# Paraxial approximation

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, namely:[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

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

- Greivenkamp, John E. (2004).
*Field Guide to Geometrical Optics*. SPIE Field Guides.**1**. SPIE. pp. 19–20. ISBN 0-8194-5294-7. - Weisstein, Eric W. (2007). "Paraxial Approximation".
*ScienceWorld*. Wolfram Research. Retrieved 15 January 2014. -
"Paraxial approximation error plot".
*Wolfram Alpha*. Wolfram Research. Retrieved 26 August 2014.

## External links

- Paraxial Approximation and the Mirror by David Schurig, The Wolfram Demonstrations Project.