Paraxial approximation

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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, namely:[1]

\sin \theta \approx \theta,\quad
\tan \theta \approx \theta
\quad \text{and}\quad\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.


  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. 
  2. ^ Weisstein, Eric W. (2007). "Paraxial Approximation". ScienceWorld. Wolfram Research. Retrieved 15 January 2014. 
  3. ^ "Paraxial approximation error plot". Wolfram Alpha. Wolfram Research. Retrieved 26 August 2014. 

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