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where the product V0A31 is positive.

For v G [0, 1/2], we obtain a/b e [0.6742, 0.6255], respectively. This hybrid configuration is displayed in Fig. 3.13 for a stainless steel alloy Fe87Cr13 where v = 0.305 ^ a/b = 0.6444. We can derive other hybrid configurations by modifying the outer rigidity and the intensity of the prismatic ring forces. This is achieved with the constancy of the ratio t^/V0. Two advantages of these configurations are a natural setting of the reacting moment at the link r = a and a full-aperture solution for Coma3 (Fig. 3.14).

3.4.4 Balance with a Tilt Mode

Geometrical properties of a coma wavefront: One defines a coma mode of order n by the form zn1 = An1 rn cos9, with n = 3,5,7,... Considering tilted planes containing a y-axis, i.e. of the form z11 = A11 rcos 9, the intersection of the two surfaces are ellipses belonging to the circular cylinder r = (A11/An1)1/(n_1).

If we consider now balanced wavefronts resulting from the co-addition of a Tilt 1 mode and Coman mode, i.e. of the form Z = A11rcos 9 - An1rn cos 9, the sections in the z = 0 plane are circles V n having the radius of the previous cylinder.

Fig. 3.14 Hybrid CTD-VTD configurations providing Coma3 mode z = A31 r3 cos 0 by two prismatic ring forces applied at r = a and r = b r=a b

Fig. 3.14 Hybrid CTD-VTD configurations providing Coma3 mode z = A31 r3 cos 0 by two prismatic ring forces applied at r = a and r = b

Since the co-addition of a Tilt 1 mode does not modify the coma image and neither the bending moment and net shearing force generating any elastic mode, this entails the following results:

^ The boundaries of a Coma 3 distorted mirror can be achieved by a rigid ring linked to it via cylindric collars of thin radial thickness:

• VTD class: a clamped center reacting with a perimeter single prismatic ring-force, applied via a thin collar, generate only Vr{a, 9}, thus providing a simply supported edge.

• CTD and hybrid classes: a pair of opposite prismatic ring-force, applied at r = a and r = a' > a via thin collars, generate Mr {a, 9} and Vr {a, 9}, thus providing to a linked edge at r=a [see (3.17d)].

These properties simplify the realization of boundary conditions in practical applications. Level fringes of the co-addition of Coma 3 mode with varying Tilt 1 mode are displayed by Fig. 3.15.

Fig. 3.15 Co-addition of Coma3 and Tilt 1 modes: Z = (3Po P — P3)cos9. (Left to right) po = 0 ^ flat center. po = ^ circles at 1/^/2, 1

Fig. 3.15 Co-addition of Coma3 and Tilt 1 modes: Z = (3Po P — P3)cos9. (Left to right) po = 0 ^ flat center. po = ^ circles at 1/^/2, 1

Note: In the representation of optical modes radial Zernike polynomials [3(b), 6, 32(a)], a given order polynomial Zm(p) is built by mode co-addition of the same subfamily up to n order, and coefficients are normalized such as Zm(0)=Zm(1)=1. For instance, since Z{ = p and Z3 = 3p3 - 2p, the Zernike representation of Coma3 mode with dimensionless radius is z31=A31p3cos 0=1 A31 (Z3 + 2Z{) cos0.

3.4.5 Coma from a Pupil and Concave Mirror System

Let us consider an object at infinity providing an incident and parallel beam passing through a pupil and reflected by a concave mirror of curvature 1 /R and conic constant k. Denote s the axial separation between the pupil and the mirror, i the incident angle of the principal ray with respect to the mirror axis, aTel the semi-aperture radius of the telescope mirror and Q = f/d = R/4aTel the aperture number. If the incident principal ray belongs to the (x, z) plane, the wavefront representation of Coma 3, derived from the Wilson equations [32(b)] of Seidel sums, is i

16 Q2aTel

This relation contains some fundamental properties of a pupil and mirror system, in particular:

^ A spherical mirror k = 0 provides a coma-free system if the pupil is located at its center of curvature s = R (Schmidt system).

^ If the pupil is on the mirror (s = 0), the size of Coma 3 is the same whatever the conic constant is.

At the focal plane of a concave mirror and in this latter case (s = 0), the linear size

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