# 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).

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. Generally, this allows three important approximations (for θ in radians) for calculation of the ray's path:

begin{align} sin theta &approx theta\ tan theta &approx theta end{align}

and

The paraxial approximation is used in Gaussian optics and first-order raytracing. 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 order approximation is

The paraxial approximation is accurate within 0.5% for angles under about 10° but its inaccuracy grows significantly 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.

### Other articles related to "paraxial approximation, approximation, paraxial":

Helmholtz Equation - Paraxial Approximation
... Further information Slowly varying envelope approximation The paraxial approximation of the Helmholtz equation is where is the transverse part of the Laplacian ... In the paraxial approximation, the complex magnitude of the electric field E becomes where A represents the complex-valued amplitude of the electric field ... The paraxial approximation places certain upper limits on the variation of the amplitude function A with respect to longitudinal distance z ...
Ray (optics) - Special Rays - Optical Systems
... A paraxial ray is a ray that makes a small angle to the optical axis of the system, and lies close to the axis throughout the system ... this definition is often reversed a "paraxial ray" is then a ray that is modeled using the paraxial approximation, not necessarily a ray that remains close to the axis ... A finite ray or real ray is a ray that is traced without making the paraxial approximation ...