Algorithm and mathematical model for geometric positioning of segments on aspherical composite mirror

In recent years, the largest terrestrial and orbital telescopes operating in a wide spectral range of wavelengths use the technology of segmented composite elements to form the main mirror. This approach allows: to expand the spectral operating range from 0.2 to 11.0 μm and to increase the diameter of the entrance pupil of the receiving optical system, while maintaining the optimal value of the exponent m_S– mass per unit area.

Two variants of adjusting the position of mirror segments are considered when forming an aspherical surface of the second order, with respect to the base surface of the nearest sphere, including geometrical and opto-technical positioning.

The purpose of the research was to develop an algorithm for solving the problem of geometric positioning of hexagonal segments of a mirror telescope, constructing an optimal circuit for traversing elements when aligning to the nearest radius to an aspherical surface, and also to program the output calculation parameters to verify the adequacy of the results obtained.

Various methods for forming arrays from regular hexagonal segments with equal air gaps between them are considered. The variant of construction of arrays through concentric rings of an equal step is offered.

A sequential three-step method for distributing mosaic segments is presented when performing calculations for aligning the aspherical surface: multipath linear; multipath point; block trapezoidal.

In the course of mathematical modeling an algorithm was developed to solve the problem of geometric positioning of flat hexagonal segments of a mirror telescope. In the Python programming language, program loops are designed to form the data array necessary to construct a specular reflective surface of a given aperture. In the software package Zemax, the convergence of optical beams from flat hexagonal elements to the central region of the aspherical surface is verified.

1. Sabelhaus P.A., Decker J.E. An overview of the James Webb Space Telescope (JWST) project. Proceeding of SPIE, 2004, vol. 5487. doi: 10.1117/12.549895

2. Olczak G., Wells C., Fischer D.J., Connolly M.T. Wavefront calibration testing of the James Webb Space Telescope primary mirror center of curvature optical assembly. Proceeding of SPIE, 2012, vol. 8450–84500R. doi: 10.1117/12.927003

3. Demin A.V. [A mathematical model of the process of aligning composite mirrors]. Instrument making, 2015, vol. 58, no. 11, pp. 901–907 (in Russian). doi: 10.17586/0021-3454-2015-58-11-901-907

4. Demin A.V., Kovalev I.A. The Mathematical model and the simulation modelling algorithm of the multitiered mechanical system. ABC Journal of Advanced Research, 2013, vol. 2 (1), iss. 3, pp. 44–48. doi: 10.18034/abcjar.v2il.427

5. Mast T., Nelson J. TMT-internal communication: MT Primary Mirror Segment Shape, TMT Report, no. 58, 2004.

6. Demin A.V., Mendeleyev L.M. [Algorithm for alignment of composite mirrors of high-aperture telescopes]. News of Higher Education. Instrument making, 2014, vol. 57, no. 1, pp. 51–56 (in Russian).

7. Poleshuk A.G., Korolkov V.P., Nasyrov R.K. [Diffraction optical elements for controlling the parameters of laser radiation and precise control of the shape of aspherical surfaces]. Interexpo Geo-Siberia, 2015, vol. 5, no. 2, pp. 232–238 (in Russian).

8. Demin A.V., Rostokin P.V. [Algorithm for align-ment of composite mirrors]. Computer Optics, 2017, vol. 41, no. 2, pp. 291–294 (in Russian). doi: 10.18287/2412-6179-2017-41-2-291-294

9. Rabysh A.Yu., Demin A.V. [Algorithm for composing composite objects (for example, a mirror)]. Scientific and Technical Herald of Information Technologies, Mechanics and Optics, Saint Petersburg State University of Information Technologies, Mechanics and Optics Publ., 2018, vol. 18, no. 4, pp. 31–36 (in Russian).

10. Strom S.E., Stepp L., Brooke G. Giant Segmented Mirror Telescope: a point design based on science drivers. Angel J.R.P. and Gilmozzi, R. editors, Future Giant Telescopes-SPIE 4840, 2003, pp. 116–128.

11. Kerley D. Distributed control of a segmented telescope mirror, master’s thesis, university of Victoria, BC, Canada, 2010.

12. Amodei D., Padin S. Mirror with regular hexagonal segments. Applied optics, 2008, vol. 42, no. 25, pp. 5130–5135.

13. Baffles, C. , Baffles C., Mast T., Nelson J., Ponslet E., Stephens V., Stepp L., Williams E. Primary Mirror Segmentation Studies for the Thirty Meter Telescope. Proceeding of SPIE, 2008, vol. 7018. doi: 10.1117/12.790206