Formation of required distributions on the basis of decomposition by vortex eigen functions of a bounded non-paraxial propagation operator
Khonina S.N.
, Volotovsky S.G., Kirilenko M.S.

 

IPSI RAS – Branch of the FSRC “Crystallography and Photonics” RAS, Molodogvardeyskaya 151, 443001, Samara, Russia;
Samara National Research University, Moskovskoye shosse, 34, 443086, Samara, Russia

Abstract:
The solution of the problem of overcoming the diffraction limit based on the representation of an optical signal in the form of a superposition of communication modes matched with the vortex eigenfunctions of a bounded (in the object and spectral regions) nonparaxial propagation operator in free space is considered. Nonparaxial propagation of laser beams is described using an expansion in terms of conic waves based on the m-th order Fourier-Hankel transform. The eigenfunctions of such an operator, which have near-unity eigenvalues, determine the number of degrees of freedom and characteristics of an optical signal transmitted without distortion over a given distance. Based on the considered approach, a parametric method was developed for solving the inverse diffraction problem, including overcoming the diffraction limit.

Keywords:
near-field diffraction zone; bounded propagation operator; vortex eigenfunctions; signal approximation; solution of the inverse problem.

Citation:
Khonina SN, Volotovsky SG, Kirilenko MS. Formation of required distributions on the basis of decomposition by vortex eigen functions of a bounded non-paraxial propagation operator. Computer Optics 2019; 43(2): 184-192. DOI: 10.18287/2412-6179-2019-43-2-184-192.

References:

  1. Lord Rayleigh. On the theory of optical images with special reference to the optical microscope. Phil Mag 1896; 5(42): 167-195.
  2. Kowarz MW. Homogeneous and evanescent contributions in scalar near-field diffraction. Appl Opt 1995; 34(17): 3055-3063. DOI: 10.1364/AO.34.003055.
  3. Katrich AB. Do evanescent waves really exist in free space? Opt Commun 2005; 255(4-6): 169-174. DOI: 10.1016/j.optcom.2005.06.012.
  4. Betzig E, Trautman JK, Harris TD, Weiner JS, Kostelak RL. Breaking the diffraction barrier: Optical microscopy on a nanometric scale. Science 1991; 251(5000): 1468-1470. DOI: 10.1126/science.251.5000.1468.
  5. Betzig E, Trautman JK. Near-field optics: microscopy, spectroscopy, and surface modification beyond the diffraction limit. Science 1992; 257(5067): 189-195. DOI: 10.1126/science.257.5067.189.
  6. Heinzelmann H, Pohl D. Scanning near-field optical microscopy. Appl Phys A 1994; 59(2): 89-101. DOI: 10.1007/BF00332200.
  7. Girard C, Dereux A. Near-field optics theories. Rep Prog Phys 1996; 59(5): 657-699. DOI: 10.1088/0034-4885/59/5/002.
  8. Khonina SN, Ustinov AV. Very compact focal spot in the near-field of the fractional axicon. Opt Commun 2017; 391: 24-29. DOI: 10.1016/j.optcom.2016.12.034.
  9. Di Francia GT. Super-gain antennas and optical resolving power. Il Nuovo Cimento 1952; 9(3): 426-438. DOI: 10.1007/BF02903413.
  10. Berry MV, Popescu S. Evolution of quantum superoscillations and optical superresolution without evanescent waves. J Phys A 2006; 39(22): 6965-6977. DOI: 10.1088/0305-4470/39/22/011.
  11. Huang FM, Zheludev NI. Super-resolution without evanescent waves. Nano Lett 2009; 9(3): 1249-1254. DOI: 10.1021/nl9002014.
  12.  Miller DAB. Communicating with waves between volumes: evaluating orthogonal spatial channels and limits on coupling strengths. Appl Opt 2000; 39(11): 1681-1699. DOI: 10.1364/AO.39.001681.
  13. Gallager RG. Information theory and reliable communication. New York: John Wiley & Sons, Inc; 1968. ISBN: 978-0-471-29048-3.
  14. Di Francia GT. Degrees of freedom of an image. J Opt Soc Am 1969; 59(7): 799-804. DOI: 10.1364/JOSA.59.000799.
  15. Slepian D, Pollak HO. Prolate spheroidal wave functions, Fourier analysis and uncertainty – I. Bell Syst Technol J 1961; 40(1): 43-63. DOI: 10.1002/j.1538-7305.1961.tb03976.x.
  16. Landau HJ, Pollak HO. Prolate spheroidal wave functions, Fourier analysis and uncertainty – II. Bell Syst Technol J 1961; 40(1): 65-84. DOI: 10.1002/j.1538-7305.1961.tb03977.x.
  17. Karoui A, Moumni T. Spectral analysis of the finite Hankel transform and circular prolate spheroidal wave functions. Journal of Computational and Applied Mathematics 2009; 233(2): 315-333. DOI: 10.1016/j.cam.2009.07.037.
  18. Yoshinobu I. Evaluation of aberrations using the generalized prolate spheroidal wavefunctions. J Opt Soc Am 1970; 60(1): 10-14. DOI: 10.1364/JOSA.60.000010.
  19. Khonina SN, Kirilenko MS, Volotovsky SG. Defined distribution forming in the near diffraction zone based on expansion of finite propagation operator eigenfunctions. Procedia Engineering 2017; 201: 53-60. DOI: 10.1016/j.proeng.2017.09.659.
  20. Kirilenko MS, Khonina SN. Formation of signals matched with vortex eigenfunctions of bounded double lens system. Opt Commun 2018; 410: 153-159. DOI: 10.1016/j.optcom.2017.09.060.
  21. Nye JF, Berry MV. Dislocations in wave trains. Proceedings of the Royal Society A 1974; 336(1605): 165-190. DOI: 10.1098/rspa.1974.0012.
  22. Bazhenov VYu, Soskin MS, Vasnetsov MV. Screw dislocations in light wavefronts. J Mod Opt 1992; 39(5): 985-990. DOI: 10.1080/09500349214551011.
  23. Gibson G, Courtial J, Padgett MJ, Vasnetsov M, Pasko V, Barnett SM, Franke-Arnold S. Free-space information transfer using light beams carrying orbital angular momentum. Opt Express 2004; 12(22): 5448-5456. DOI: 10.1364/OPEX.12.005448.
  24. Wang J, Yang J, Fazal IM, Ahmed N, Yan Y, Huang H, Ren Y, Yue Y, Dolinar S, Tur M, Willner AE. Terabit free-space data transmission employing orbital angular momentum multiplexing. Nat Photonics 2012; 6(7): 488-496. DOI: 10.1038/nphoton.2012.138.
  25. Khonina SN and Golub I. How low can STED go? Comparison of different write-erase beam combinations for stimulated emission depletion microscopy. J Opt Soc Am A 2012; 29(10): 2242-2246. DOI: 10.1364/JOSAA.29.002242.
  26. Paterson L, MacDonald MP, Arlt J, Sibbett W, Bryant PE, Dholakia K. Controlled rotation of optically trapped microscopic particles. Science 2001; 292(5518): 912-914. DOI: 10.1126/science.1058591.
  27. Šiler M, Jákl P, Brzobohatý O, Zemánek P. Optical forces induced behavior of a particle in a nondiffracting vortex beam. Opt Express 2012; 20(22): 24304-24319. DOI: 10.1364/OE.20.024304.
  28. Hamazaki J, Morita R, Chujo K, Kobayashi Y, Tanda S, Omatsu T. Optical-vortex laser ablation. Opt Express 2010; 18(3): 2144-2151. DOI: 10.1364/OE.18.002144.
  29. Syubaev S, Zhizhchenko A, Kuchmizhak A, Porfirev A, Pustovalov E, Vitrik O, Kulchin Yu, Khonina S, Kudryashov S. Direct laser printing of chiral plasmonic nanojets by vortex beams. Opt Express 2017; 25(9): 10214-10223. DOI: 10.1364/OE.25.010214.
  30. Vinogradova MB, Rudenko OV, Sukhorukov AP. Wave theory [In Russian]. 2nd ed. Moscow: “Nauka” Publisher; 1979.
  31. Khonina SN, Ustinov AV, Kovalyov AA, Volotovsky SG. Near-field propagation of vortex beams: models and computation algorithms. Opt Mem Neural Networks 2014; 23(2): 50-73. DOI: 10.3103/S1060992X14020027.
  32. Khonina SN, Volotovskij SG, Sojfer VA. A method of eigenvalue calculation of zero-order prolate spheroidal functions [In Russian]. Doklady Akademii Nauk 2001; 63(1): 30-33.

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