Astigmatism of Skew Pencils in Qptical Systems Containing Toric Surfaces

Toric refracting surfaces are used chiefly in two classes of optical systems, spectacle lenses, and anamorphotic photographic and projection objectives. In spectacles the toric surfaces are arranged to correct the axial astigmatism of t he eye, and in anamorphotic systems two or more toric (possibly cylindrical) surfaces are included to give stigmatic imagery with different magnifications in different directions in t he image plane. "When the axial pencil is refracted by a toric,3 some astigmatism is in trodu ced; if a pencil is incident obliquely, either cent rally or eccentrically , astigmatism of different magnitude is in troduced. It is therefore desirable in the case of both types of optical systems mentioned above to determine the astigmatism of obliquely incident pencils, in order to assess the imagery in the outer par ts of the field. This is t he problem which was studied in the work to be described. The problem arose in connection with spectacle lenses and may be outlined as follows: Figure 1 shows fron t and side views of a spectacle lens wit h toric surface ; 0 is the center of the lens, C is t he center of rotation of the eye, and PI, P2 , P3 are points of incidence of the chief rays of oblique pencils. L et the meridian and equatorial sections of the toric be, respectively , in the ver tical and horizontal planes OPt and OP3 • Then the problem for chief rays passing through the axis and tlu'ough PI and P 3 is t rivial. The chief ray itself can be traced by the ordinary ray-tracing formulas for meridian rays and spherical surfaces, and the astigmatism can be determined by the usual methods for close sagittal and tangen tial fans, with the sligh t difference that the curva tures of the refracting surface in tbe two directions are no t eq Llal (see eq (8) ).