Smith-Helmholtz invariant

In optics the Smith-Helmholtz invariant is an invariant quantity for light propagating through an optical system. It is defined by[1][2]

${\displaystyle H=nyu}$,

where ${\displaystyle u}$ is the marginal ray angle, ${\displaystyle y}$ is the chief ray height, and ${\displaystyle n}$ is the refractive index. For a given optical system, this quantity is invariant under refraction, and implies a variety of other interesting relationships.

It is closely connected to the Abbe sine condition. The paraxial version of the sine condition is satisfied if the ratio ${\displaystyle u/u'}$ is constant, where ${\displaystyle u'}$ is the marginal ray angle in image space. The Smith-Helmholtz invariant implies that the lateral magnification, ${\displaystyle y/y'}$ is constant if and only if the sine condition is satisfied.[3]

See also

• Etendue
• Lagrange invariant
• Abbe sine condition

References

1. Born, Max; Wolf, Emil. Principles of optics : electromagnetic theory of propagation, interference and diffraction of light (6th ed.). Pergamon Press. pp. 164–166. ISBN 978-0-08-026482-0.
2. "Technical Note: Lens Fundamentals". Newport. Retrieved 16 April 2020.
3. Jenkins, Francis A.; White, Harvey E. Fundamentals of optics (4th ed.). McGraw-Hill. pp. 173–176. ISBN 0072561912.