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In geometric optics, the paraxial approximation is an 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]

 \sin(\theta) \approx \theta
 \tan(\theta) \approx \theta


 \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". The approximations above for sine and tangent are already accurate to second order in θ, but the approximation for cosine needs to be expanded by including the next term in the Taylor series expansion. The third approximation then becomes

 \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.


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

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