## Gaussian Optics and Conjugate Distances

Considering a curved surface (diopter) separating two optical media of refractive indexes n in the object space and n' in the image space, and an optical axis z taken perpendicular to this diopter with the origin O onto its surface, the first-order optics theory based on Snell's law allows determining the axial positions of the focii F and F' and of the conjugate distances of an object and its real or virtual image.

The general form of a local surface is defined by its two principal curvatures. We may set the section plane of these curvatures along x and y directions of the x, y, z frame. The diopter shape is represented by the power series z = 2 cxx2 + 2 cyy2 + O(xp,yq), (1.15)

where p and q are integers greater than 2.

The determination of focus positions and conjugate distances in each x, z or y, z plane leads to expressions of similar forms in cx and cy (see for instance Sect. 3.5.7). Then, we do not restrict the analysis by considering cx=cy and a diopter represented by z = 1 cr2 + O(xp,yq). (1.16)

K.F. Gauss [66], in a celebrated memoir of 1841 (see comments by Wilson [171]) and with complete generality, demonstrated that the higher order terms O, do not enter into the determination of the focus positions and conjugate distances. These are obtained from the first-order optics theory or Gaussian theory, also called paraxial theory. In Gaussian optics, the representation of any surface of an optical system reduces to the quadratic form, zt = Zo,t + 1 ct r2, t = 1,2,3,..., (1.17)

where all the axisymmetric surfaces, individually numbered t, have the same axis. Such a system is usually called a centered system.

Hence, considering a refractive or reflective spherical surface, we in fact cannot distinguish it from a paraboloid or another conicoid of the same curvature because its shape only differs by the next order term. The Gaussian theory concerns the analysis with the so-called paraxial rays, i.e. rays which are in the immediate neighborhood of the axis. In this theory the sine expansion of Snell's law, which relates the incident angle i in medium n to the conjugate emerging angle i ' in medium n ', n '( i ' - 3- i '3 +...) = n(i - 3 i3 +...) , (1.18)

• Sign convention: Because of the small angles considered, a paraxial drawing may have difficulty showing simultaneously the focal points and the curvature of a diopter. This curvature is schematically represented by a bracket with ends turned towards the center of curvature. We use the Cartesian sign convention: a positive curvature corresponds to a surface whose concavity is turned towards z positive.

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