Table 3.6. Third order aberrations and associated relations for a 2-mirror telescope in afocal form

Table 3.6. Third order aberrations and associated relations for a 2-mirror telescope in afocal form

Before considering the application of the above third order theory to various forms of 2-mirror telescope, we must deal briefly with the effects of higher order aberrations and the practical evaluation of systems with ray-trace programs.

3.2.5 Higher order aberrations and system evaluation

3.2.5.1 General definition of aspheric surfaces. In § 3.1 we gave the general definition of conic sections in terms of the Schwarzschild (conic) constant bs, as represented by Eq. (3.11). The theory of § 3.2.4 above is limited to the second term of this equation, known as the third order, or Seidel, approximation. With steep apertures or significant field sizes, the third order approximation remains of immense value for understanding the basic correction potential of a given design, but is no longer adequate to describe the final image quality. Furthermore, departures may be necessary from the strict conic sections defined by (3.11) in order to compensate higher order aberrations. This requires a more general definition of aspheric surfaces.

3.2 Characteristic Function and Seidel (3rd order) aberrations 83 From Eq. (3.4) we had for the general equation of an ellipse

The longer semi-axis a can be replaced by the vertex curvature c from (3.6) and (3.10):

From Eq. (3.12), this expression is general for any conic section. If, now, both sides of (3.71) are multiplied by y2 /2

we have the form fi y2

From (3.72) and b2 = a2(1 + bs) , (3.74) this reduces to z =-—-W2 (3.75)

This form still describes a strict conic section and can be expanded into the polynomial of Eq. (3.11). If bs = -1, Eq. (3.75) reduces to the parabolic first term of (3.11):

Most optical design programs allow a modification of any desired conic section defined by bs in (3.75) by adding aspheric terms of higher orders whose coefficients As, Bs, Cs, Ds can be chosen at will, giving the general form:

Frequently, the designer will find it convenient to define the conic section as a parabola with bs = -1, so that the supplementary coefficients As, Bs,... give the supplement to the parabola of (3.76). They can then be compared with the coefficients of (3.11) for any other conic defined by bs. For the third order and fifth order terms, the comparison gives z

If the coefficients As, Bs,... are referred to the sphere, then, of course, the terms -1 in Eq. (3.78) vanish. In this way, it is a simple matter to determine the departure of a given surface from a conic form 6s defined by the third order coefficient As. From (3.11) it is clear that strict conics have only positive terms, but it should be borne in mind that both c and (1 + 6s) can have positive or negative values. Sign reversals of terms compared with Eq. (3.11) will imply, if the contributions to z are significant, major departures in the higher orders from strict conic sections.

Another form, used by Laux [3.8(a)], defines the conic parameter as 6s by

From (3.71), this gives the general form for a strict conic section as

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