TOROIDAL RESPONSE IN DIELECTRIC METAMATERIALS Ε. Ν. Εconomou IESL, FORTH, Dept of Physics, U of Crete Spetses, June 4, 2015

WORK BY A. A. Basharin 1,2, M. Kafesaki 1,3, E. N. Economou 1,4, C. M WORK BY A. A. Basharin 1,2, M. Kafesaki 1,3, E. N. Economou 1,4, C. M. Soukoulis 1,5, V. A. Fedotov 6, V. Savinov 6, and N. I. Zheludev 6,7 1Institute of Electronic Structure and Laser (IESL), Foundation for Research and Technology Hellas (FORTH), P.O. Box 1385, 71110 Heraklion, Crete, Greece 2National Research University “Moscow Power Engineering Institute,” 112250 Moscow, Russia 3Department of Materials Science and Technology, University of Crete, 71003 Heraklion, Greece 4Department of Physics, University of Crete, 71003 Heraklion, Greece 5Ames Laboratory-USDOE and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA 6Optoelectronics Research Centre and Centre for Photonic Metamaterials, University of Southampton, Highfield, Southampton SO17 1BJ, United Kingdom 7Centre for Disruptive Photonic Technologies, Nanyang Technological University, Singapore 637371, Singapore

OUTLINE Toroidal dipole/polypoles Outline of derivation Metallic MMs with toroidal dipole Polar dielectric MMs with toroidal dipole Resonances, results and comments

In books of EM the radiation energy per unit time, J, is given in terms of electric and magnetic polypoles: The dipoles P M The quadropoles

There is though a third type of polypoles: The toroidal Toroidal dipole Toroidal dipole: There is one more term of O(1/c5)

Unusual electromagnetic phenomena High Q-factor Optical activity & circular dichroism Negative refraction and backward waves Generation of waves with gauge irreducible vector potentials with no EM fields Aharonov-Bohm Effect The Radiation pattern P=T

Toroidal static dipole in nature Why toroidal dipole? Correct characterization of toroidal objects Interaction with electromagnetic fields Sensitive sensors of toroidal objects Fundamental interest Y. B. Zel'dovich, 1958, Parity I. Naumov, at al., 2004, f-e-n M. Kläui at al., 2003, f-m-r Y. F. Popov at al., 1998, m-e Y. V. Kopaev at al., 2009, t-Cr L. Ungur at al., 2012, t in mo A. Ceulemans at al., 1998, )) A. Karsisiotis at al., 2013 For potentials without fields and Aharonov-Bohm see G. N. Afanasiev and Yu. P. Stepanovsky, J. Phys. A: Math. Gen. 28 (1995) 4565-4580

Wave Equation:

The charge density ρ(r,t) and the current density j(r,t) can be expanded in terms of a complete orthonormal scalar and vector sets1: 1 See E. E. Radescu and G. Vaman, Phys. Rev. E 65, 046609 (2002) and their comments

T. Kaelberer et al, Science 330, 1510 (2010) Yuancheng Fan, et al. , Phys. Rev, B 87, 115417 (2013)

A. A. Basharin et al. Phys. Rev. X 5, 011036 (2015)

Polaritonic (phonon-polariton) materials Polar crystals (e.g. NaCl) in which incident radiation excites crystal vibrations (acoustic waves) - resonant in the THz regime + + - - - E k Phonon-polariton modes? Electromagnetic waves coupled to transverse crystal vibrations (phonons)

Easily obtained by eutectics self-organization approach Polaritonic systems Alkali-halide (2D periodic) systems of rods in a host , e.g. LiF rods in KCl (6%LiF) in NaCl (25% LiF) μm scale systems (lattice const from ~2 to 30 μm) Easily obtained by eutectics self-organization approach Coronado et. al., Opt. Express 20, 14663 (2012) Basharin et. al., Phys. Rev. B 87, 155130 (2013) Massaouti et. al., Opt. Lett. 38, 1140 (2013)

Polaritonic material permittivity ωΤ : transverse bulk eigenfrequency ωL: longitudinal bulk eigenfrequency For LiTaO3 we have LiTaO3/LiNbO3- very promising candidates for toroidal dielectric metamaterials in THz due to high permittivity ε ~ 40 and low losses up to half the transverse eigenfrequency.

Magnetic moment due to Mie-resonance in high-index dielectric cylinder Ez k y x m j, Ez A. A. Basharin et al. Phys. Rev. X 5, 011036 (2015) Mie- resonance frequency of the single cylinder when at 2.18 THz

(a) (b) (c) Ez |H| j -2,3 2,3 0 10.4 -93.3 93.3 A. A. Basharin et al. Phys. Rev. X 5, 011036 (2015); f=1.89 THz

A. A. Basharin et al. Phys. Rev. X 5, 011036 (2015) Normalized power reflected by multipoles, a.u. A. A. Basharin et al. Phys. Rev. X 5, 011036 (2015)

Magnetic quadrapole, (a) (b) (c) Ez |H| j -4.4 4.4 0 21.1 -175.2 175.2 -4.4 4.4 0 21.1 -175.2 175.2 A. A. Basharin et al. Phys. Rev. X 5, 011036 (2015); f=1.95 THZ

Dependence on the angle of incidence A. A. Basharin et al. Phys. Rev. X 5, 011036 (2015); fres. tor. =white bar

Cylinders of finite length (a) (b) (c) Ez |H| j -2,2 2,2 0 8.7 -89.3 89.3 A. A. Basharin et al. Phys. Rev. X 5, 011036 (2015); at f=1.87 THz

TE polarization A. A. Basharin et al. Phys. Rev. X 5, 011036 (2015); f=1.95 THZ

CONCLUSIONS Resonance (approximate eigenmode) in a single cylinder for E//axis y x m This mode suppresses electric dipole and boosts the magnetic dipole

Four coupled modes One of them is toroidal

(a) (b) (c) Ez |H| j -2,3 2,3 0 10.4 -93.3 93.3 A. A. Basharin et al. Phys. Rev. X 5, 011036 (2015); f=1.89 THz

Thank you for your attention