Abbe sine condition

The Abbe sine condition (short: sine condition or, more rarely, the Abbe condition ) is a matter of geometrical optics and was formulated by Ernst Karl Abbe . It is a necessary condition in order to display a small surface element close to the axis and perpendicular to the axis free of image defects .

For this purpose, consider rays that emanate from the axis point of said surface element. The angle of intersection of such a beam with the optical axis is in the object space and the image space. The refractive index in the object space is , in the image space . The sine condition is: ${\displaystyle \sigma }$ ${\displaystyle \sigma '}$ ${\displaystyle n}$ ${\displaystyle n'}$

${\displaystyle {\frac {n\sin \sigma }{n'\sin \sigma '}}=\beta '={\text{const.}}}$

The quotient on the left is therefore for all of the above. Rays are equal, and this constant is the paraxial image scale . However, the sine condition is not sufficient. Only when the opening error for the axis point of the surface element in question has also been eliminated is it actually mapped without image errors. ${\displaystyle \beta '}$

With an infinite object distance, the above Rays on the object side are not defined by the angle of intersection, since they all run parallel to the optical axis, but rather by their distance from the optical axis. The sine condition then changes into: ${\displaystyle h}$

${\displaystyle {\frac {h}{\sin \sigma '}}={\text{const.}}=f'}$

with the image-side focal length . ${\displaystyle f'}$

The sine condition must also be fulfilled outside of the paraxial area in optical systems that map a small field with a large aperture . This applies e.g. B. for microscope objectives. A single refractive or reflective spherical surface has three pairs of points that meet the sine condition and map them without opening errors ( aplanatic pairs of points ).