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3.2 Characteristic Function and Seidel (3rd order) aberrations 59 Now it can easily be shown that the vertex curvature is c = 72 = n 1 ' (3-6)

where e is the eccentricity. Then

;z" = >2+ 8>+iCs«2 y'+■■■ <">

A hyperbola gives the same result except for alternating negative signs:

Considering now the three equations (3-3), (3-7) and (3-8) and following Schwarzschild [3-1], they can all be written from (3-6) in the form:

Z =2y2 + C3 (1 — e2)y4 + 1j(1 — e2)2y' + ... (3-9)

or, introducing the Schwarzschild (conic) constant bs [3-1] as bs = —e2 (3-10)

c c3 c5

z = 2y2 + 8 (1 + bs)y4 + 16(1 + bs)2y' + ••• (3-11)

To the third order (second term), all surfaces have conic sections, uniquely defined by bs as follows:

bs = —1, e = 1, parabola — 1 <bs < 0, 0 < e < 1, ellipse bs < —1, e > 1, hyperbola

A further case is of interest, as it can be useful in optical systems:

bs > 0, e imaginary, oblate spheroid (minor axis of an ellipse) (3-13)

The formulation of Eqs- (3-11) and (3-12) is the essential basis of aberration theory for telescopes.

3.2 Characteristic Function and Seidel (3rd order) aberrations: aberration theory of basic telescope forms

3.2.1 The Characteristic Function of Hamilton

In 1833, Hamilton [3-2] published one of the most profound and elegant analyses in the history of geometrical optics: the Characteristic Function- Based solely on the property of symmetry of a centered optical system about its axis, he deduced the general form of the aberration function in terms of three fundamental parameters: the aperture radius (p) (normalized to 1 at the edge), the field radius (a) (normalized to 1 at the edge), and the azimuth angle of the plane containing the ray and principal ray in the image forming wavefront. Following Hopkins [3.3], we will define these normalized parameters as p, a and 0, the dashes of p',a' and 0', denoting the image space, being omitted for simplicity. Hamilton showed that, because of symmetry, these parameters can only appear in the general aberration function in the forms

The aberration function must take the form, expressed as wavefront aberration:

W (a,p,0) = okoo + (0^20 p2 + ikn ap cos 0 + 2kooa2)

It is easily shown that some of these terms must be zero because of the definitions of Gaussian optics. All terms 2k00a2, 4k00a4 ... are zero; also the constant term okoo must be zero with the normal definition of the wavefront. The first terms that remain are then

W(a,p,0) = ok2op2 + ikiiap cos 0 + ok40p4 + ik3iap3 cos 0 (3 15)

This can be expressed as the general function

W(a, p, 0) = E(i+„)k(m+„),„a(!+")p(m+") cos" 0 , (3.16)

in which l, m are even positive integers or zero, and n is any positive integer or zero. However, the terms with m = n =0

are excluded if the Gaussian conditions are defined as zero. In telescope optics, this is not, in general, the case for the latter two of these conditions.

The type of aberration depends on the functions of p and cos 0, whereas the function of a shows how the effect varies in the field.

Table 3.1 shows the first, third and fifth order aberrations of this general function.

Table 3.1. Aberration types from the Characteristic Function

m

n = 0

n = 1

n = 2

n = 3

Order

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