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SPHERICAL ABERRATION 4f'3/x4

SPHERICAL ABERRATION 4f'3/x4

Fig. 9.2 Variation of Sphe 3 of a thin lens with the shape variable B. The full line parabolas are for opposite conjugates, C = 0 i.e. a transverse magnification M = -1, and for the refractive indices indicated. For other values of C the curves are translated; for example, the curve for n = 1.5 is moved so that its vertex stays on the dashed line, the appropriate value of C being labelled [1]

Fig. 9.2 Variation of Sphe 3 of a thin lens with the shape variable B. The full line parabolas are for opposite conjugates, C = 0 i.e. a transverse magnification M = -1, and for the refractive indices indicated. For other values of C the curves are translated; for example, the curve for n = 1.5 is moved so that its vertex stays on the dashed line, the appropriate value of C being labelled [1]

• Lenses with a conjugate at infinity: If a conjugate is at infinity, C = ±1, the lens with Sphe 3 minimal must have the shape n2- 1

which leads to CiC2 < 0 up to a refractive index n < (a/33 - 1)/2 = 2.372. For this index, B2 = 1, so the minimum aberration lens would be with a flat surface. Returning to (9.8) and substituting B into (9.4), the curvature ratio of such lenses for an object at infinity, C = -1, is

and is the reciprocal for an image at infinity, C = 1. If n = 3/2, we obtain the curvature ratios c2/c1 = -4/21 and -21/4, respectively. In addition, the following rule must be applied:

^ If a lens is with spherical surfaces, then the Sphe 3 aberration is minimal when the surface of higher curvature is towards the conjugate at infinity.

Stigmatic lenses in a convergent beam: For conjugates where one of them, is virtual, equations (9.7) give the values of B2 and C2 for SI = 0. Since, from (9.6), these variables must have opposite signs, the conditions are

which corresponds to meniscus lenses since \B\ > 5v/377 = 3.273 for usual materials n > 3/2.

9.1.2 Stigmatic Lens with Descartes Ovoid and Spherical Surface

If one of the surfaces of a lens is a sphere centered on one of the Gaussian conjugate points, Descartes [5] demonstrated that the sigmatism at all orders is achieved if the surface of the other diopter is an ovoid, known as Descartes ovoid. Creating analytic geometry for this purpose, Descartes also gave a classical construction of the ovals with the marked straight edge and a string (cf. Sect. 1.1.4 and Fig. 1.4).

Let us consider a source point P in a medium of refractive index unity, and its conjugate P' in a next medium n. The locus of points M determining the surface separation of the two media can be derived from the sum of light propagation times AtPM + AtMP/ which is a constant. For isotropic media, these terms are proportional to PM/c and nMP'/c respectively, where c is the velocity of light. Hence the stig-matism condition is achieved if the optical path is stationary,

where a is the optical path constant. Considering a polar coordinate system r, 6 whose origin is P with 6 = PP', PM and denoting the axial distance PP' = 2b, the above relation writes

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