Saint-Petersburg State Institute of fine mechanics and optics,
Department of Applied and computer optics

The precision testing technique development and the investigations of quality of the main BTA mirror reflection surface

The work completed in the Saint-Petersburg,
under the supervision of prof. Sergey A. Rodionov
part of 1997-98.

Phe page prepared by:
Ph.D., Tolstoba N.D.
http://aco.ifmo.ru/~nadinet

Hartmanogramm scale and orientation definition.
Spot identification

For the BTA primary mirror control the classical radial screen Hartmann with 8 radial lines, 32 points in each (Fig. 1) is used. Hartmann mask parameters, so and the coordinates of all its apertures, are known and will be used for transverse aberrations definition, calculated as a difference between spots coordinates on real and ideal hartmanogramm.

Fig. 1. The hartmanogramm.

The preliminary hartmanogramm analysis consists in of all sample intensity values gravity centre determination on hartmanogramm. After the centre of gravity is chosen as a beginning of coordinate system, with the help two reper points in the diagram top right part it is necessary to define hartmanogramm scale and orientation. On known Hartmann mask parameters now it is easy to define scale of the hartmanogramm real centre and it's displacement from ideal.

Besides reper points position gives the hartmanogramm orientation unequivocally. The identification of spots carry out under the accepted order of apertures numbering on mask, that is conformity of spots to certain apertures is made. Thus, determining the order of mask apertures detour.

Approached values of spots centers coordinates as of gravity centres (geometric average) area allocated on hartmanogramm is obtained at first. Then for achievement of required accuracy sample of intensity on separated area aproximated as Haussoide my means of the least squares method, and then the coordinates of spot centre are defined as coordinates of approximated function maximum. Development of a method with the help of use Haussoide of the higher order is hereinafter supposed.

The spot centers definition

In this case the method of definition of the energy center of the spot on a weight center of the chosen area is traditional. The approached value of coordinates of the spot centers as weight centers (geometrical average) area allocated on hartmanogram.

Fig. 2. The one spot area.

For the checking of the hartmannogram recognition accurasy, the computer simulation of the hartmannogram making completed.

The CCD receiver parameters

Amount	2048 x 2048 elements
the element size	18 x 18 mkm
The photosensitive area size	36.8 x 36.8 mm

Mathematical apparatus telescope main mirror deformation restoration

In astronomical systems the large importance has decomposition on Zernike polynomials, orthogonal on a annular area with given factor of central obsquration . For the approximation of wave aberration function the most reasonable will be the use of basis, orthogonal on ring area.

Construction of the approximation basis

The orthogonal Zernike polynomials dependence from polar coordinates in general is::

where m,n - integer positive numbers, have restriction as: , moreover the difference () should be even number.

The most effective way of Zernike polynomials values calculation consists in use Forsythe three-member expressions:

, ,
.

Factors of these expressions , are calculated proceeding from the requirements of orthogonality area form (circle or ring). The parameters of a ring are determined by obsquration. Integrals for calculation of factors and norm are obtained numerically:

, ,
, .

Directly for approximation transverse aberrations not only polynomials calculation is required, but also their private derivatives on initial coordinates, as:

;

With use of mentioned expressions we have an opportunity to receive values as Zernike polynomials, and their private derivatives, orthogonal on a ring and on a circle.

The aberration approximation

It is clear, that for calculation of wave aberration decomposition coefficients on Zernike polynomials the following data are necessary: ╥ collection of transverse aberrations values and in various points of pupil, received after test results processing,
╥ testing points coordinates,
╥ collection of Zernike polynomials derivatives values, the testing points calculated for.

For decomposition factors determination it is necessary to present initial data in such a way, defined by a calculation technique used in this work.

If to present equations system in the matrix form, the approximation coefficients will be the decision of linear equations system, looks like:

where - task constructional matrix. The matrix structure is shown below:

-
aberration coefficients vector,

-
transverse aberrations values vector, it's structure corresponds to P matrix:

Accordingly to the least squares method the equations system (16) should be transformed in such a way:

, and this expression can be written down as:

The system decision in a general kind will look as this:

or:

For this decision realization we shall use a Gram-Schmidt orthogonalization procedure, with help of what we shall receive values of coefficients. Applying this method, we orthogonalize matrix rows and shall transform that to an orthogonal matrix , which has property:

In orthogonalization process, carried out on optimum strategy, optimum choice of rows in matrix , ensuring minimum conditionality, set as the relative admission on line norm size is made. Besides an opportunity to make the regression analysis of a norms vector by one of statistical criteria (Fisher's criterion, criterion and etc.) is available. After processing the final matrix can be used for the task solving, using orthgonatily property:

The orthogonalization technique advantages:

The fastest technique
It provides the automated choosing of the optimal data in the solving process, that garantees the minimal task conditionality.
The ability of the regressive analysis application in the most simple form.

REFERENCES

1. G.A. Korn, Th.M.Korn Mathematical Handbook for Scientists and Engineers, - McGraw - Hill, New York, 1968, 832 pp.
2. Optical shop testing / edited by Daniel Malacara. - A Wiley - Interscience Publication, John Wiley & Sons, Inc, 1992, 773 pp.
3. Zverev V.A., Rodionov S.A., Sokolsky M.N., Usoskin V.V. The BTA main mirror testing in observatory conditions, - Soviet Journal of Optical Technology, 1977, 4.
4. Zverev V.A., Rodionov S.A., Sokolsky M.N., Usoskin V.V. Mathematical bases of the BTA main mirror Hartmann test, - Soviet Journal of Optical Technology, 1977, 2.
5. Zverev V.A., Rodionov S.A., Sokolsky M.N., Usoskin V.V. Technological control of by Hartmann technique, - Soviet Journal of Optical Technology, 1977, 3.

Saint-Petersburg State Institute of fine mechanics and optics, Department of Applied and computer optics