"JOURNAL OF RADIO ELECTRONICS" (Zhurnal Radioelektroniki ISSN 1684-1719, N 12, 2018

contents of issue      DOI  10.30898/1684-1719.2018.12.11     full text in Russian (pdf)  

Theory and practical application of Riemann-Zilberstein vectors for the tasks of antenna technology

 

B. L. Kogan, I. V. Belkovich

National Research University Moscow Power Engineering Institute, Krasnokazarmennaya 14, Moscow 11250, Russia

 

The paper is received on October 5, 2018

 

Abstract. Exact vector solution of Maxwells equations describing an electromagnetic field in electrically large tasks as large antennas or beam-waveguides are always time-consuming, requiring substantial computational resources. There exist many asymptotic methods simplifying the problem but having significant assumptions and inaccuracies. In this paper, a theoretical foundation and solution of some practical problems utilizing unconventional electromagnetics vectors are presented. The vectors are the Riemann-Silberstein vectors (the RS vectors), which are a linear combination of the electric and magnetic field vectors. In homogeneous space, utilizing the RS vectors Maxwells Equations are converted into a system of two independent equations, with each vector describing the total electromagnetic field of an ideal circular polarization. One of the most effective methods of rigorous numerical and analytical solution of Maxwell's equations is multipole series expansion. The method of electromagnetic field expansion into a series of vector spherical harmonics of the Riemann-Silberstein vectors in helical coordinates is presented in the paper. Such expansion has a simpler mathematical representation compared to the traditional multipole expansion, it is symmetric and has a clear physical meaning. Solution of field scattering and diffraction problems of aperture antennas is described. The amount of computational work compared to the traditional method of TE and TM multipole expansion is reduced due to the initial independence of the Riemann-Silberstein vectors and the direct vector spherical expansion.

Key words: antennas, aperture antennas, computational electromagnetics, electromagnetic fields, electromagnetic radiation, spherical harmonics, spherical wave expansion.

References

1. L.Silberstein. Elektromagnetische Grundgleichungen in bivectorieller Behandlung (Basic electromagnetic equations in bivectorial form). Ann. d. Phys., 1907, Vol. 327, pp. 876880.

2. I. Bialynicki-Birula. Photon wave function. Progress in Optics, 1996, Vol.  36, pp. 245294

3. I. Bialynicki-Birula. The role of the Riemann-Silberstein vector in classical and quantum theories of electromagnetism. Journal of Physics A: Mathematical and Theoretical Physics, 2012, Vol. 46(15).

4. H. Weber. Die partiellen Differential-Gleichungen der mathematischen Physik nach Riemanns Vorlesungen bearbeitet von Heinrich Weber. Braunschweig: Friedrich Vieweg und Sohn, 1901, 348 p.

5. E. Beltrami. Considerazioni idrodinamiche. Rendiconti del reale Instituto Lombardo, Milano, 1889, t.XXII, p. 121-130.

6. I.S.Gromeka. Some cases of movement of incompressible fluid. Proceedings of Kazan University, 1881. See also Gromeka I.S. Collected Works. Moscow, Academy of Sciences Publishing, 1952, pp. 76-148 (In Russian)

7. Max von Laue. Die Relativitätstheorie. Zweiter Band : Die Allgemeine Relativitätstheorie Und Einsteins Lehre Von Der Schwerkraft, Friedr. Vieweg & Sohn, Braunschweig, 1921 and 1923.

8. Hermann Minkowski. Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern.  Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: 53111, 1908.

9. H. Bateman. The mathematical analysis of electrical and optical wave-motion. Cambridge: at the University Press, 1915.

10. L. Lewin. Theory of waveguides. London: Butterworth and Co Ltd., 1975.

11. Rumsey V.H. A New Way of Solving Maxwells Equations. IRE Transactions on Antennas and Propagation, Sept. 1961, 461-465

12. A. Lakhtakia. Beltrami Fields in Chiral Media. Singapore: River Edge, N.Y.: World Scientific, 1994.

13. A. Lakhtakia. Vector spherical wavefunctions for orthorhombic dielectric-magnetic material with gyrotropic-like magnetoelectric properties. Journal of Optics, 2012, Vol. 41, Issue 4, pp. 201-213

14. B. L. Kogan. Application of Maxwells equations in diagonal form to the investigation of polarization and diffraction properties of reflector antennas. Proceedings of international symposium on satellite communications and remote sensing, (SCRS99), Yandai, China, - October 26-29, 1999. - p. 113-114.

15. B. L. Kogan. Polarization Characteristic of Reflector Antennas Zhurnal Radioelektroniki Journal of Radio Electronics, 1999, No. 9, URL http://jre.cplire.ru/jre/sep99/2/text.html (In Russian)

16. B.L.Kogan. On sources of cross-polarized radiation. Journal of Communications Technology and Electronics, 2004, Vol. 49, No. 4, pp. 389-397.

17. B.L.Kogan. Electromagnetic fields of circular polarization, Moscow, MPEI, 2004. (In Russian)

18. B.L.Kogan. About vector spherical harmonics of circular polarization. Antenny Antennas, 2004, Vol. 81, No. 2, pp 59-63. (In Russian)

19. B. L. Kogan. Application of Faradays vectors in antenna theory. 1st Eur. Conf. Ant. and Propag. (EuCAP), Nice, France, Nov. 2006.

20. B.L.Kogan. Applications of Faraday's vectors to antenna theory... . . Zhurnal Radioelektroniki Journal of Radio Electronics, 2008, No. 7, URL:   http://jre.cplire.ru/mac/jul08/1/text.html

21. I. V. Belkovich, B. L. Kogan. Utilization of Riemann-Silberstein Vectors in Electromagnetics. Progress In Electromagnetics Research B, 2016, Vol. 69, p. 103116.

22. Igor V. Belkovich ;  Boris L. Kogan. The Riemann-Silberstein vectors theory and vector spherical expansion. Progress In Electromagnetics Research Symposium - Spring (PIERS) 2017, 22-25 May 2017, St. Petersburg, Russia

23. Igor V. Belkovich ;  Boris L. Kogan. Application of the Riemann-Silberstein vectors for the analysis of electromagnetic fields in reflector antennas. Radiation and Scattering of Electromagnetic Waves (RSEMW), 2017, 26-30 June 2017, Divnomorskoe, Russia.

24. Fresnel A. M´emoire sur la diffraction de la lumi`ere. In Œuvres compl`etes dAugustin Fresnel, Tome premier: Th´eorie de la Lumi`ere (eds De S´enarmont H, Verdet ´E, and Fresnel L), pp. 247382, Imprimerie imp´eriale, Paris, 1818.

25. Kirchhoff G. Mathematische Optik (Vorlesungen uber mathematische Physik, 2. Band), Teubner Verlag, Leipzig, 1891.

26. Rubinowitz A. Die Beugungswelle in der Kirchhoffschen Theorie der Beugungserscheinungen. Annalen der Physik. IV Folge, Band 53, Heft 12, 257-278.

27. F. Kottler. Electromagnetische theorie der Beugung an schwarzen Schirmen.  Ann. d. Phys. 70. S. 405. 1923.

28. Larmor J. On the mathematical expression of the principle of Huygens. Proc. London Math. Soc., 1, 113.

29. A. E. H. Love. The Integration of Equations of Propagation of Electric Waves. Phil Trans. Roy. Soc. London, Ser. A, 197, 1901, pp. 1-45.

30. W. V. Ignatowsky. Reflexion elektromagnetisches Wellen an einem Draht. Annalen der Physik, Vol. 323, Issue 13, pp.495-522.

31. H. M. Macdonald, Proc. Lond. Math. Soc., 1911, Vo. 2, No. 10, p. 91.

32. Schelkunoff S.A. Some equivalence theorems of electromagnetics and their application to radiation problems. Bell System Technical Journal, 1936, Vol. 15, pp. 92-112

33. Schelkunoff S.A. On diffraction and radiation of electromagnetic waves. Physical Review, 1939, Vol. 56, pp. 308-316.

34. Stratton J.A. and Chu L.J. Diffraction theory of electromagnetic waves. Physical Review, 1939, Vol. 56, pp. 99-107.

35. Stratton J.A. Electromagnetic Theory, McGraw-Hill, 1941.

36. J.D. Jackson. Classical Electrodynamics. John Wiley & Sons Ltd. 1962.

37. B.C. Brock. Using Vector Spherical Harmonics to Compute Antenna Mutual Impedance from Measured or Computed Fields. 2001 URL: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.457.8300&rep=rep1&type=pdf

38. Wood P.J. Spherical waves in antenna problems. Marconi Rev. 1971, Vol. 34, pp.149-172

39. Wood P.J. Reflector antenna analysis and design. P.Peregrinus Lmt. on behalf of the IEE, 1980. ( : . . .: , 1984. 207 .).

40. Gelfand I.M., Shapiro Z.Ya. Representations of the group of rotations of three-dimensional space and their applications. Uspekhi matematicheskikh nauk - Russian Mathematical Surveys, 1952. Vol. 7:1, No. 47, pp. 3-117. (In Russian)

41. Gelfand I.M., Minlos R.A., Shapiro Z.Ya. Predstavleniya gruppy vrascheniy I gruppy Lorentsa [Representations of a rotation group and a Lorentz group]. Moscow, Fizmatlit Publ., 1958. 368 p. (In Russian)

42. Vilenkin N.Ya. Spetsialnye funktsii i teoriya predstavleniy grupp [Special functions and the theory of group representations]. Moscow, Nauka Publ., 1965, 588 p. (In Russian)

43. D.A.Varshalovich, A.N.Moskalev, V.K.Khersonskiy. Kvantovaya teoriya uglovogo momenta [Quantum theory of angular momentum]. Leningrad, Nauka Publ., 1975 (In Russian)

44. A.S. Marathay and J.F. McCalmont. Vector diffraction theory for electromagnetic waves Journal of the Optical Society of America A, 2001,Vol. 18, Issue 10, pp. 2585-2593

45. Hansen J.E. Spherical near-field measurement. IEE Electromagnetics Wave Series, Peter Peregrinus Ltd., UK, 26 - 1988.

46. R. F. Harrington. On the gain and beamwidth of directional antennas. IRE Trans., 1958, AP-6, p. 219

 

For citation:
B. L. Kogan, I. V. Belkovich. Theory and practical application of Riemann-Zilberstein vectors for the tasks of antenna technology. Zhurnal Radioelektroniki - Journal of Radio Electronics. 2018. No. 12. Available at http://jre.cplire.ru/jre/dec18/11/text.pdf

DOI  10.30898/1684-1719.2018.12.11