## Gqx

For n = 3/2, we obtain c2/c1 = -(1/9)±1. If the conjugate at infinity is the object, C = -1, and from (9.3) the curvatures are

^ A Coma 3 corrected lens is obtained when the surface of higher curvature is towards the conjugate at infinity.

^ From (9.15), if n = (1 + V5)/2, then a Coma 3 corrected lens becomes convexo-plane or plano-convex for objects at t™, and the curvature of the convex surface is c = ± nK.

These latter cases, c1 c2 = 0, are the limit cases between biconvex and meniscus isoplanatic lenses, i.e. c1c2 < 0 or > 0 respectively. Since n2 = n + 1, if the object is at infinity, then c1 = -nK and c2 = 0. From (1.78), (1.45), and (1.46), the radii of curvature of the Petzval, sagittal, mean and tangential surfaces are RP = -(n + 1)R1, Rs = -R1, Rm = -R1/n, and Rt = -R1/(2n - 1), respectively (Fig. 9.6).

Lens and remote pupil: The amount of Coma 3, Astm 3, and Dist 3 of a lens are also functions of the location of the entrance pupil or stop, whilst the Sphe 3 and Petz 3 amounts remain unchanged. The stop-shift effect can be introduced into the analytic representation of the Seidel sums Sii, Siil, and SV (cf. for instance Welford [1]).

As a basic example of an isoplanatic arrangement with a remote pupil, let us consider an extended object at infinity whose optical beams first pass through an aperture stop (Fig. 9.6). A plano-convex thin lens, c1 = 0, is then located at a distance d from the stop so a ray passing at the center of the stop is perpendicular to the second surface of the lens, i.e. dc2 = 1/n if we neglect the thickness of the

focal surface focal surface

Fig. 9.6 Isoplanatic lenses (Sj = 0, Sjj = 0) with spherical surfaces for an object at infinity. Left: Convexo-plane lens of refractive index n = (1 + V5)/2 = 1.618 with entrance pupil at the lens (Sjjj = 0). Right: Entrance pupil and remote plano-convex lens; the center of the entrance pupil -or aperture stop - is located at the conjugate of the center of curvature of the lens second surface with respect to its plane first surface (hence Sjjj = 0 also)

focal surface focal surface

Fig. 9.6 Isoplanatic lenses (Sj = 0, Sjj = 0) with spherical surfaces for an object at infinity. Left: Convexo-plane lens of refractive index n = (1 + V5)/2 = 1.618 with entrance pupil at the lens (Sjjj = 0). Right: Entrance pupil and remote plano-convex lens; the center of the entrance pupil -or aperture stop - is located at the conjugate of the center of curvature of the lens second surface with respect to its plane first surface (hence Sjjj = 0 also)

lens. The result is that Coma 3 is corrected, Sn = 0. In addition, this system is also free from Astm 3, (Sni = 0), which provides interesting properties for very wide field systems if the chromatism and Sphe 3 are corrected by additional lenses.

• Photographic objectives: The remote pupil principle is a key feature in the development of photographic objectives. This was implicitly used in 1840 by Joseph Max Petzval in the first calculation of a portrait objective which consisted of a cemented lens plus a pair of air-spaced elements. Some important accounts of the design and evolution of the various forms of photographic objectives are given, for instance, by Zeiss [6], Chretien [2], Kingslake [7] and Laikin [8].

### 9.1.5 Aspheric Lenses in the Third-Order Theory

Instead of considering a lens where one of the surfaces is a Descartes ovoid, we may fall back to the approximation of the optics third-order theory by representing one of its surfaces or both by the two first terms of a conicoid (cf. Sect. 1.7.1). The equation of first surface is then

whilst the second surface is with the subscript 2.

It can be shown (cf. for instance Born and Wolf [3]) that, for an object at infinity and the entrance pupil on the lens, the condition which cancels Sphe 3 of the lens, i.e. Sj = 0, is satisfied if

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