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This equation may be checked when both conjugates are at a finite distance.1

When the source point I is close to the Gaussian focus G, then M is small. When M = 0 and given a mirror f-ratio Q = f /d = R/4rmax, the amplitude of the r10 coefficient corresponding to the Rayleigh criterion A/4 may be characterized by a couple (Q;d) where Q = [193d/224A ]1/9. For instance at A = 550nm, such couples are (f/1.2; 0.25), (f/1.4; 1.00) or (f/1.6; 3.30), where the second number in the parenthesis is the beam aperture diameter d at z = 0 expressed in meters.

4.1.4 Optical Design of Correctors - Preliminary Remarks

On-axis refractive correctors as well as tilted reflective correctors - providing or not diffraction limited beams for spectroscopy or imaging - have been found useful for wide-field studies. From the previous wavefront equation Zw(r,M), a formal analysis could be carried out for the determination of the best corrector shape Zopt. In the case where the magnification M is of low value, optical ray tracings show that such an analysis is unnecessary. Let us consider the case of a 100% obstruction reflective corrector having its figure straightforwardly defined by ZOpt = Zw/2: at f/1.5 and with M e [0;0.03] - which includes the possibility of having a null power zone outside of the clear aperture -, the result from ray tracing shows that the size of the axial blur image is smaller than 1/50 arcsec at the focal position defined by Li (M). The size of the axial blur image is found similarly sharp with a refractive corrector such as ZOpt = —Zw/ (N-1) at the wavelength corresponding to the refractive index N. This result can be extended to all practical designs based on a Schmidt concept.

1 As a verification, the first coefficients of the wavefront (4.6) give a spherical shape for the particular cases where M — 1 and M — -1, corresponding to point S at V and point S at C, respectively.

^ For correctors located at the center of curvature of a spherical mirror up to f/\.7 — f/\.5 acting in the range M e [0;0.03], i.e. for objects at infinity, the magnification M becomes a positioning parameter for the wavefront null-power zone, and

(1) all types of axisymmetric optical correctors can be accurately derived from an affinity of the wavefront equation, i.e. ZOpt = constant x Zw(r, M),

(2) all types of tilted optical correctors can be accurately derived from an affinity of the wavefront equation, i.e. ZOpt = constant x Zw(x, y, M).

These affinity rules allow one to avoid the optical path analytical problem of determining the corrector optical shape ZOpt in each case - refractive or reflective, and diffractive or not - from the wavefront Zw. Furthermore, these rules allow a fine set out of aspherical coefficients for iterative optical codes. With reflective Schmidts, these rules also allow the set up of convenient over or under corrections in order to control the image balance in the field.

4.1.5 Object at Infinity - Null Power Zone Positioning

In this Section and the following, only the case where point T now becomes an object at infinity (see Fig. 4.4) will be considered, so that its conjugate image point S is close to the spherical mirror Gaussian focus G. For S at G, the magnification parameter is M = 0 and coefficient A2 = 0, i.e. the aspherical wavefront does not present any curvature at point C (z = 0). For instance, when optimizing the corrector for reducing off-axis aberrations or when minimizing the on-axis sphero-chromatism variation of refractive elements, one needs a coefficient A2 of sign opposite to that of higher coefficients. These wavefronts show an inflexion point, and for a radial distance r0, the propagation is parallel to the z-axis. If rm is the radius of the clear aperture beam, the radius r0 for which a ray propagates in a parallel direction can be characterized by the null-power zone ratio Vk defined from

Let us define the f-ratio f/d by Q = |R/4rm | where R is still the radius of curvature of the spherical Schmidt mirror. Since parameter M is necessarily small, a first approximation from the derivation of the two first terms of (4.6) provides the link between M and k,

For k = \, the rays are parallel to the z-axis at the height r0 = rm i.e. at the full aperture edge. As will be shown, the optical optimizations lead mainly to study cases where 0 < k < 2. With the two previous relations, and denoting p = r/rm a reduced radius with respect to the full aperture, a first third-order approximation of the wavefront lying at the mirror center of curvature is Zw = Zw/R = (2kp2 — p4)/210. The various meridian profiles by respect to k are displayed by Fig. 4.5.

Jfc=3/4

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