1
NOTE ON ASPHERICAL LENS SYSTEMS BY WILLARD J. FISHER In the paper by L. Silberstein, on aspherical lens systems, J.O.S.A., 11, p. 479-494; 1925, the statement is made, p. 483, that to his knowl- edge the only previous method of closing a simple lens, whose front surface is generated by revolving a Cartesian oval, is by means of a spherical concave surface centered at the focal point of the ovaloid. In Eq. (7), p. 486, this case corresponds to makingfi=f 2 . But if lifi =f2, the second surface is again a hollow sphere, with one of its aplanatic points at F 1 , the other at F 2 , (Fig. 5, though without limitation to object at infinity). This combination of surfaces is not aplanatic, by Eq. (15) and the paragraph following. The aplanatic condition being /sin = const., the deviation from aplanatism for the front surface is proportional to the departure from constancy of 1/(1 -cos u/ju), in whichu 1 = 0 1 F 1 P 1 . If the back surface is the aplanatic spherical surface mentioned, and U 2 is the angle at F 2 , then /.t sin u 2 =sin . The aplanatic condition can equally well be written sin u/l = const. Then the departure from constancy is proportional to cos u/A= _Vtu. 2 - sin 2 u 2 . For quartz glass and G light, 4340, ,u= 1.467. If cos u changes from cos u=1, =0 0 , to cos u=O.99, =8'06', then 1-cos u,u changes from 0.320 to 0.327, or 2.2 per cent. Simultaneously u 2 changes from 0° to 530'. 2 tan 530'=f/5.2, roughly. Such a quartz lens, computed for the transmission band of a silver film, might be of use in eclipse photography. It may be mentioned that the writer has not seen any mention of the fact that a hollow sphere has aplanatic points. The above is a demon- stration of the theorem, which was however discovered in an attempt to invert the Weierstrass construction, some years ago. As the theorem for a solid sphere is of great importance in microscopy, its inverse may equally become of technical importance. 290

NOTE ON ASPHERICAL LENS SYSTEMS

Embed Size (px)

Citation preview

Page 1: NOTE ON ASPHERICAL LENS SYSTEMS

NOTE ON ASPHERICAL LENS SYSTEMS

BY WILLARD J. FISHER

In the paper by L. Silberstein, on aspherical lens systems, J.O.S.A.,11, p. 479-494; 1925, the statement is made, p. 483, that to his knowl-edge the only previous method of closing a simple lens, whose frontsurface is generated by revolving a Cartesian oval, is by means of aspherical concave surface centered at the focal point of the ovaloid.In Eq. (7), p. 486, this case corresponds to makingfi=f 2 .

But if lifi =f2, the second surface is again a hollow sphere, with oneof its aplanatic points at F1, the other at F2 , (Fig. 5, though withoutlimitation to object at infinity).

This combination of surfaces is not aplanatic, by Eq. (15) and theparagraph following. The aplanatic condition being /sin = const.,the deviation from aplanatism for the front surface is proportional tothe departure from constancy of 1/(1 -cos u/ju), in whichu1 = 01F1P1.If the back surface is the aplanatic spherical surface mentioned, andU2 is the angle at F2 , then /.t sin u 2 =sin .

The aplanatic condition can equally well be written sin u/l = const.Then the departure from constancy is proportional to cos u/A=_Vtu. 2 - sin2u2.

For quartz glass and G light, 4340, ,u= 1.467. If cos u changes fromcos u=1, =00 , to cos u=O.99, =8'06', then 1-cos u,u changesfrom 0.320 to 0.327, or 2.2 per cent. Simultaneously u2 changes from0° to 530'. 2 tan 530'=f/5.2, roughly.

Such a quartz lens, computed for the transmission band of a silverfilm, might be of use in eclipse photography.

It may be mentioned that the writer has not seen any mention of thefact that a hollow sphere has aplanatic points. The above is a demon-stration of the theorem, which was however discovered in an attemptto invert the Weierstrass construction, some years ago. As the theoremfor a solid sphere is of great importance in microscopy, its inverse mayequally become of technical importance.

290