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Lengths in [mm]. Associated parameters are = 0.25° = 0.00436 rad, ro = 100, a\ = 0.0256rad = 1.467°, a2 = 0.0584rad = 3.346°, f = 967, L1 = 50, L2 = 67.

Lengths in [mm]. Associated parameters are = 0.25° = 0.00436 rad, ro = 100, a\ = 0.0256rad = 1.467°, a2 = 0.0584rad = 3.346°, f = 967, L1 = 50, L2 = 67.

An example of an optical data set for a telescope that images a field of view corresponding to the angular diameter of the Sun is given in Table 10.1. The best focused blur images from Zemax code are shown in Fig. 10.3.

Following the first Wolter study, Mangus and Underwood [2] designed a PH Wolter Type I telescope and evaluated the performance of both laboratory prototype and rocket flight telescope. VanSpeybroeck and Chase [3] elaborated an optical code for the determinations of the field aberrations of these telescopes. Werner [4] attempted to achieve a flat field from raytrace optimizations with spheroid surfaces. Malina, Bowyer et al. [5] designed field optimized instruments by focusing balance. Aschenbach [6] optimized the mirror geometries of the first wide-field X-ray telescope. Nariai [7] analytically investigated the geometrical aberrations and concluded that high-order terms are significant.

In an anastigmatic two-mirror system, the Petzval condition for flat field is 1/R2 - 1/R1 = 0 (cf. Sect. 1.10.1). For a two-mirror system where the astigmatism is small, it is well known that the field curvature of least confusion imaging is significantly reduced when R1 and R2 have closed values. For a PH Wolter Type I telescope only R2 somewhat larger than R1 can be achieved.

A conclusion is that, whatever the optimization with the free parameters R1, R2, and k2 in Table 10.1, a grazing incidence PH telescope always shows coma, astigmatism, and low field curvature.

Fig. 10.3 Through focus spot diagram resulting from optical parameters in Table 10.1. The rms angular diameter of the blur image at qm = 15 arcmin semi-field edge is drms = 6.3 arcsec

10.1.3 Sine Condition and Wolter-Schwarzschild (WS) Telescopes

• Coma aberration: It is well known that for parallel field rays which lie on a circle C located in a plane of the input pupil, the gaussian image of this tube-ray is a small circle c. In addition, when each of the rays describes the whole circle C, then circle c is described two times. If the circle C is the edge of the input pupil, then the diameter of circle c is the tangential size of the coma aberration. Although not showing in a meridian direction the typical "V"-pattern of coma because of the annular entrance pupil, the basic grazing incidence PH telescope suffers from coma aberration.

In a second 1952 paper, Wolter [8] attempted to formulate the equation of the two mirrors for his three system types when strictly satisfying Abbe's sine condition. When free from spherical aberration and coma, the three Wolter grazing incidence systems are known as Wolter-Schwarzschild (WS) telescopes. However, only the Wolter Type I form has been of general use in X-ray astronomy. This WS telescope shows a similar geometry to that of a grazing incidence PH telescope (see Fig. 10.2).

• Implication of sine condition: For the axial beam, the Abbe sine condition implies that the locus of the intersection points of any ray from infinity with its conjugate passing through the final focus is a sphere centered at the focus, the Abbe sphere (cf. Sect. 1.9.2).

H. Chretien [9] was the first to derive a mirror parametric equation set for solving the general case of the two-mirror telescope family satisfying the sine condition. From these equations, he established the well known and accurate conicoid approximations which represent the mirrors in the quasi-normal incidence Cassegrain and Schwarzschild forms. Although Chretien also derived the fifth-order polynomial coefficients for the two latter forms, it appears that, in practice, these more accurate approximations have negligible effects either in the RC form or aplanatic Gregory form.

The design of a RC telescope strictly satisfying the sine condition is also that of a WS telescope. In the grazing incidence region, this design corresponds to the Wolter Type II. Whereas, even for all large existing RC telescopes, it is always possible to accurately approximate the shape of the mirrors by conicoids, this approximation does not apply to a WS telescope because very-high-order terms for linear coma (cf. eq. (1.66) in Sect. 1.9.2) cannot be accurately cancelled with con-icoid surfaces. In a WS telescope and whatever the Wolter type, the importance of these high-order terms may be glimpsed by considering the grazing incidence regions of the mirrors of an extremely fast Cassegrain telescope satisfying the sine condition, such as determined by Lynden-Bell [10] (Fig. 10.4).

Since low-order polynomial approximations are inaccurate in representing grazing incidence mirrors of WS telescopes, each of Wolter Type-I and -III systems must be determined by a similar approach to that of Chrétien mirror parametric representations.1

1 Following Chretien [9], a similar parametric representation of a Cassegrain two-mirror telescope satisfying the sine condition was reformulated by Korsch [11] for near-normal (RC telescope) to

Fig. 10.4 Mirror parametric representations of a Cassegrain telescope strictly satisfying the sine condition up to a very fast f-ratio [10]. Although Wolter Type I uses the reverse side of the second optical surface, somewhat similar "S"-shapes exist on the mirrors of grazing incidence WS telescopes

Fig. 10.4 Mirror parametric representations of a Cassegrain telescope strictly satisfying the sine condition up to a very fast f-ratio [10]. Although Wolter Type I uses the reverse side of the second optical surface, somewhat similar "S"-shapes exist on the mirrors of grazing incidence WS telescopes

• Chase and VanSpeybroeck design of WS telescopes: The first design of a WS telescope in the Wolter Type I form - which is the only form of interest for X-ray astronomy - were derived in 1972 by Chase and VanSpeybroeck [13] by use of mirror parametric representations in a study that has become a classic reference. The case of grazing incidence microscopes satisfying the sine condition was subsequently investigated by Chase [14]. A generalized parametric representation for WS telescopes was obtained by Saha [15] and a dedicated raytrace code for this purpose was elaborated by Thompson and Harvey [16].

Although not entering into detail, Chase and VanSpeybroeck also computed the exact figures of the two mirrors of a WS telescope showing that both mirrors may have two inflexion zones. In a systematic study of many WS wide field telescopes, these authors showed that the effect of the field curvature is small and that the astigmatism is dominant.

Denoting dms the rms angular diameter of the least confusion astigmatism blur at a semi-field angle (pm, Chase and VanSpeybroeck derived a relation expressing - here in a somewhat different form - the angular resolution of a WS telescope, grazing incidence (Wolter Type II). Unfortunately, because of the requirement for much larger ray deviation angles than with Wolter Type I, this system was never adopted for X-ray astronomy. Korsch also investigated the case of grazing incidence three-mirror systems [12].

V 02) r0 rm where the angles are in radians and the quantities a1, a2, r0, L1, (pm are defined in Sect. 10.1.2 with grazing angles a1 and a2 of the same sign. The validity of this relation is in the region defined by

- < — < 4, - < — < 6, 0 < (m < 30arcmin. (10.10)

In fact the original Chase-VanSpeybroeck relation for the resolution and associated inequalities included three geometrical ratios, but these can be readily reduced to the following two ratios: the grazing angle ratio a1/a2, and the primary mirror aspect ratio L1 /r0, so the above relations are totally valid for the small grazing angles that are required in X-ray telescopes.

Applying these results to a WS telescope having the same optical parameters as those of the PH telescope in Table 10.1, we find that a\/a2 = 0.438 and L1/r0 = 1/2 satisfy the above inequalities. If we also consider the same maximum semi-field angle ( m = 15 arcmin, the rms diameter of the image at the field-edge, from (10.9), is now drms = 14.8 x 10-6rad — 3.1 arcsec. Comparing with the spot-diagram in Fig. 10.3, where drms = 6.3 arsec for the PH telescope, the gain in rms diameter of field aberrations is here a factor of two in favor of the WS telescope.

To summarize, in a grazing incidence WS telescope the mirror parametric equations allow one to cancel both spherical aberration at all orders and linear coma terms, and to reduce the field curvature to a low amount. However, an important amount of astigmatism remains which, from (10.9), is quadrati-cally dependent on the field angle. Generally the graze angles are set such that a2 > ai; therefore this aberration can be only decreased if the Li /r0 ratio is decreased, i.e. for a short axial length and a large aperture radius of the primary mirror. The primary mirror aspect ratio L1 /r0 appears as the fundamental parameter that governs the dominating residual blur of all grazing incidence Wolter-Schwarzschild telescopes.

^ In a Wolter-Schwarzschild telescope where the primary and secondary mirrors are joined at the aperture radius r0, ifL1 is the axial length of the primary, then the amount of astigmatism is linear with the primary mirror aspect ratio L1 / rQ.

10.1.4 Aberration Balanced Hyperboloid-Hyperboloid (HH) Telescopes

The complex mirror shapes resulting from the use of parametric equations in grazing incidence WS telescopes, which moreover are not astigmatism-free and also present considerable difficulties in their execution, led to another alternative concept developed for the optical design of Wolter Type I telescopes.

This concept, proposed and developed by Harvey et al. [17, 18], consists of the use of the optimization capability of conventional raytrace codes with mirrors that are pure conicoid surfaces, so the sine condition is no longer applied. Starting from the stigmatic design of a PH telescope, the optimization process is now carried out with the four parameters R1, k1, R2, k2 and also slight variations - or despaces -of the axial separations V1F1 and F2V2 (cf. Fig 10.1). In addition the axial lengths L1, L2 of the mirrors and their gap must be specified. Given an intermediate field angle pi, such that 0 < pi < pm, for which the aberrations must be minimized, ray-trace optimizations provide aberration balanced fields where both mirrors are hyperboloids. Hence these systems are called grazing incidence HH telescopes.

Repeating the iteration process for various values of pi, Harvey et al. plotted the rms diameter of the total aberration vs field of view and obtained a grid showing the aberration evolution (Fig. 10.5). They noticed that the locus of the minima pi is a straight line and interpreted these raytrace results - shown on the figure as a shaded region - as the uncorrectable linear coma for those field angles. This conclusion is consistent with the results in Sect. 10.3.1 where the sine condition necessarily entails use of mirrors represented by parametric equations which cannot be accurately approximated in the conicoid class.

Finding the optimal ratio pi/pm in this grid corresponds to finding the balanced field for which drsm(0) = drms(pm). For a maximum semi-field angle pm < 20 ar-cmin, this ratio is not a constant but a function of pm; its value is included in the range [0.75, 0.90]. For a very small semi-field of view, say pm = 1 arcmin, this ratio is close to 0.9.

Fig. 10.5 Angular blur-image rms radius vs semi-field angle p for a grazing incidence HH telescope. Each curve is derived by Harvey et al. from a particular value ( of the intermediate design angle for which the aberrations are minimized [18]

In the optical design of grazing incidence HH telescopes with a solar-like field angle and plane detector, the spherical aberration, astigmatism and next-order aberrations - such as cubic coma and linear triangle terms - are substantially reduced whereas the linear coma remains and astigmatism still dominates.

10.1.5 Aberration Balanced Spheroid-Spheroid (SS) Telescopes

As initiated by Werner (10.5) for wide field telescopes, it is possible to consider mirrors represented by polynomial expansions that roughly approximate the sine condition and reduce astigmatism. As for all previous designs, the field curvature of least confusion imaging is obtained when R2 is little larger than R1 (cf. Sect. 10.1.2).

Each mirror equation that generates the polynomial expansion is obtained from a generalization of (10.4). Setting a new origin z = 0 at the intersection plane of the two mirrors, if the radius of this circular intersection is r0, then the mirror surfaces z1 and z2 may be represented by aZ + 2bjRjZi - r2 + r2 = 0, (10.11)

where Ri is the radius of curvature of the primary or secondary, with the respective suffix i = 1 or 2, and ai, bi four free dimensionless coefficients. The above implicit equations lead to expansions zi(r) which take into account high order terms. The shape (10.11) of these mirrors is included in the general class of spheroids, hence such systems may be called grazing incidence SS telescopes.

This analytic representation, however implicitly expressed from the above trinomial form of r2(zi), was proposed and investigated by Burrows et al. [19] with a dedicated raytrace code called Osac [20]. These latter authors and Concini and Campana [21] introduced a specialized merit function for the image tolerancing.

Subsequently, Saha and Zhang [22] investigated an SS telescope design which was called "equal curvature." This led the authors to define the mirror shapes as a co-addition of simple quadratic sags to the generatrices of a cone-cone telescope. From Osac's Legendre polynomials, a z-shifted quadratic correction can be added for each surface. However, it is unclear whether each mirror is a region (or not) of a torus, such as they state. A torus does belong to the spheroid class.

For a moderate field of view and plane detector, the imaging performance with a grazing incidence SS telescope is somewhat superior to that of the HH telescope design. On the other hand, this latter design seems more successful for a solar-like field. The SS design shows that the residuals for spherical aberration, coma, and dominating astigmatism remain.

10.1.6 Existing and Future Grazing Incidence X-ray Telescopes

Nearly all realized telescopes for X-ray astronomy are in the Wolter Type I form. Because of the short length of the mirrors generally adopted - primary mirror aspect ratio Li/ro < 1/2 - a particular feature is that one or several additional smaller aperture telescopes can be nested into the first, thus allowing, in a single volume, simultaneous observation in imaging and spectroscopic modes.

• PH telescopes: The early imaging systems were designed with grazing incidence paraboloid and hyperboloid. Such was the case, for instance, for the Einstein-Heao 2 telescope, Rosat telescope, and Chandra-AXAF telescope (cf. Table 1.1 in Chap. 1).

• WS telescopes: The mirrors defined by the parametric equations for WS telescopes have been found to be extremely difficult to fabricate and test. Each of their mirrors may have two inflexion zones (cf. Sect. 10.1.3 and Fig. 10.13). Compared to the HH or SS designs, their superiority is for a very high angular resolution imaging of small fields. Up to now, astronomical programs for such researches have not been investigated.

• SS telescopes: The grazing incidence SS telescope design is attractive for moderate fields of view, say 2qm < 20 arcmin. The large and high-resolution imaging Chandra-AXAF telescope (see Table 1.1) was originally designed in both PH and SS forms2 with two nested coaxial mirror pairs (Fig. 10.6). Examples of designs underway are the Nasa projects Wfxt and Xrt.

• HH telescopes: The HH design is attractive for a wide-field, say 2qm > 30 ar-cmin. It implies mirrors that have simple shapes included in the conicoid class and which thus can be readily optimized with high order corrections by the usual raytrace codes. For instance, this design has been adopted for the Solar X-ray Imager (Sxi).

• Large segmented telescopes: Similarly as for large ground-based telescopes in the visible and infrared, future large space-based X-ray telescopes will require use of segmented and active mirrors. Examples of such projects under investigation are the Sxt [23] and Xeus [24].

Fig. 10.6 (Left) Chandra X-ray Observatory, launched in 1999, includes a 1 arcsec resolution imaging telescope and a spectrometer using two transmission gratings (credit NASA). (Right) X-ray image of the Crab Nebula obtained with Chandra; the dot at the center is the neutron star - or pulsar -spinning 30 times per second (credit Nasa/Cxc/Sao/J. Hester et al.)

Fig. 10.6 (Left) Chandra X-ray Observatory, launched in 1999, includes a 1 arcsec resolution imaging telescope and a spectrometer using two transmission gratings (credit NASA). (Right) X-ray image of the Crab Nebula obtained with Chandra; the dot at the center is the neutron star - or pulsar -spinning 30 times per second (credit Nasa/Cxc/Sao/J. Hester et al.)

2 Stephen O'Dell and Martin Weisskopf pointed out that, at the urging of Riccardo Giaconni, the Axaf design was modified from PH to SS polynomials. However, in the end, the PH prescription was selected. Thus, Chandra-AXAF should be included with Einstein and Roast in the PH telescope family.

10.2 Elasticity Theory of Axisymmetric Cylindrical Shells

10.2.1 X-ray Mirrors and Super-Smoothness Criterion

Whatever the design option of an axisymmetric two-mirror telescope, the fabrication of the mirrors is difficult. The super-smoothness of their surface is without doubt one of the most important features because slope errors due to ripples entail absorption and scattering effects which may severely degrade the performances. The X-ray energy domain is [10-0.1 keV] which corresponds in round number to the wavelength range XX[0.2-20 A]. Hence, due to the extreme difficulty of surface testing by X-ray in a long tunnel lab, it is presently admitted that the surface roughness of the mirrors must not exceed a value of 2-3 A.

For this reason, active optics figuring with rigid lap segments - whose curvature is constant along the mirror axial direction - would greatly improve the performance of tubular mirrors since then avoiding ripple polishing errors. Active optics aspher-ization can be applied either by mirror stressing or by mandrel stressing which generates a replica mirror.

The stress figuring of long rectangular mirrors is already applied to the aspher-ization by a perimeter distribution of the bending moments. For such X-ray mirrors, an optimal surface geometry is further achieved by in-situ actuators. This two-stage process is of current use in synchrotron laboratory applications. For instance, some active optics processes applied to long rectangular mirrors were presented by Underwood et al. [26] and Ferme [8].

The stress figuring of grazing incidence mirrors requires use of the elasticity theory of shells. Starting hereafter from investigations with the theory of axisymmet-ric cylindrical shells, we shall then present an appropriate theory of axisymmetric weakly conical shells.

10.2.2 Elasticity Theory of Thin Axisymmetric Cylinders

Because of the very small amount of maximum optical sag of X-ray mirrors, we assume hereafter that the elastic relaxation process provides the complete optical sag distribution with respect to the geometry of a cylinder or a cone, both having straight generatrices during the stress figuring.

The elasticity theory of thin axisymmetric cylinders involves radial extension and retraction of the mid-thickness surface. Considering the static equilibrium of a cylinder element (Fig. 10.7), we hereafter follow the analysis by Timoshenko [27].3

3 Stephen Timoshenko obtained important results in this field. Noticeably, he derived the critical load for the buckling of thin shells [28], results which were applied to improve the strength of large ships. For instance in the case of a cylindrical shell, he established that the critical axial force is Fcr = 2nEt2/v5IT — v2). Hence, unlike the length-dependent column buckling (cf. Sect. 1.13.1), the axial shell buckling is independent of the shell length.

Fig. 10.7 Static equilibrium at the mid-thickness surface of a cylinder element of lengths dx and ady. The uniform load q - not shown - is normal to the surface

Let us denote a the radius of the middle surface of the cylinder and q the intensity of an external uniform load, per unit area, distributed all over the surface. In a small element of this surface, set an x-axis parallel to the cylinder axis and a z-axis passing through this latter axis and positive towards it. The force components involved at this element are the uniform load q, the bending moments Mx and Mw, the forces Nx and Ny, the shearing force Qx, and the x-variation of Mx, Nx, and Qx. From axial symmetry, the force Nw and the bending moment Mw are constant along the circumference. Writing the equations of equilibrium at the center of the element, the forces in x- and z-directions and the moment about the y-axis, are respectively, after division by the element area adxdy,

The first equation implies that the force Nx is a constant. We will assume that no load is applied to the cylinder ends in the x-direction, and thus Nx = 0.

Similarly as in Sect. 1.13.3, let us denote u = ux, v = uy and w = uz the components of the displacement vector. Since both the shell and load are axisymmetric, the flexure also is, thus v = 0. The normal strain components are du w

and all three shear strains are £yv = £wx = £xy = 0. From Hooke's law, the formulation of the axial and tangential forces leads to

where E is the Young modulus, v Poisson's ratio, and t(x) the thickness of the cylinder. From the first equation, we obtain du/dx = vw/a, which entails

0 0

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