(46-2) 01 * << * >> * Russian * English * Content * All Issues
Method for calculating the eikonal function and its application to design of diffractive optical elements for optical beam shaping
L.L. Doskolovich 1,2, A.A. Mingazov 1,2, E.V. Byzov 1,2, D.A. Bykov 1,2, E.A. Bezus 1,2
1 IPSI RAS – Branch of the FSRC "Crystallography and Photonics" RAS,
443001, Samara, Russia, Molodogvardeyskaya 151,
2 Samara National Research University, 443086, Samara, Russia, Moskovskoye Shosse 34
PDF, 3324 kB
Full text of article: Russian language.
We develop a method for calculating the eikonal function (or the phase function) of the light field, ensuring the formation of a prescribed irradiance distribution in the geometrical optics approximation. In the proposed method, the problem being solved is formulated in a semi-discrete form as a problem of the maximization of a concave function. For finding the solution to the latter problem, a gradient method is used, with analytical expressions obtained for the gradient. Using the developed method, we calculate an eikonal function that provides the formation of a “discontinuous” hexagram-shaped irradiance distribution. We demonstrate that the use of the solution obtained in the framework of the geometrical optics as an initial approximation in iterative Fourier transform algorithms allows one to calculate diffractive optical elements having a quasi-regular microrelief.
geometrical optics, inverse problem, eikonal, diffractive optical element, Fresnel approximation, Gerchberg-Saxton algorithm.
Doskolovich LL, Mingazov AA, Byzov EV, Bykov DA, Bezus EA. Method for calculating the eikonal function and its application to design of diffractive optical elements for optical beam shaping. Computer Optics 2022; 46(2): 173-183. DOI: 10.18287/2412-6179-CO-1029.
The development of the gradient method and the design of the geometrical-optics phase function was funded by Russian Science Foundation under project 18-19-00326, the design of DOEs in the framework of the scalar diffraction theory was funded by Russian Federation Ministry of Science and Higher Education (State contract with the “Crystallography and Photonics” Research Center of the RAS under agreement 007-GZ/Ch3363/26), and the development of the software implementing the iterative Fourier transform algorithms was funded by Russian Federation Ministry of Science and Higher Education in the framework of the research performed by the laboratory "Photonics for a smart home and smart city" (State contract with the Samara University) (project FSSS-2021-0016).
- Zhang J, Pégard N, Zhong J, Adesnik H, Waller L. 3D computer-generated holography by non-convex optimization. Optica 2017; 4(10): 1306-1313. DOI: 10.1364/OPTICA.4.001306.
- Schmidt S, Thiele S, Toulouse A, Bösel C, Tiess T, Herkommer A, Gross H, Giessen H. Tailored micro-optical freeform holograms for integrated complex beam shaping. Optica 2020; 7(10): 1279-1286. DOI: 10.1364/OPTICA.395177.
- Banerji S, Meem M, Majumder A, Vasquez FG, Sensale-Rodriguez B, Menon R. Imaging with flat optics: metalenses or diffractive lenses? Optica 2019; 6(6): 805-810. DOI: 10.1364/OPTICA.6.000805.
- Banerji S, Sensale-Rodriguez B. A computational design framework for efficient, fabrication error-tolerant, planar THz diffractive optical elements. Sci Rep 2019; 9: 5801. DOI: 10.1038/s41598-019-42243-5.
- Pégard NC, Mardinly AR, Oldenburg IA, Sridharan S, Waller L, Adesnik H. Three-dimensional scanless holographic optogenetics with temporal focusing (3D-SHOT). Nat Commun 2017; 8: 1228. DOI: 10.1038/s41467-017-01031-3.
- Soifer VA, Kotlyar VV, Doskolovich LL. Iterative meth-ods for diffractive optical elements computation. London: CRC Press; 1997. ISBN: 978-0-7484-0634-0.
- Gerchberg RW, Saxton WO. A practical algorithm for the determination of phase from image and dif-fraction plane pictures. Optik 1972; 35(2): 237-246.
- Fienup JR. Phase retrieval algorithms: a comparison. Appl Opt 1982; 21(15): 2758-2769. DOI: 10.1364/AO.21.002758.
- Shechtman Y, Eldar YC, Cohen O, Chapman HN, Miao JW, Segev M. Phase retrieval with application to optical imaging. IEEE Signal Process Mag 2015; 32(3): 87-109. DOI: 10.1109/MSP.2014.2352673.
- Latychevskaia T. Iterative phase retrieval in coherent diffractive imaging: practical issues. Appl Opt 2018; 57(25): 7187-7197. DOI: 10.1364/AO.57.007187.
- Ripoll O, Kettunen V, Herzig HP. Review of iterative Fourier transform algorithms for beam shaping applications. Opt Eng 2004; 43(11): 2549-2556. DOI: 10.1117/1.1804543.
- Feng Z, Froese BD, Liang R. Composite method for precise freeform optical beam shaping. Appl Opt 2015; 54(31): 9364-9369. DOI: 10.1364/AO.54.009364.
- Yang L, Badar I, Hellmann C, Wyrowski F. Light-shaping design by a fourier pair synthesis: the homeomorphic case. Opt Express 2021; 29(3): 3621-3630. DOI: 10.1364/OE.415649.
- Bösel C, Gross H. Ray mapping approach for the efficient design of continuous freeform surfaces. Opt Express 2016; 24(13): 14271-14282. DOI: 10.1364/OE.24.014271.
- Benamou JD, Froese BD, Oberman AM. Numerical solutionof the optimal transportation problem using the Monge-Ampère equation. J Comput Phys 2014; 260: 107-126. DOI: 10.1016/j.jcp.2013.12.015.
- Prins C, Beltman R, ten Thije Boonkkamp J, Ijzerman W, Tukker T. A least-squares method for optimal transport using the Monge-Ampère equation. SIAM J Sci Comput 2015; 37(6): B937-B961. DOI: 10.1137/140986414.
- Doskolovich LL, Mingazov AA, Bykov DA, Andreev ES, Bezus EA. Variational approach to calculation of light field eikonal function for illuminating a prescribed region. Opt Express 2017; 25(22): 26378-26392. DOI: 10.1364/OE.25.026378.
- Mingazov AA, Bykov DA, Bezus EA, Doskolovich LL. On the use of the supporting quadric method in the problem of designing double freeform surfaces for collimated beam shaping. Opt Express 2020; 28(15): 22642-22657. DOI: 10.1364/OE.398990.
- Mérigot Q. A multiscale approach to optimal transport. Comput Graph Forum 2011; 30(5): 1583-1592. DOI: 10.1111/j.1467-8659.2011.02032.x.
- Bleistein N, Handelsman RA. Asymptotic expansions of integrals. New York: Dover Publications Inc; 1986. ISBN: 0-486-65082-0.
© 2009, IPSI RAS
151, Molodogvardeiskaya str., Samara, 443001, Russia; E-mail: email@example.com ; Tel: +7 (846) 242-41-24 (Executive secretary), +7 (846) 332-56-22 (Issuing editor), Fax: +7 (846) 332-56-20