairly accurate for angles under about 10°, but is inaccurate for larger angles.