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

,

where 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 th conjugate image point with height and refracted axial ray with angle in medium with index of refraction we have . 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 is constant, where and are the axial ray angle and refractive index in object space and and are the corresponding quantities in image space. The Smith–Helmholtz invariant implies that the lateral magnification, 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 and respectively. Applying the invariant to the object and image points implies the product of these magnifications is given by[5]

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

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