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10.4 Active Optics Aspherization of X-ray Telescope Mirrors 10.4.1 Thickness Distributions for Monolithic Tubular Mirrors

Among the Wolter Type I telescope family reviewed in Sect. 10.1, the WolterSchwarzschild (WS) form is the most performing for high angular resolution imaging in X-ray. Although limited to a few arcmin field of view, the fulfilment of the

Abbe sine condition potentially allows an optical imaging towards the diffraction limited tolerance at these very short wavelengths. In counterpart, compared to the PH, SS or HH forms of this family, the shape of WS telescope mirrors is the most difficult to obtain; this fully explains why this telescope design has only been built in one or two laboratory samples - and seems to have never been used in space up to now.

The geometry of a truncated conical shell of small slope angle for any grazing incidence mirror - hereafter simply called tubular mirror - can be readily determined from the above linear product law.

• Linear product law and tubular mirror design : Whatever the form of a Wolter Type I telescope - PH, WS, SS, or HH forms (cf. Sect. 10.1) -, relation (10.71) allows deriving the associated thickness distributions T1, T2 of the primary and secondary mirrors provided the flexure W1, W2 for these mirrors have been set such as the radial retraction (or extension) is always of monotonic sign all along the mirror surface. This is achieved by convenient choices of the a2 and a2 values and sign before them in (10.71).

• Application to grazing incidence WS telescopes : As an application example of active optics aspherization, let us investigate the most difficult case of WS telescope mirrors. We have seen in Sect. 10.1.3 that Chase and VanSpeybroeck [11] derived the mirror parametric representations for the WS telescope (i.e. a telescope form strictly satisfying the sine condition). These authors also gave the resulting shapes for the construction and X-ray tests of a prototype WS telescope. The parameters of the primary mirror are the length L = 165 mm, low-angle slope corresponding to i = 1.5°, and inner radius rj = 152 mm at the joint of the mirrors which gives r0 = 154.16mm at the middle of the mirror (Fig. 10.14).

Using the geometrical parameters corresponding to Fig. 10.14 and introducing, at x = 0, a thickness t0 = 12.33 mm, we obtain a central radius of the mid-surface a0 = r0 + to/2 = 160.32mm providing a shell thickness-ratio L/a0 = 1/13. The edge abscissa x = ±5 is then with ¡5 = L/2a0 = 0.5146. A polynomial representation of the WS primary mirror in Fig. 10.14 where ZAnx" includes terms up to n = 8 allows to obtain, from (10.71), the thickness and complete geometry of this mirror when aspherized with the best circle-segment fit (Fig. 10.15).

In the previous example, we assumed that the aspherization is generated from the best circle-segment fit. However, both curvature and aspherization can be also

Fig. 10.15 Shell geometry and thickness distribution of the WS primary mirror in Fig. 10.14 as-pherized by uniform load and best circle fit. (Left) Inner and outer lines Li, Lo, of the shell meridian section. (Right) Enlarged scale of the outer line Lo after origin change and removal of the slope angle component at Mo

Fig. 10.15 Shell geometry and thickness distribution of the WS primary mirror in Fig. 10.14 as-pherized by uniform load and best circle fit. (Left) Inner and outer lines Li, Lo, of the shell meridian section. (Right) Enlarged scale of the outer line Lo after origin change and removal of the slope angle component at Mo achieved from the best straight-segment fit. In any case, a convenient set up of the constant a2 and thickness T(0) =t0/a0 must be done. If these two quantities are increased, then the amplitude of the outer surface with respect to a straight line is decreased, but the intensity of the load is increased.

• Note : Slight deviations may affect the accuracy of the flexure from the thickness distribution T(x) as given by (10.71). The axial reaction Rq from the axial component of the uniform load q provides slight displacement effects; the displacements in the axial direction may be assumed negligible whilst those in the radial direction are denoted AW1(x). Since we assume here that the mid-thickness surface is a perfect conical shape - i.e. a(x) is linear -, the thickness variation which is required for the aspherization entails that the middle surface is not exactly a cone; the corresponding deviation is AW2(x). The linear approximation in the expansion of ax also causes a slight deviation AW3(x). The correction of these three deviations to the requested flexure can be accurately achieved by finite element code analysis in slightly modifying the non-optical surface of the shell.

10.4.2 Boundaries for Segment Mirrors of Large Tubular Telescopes

Beside increasing the angular resolution of X-ray telescopes - presently limited by technological difficulties in the obtention of mirror surfaces having an extremely high smoothness -, astrophysical programs for faint object studies require the development of X-ray telescopes with much larger surface areas. Future space-based

Wolter Type I telescopes will be designed with segmented mirrors allowing construction of large tubular mirrors [23, 24]. The extremely high precision of the tubular surfaces to be figured and aligned will require an extensive use of active optics methods.

Active optics shall be mostly elaborated as well for the stress figuring of the submirrors as for the in situ control of the two successive primary and secondary mirror surfaces resulting from segment assemblies.

Let us consider hereafter the execution by stress figuring of a mirror segment whose contour lines are plane cuts defined by two angular planes passing through the mirror general axis and two parallel planes perpendicular to this axis. Similarly to the second deformation case of a cylinder element (Fig. 10.7), we assume that the static equilibrium of the conical shell segment to a uniform load q is realized by self-compression or self-extension only, i.e. without any radial boundary forces. Then, no shearing force is generated through the shell, Qx = 0, and the bending moments along its contour are null, Mx = My = 0 whatever x, y or x, y. This means that the load q generates a pure extension - or a pure retraction - of the shell segment. Therefore, the linear product law (10.70) applies and the thickness distribution can be easily derived from this law.

For instance, when the stress figuring is operated on the inner surface of a segment by a pressure load q which acts on the outer surface of low slope angle o -which slightly differs from the inner slope angle i -, the boundaries at a segment reduce to the three following conditions (Fig. 10.16).

• Boundary condition C1: The two facets ±y = constant of a segment must be supported by a normal pressure p as qdAq cos(atano)+2pdAp y = 0, (10.72)

where dAq and dAp are infinitesimal areas, of length dx, of the segment outer surface and of one of these facets. For finite values this gives a pressure distribution p (x) = Nw/1(x) which is linear only for a strictly conical thickness. However, from axial symmetry, when passing from the whole tubular cone to the segment cut, the p (x) distribution in the harness support is implicitly conserved if the segment is supported by two planes passing through the mirror axis and forming a dihedral angle 2y.

Boundary condition C2: In the dihedral planes ±y = constant, these two facets must be free to slide in the radial directions.

Boundary condition C3: The facet end x = constant corresponding to the largest diameter of the loaded area must receive the axial reaction from the uniform load q. The resultant force Rq of this reaction is given by qAq sinatan < o > +Rq = 0, (10.73)

where Aq is the segment surface area receiving the load q and < o > a mean-value of this surface slope angle.

10.4.3 Concluding Remarks on the Aspherization Process

Subject to the above conditions of weakly conical shells the active optics aspher-ization of grazing incidence mirrors can be carried out by a simple uniform load applied to a closed shell with radially sliding ends. Whatever the figuring option -best straight-line fit or best circle-line fit - the rigidity of the surfacing tools must be much higher in its axial direction than in the tangential cross-section. This can be also realized by contiguous lapping segments with some freedom to radial motions.

For a large segmented telescope the elasticity design of a mirror segment is simply a cut-out of the elasticity design of the complete tubular mirror. Hence, whether we have a monolithic tubular mirror or simply a segment of it, the thickness distribution which provides the convenient radial displacement distribution can be directly derived from the same linear product law.

In a final design, a 3-D finite element code optimization would provide some slight corrections of the mirror substrate geometry such as taking into account the small effect of the load reaction acting at the larger end because of the mirror slope.

Stress figuring of a mirror segment requires use of a support - or "harness" -which must be designed with high stiffness and adequate setups to accurately satisfy the above three boundary conditions, C1, C2, C3, during the process.

Compared to the aspherization of a mirror segment for a large quasi-normal incidence telescope (cf. Chap. 7), the boundary conditions for a weakly conical shell are greatly simplified since the above process avoids the complication of applying bending moments at the mirror contour.

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