only even powers appearing because of axial symmetry. The quantity c is the curvature, the reciprocal of r, the radius of curvature, while ai,a2... are constants. Wavefronts emerging from axially placed object points and passing through the system would also be expressed by Eq. (2.13). For object points in the principal section which are not on the axis (y = 0), the central ray of the beam (the principal ray to be defined in connection with pupils, below) will define an oblique line in the principal section as the equivalent of the z-axis, to which a polynomial can also be applied for wavefronts centered on such a principal ray. Because of the asymmetry of such an object point to the axis, this polynomial will be an extended form of Eq. (2.13) which includes odd terms as well as even ones. It should be noted, however, that the only odd term of lower order than the first term of Eq. (2.13) is a linear term, implying a tilt of the wavefront. But this implies an incorrect image height: if this height is chosen correctly, the linear term is zero and the first quadratic term of (2.13) also remains the first for off-axis object points.
Gauss [2.4] first developed a complete theory of image formation based only on the approximation of the first, quadratic, term in Eq. (2.13). This is also called the parabolic term since it defines a parabola if all other terms are zero. Further properties of Eq. (2.13) will be discussed in § 3.1: at this stage we are concerned with the significance of the Gaussian approximation, whereby only the first term is considered.
Higher order terms than the first can only be negligible if the height y above the axis is very small compared with the radius of curvature r. This domain of validity is called the paraxial region. By definition, the difference between a parabola and a sphere, or any other conic section (see Chap. 3), is negligible in the paraxial region. The definition therefore excludes all qualitative influences of the precise form of refracting or reflecting surfaces, since the wavefronts are also defined in a way that ignores the effects of differing forms. In other words, the Gaussian or paraxial region is concerned, by definition, with ideal image formation. It is solely concerned with the position and nominal size of an image, not with its quality.
Since the heights y of the ray intersections are small, it follows that the angles u, u of the rays with the axis, and the angles of incidence and refraction i, i with the refracting (or reflecting) surfaces are small. This leads to another important interpretation of the paraxial region. Snell's law (Eq. (2.12)) can be written, expanding the sines by Maclaurin's theorem
To the Gaussian approximation all terms after the first are negligible, so that Snell's law for the paraxial region becomes simply a simple linear equation.
The basic tool of optical design is the exact tracing of the path of optical rays through the system. The general formulation for doing this is given in specialist works on optics, e.g. Welford [2.3], Herzberger [2.7]. Although it is normally expressed in a vectorial, algebraic form adapted to modern computers, the exact formulation corresponds to the exact form of Snell's law of Eq. (2.12). "Paraxial rays" are traced with a reduced, simplified formulation based on the linear equation (2.15). All the equations involved are also linear and are given below. This linearity has the following important consequence: although in physical reality the heights y and angles u and i of a paraxial ray (or its diagrammatic representation) must be very small if the results are to be valid, the calculated results for the position and size of paraxial images are independent of the input conditions. Thus, if for a real physical lens surface whose diameter far exceeds the paraxial region, a ray is traced from an axial object point to the aperture edge with the exact formulae and then for the same aperture height with the paraxial formulae, the difference gives directly the effect of the terms in Eq. (2.13) beyond the first. These terms determine the quality of the image, in other words, the extent to which it is affected by aberrations. This is the subject of Chap. 3.
The laws of Gaussian optics completely define the position and size of the image in any telescope system, irrespective of the number and nature of its elements. The paraxial form of Snell's law is the basic law. We shall now deduce some other relationships.
In Fig. 2.4 the object medium has index n, the image medium n . In the paraxial sense, the principal planes PPr and P PH can be considered as wavefronts of the rays r\ and r2. For r\, the optical path FPrPrFr must be equal to FPP F or
F Medium n
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