Paraxial approximation

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

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]

${\displaystyle {\begin{aligned}\sin(\theta )&\approx \theta \\\tan(\theta )&\approx \theta \end{aligned}}}$

and

${\displaystyle \cos(\theta )\approx 1}$

The paraxial approximation is used in first-order raytracing and Gaussian optics.^[1] Ray transfer matrix analysis is one method that uses the approximation.

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 approximation becomes

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

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

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. Greivenkamp, John E. (2004). Field Guide to Geometrical Optics. SPIE Field Guides vol. FG01. SPIE. pp. 19–20. ISBN 0-8194-5294-7.

External links

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