Measurement of Convergent Mirror of Millimetron Telescope Using Computer-Generated Hologram
The main geometric characteristics of convergent mirror:
• optical diameter – 542.13 mm;
• shape of reflective surface – convex hyperboloid;
• equation of the generator of reflective surface:
у 2 = 509.4674 z + 0.147452 z 2.
The main geometric characteristics of composite primary mirror:
• optical diameter – 10 m;
• shape of reflective surface – paraboloid;
• equation of the generator of reflective surface:
у 2 = 9600 z.
The major peculiarity of telescope operation in this observatory consists in cryogenic conditions (~4K). It extremely complicates the process of shaping and measurement of operating surfaces of telescope mirrors and attestation process of these operating surfaces in the chambers imitating the space environment by temperature and vacuum. Therefore, during design and fabrication of mirror operating surfaces it is required to introduce modifications compensating the alteration of their shape during the transition from normal temperature (shop floor conditions of shaping) to operating cryogenic temperatures. The following significant stage should include the experimental check of correctness of introduced modifications.
Possible variant of solution of this problem is suggested below in relation to convex convergent mirror on the basis of use of computer-generated hologram executed on plane surface of plane-convex Zerodur blank in the capacity of optical compensator in the layout of unequal-arm laser interferometer of Twyman-Green type  shown in Fig. 2.
It should be noted that this optical testing layout can be used under production conditions for the technological control of convergent mirror operating surface in the process of its shaping and for the attestation control of this surface under the conditions imitating the space environment by temperature and vacuum.
The unique properties of Zerodur are implemented in this technical solution: extremely low values of its coefficient of thermal expansion (CTE) and optimization of CTE dependence on temperature [4, 5]. Dependence of Zerodur CTE on temperature is such that it is possible to select its specific brand with the values of CTE, which are close for two temperature ranges – in the neighborhood of 4K and 295K. At the same time, the negative properties of Zerodur – low heat conductivity, modulus of elasticity, rigidity, Maksutov coefficient  – are not significant in this case.
Developers of the optical system of Millimetron observatory give preference to silicon carbide as the material for convergent mirror because it significantly excels all other materials used in space telescope construction, even Zerodur, in Maksutov coefficient.
Selected variant of optical testing layout (see Fig. 2) has the following parameters:
• the wavelength of laser light source l = 632.8 nm;
• distance from "point" radiation source (plane of diaphragm 5) to on-axis computer-generated hologram 6 is ~ 3000 mm;
• parameters of on-axis computer-generated hologram 6:
• blank – plane-convex;
• material – Zerodur (possibly, astro-glassceramics);
• optical diameter 980 mm;
• curvature radius of convex surface 1300 mm;
• diffraction structure is fabricated on plane surface;
• maximum spatial frequency nмax = 850 mm–1.
Experience has proven that it is appropriate to implement this optical testing layout in interferometer with vertical orientation of interfering rays.
When performing attestation control of operating surface of convergent mirror in the thermal vacuum chamber imitating the space environment, it is appropriate to locate only computer-generated hologram 6 and controlled convergent mirror 7 in this chamber; all other elements of interferometer should be placed outside the chamber.
Currently, there is circular ruling engine of MDG type providing the production of on-axis computer-generated holograms with the diameter up to 600 mm at JSC "NPO "State Institute of Applied Optics", and possible technical solutions for bringing the threshold diameter of on-axis computer-generated holograms up to 1000 mm, including the creation of new circular ruling engine, are being developed.
It should be noted that creation of such circular ruling engine is relevant from the point of view of necessity of solution of the other significant problem of telescope construction – precision laser-holographic control of the adjustment process of two-mirror telescopes of Cassegrain and Ritchey-Chretien types [7, 8].
It is understood that the physical investigations using the models of convergent mirror with the diameter up to 200 mm and technological equipment and laser-holographic measuring instrumentation, which exist at JSC "NPO "State Institute of Applied Optics", must precede the practical implementation of suggested technical solution. These studies will take into account the changes in the optical parameters of actually used material modifications (Zerodur and silicon carbide) during the transition from normal to cryogenic temperatures and thereby ensure the given interferometric accuracy of the measurement the shape of the operating surface of the full-size staff convergent mirror.