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Table 7.5 Grating substrate coefficients (maximum flexures in ^m) 

Grating A40 A60

Table 7.6 MDM coefficients and incidence angle 

Grating A31 A33 A42 i mdm

^ In x, the Cos spectral resolution X/SX is increased by a factor 10, ^ In y, the limiting magnitude is increased by 1-1.2 magnitude 

The Fwhm of the blur image at 1,300A is 2.5 x 88 /m2. The diffraction will increase the width in the direction of dispersion up to 3.8 x 88/m2. Compared to the results of imaging and resolution tests by the Cos team , with a 3,800l/mm classically Astm3 corrected grating at 1,284A showing the Fwhm image of 38 x 264/m2, the high-order corrected grating provides images 10-times better in dispersion and 2.8-times better in cross direction.

The resolving power of the f/24 Hst images at the input of Cos is 1.22 X f /d = 3.8/m at 1,300 A and the concave gratings provide a magnification ~ — 1. The images given by gratings #1 and #2 are diffraction limited with regard to the resolution over the main part of the spectral range, and are nearly diffraction limited at the lower wavelength. with grating #3, of low dispersion, both the widths and heights of the images are diffraction limited. Fig. 7.14 Spot-diagram of COS with grating#1,3,800l.mm . For each color, A is defocussing onto the principal ray with respect to the Rowland circle. A positive A means an increase of the image distance to the grating vertex. The pixel size of the COS detector is 2.4 x 33 /m2. Compared to the design by Osterman et al. , the gain is 28 in light concentration (after Duban )

Fig. 7.14 Spot-diagram of COS with grating#1,3,800l.mm . For each color, A is defocussing onto the principal ray with respect to the Rowland circle. A positive A means an increase of the image distance to the grating vertex. The pixel size of the COS detector is 2.4 x 33 /m2. Compared to the design by Osterman et al. , the gain is 28 in light concentration (after Duban ) Laser source

Fig. 7.15 Basic recording mounting. The recording angles a and P of the principal rays at the vertex O of the grating from laser source points L\ and L2 respectively are given by Table 7.4. For COS grating #1, the a and P value are shown, the incidence angle at the vertex M of the MDM is ¿mdm = 29.96°; the Rowland circle optical paths are L1O = Rcos a and L2O = RcosP with R = 1,652mm and L1M = 1,100mm (after Duban )

The recording distance from the laser source 1 to the MDM is 1,100 mm for gratings #1 and #2, and 1,000 mm for grating #3 (Fig. 7.15).

The f/24 Hst projects central beam areas for all three gratings that are contained in a 73.2 mm circle since i < 20° (see Table 7.4); for recording 80-mm circular aperture Cos gratings, the corresponding size of the recording projected beam at the MDM is 42.9 x 53.3 mm2 for #1 and #2, and a little smaller for #3.

7.8.3 Elasticity Design of a Six-Arm MDM as Recording Compensator

Plane six-arm MDMs have been developed and experimented by Lemaitre [37, 39]. These mirrors can provide the co-addition of six Clebsch-Seidel modes but only three modes have to be generated to achieve the recording of the Cos-Hst gratings. Aberration corrections achieved by an active MDM are straightforwardly obtained though the other alternative based on the use of a digital computergenerated interferogram is also of interest [14, 26].

Compared to glass or vitroceram materials, metal mirrors present several features that are of interest for the achievement of high curvature variations and high asphericity variations as well. The gain in flexibility-ratio fflim/E, is larger than 100. This is basically due to the much higher yield strength alim of metal alloys.

A second selective criterion for metal substrates is a perfect stress-strain linearity in the sense of Hooke's law. With respect to these criteria, the Fe87Cr13 alloy is well known as having a large linear range and no polishing problems. Otherwise metal alloys such as Cu62Ni18Zn20 or Ti90Al6 V4 have not been characterized but might show good stress-strain linearities on large linear ranges while more flexible Al predominant alloys show more restricted linear ranges and need a Ni overcoat for polishing. The selected metal alloy was Fe87Cr13 - referenced in the series Aisi 420 - brought in a quenched state to Brinell hardness BH = 300. This material used for a long time at Loom provides very smooth polished surfaces. The optimization for a convenient flexibility has been done by determining the rigidities Di and D2 i.e. the thicknesses ti and t2, with respect to the maximum stress given by a fifth-order triangle aberration Tri5 as defined by the A33 coefficient of the Cos grating #1 (Table 7.6). The maximum stress has been kept lower than the yield strength of the Fe87Cr13 alloy which is 1,200N/mm2. In order to ensure the best 3D-homogeneity of the substrate, the vase form and radial arms were machined out from the mass of a disk. Nine differential screws linked between the MDM and its support allow generating the flexure while the three remaining screws located at 9 = 0, ±2n/3 and r = a, define the reference plane of the deformations. A six-arm MDM was appropriate as a Cos grating recording compensator (Fig. 7.16).

The axial distribution of forces Fa,k and Fc,k applied to the MDM has been determined for each of six Clebsch-Seidel modes having a ptv deformation of at r = a = 40mm over 9 e [0,2n]. These forces and the associated geometrical parameters of the built MDMs are given in Table 7.7.

Five flexure modes were evaluated from He-Ne interferograms with respect to a plane (Fig. 7.17).

With the above optical designs of Cos, the grating substrates must also be ax-isymmetric aspherics - caused by the Sphe 3 residual of the Hst primary mirror -and the holographic recording requires the co-addition onto the MDM of the three A31, A33, and A42 Clebsch-Seidel modes such as given in Table 7.5. For Cos grating #1, the MDM geometry of the holographic recording compensator and Fa,k and Fc,k forces that generate those flexures are listed in Table 7.8. Fig. 7.16 Six-arm vase-form MDM as recording compensator for aberration corrected holographic gratings. Clear aperture diameter 2a = 80mm. (Left) True proportion design. (Right) View in its stressing cell [Loom]

Table 7.7 Six-arm plane MDM. Force distributions Fak and FcA- derived from Mr(b) and V(b). MDM geometry ti = 5mm. 7 = (/1//2)3 = 1 /27, a = 40mm. b/a = 1.35. c/a = 2. Fe87 Crl3 alloy E = 205 x 109 Pa and v = 0.305. Forces generating Zum — Anm sj! cosïiiO modes with a ptv flexure of 1/im at a — 40mm corresponding to Clebsch-Seidel coefficients A2o = 6.250 x 10~7,A40 = 3.906 x 10~10,A22 =A2o/2,A3i =7.812x 10~9,A33 =A3i andA42 =A40/2inmm1~". Clear aperture 2a. [Units: daN]

Table 7.7 Six-arm plane MDM. Force distributions Fak and FcA- derived from Mr(b) and V(b). MDM geometry ti = 5mm. 7 = (/1//2)3 = 1 /27, a = 40mm. b/a = 1.35. c/a = 2. Fe87 Crl3 alloy E = 205 x 109 Pa and v = 0.305. Forces generating Zum — Anm sj! cosïiiO modes with a ptv flexure of 1/im at a — 40mm corresponding to Clebsch-Seidel coefficients A2o = 6.250 x 10~7,A40 = 3.906 x 10~10,A22 =A2o/2,A3i =7.812x 10~9,A33 =A3i andA42 =A40/2inmm1~". Clear aperture 2a. [Units: daN]

Angle

Arm

Cv 1

0 0