Advertisement

Russian Physics Journal

, Volume 62, Issue 5, pp 794–799 | Cite as

Formulation of the Snell–Descartes Laws in Terms of Geometric Algebra

  • V. V. FisanovEmail author
Article

A formulation of the laws of reflection and refraction of optical rays by the interface of two isotropic media using the mathematical apparatus of geometric algebra is given. The cases of specular reflection, positive and negative refraction, and refraction in the presence of a metasurface are considered.

Keywords

isotropic media metamaterials forward and backward waves reflection refraction geometric algebra 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    E. R. Tkaczyk, Opt. Lett., 37, No. 5, 972–974 (2012).ADSCrossRefGoogle Scholar
  2. 2.
    G. A. Mozharov and A. A. Panov, Izv. Vyssh. Uchebn. Zaved. Geod. Aerial Photogr., No. 3, 112–114 (2013).Google Scholar
  3. 3.
    P. R. Bhattacharjee, Optik, 125, No. 16, 4257–4261 (2014).ADSCrossRefGoogle Scholar
  4. 4.
    V. V. Fisanov, Russ. Phys. J., 58, No. 8, 1074–1079 (2015).CrossRefGoogle Scholar
  5. 5.
    G. Martínez-Borreguero, Á. L. Pérez-Rodríguez, M. I. Suero-López, and F. L. Naranjo-Correa, Revista Brasileira de Ensino de Fisica, 40, No. 3, e3401-1–e3401-5 (2018).Google Scholar
  6. 6.
    P. R. Bhattacharjee, Optik, 172, 1187–1192 (2018).ADSCrossRefGoogle Scholar
  7. 7.
    V. V. Fisanov, Russ. Phys. J., 57, No. 5, 691–696 (2014).MathSciNetCrossRefGoogle Scholar
  8. 8.
    D. Smith, M. Campbell, and A. Bergmann, Ann. Phys., 361, 184–189 (2015).CrossRefGoogle Scholar
  9. 9.
    V. V. Fisanov, Russ. Phys. J., 59, No. 8, 1181–1186 (2016).CrossRefGoogle Scholar
  10. 10.
    S. Kornblit, Microwave Optics [in Russian], Svyaz’, Moscow (1980).Google Scholar
  11. 11.
    J. Hanlon and H. Ziock, Using geometric algebra to understand pattern rotations in multiple mirror optical systems, Los Alamos National Lab., Report LA-UR-97-721 (1997).Google Scholar
  12. 12.
    Q. M. Sugon, Jr. and D. J. McNamara, Am. J. Phys., 72, No. 1, 92–97 (2004).ADSCrossRefGoogle Scholar
  13. 13.
    H. Zhang, C. Zhu, Q. Peng, and J. X. Chen, Comput. Sci. Eng., 8, No. 3, 68–75 (2006).CrossRefGoogle Scholar
  14. 14.
    D. Hestenes, Am. J. Phys., 71, No. 2, 104–121 (2003).ADSCrossRefGoogle Scholar
  15. 15.
    D. Hestenes, Adv. Appl. Clifford Algebras, 27, No. 1, 351–379 (2017).MathSciNetCrossRefGoogle Scholar
  16. 16.
    L. Dorst and S. Mann, IEEE Computer Graphics and Applications, 22, No. 3, 24–31 (2002).CrossRefGoogle Scholar
  17. 17.
    S. Mann and L. Dorst, IEEE Computer Graphics and Applications, 22, No. 4, 58–67 (2002).CrossRefGoogle Scholar
  18. 18.
    D. Fontijne and L. Dorst, IEEE Computer Graphics and Applications, 23, No. 2, 68–78 (2003).CrossRefGoogle Scholar
  19. 19.
    J. M. Chappell, S. P. Drake, C. L. Seidel, et al., Proc. IEEE, 102, No. 9, 1340–1363 (2014).CrossRefGoogle Scholar
  20. 20.
    K. J. Sangston, IEEE Aero. El. Sys. Mag., 31, No. 3, 32–69 (2016).CrossRefGoogle Scholar
  21. 21.
    J. M. Chappell, A. Iqbal, J. G. Hartnett, and D. Abbott, IEEE Access, 4, 1997–2004 (2016).CrossRefGoogle Scholar
  22. 22.
    A. Arsenovic, IEEE Access, 5, 19920–19941 (2017).CrossRefGoogle Scholar
  23. 23.
    G. Kazanova, From Clifford Algebra to the Hydrogen Atom [in Russian], Platon, Volgograd (1997).Google Scholar
  24. 24.
    N. Igosheva and A. Suromyasov, Mordovia Univ. Bull., 13, No. 3–4, 128–134 (2003).Google Scholar
  25. 25.
    V. I. Tarkhanov, Geometric Algebra – The Language of Creative Thought [in Russian], Sankt Peterburg (2004); http://plotnikovna. narod.ru/ga.pdf; https://docplayer.ru/39145998-Geometricheskaya-algebra-yazyktvorcheskogo-myshleniya.html.
  26. 26.
    O. A. Mornev, Hypercomplex Numbers in Geometry and Physics, 6, No. 2(12), 92–137 (2009).Google Scholar
  27. 27.
    N. Yu, P. Genevet, M. A. Kats, et al., Science, 334, No. 6054, 333–337 (2011).ADSCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.V. D. Kuznetsov Siberian Physical-Technical Institute at Tomsk State UniversityTomskRussia
  2. 2.National Research Tomsk State UniversityTomskRussia
  3. 3.Institute of Physical Material Science of the Siberian Branch of the Russian Academy of SciencesUlan-UdeRussia

Personalised recommendations