Descartes Ovoids And The Aplanatic Sphere

Fig. 9.4 A limit case of ovoids is a sphere (thick line) providing anastigmatic conjugates. This sphere, obtained for a = 0 and which center C is located at PC/P'C = n2, satisfies the all-order correction of spherical aberration and coma, i.e. the Abbe sine condition

An all-order anastigmatic thick lens for one wavelength is a monocentric thick lens of index n with front and back spherical radii R/n and R centered in C. If the emerging beams virtually converge behind C on a sphere of radius nR also centered at C, then all the incident beams converge on the lens front-surface of radius R/n (Fig. 9.5). The Abbe sine condition (cf. Sect. 1.9.2) is satisfied and due to the monocentric symmetry, the astigmatism is also removed at all orders.

An aplanatic thick lens with spherical surfaces can be obtained with conjugates at similar locations R'/n, nR' from the center of a sphere of radius R', where the beam is refracted, whilst the other surface is in normal incidence. Setting the third-order aplanatism condition for a thin lens as Sj = Sjj = 0 in (9.5a) and (9.5b) allows solving for the B and C variables. After simplifications, we find

If n = 3/2, this gives B = ±4 and C = ^5, i.e. a meniscus lens and a virtual conjugate for both cases.

Substituting the C variable into (9.4), the transverse magnification is

0 0

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