Vol. 37, issue 04, article # 8

Abstract:

Due to the importance of using the structure function for problems of optical radiation propagation in a turbulent atmosphere, the task of determining this function from the known mode coefficients of wavefront expansion was set. New formulas have been derived for the exact analytical calculation of the wavefront phase structure function on a circular aperture. Unlike the previously published analytical method, the proposed approach correctly accounts the entire domain, including the area near the edge of the aperture. The new method is compared to the published one and a numerical calculation with the distretization chosen sufficiently fine. The test samples comprised Kolmogorov wavefronts and Zernike polynomials and Karhunen–Loѐve functions corresponding to the Kolmogorov turbulence model. The deviations of the results of the published before method from the new one and from the numerical calculation are provided. The advantages and generality of the new method are stated and explained. The result will make it possible to accurately determine the structure function of the wavefront by its mode coefficients in problems of optical radiation propagation in randomly inhomogeneous media.

Keywords:

adaptive optics, wavefront, structure function, Kolmogorov turbulence, analytical calculation, Zernike polynomials, Karhunen–Loѐve functions

References:

1. Tatarskii V.I. Rasprostranenie voln v turbulentnoi atmosfere. M.: Nauka, 1967. 548 p.
2. Shishakov K.V., Shmal'gauzen V.I. Approksimatsiya strukturnoi funktsii fazy volnovogo fronta // Optika atmosf. 1989. V. 2, N 2. P. 160–162.
3. URL: https://arxiv.org/abs/2402.04826.
4. Schmidt J.D. Numerical simulation of optical wave propagation with examples in MATLAB. Bellingham: SPIE, 2010. 199 p.
5. Herweijer J.A. The small-scale structure of turbulence. [Phd Thesis 1 (Research TU/e / GraduationTU/e), Chemical Engineering and Chemistry]. Technische Universiteit Eindhoven, 1995.
6. He L. Optical surface characterization with the structure function. PhD dissertation. The University of North Carolina at Charlotte, 2013. 166 p.
7. Kreis T., Burke J., Bergmann R.B. Surface characterization by structure function analysis // J. Europ. Opt. Soc. Rap. Public. 2014. V. 9, N 14032. 8 p.
8. Fried D.L. Optical resolution through a randomly inhomogeneous medium for very long and very short exposures // J. Opt. Soc. Am. 1966. V. 56, N 10. P. 1372–1379.
9. Vorontsov M.A., Shmal'gauzen V.I. Printsipy adaptivnoi optiki. M.: Nauka, 1985. 336 p.
10. Handbook of Optics. New York: McGraw-Hill, 2010. Vol. V. 1280 p.
11. Guang-ming Dai. Modal compensation of atmospheric turbulence with the use of Zernike polynomials and Karhunen–Loѐve functions // J. Opt. Soc. Am. A. 1995. V. 12. P. 2182–2193.
12. Hvisc A.M., Burge J.H. Structure function analysis of mirror fabrication and support errors // Proc. SPIE. 2007. V. 6671. P. 1–10.
13. URL: https://wp.optics.arizona.edu/visualopticslab/wp-content/uploads/sites/52/2016/08/ Zernike-Notes-15Jan2016.pdf (last access: 31.08.2023).
14. Roddier N. Atmospheric wavefront simulation using Zernike polynomials // Opt. Eng. 1990. V. 29, N 10. P. 1174–1180.
15. Guang-ming Dai. Wavefront simulation for atmospheric turbulence // SPIE. 1994. V. 12. P. 62–72.
16. Kellerer A. A Cook Book of Structure Functions // arXiv:1504.00320vl. 2015. 11 p.