Annales Henri Poincaré

, Volume 18, Issue 5, pp 1789–1831 | Cite as

Derivation of Ray Optics Equations in Photonic Crystals via a Semiclassical Limit

Article

Abstract

In this work, we present a novel approach to the ray optics limit: we rewrite the dynamical Maxwell equations in Schrödinger form and prove Egorov-type theorems, a robust semiclassical technique. We implement this scheme for periodic light conductors, photonic crystals, thereby making the quantum-light analogy between semiclassics for the Bloch electron and ray optics in photonic crystals rigorous. One major conceptual difference between the two theories, though, is that electromagnetic fields are real, and hence, we need to add one step in the derivation to reduce it to a single-band problem. Our main results, Theorem 3.7 and Corollary 3.9, give a ray optics limit for quadratic observables and, among others, apply to local averages of energy density, the Poynting vector and the Maxwell stress tensor. Ours is the first rigorous derivation of ray optics equations which include all subleading-order terms, some of which are also new to the physics literature. The ray optics limit we prove applies to photonic crystals of any topological class.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Allaire, G., Palombaro, M., Rauch, J.: Diffraction of Bloch wave packets for Maxwell’s equations. Commun. Contemp. Math. 15, 1–36 (2013)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bellissard, J., Rammal, R.: An algebraic semi-classical approach to Bloch electrons in a magnetic field. J. Phys. Fr. 51, 1803–1830 (1990)CrossRefGoogle Scholar
  3. 3.
    Bergmann, E.E.: Electromagnetic propagation in homogeneous media with hermitian permeability and permittivity. Bell Syst. Tech. J. 61, 935–948 (1982)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bliokh, K.Y., Bekshaev, A.Y., Nori, F.: Dual electromagnetism: helicity, spin, momentum and angular momentum. New J. Phys. 15, 033026 (2013)ADSCrossRefGoogle Scholar
  5. 5.
    Bliokh, K.Y., Bliokh, Y.P.: Topological spin transport of photons: the optical Magnus effect and Berry phase. Phys. Lett. A 333, 181–186 (2004)ADSCrossRefMATHGoogle Scholar
  6. 6.
    Bliokh, K.Y., Kivshar, Y.S., Nori, F.: Magnetoelectric effects in local light-matter interactions. Phys. Rev. Lett. 113, 033601 (2014)ADSCrossRefGoogle Scholar
  7. 7.
    Bliokh, K.Y., Rodríguez-Fortuño, F.J., Nori, F., Zayats, A.V.: Spin–orbit interactions of light. Nat. Photonics 9, 796–808 (2015)ADSCrossRefGoogle Scholar
  8. 8.
    De Nittis, G., Lein, M.: Applications of magnetic \(\Psi \)DO techniques to SAPT—beyond a simple review. Rev. Math. Phys. 23, 233–260 (2011)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    De Nittis, G., Lein, M.: Effective light dynamics in perturbed photonic crystals. Commun. Math. Phys. 332, 221–260 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    De Nittis, G., Lein, M.: On the role of symmetries in photonic crystals. Ann. Phys. 350, 568–587 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    De Nittis, G., Lein, M.: The perturbed Maxwell operator as pseudodifferential operator. Doc. Math. 19, 63–101 (2014)MathSciNetMATHGoogle Scholar
  12. 12.
    De Nittis, G., Lein, M.: On the role of symmetries and topology in classical electromagnetism (in preparation) (2016)Google Scholar
  13. 13.
    Esposito, L., Gerace, D.: Topological aspects in the photonic crystal analog of single-particle transport in quantum Hall systems. Phys. Rev. A 88, 013853 (2013)ADSCrossRefGoogle Scholar
  14. 14.
    Figotin, A., Vitebskiy, I.: Frozen light in photonic crystals with degenerate band edge. Phys. Rev. E 74, 066613 (2006)ADSMathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Gat, O., Lein, M., Teufel, S.: Semiclassics for particles with spin via a Wigner–Weyl-type calculus. Ann. Henri Poincaré 15, 1967–1991 (2014)ADSMathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Jackson, J.D.: Classical Electrodynamics. Wiley, Hoboken (1998)MATHGoogle Scholar
  17. 17.
    Kuchment, P., Levendorskiî, S.: On the structure of spectra of periodic elliptic operators. Trans. Am. Math. Soc. 354, 537–569 (2001)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Littlejohn, R.G., Flynn, W.G.: Geometric phases and the Bohr–Sommerfeld quantization of multicomponent wave fields. Phys. Rev. Lett. 66, 2839–2842 (1991)ADSMathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Longhi, S.: Quantum-optical analogies using photonic structures. Laser Photon. Rev. 3, 243–261 (2009)CrossRefGoogle Scholar
  20. 20.
    Morame, A.: The absolute continuity of the spectrum of Maxwell operator in periodic media. J. Math. Phys. 41, 7099–7108 (2000)ADSMathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Onoda, M., Murakami, S., Nagaosa, N.: Hall effect of light. Phys. Rev. Lett. 93, 083901 (2004)ADSCrossRefGoogle Scholar
  22. 22.
    Onoda, M., Murakami, S., Nagaosa, N.: Geometrical aspects in optical wave-packet dynamics. Phys. Rev. E 74, 066610 (2006)ADSMathSciNetCrossRefGoogle Scholar
  23. 23.
    Panati, G., Spohn, H., Teufel, S.: Effective dynamics for Bloch electrons: Peierls substitution and beyond. Commun. Math. Phys. 242, 547–578 (2003)ADSMathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Panati, G., Spohn, H., Teufel, S.: Space-adiabatic perturbation theory. Adv. Theor. Math. Phys. 7, 145–204 (2003)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Panati, G., Teufel, S.: Propagation of Wigner functions for the Schrödinger equation with a perturbed periodic potential. In: Blanchard, P., Dell’Antonio, G. (eds.) Multiscale Methods in Quantum Mechanics. Birkhäuser, Boston (2004)Google Scholar
  26. 26.
    Perlick, V.: Ray Optics, Fermat’s Principle, and Applications to General Relativity. Lecture Notes in Physics, vol. 61. Springer (2000)Google Scholar
  27. 27.
    Pfeifer, R.N.C., Nieminen, T.A., Heckenberg, N.R., Rubinsztein-Dunlop, H.: Colloquium: momentum of an electromagnetic wave in dielectric media. Rev. Mod. Phys. 79, 1197–1216 (2007)ADSCrossRefGoogle Scholar
  28. 28.
    Pozar, D.M.: Microwave Engineering. Wiley, Hoboken (1998)Google Scholar
  29. 29.
    Raghu, S., Haldane, F.D.M.: Analogs of quantum-Hall-effect edge states in photonic crystals. Phys. Rev. A 78, 033834 (2008)ADSCrossRefGoogle Scholar
  30. 30.
    Rangarajan, A., and Gurumoorthy, K. S.: A Schrödinger wave equation approach to the Eikonal equation: application to image analysis. In: D. Cremers, Y. Boykov, A. Blake, F. Schmidt (eds.) Energy Minimization Methods in Computer Vision and Pattern Recognition. Lecture Notes in Computer Science, vol. 5681, pp. 140–153. Springer, Berlin (2009)Google Scholar
  31. 31.
    Rechtsman, M.C., Zeuner, J.M., Plotnik, Y., Lumer, Y., Podolsky, D., Dreisow, F., Nolte, S., Segev, M., Szameit, A.: Photonic Floquet topological insulators. Nature 496, 196–200 (2013)ADSCrossRefGoogle Scholar
  32. 32.
    Robert, D.: Autour de l’Approximation Semi-Classique. Birkhäuser, Basel (1987)MATHGoogle Scholar
  33. 33.
    Siddiqi, K., Tannenbaum, A., Zucker, S. W.: A Hamiltonian approach to the Eikonal equation. In: E. Hancock, M. Pelillo (eds.) Energy Minimization Methods in Computer Vision and Pattern Recognition. Lecture Notes in Computer Science, vol. 1654, pp. 1–13. Springer (1999)Google Scholar
  34. 34.
    Someda, C.G.: Electromagnetic Waves. CRC Press Inc, Boca Raton (1998)Google Scholar
  35. 35.
    Stiepan, H.-M., Teufel, S.: Semiclassical approximations for Hamiltonians with operator-valued symbols. Commun. Math. Phys. 320, 821–849 (2013)ADSMathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Sundaram, G., Niu, Q.: Wave-packet dynamics in slowly perturbed crystals: gradient corrections and Berry-phase effects. Phys. Rev. B 59, 14915–14925 (1999)ADSCrossRefGoogle Scholar
  37. 37.
    Suslina, T.: Absolute continuity of the spectrum of periodic operators of mathematical physics. Journées Équations aux dérivées partielles 1–13, 2000 (2000)MathSciNetMATHGoogle Scholar
  38. 38.
    Teufel, S.: Adiabatic Perturbation Theory in Quantum Dynamics. Lecture Notes in Mathematics, vol. 1821. Springer (2003)Google Scholar
  39. 39.
    Wang, Z., Chong, Y.D., Joannopoulos, J.D., Soljačić, M.: Reflection-free one-way edge modes in a gyromagnetic photonic crystal. Phys. Rev. Lett. 100, 013905 (2008)ADSCrossRefGoogle Scholar
  40. 40.
    Wang, Z., Chong, Y.D., Joannopoulos, J.D., Soljačić, M.: Observation of unidirectional backscattering-immune topological electromagnetic states. Nature 461, 772–775 (2009)ADSCrossRefGoogle Scholar

Copyright information

© Springer International Publishing 2017

Authors and Affiliations

  1. 1.Facultad de MatemáticasPontificia Universidad Católica de ChileSantiagoChile
  2. 2.Advanced Institute of Materials ResearchTohoku UniversitySendaiJapan

Personalised recommendations