Paraxial approximation

In

A paraxial ray is a ray that 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]

${\displaystyle \sin \theta \approx \theta ,\quad \tan \theta \approx \theta \quad {\text{and}}\quad \cos \theta \approx 1.}$

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

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

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

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