Helmholtz-Lagrangian invariant

The Helmholtz-Lagrange invariant (according to Helmholtz and Lagrange ) represents the relationship between image scale and angle ratio in paraxial optics :

${\displaystyle {\begin{aligned}y\cdot n\cdot \sigma &=y'\cdot n'\cdot \sigma '\\\Leftrightarrow {\frac {n}{n'}}\cdot {\frac {\sigma }{\sigma '}}&={\frac {y'}{y}}\end{aligned}}}$

With

• y or y '= object or image size (each' for the image side)
  • ${\displaystyle {\frac {y'}{y}}}$ = Reproduction scale
• n or n '= object or image-side refractive index
• σ or σ '= object or image-side beam angle.

In the case of optical imaging , the product of the object / image size, the refractive index and the beam angle is therefore the same on the object and the image side.

A similar basic statement of paraxial optics is the Abbe invariant .

Individual evidence

1. ↑ ^{a b} The angular relationship exists between the angles at which a light beam before and after a refracting surface intersects the optical axis .
2. ↑ ^{a b} Fritz Hodam: Technische Optik , VEB Verlag Technik, 2nd edition, 1967, p. 52
3. ↑ Heinz Haferkorn: Optics: Physikalisch-Technischen Basics and Applications , Barth, 1994, ISBN 3-335-00363-2 , S. 187