Alexandr P. Semenov*, Magomed A. Abdulkadyrov, Sergey P. Belousov, Alexandr N. Ignatov, Vladimir E. Patrikeev, Vitaliy V. Pridnya, Andrey V. Polyanchikov, Victor V. Rumyantsev, Anatoly V. Samuylov, Yury A. Sharov

JSC LZOS under the contract with Carl Zeiss Jena, Germany produced 645 mm (F/2.5) secondary mirror for TTL project (Telescope Technologies Limited, Great Britain) during 1999-2001. The asphericity is approximately 12 m from the nearest sphere. The system of primary and secondary mirrors has 80% encircle energy within less than 0.2 arcsec. The surface error RMS is about 9 nm. 753 mm (F/2) secondary mirror for NOA project (Astronomical Institute – National Observatory of Athens, Greece) was produced. The asphericity is approximately 26 m. The surface error RMS is about 12 nm. The telescope field of view is approximately 1.04 degrees. The system of primary and secondary mirrors has 80% encircle energy within less than 0.3 arcsec. 938 mm (F/2.3) secondary mirror for VST project (VLT Survey Telescope, Osservatorio Astronomico di Capodimonte Napoli) was produced. The asphericity is approximately 100 m. The telescope field of view with corrector will be 1.5 degrees. Three Hindle sphere 1610 mm, 1640 mm and 1985 mm in diameter with radii of 6300 mm, 3995 mm and 2708 mm were using to test three secondary mirrors. Each convex hyperbolic mirror was tested by using two Hindle spheres. The wavefront of tested mirror was determined by wavefront superposition method.

Keywords: telescopes, optical fabrication, optical testing, aspheres

* Correspondence: e-mail: lastro@comail.ru; Telefone: 007-095-552-15-72; Fax 007-095-552-15-86

During 1999-2000, as described in the previous article, JSC LZOS produced under the contract with Carl Zeiss (Germany) the secondary mirror 645 mm of Sitall ÑÎ-115Ì for the TTL (Telescope Technologies Limited, Great Britain) project. Secondary mirror has D/f = 1:2.5. The telescope field of view is approximately 40 arcmin. In 2000 the set of optics for the telescope NOA (Astronomical Institute - National Observatory of Athens, Greece) with the secondary mirror 753 mm was completed. The field of view of the whole telescope is about 1.04^o. The telescope will be installed at Balkans. In 1999-2001 the set of optics for survey telescope VST¹ (VLT Survey Telescope, Osservatorio Astronomico di Capodimonte Napoli) with secondary mirror 938 mm was produced. With field corrector the telescope will have field of vision 1.5^o. Installation location - Observatory Paranal in Chile.

Secondary mirrors fabricated for TTL, NOA and VST projects have the following parameters:

specifications	project
	TTL	NOA	VST
material	Sitall CO-115M
shape	convex, hyperbolic
outer diameter	645 mm	753 mm	938 mm
central screening	59 mm	139 mm	176 mm
thickness	110 mm	115 mm	130 mm
clear aperture	617 mm	740 mm	900 mm
curvature	4813.19 ± 10 mm	4602.2 - 2 mm	4374.46 ± 2 mm
conic constant	-4.179	-4.2087	-5.421864
maximum asphericity	12 m	24 m	98 m

The following out parameters of telescope optics are required according to specification:

TTL project:
light concentration (80%) 0.2 arcsec diameter.
NOA project:
image quality (80% encircled energy)
on axis uncorrected: 0.35 arcsec
on axis corrected: 0.35 arcsec
off axis to 5 arcmin radius, uncorrected: 0.5 arcsec
off axis to 20 arcmin radius, corrected: 0.5 arcsec
VST project:
Geometrical spot concentration (80%) on axis, two mirror system, within 0.30 arcsec diameter, after removal of the coefficients constant, focus, decentering coma.
Geometrical spot concentration (80%) on axis, two mirror system, within 0.15 arcsec diameter, after additional removal of the coefficients 3^rd order spherical aberration, 3^rd order astigmatism, triangular coma, quadratic astigmatism.

Several options of testing design including classical Hindle schema were considered and analyzed for fabrication of secondary mirrors. But it is hard to implement testing of the secondary mirror with one Hindle sphere due to its technical complexity and high price. Calculation complexity is caused by necessity of testing of high aperture mirrors with different parameters and with small central screening. In particular, it would be necessary to produce Hindle mirror of 2800 mm diameter for testing of these secondary mirrors.

As the result of the analysis the scheme of testing of each hyperboloid using two Hindle spheres was selected². One sphere of 1610 mm diameter (diameter of opening 230 mm) with radius of curvature 6300 mm (Ì1) was available for the previously fabricated secondary mirrors. Two new spheres were manufactured for the testing. One sphere has diameter 1640 mm and curvature radius 3995 mm (Ì 2), the other one of diameter 1985 mm has curvature radius 2708.15 mm (Ì3). Deviation of wavefront 0.05 (RMS) was obtained on sphere with diameter 1640 mm and curvature radius 3995 mm. Certain hazard was connected with principal possibility of fabrication of the second large size high aperture sphere of 1985 mm diameter and curvature radius 2708.15 mm (f/D=1:0.7). But the problem was solved successfully, deviation of wavefront 0.07 (RMS) was obtained (Fig. 1).

Figure.1 Spherical mirror of 1985 mm diameter and curvature radius 2708.15 mm.

Fig. 2-4 shows the accepted schema of testing of convex hyperbolical mirrors with the use of two additional mirrors Ì 1, diameter 1610 mm, Ì 2, diameter 1640 mm and Ì 3, diameter 1985 mm. Each secondary mirror is tested in tern with two additional mirrors. While testing with mirror Ì 3 zone 270-900 mm for the secondary mirror VST was tested and 309-740.6 mm for the secondary mirror NOA; correspondingly, while testing with Ì 2, we have zones 112-500 mm and 128-600 mm. In both cases we have an overlapping zone (zone which is tested both with Ì 2 and with Ì 3 mirrors) of about 200 mm and more. Existence of this zone enables to construct topography by the method of overlapping of wavefronts, obtained after description of the results of interferential testing with the use of mirrors Ì 2 and Ì 3 and to construct the general topography of tested secondary mirror. Testing of the secondary mirror of the telescope TTL with diameter 645 mm was made according to the similar schema. The external clear zone of the secondary mirror 123-617 mm is tested mirror M 2 and the central part 59-353 mm is tested with mirror M 1. Implementation of this method reduces the expenses of testing significantly and shortens the terms of fabrication of the secondary mirrors described above.

Complexity of computer controlled polishing processing of convex large-size optical surfaces lies first of all in acquisition of the true information about the shape of surface on the base of its testing data. The method of wavefront shape acquisition by the method of overlapping with the use of Zernike polynomials is known. But in such description of the waveform the local errors are smoothened. Therefore the method of topographical map construction by overlapping useful for automatic finishing of the given surface was developed.

On fig. 5 interferograms for testing of NOA mirror with two spherical mirrors are shown. On the left side there is an interferogram of the mirror outer part (area 1) of 741 mm in diameter (shaded zone 309 mm) and on the right side there is an interferogram of the mirror inner part of 600 mm in diameter (shaded zone 128 mm) being tested with the other sphere (area 2). Common area of the diameter from 309 to 600 mm on the both interferograms (on the left one - the central part, on the right one - the upper part) is used for wavefronts conjugation (area 1,2). Surface topography calculation is performed as follows. Let all the mirror surface be divided into rectangular net with coordinates X_n, Y_m, where m = 1,2,...,M, n = 1,2,...,M. Normal shape deviations from the nearest surface are calculated in each area. Hence we calculate normal deviations W_m,n^(k)(X_n,Y_m) in k area. It is necessary to bind areas in order to build common topography in relation to the single nearest surface for comparison.

Figure.5 Final interferograms of twice reflected wavefront of the secondary NOA mirror of 753 mm in diameter at the control with two Hindle spheres of 1985 and 1640 mm in diameter.

We determine normal deviations in the intersection zone of areas 1 and 2:

W_1,2⁽¹⁾ = A_1,2⁽¹⁾ + B_1,2⁽¹⁾X_n + C_1,2⁽¹⁾Y_m + D_1,2⁽¹⁾(X_n² + Y_m²) - W₁           (1)

where W_1,2⁽¹⁾ - are the deviations in the intersection zone for areas 1 and 2 got by the deviations for the first zone W₁, for points m, n in the intersection zone. It must be noted that in this case the nearest surface is spherical. Just the same:

W_1,2⁽²⁾ = A_1,2⁽²⁾ + B_1,2⁽²⁾X_n + C_1,2⁽²⁾Y_m + D_1,2⁽²⁾(X_n² + Y_m²) - W₂           (2)

where W_1,2⁽²⁾ - are the deviations in the intersection zone for areas 1 and 2 got by the declinations for the second zone W₂, for points m, n also caught into the intersection zone. Hence we can determine RMS_1,2⁽¹⁾ è RMS_1,2⁽²⁾ for the intersection part of areas 1 and 2. It is clear that if there are no errors of the interferogram processing errors overlapping method the condition:

RMS_1,2⁽¹⁾ RMS_1,2⁽²⁾,           (3)

àis to be complied, and the difference of topographies (1) and (2) is zero. By topography difference RMS for the intersection zone we can determine the accuracy of the control method in each case. Using the coefficients of equations  (1) and (2) we determine normal deviations of areas 1 and 2 in relation to the common nearest surface:

W₁' = A_1,2⁽¹⁾ + B_1,2⁽¹⁾X_n + C_1,2⁽¹⁾Y_m + D_1,2⁽¹⁾(X_n² + Y_m²) - W₁           (4)

where X_n, Y_m belong to area 1;

W₂' = A_1,2⁽²⁾ + B_1,2⁽²⁾X_n + C_1,2⁽²⁾Y_m + D_1,2⁽²⁾(X_n² + Y_m²) - W₂           (5)

where X_n, Y_m belong to area 2. If condition (3) is complied with the appropriate accuracy then we can determine the common sphere of comparison for zones 1 and 2:

W_1,2 = 0.5(W₁’ + W₂’)

By solving of the set of equations:

A_1,2 + B_1,2X_n + C_1,2Y_m + D_1,2(X_n² + Y_m²) = W_1,2

we calculate the deviations in area 1 - 2:

W_1,2' = A_1,2 + B_1,2X_n + C_1,2Y_m + D_1,2(X_n² + Y_m²) - W_1,2

for points m,n attached to all area 1 - 2. If the difference of wavefronts is determined:

S_1,2 = W_1,2⁽¹⁾ - W_1,2⁽²⁾,

with the results obtained by solving the equations (1) and (2) for the intersection part of area 1 and 2 then we’ll get the estimation of the error for determination of common surface of areas 1-2 RMS_1,2. In ideal case RMS_1,2 must tend to zero. The main sources of overlapping method errors are:

• errors of calculations of interferograms;
• error caused by the own deformations of the tested mirror during change-over of the control equipment from one sphere to the other one and the turn of the controlled hyperbola.

That’s why it is necessary to get the difference of wavefronts of common overlapping zone for the estimation of control scheme errors. During final control of hyperbola mirror surface with two spheres the errors of the spherical mirrors wavefront were subtracted from total wavefront of the mirror + hyperbola on visible part of sphere and hyperbola that was necessary condition to get required accuracy of hyperbolic mirror manufacturing.

Secondary mirrors pre-shaping process for TTL, NOA and VST projects was performed according to the classical methods. Firstly the nearest sphere was fabricated and then the surface aspherization was performed by grinding and further polishing. At the stage of aspherization the surface profile was measured by the set of spherometers according to special algorithm. This method provides successful surface aspherization with the deviation of the set profile up to 1 - 2 mm. After previous surface processing each mirror was put into a horizontal test schema with spherical mirrors (fig. 6, fig. 7) and was tested with interferometer. Testing on the stage of prime and final finishing was performed with CCD camera. Two positions of hyperbolic mirror with two spherical mirrors were tested. In one position the outer part of the mirror was tested, and in the other - the inner one. Two positions have the intersection part of hyperboloid that is the base for wavefront conjunction and the full required topography of the tested mirror construction.

Figure.6 Test set-up for secondary mirror with two Hindle spheres.

Figure.7 Secondary mirror of VST.

Figure.8 Secondary mirror processing.

When the deviations from the hyperboloid surface became comparable with spherical mirrors deviations the subtraction of spherical mirrors wavefronts was done in order to get he real shaping map for hyperboloid surface. It must be noted that controlled total wavefront includes twice reflected wavefront of hyperbolic mirror and wavefront of spherical Hindle mirror. So controlled hyperbola errors are duplicated in comparison with the errors of spherical mirror. Wavefronts subtraction was done after twice reflection of the total wavefront from the hyperboloid surface with the error about 0.15 (RMS).

For finishing of secondary mirrors surface in TTL, NOA and VST projects the computer controlled polishing (fig. 8) described in previous article was applied³. Finishing of the secondary mirror TTL was performed within a month (Fig. 9). Surface with RMS about 11 nm (RMS of wavefront with single reflection 0.035) was obtained. On the secondary NOA mirror the error was RMS = 15 nm (0.048 on the wavefront).

Period of final automated finishing is about two months. On the secondary VST mirror the error was RMS = 13 nm. This mirror was the most difficult for processing because its asphericity was about 100 m on the diameter 900 mm.
Fig.5 shows interferograms of NOA mirror got with spheres M2 and M3.
Fig.9 shows interferograms of the secondary TTL mirror got with spheres M2 and M3.
Fig.10 shows interferograms of the secondary VST mirror got with spheres M1 and M2.

So we have shown the possibility of manufacturing of high aperture hyperbolic mirror with the help of two Hindle spheres. The surface of secondary mirror for TTL telescope of 645 mm in diameter with the precision of 11 nm RMS. On the NOA mirrors surface the error was 15 nm (RMS). On the secondary VST mirror the error was 13 nm (RMS). System of three spherical mirrors can be applied for manufacturing of the analogous high aperture details with the diameter up to 1 m. Now it is applied for manufacturing of he hyperbolic mirrors in other projects.

After manufacturing of the primary and secondary mirrors the total energy concentration on the axis of two mirrors system was determined. For TTL telescope it was less than 0.2 arcsec, for NOA telescope - less than 0.35 arcsec.

For VST mirror system:
Geometrical spot concentration (80%) on axis, two mirror system, within 0.20 arcsec diameter, after removal of the coefficients constant, focus, decentering coma. Geometrical spot concentration (80%) on axis, two mirror system, within 0.12 arcsec diameter, after additional removal of the coefficients 3^rd order spherical aberration, 3^rd order astigmatism, triangular coma, quadratic astigmatism.

REFERENSES

1. D. Mancini, G. Sedmak, M. Brescia, F. Cortecchia, D. Fierro, V. Fiume Garelli, G. Marra, F. Perrotta, F. Rovedi, P. Schipani, VST project: technical overview. Proceedings of SPIE, 4004, pp. 79-90, 2000.
2. M. A. Abdulkadyrov, S. P. Belousov, A. N. Ignatov, V. V. Rumyantsev, Non-traditional technologies to fabricate lightweighted astronomical mirrors with high stability of surface shape. Proceedings of SPIE, 3786, pp. 468-473, 1999.
3. A. P. Semenov, V. E. Patrikeev, A. V. Samuylov, Y. A. Sharov, Computer-controlled fabrication of large-size ground and space-based optics from glass ceramic Sitall CO-115M. Proceedings of SPIE, 3786, pp. 474-479, 1999.