Chief nodal axes of a heterocentric astigmatic eye and the Thibos–Bradley achromatic axis

Two kinds of axes are described as achromatic in the vision science literature: those of Le Grand and Ivanoff, originally proposed in the 1940s, and those of Thibos and Bradley proposed in the 1990s. Thibos-Bradley axes are based on chief nodal rays, that is, nodal rays that intersect the pupil at its center. By contrast Le Grand-Ivanoff achromatic axes are pupil independent. The purpose of this paper is to develop the linear optics of Thibos-Bradley achromatic axes and to examine the sense in which such axes can be said to be achromatic. Linear optics is used to define the chief nodal ray of an arbitrary optical system whose refracting elements may be heterocentric and astigmatic and with nonaligned principal meridians. The incident segment of the ray then defines what is called here the incident chief nodal axis and the emergent segment the emergent chief nodal axis. When applied to an eye they become the external and retinal chief nodal axes of the eye. The axes are infinite straight lines. Equations are derived, in terms of the properties of the eye, for the inclination and transverse positions of both axes at incidence onto the eye. An equation is also derived for the position of the retinal chief nodal axis at the retina. The locations of the axes are calculated for a particular model eye in Appendix A. The equations are specialized for the system consisting of an eye and a pinhole in front of it. For a reduced eye the external and retinal chief nodal axes coincide and are independent of the frequency of light; and, hence, the Thibos-Bradley axes are strictly achromatic for that eye. However for more complicated eyes this is not usually the case; the external and retinal axes are usually distinct, dependent on frequency, and, hence, not strictly achromatic. It seems appropriate, therefore, to reserve the term achromatic axis for axes of the Le Grand-Ivanoff type and generalizations thereof, and to call Thibos-Bradley axes chief nodal axes.