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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.

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^{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. **^**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.

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