# Derivation of conventional formula of the third-order aberration for off-axial optical system

- 14 Downloads

**Part of the following topical collections:**

## Abstract

A formula of the third-order aberration based on the primary aberration expansion for an off-axial optical system is investigated. The point eikonal analysis and the coordinate system using the surface normal vector lead to a simple wavefront-based expression. Three-ray system employing the central, the principal and an arbitrary ray is examined for the derivation. The rays are forced to reach from the last surface to a virtual image point. Therefore, the contradiction of the optical path lengths brings the wavefront aberration. The extracted third-order aberrations are not strictly the same expressions as the conventional formula but show clearly the features of the five kinds of the primary aberrations along with the term originated by the off-axial layout. The relation between the second-order term and the expanded gaussian matrix for paraxial ray tracing and a term related to field curvature are discussed for off-axial systems.

## Notes

## References

- 1.Herzberger, M.: Modern Geometrical Optics. Interscience Publishers, New York (1958)zbMATHGoogle Scholar
- 2.Araki, K.: Analysis of off-axial optical systems (1). Opt. Rev.
**7**, 221 (2000)CrossRefGoogle Scholar - 3.Araki, K.: Paraxial and aberration analysis of off-axial optical systems. Opt. Rev.
**12**, 219 (2005)CrossRefGoogle Scholar - 4.Stone, B.D., Forbes, G.W.: Foundations of first-order layout for asymmetric systems: an application of Hamilton’s methods. J. Opt. Soc. Am. A
**9**, 96 (1992)ADSCrossRefGoogle Scholar - 5.Stone, B.D., Forbes, G.W.: Characterization of first-order optical properties for asymmetric systems. J. Opt. Soc. Am. A
**9**, 478 (1992)ADSCrossRefGoogle Scholar - 6.Araki, K.: Analysis of off-axial optical systems (2). Opt. Rev.
**7**, 326 (2000)CrossRefGoogle Scholar - 7.Wakazono, T., Yatagai, T., Araki, K.: Third-order aberration analysis of an off-axial optical system. Opt. Rev.
**23**, 61–76 (2016)CrossRefGoogle Scholar - 8.Buchdarl, H.A.: System without symmetries: foundations of a theory of Lagrangian aberration coefficients. J. Opt. Soc. Am.
**63**, 1314 (1972)ADSCrossRefGoogle Scholar - 9.Stone, B.D., Forbes, G.W.: Foundations of second-order layout for asymmetric systems. J. Opt. Soc. Am. A
**9**, 2067 (1992)ADSMathSciNetCrossRefGoogle Scholar - 10.Oleszko, M., Hamback, R., Gross, H.: Decomposition of the total wave aberration in generalized optical systems. J. Opt. Soc. Am. A
**34**, 1856 (2017)ADSCrossRefGoogle Scholar - 11.Fuerschback, K., Rolland, J.P., Thompson, K.P.: Theory of aberration fields for general optical systems with freeform surfaces. Opt. Express
**22**, 26585 (2014)ADSCrossRefGoogle Scholar - 12.Thompson, K.: Description of the third-order optical aberrations of near-circular pupil optical systems without symmetry. J. Opt. Soc. Am. A
**22**, 1389 (2005)ADSCrossRefGoogle Scholar - 13.Matsui, Y.: Lens design method (in Japanese). Kyoritsu Publication, Tokyo (1972)Google Scholar
- 14.Born, M., Wolf, E.: Principles of Optics, 6th edn. Cambridge University press (1980)Google Scholar