Smith–Helmholtz invariant

In optics the Smith–Helmholtz invariant is an invariant quantity for paraxial beams propagating through an optical system. Given an object at height ${\bar {y}}$ and an axial ray passing through the same axial position as the object with angle $u$ , the invariant is defined by^[1]^[2]^[3]

H=n{\bar {y}}u

where $n$ 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 $i$ th conjugate image point with height ${\bar {y}}_{i}$ and refracted axial ray with angle $u_{i}$ in medium with index of refraction $n_{i}$ we have $H=n_{i}{\bar {y}}_{i}u_{i}$ . 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 $nu/n'u'$ is constant, where $u$ and $n$ are the axial ray angle and refractive index in object space and $u'$ and $n'$ are the corresponding quantities in image space. The Smith–Helmholtz invariant implies that the lateral magnification, $y/y'$ is constant if and only if the sine condition is satisfied.^[4]

The Smith–Helmholtz invariant also relates the lateral and angular magnification of the optical system, which are $y'/y$ and $u'/u$ respectively. Applying the invariant to the object and image points implies the product of these magnifications is given by^[5]

{\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.

References

^ 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.
^ "Technical Note: Lens Fundamentals". Newport. Retrieved 16 April 2020.
^ Kingslake, Rudolf (2010). Lens design fundamentals (2nd ed.). Amsterdam: Elsevier/Academic Press. pp. 63–64. ISBN 9780819479396.
^ Jenkins, Francis A.; White, Harvey E. Fundamentals of optics (4th ed.). McGraw-Hill. pp. 173–176. ISBN 0072561912.
^ 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.

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