# Abbe invariant

The Abbe invariant (after Ernst Abbe ; also invariant of refraction , zero variant ) represents in paraxial optics the relationship between the object-side and image-side back focus of light rays that are refracted on a surface :

${\ displaystyle n \ left ({\ frac {1} {r}} - {\ frac {1} {s}} \ right) = n '\ left ({\ frac {1} {r}} - {\ frac {1} {s'}} \ right)}$

With

• n, n '= refractive index before or after the refracting surface (each' for the image side)
• r = radius of curvature of the refracting surface
• s, s' = object-side or image-side focal length.

The equation says that the linear relationship between the index of refraction, the radius of curvature and the back focus remains constant before and after the refraction.

This invariant is a basis for the derivation of all the laws of optical imaging in the area close to the axis.

Another basic statement in this regard is the Helmholtz-Lagrangian invariant .

## Derivation

Derivation of the Abbe invariant

In the triangles ACO and ACO ', the following relationships exist according to the sine law :

${\ displaystyle {\ frac {\ sin \ epsilon} {\ sin (180 ^ {\ circ} - \ phi)}} = {\ frac {sr} {l}}}$   and
${\ displaystyle {\ frac {\ sin \ epsilon '} {\ sin (180 ^ {\ circ} - \ phi)}} = {\ frac {s'-r} {l'}}}$  .

The first divided by the second relationship:

${\ displaystyle {\ frac {\ sin \ epsilon} {\ sin \ epsilon '}} = {\ frac {l' (sr)} {l (s'-r)}}}$  .

With the law of refraction   n sinε = n 'sinε':

${\ displaystyle n \ left ({\ frac {sr} {l}} \ right) = n '\ left ({\ frac {s'-r} {l'}} \ right)}$  .

In the paraxial area, the angles σ and σ 'are so small that the intercept lengths s and s' can be set for the beam lengths l and l'. This gives:

${\ displaystyle n \ left ({\ frac {1} {r}} - {\ frac {1} {s}} \ right) = n '\ left ({\ frac {1} {r}} - {\ frac {1} {s'}} \ right)}$  .

## Individual evidence

1. Lexicon of Physics, Abbesche Invariante. Spektrum.de, accessed on April 6, 2014 .
2. Heinz Haferkorn: Optics: Physikalisch-Technischen Basics and Applications , Barth, 1994, ISBN 3-335-00363-2 , p. 185/86
3. a b Fritz Hodam: Technical Optics , VEB Verlag Technik, 2nd edition, 1967, p 42