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In optics the Smith–Helmholtz invariant is an invariant quantity for paraxial beams propagating through an optical system. Given an object at height ${\displaystyle {\bar {y}}}$ and an axial ray passing through the same axial position as the object with angle ${\displaystyle u}$, the invariant is defined by

${\displaystyle H=n{\bar {y}}u}$,

where ${\displaystyle n}$ is the refractive index. For a given optical system and specific choice of object height and axial ray, this quantity is invariant under refraction. Therefore, at the ${\displaystyle i}$th conjugate image point with height ${\displaystyle {\bar {y}}_{i}}$ and refracted axial ray with angle ${\displaystyle u_{i}}$ in medium with index of refraction ${\displaystyle n_{i}}$ we have ${\displaystyle H=n_{i}{\bar {y}}_{i}u_{i}}$. Typically the two points of most interest are the object point and the final image point.

The Smith–Helmholtz invariant has a close connection with the Abbe sine condition. The paraxial version of the sine condition is satisfied if the ratio ${\displaystyle nu/n'u'}$ is constant, where ${\displaystyle u}$ and ${\displaystyle n}$ are the axial ray angle and refractive index in object space and ${\displaystyle u'}$ and ${\displaystyle n'}$ are the corresponding quantities 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.

The Smith–Helmholtz invariant also relates the lateral and angular magnification of the optical system, which are ${\displaystyle y'/y}$ and ${\displaystyle u'/u}$ respectively. Applying the invariant to the object and image points implies the product of these magnifications is given by

${\displaystyle {\frac {y'}{y}}{\frac {u'}{u}}={\frac {n}{n'}}}$

The Smith–Helmholtz invariant is closely related to the Lagrange invariant and the optical invariant. The Smith–Helmholtz is the optical invariant restricted to conjugate image planes.

See also

• Etendue
• Lagrange invariant
• Abbe sine condition

References

1. Born, Max; Wolf, Emil (1980). 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. Kingslake, Rudolf (2010). Lens design fundamentals (2nd ed.). Amsterdam: Elsevier/Academic Press. pp. 63–64. ISBN 9780819479396.
4. Jenkins, Francis A.; White, Harvey E. (3 December 2001). Fundamentals of optics (4th ed.). McGraw-Hill. pp. 173–176. ISBN 0072561912.
5. Born, Max; Wolf, Emil (1980). 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.

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