Fifth-order theory of astigmatism of thin spherical spectacle lenses

To demonstrate and analyze the fifth-order theory of oblique astigmatism of a thin spherical spectacle lens and make a comparison with the third-order theory and exact ray tracing.

Fifth-order equations were derived and used for analysis of oblique astigmatism of a spherical spectacle lens to calculate analytically the shape of the lens with corrected oblique astigmatism for large angles of field of view. These results were compared with those of finite ray tracing and the third-order aberration theory.

Formulas for the calculation of oblique astigmatism of a thin spherical spectacle lens were derived. These formulas analytically express oblique astigmatism of the third and fifth order. The theory presented generalizes the third-order description of astigmatism of the spherical spectacle lens and derived equations enable calculation of the shape of the spectacle lens with corrected astigmatism even for a large field of view. The fifth-order solution is compared with the third-order theory and the exact solution found by ray tracing. Differences between the third- and fifth-order theory are <0.05 D for spherical lenses, which is negligible clinically.

The presented fifth-order equations, which are a generalization of the third-order formulas for the description of oblique astigmatism, can be used for the analytical expression of the fifth-order astigmatism of the spherical lens. They can simply be applied for the initial design of lenses with corrected astigmatism for large angles of view, something not possible using the third-order theory. We conclude that astigmatism of the fifth order has little effect on the image quality of the spectacle lens, and the third-order theory is satisfactory for practical calculations in optometry.