# Paraxial approximation

The error associated with the paraxial approximation. In this plot the cosine is approximated by 1 - θ2/2.

In geometric optics, the paraxial approximation is a small-angle approximation used in Gaussian optics and ray tracing of light through an optical system (such as a lens).[1] [2]

A paraxial ray is a ray which makes a small angle (θ) to the optical axis of the system, and lies close to the axis throughout the system.[1] Generally, this allows three important approximations (for θ in radians) for calculation of the ray's path:[1]

\begin{align} \sin \theta &\approx \theta\\ \tan \theta &\approx \theta \end{align}

and

$\cos \theta \approx 1$

The paraxial approximation is used in Gaussian optics and first-order ray tracing.[1] 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

$\cos \theta \approx 1 - { \theta^2 \over 2 } \ .$

The second-order approximation is accurate within 0.5% for angles under about 10°, but its inaccuracy grows significantly for larger angles.[3]

For larger angles it is often necessary to distinguish between meridional rays, which lie in a plane containing the optical axis, and sagittal rays, which do not.

## References

1. ^ a b c d Greivenkamp, John E. (2004). Field Guide to Geometrical Optics. SPIE Field Guides 1. SPIE. pp. 19–20. ISBN 0-8194-5294-7. edit
2. ^ Weisstein, Eric W. (2007). "Paraxial Approximation". ScienceWorld. Wolfram Research. Retrieved 15 January 2014.
3. ^ [{%28x+Deg+-+Sin[x+Deg%29%2FSin[x+Deg]%2C+%28Tan[x+Deg]+-+x+Deg%29%2FTan[x+Deg]%2C+%281+-+Cos[x+Deg]%29%2FCos[x+Deg]%2C%281-%28x+Deg%29^2%2F2-cos[x+Deg]%29%2FCos[x+Deg]}%2C+{x%2C+0%2C+15}] "Paraxial approximation error plot"]. Wolfram Alpha. Wolfram Research. Retrieved 15 January 2014.