Electromagnetic toroidal excitations in matter and free space

Journal name:
Nature Materials
Volume:
15,
Pages:
263–271
Year published:
DOI:
doi:10.1038/nmat4563
Received
Accepted
Published online

Abstract

The toroidal dipole is a localized electromagnetic excitation, distinct from the magnetic and electric dipoles. While the electric dipole can be understood as a pair of opposite charges and the magnetic dipole as a current loop, the toroidal dipole corresponds to currents flowing on the surface of a torus. Toroidal dipoles provide physically significant contributions to the basic characteristics of matter including absorption, dispersion and optical activity. Toroidal excitations also exist in free space as spatially and temporally localized electromagnetic pulses propagating at the speed of light and interacting with matter. We review recent experimental observations of resonant toroidal dipole excitations in metamaterials and the discovery of anapoles, non-radiating charge-current configurations involving toroidal dipoles. While certain fundamental and practical aspects of toroidal electrodynamics remain open for the moment, we envision that exploitation of toroidal excitations can have important implications for the fields of photonics, sensing, energy and information.

At a glance

Figures

  1. Toroidal structures at different length scales.
    Figure 1: Toroidal structures at different length scales.

    Toroidal topology, encountered very often in both artificial and naturally occurring objects, provides an indication for the presence of spontaneous or induced toroidal moments. Top row, from left to right: solenoidal currents lead to a toroidal moment in the atomic nucleus6; quaternary structure of archaeon (S. solfataricus) Cas4 (ref. 84); red blood cells take a biconcave, torus-like shape. Bottom row from left to right: benzene (left), hexaphenylbenzene (centre) and a toroidal carbon cage consisting of 120 carbon atoms (right) are organic molecules with elements of toroidal symmetry; perovskite (BaTiO3) nanotori87 are examples of artificial toroidal structures. Figure reproduced from: top row, middle, ref. 84, American Chemical Society; top row, right, © PhonlamaiPhoto/iStock/Thinkstock; bottom row, left, cage image, ref. 9, APS; bottom row, right, ref. 87, APS.

  2. The 'multipole zoo'.
    Figure 2: The 'multipole zoo'.

    Electric multipoles represent charge configurations (far left column), whereas magnetic multipoles correspond to current sources (second column from left). The (magnetic) toroidal multipole family (second column from the right) corresponds to current distributions that cannot be represented by electric and magnetic multipoles. Same order members of each multipole family have identical power radiation patterns of corresponding oscillating multipoles (far right column). Electric and toroidal dipoles also have identical radiated field patterns as indicated by the same colour (red) arrows. Figure reproduced from ref. 50, APS.

  3. Toroidal metamaterials.
    Figure 3: Toroidal metamaterials.

    a, Artistic drawing of the metamaterial unit cell used for the first demonstration of a dynamic toroidal dipole absorption resonance51. b, A planar low-loss split-ring metamaterial on a dielectric substrate supports toroidal modes of excitation57. c, A scaled-down version of the metamaterial presented in a shows plasmonic toroidal response at optical wavelengths61. d, An optical toroidal metamaterial exploiting resonant plasmonic response62. e, A spoof plasmon structure supports a toroidal dipole excitation at oblique angles of incidence67. f, Plasmonic oligomers consisting of voids in metallic films exhibit toroidal response at visible wavelengths and can be excited by a free-electron beam68. g, Low-loss toroidal metamaterial consisting of dielectric cylinders70. h, Interference of induced electric and toroidal dipoles in a resonantly transparent metamaterial consisting of dumbbell-shaped apertures leads to a non-radiating configuration73. i, Near-field signature of toroidal dipole excitation in a dielectric nanoparticle48. Figure reproduced from: b, ref. 57, APS; d, ref. 62, AIP; e, ref. 67, OSA; f, ref. 68, American Chemical Society; i, ref. 48, NPG.

  4. Non-radiating configurations.
    Figure 4: Non-radiating configurations.

    Such configurations consist of a toroidal dipole, represented by a solenoid with oscillating poloidal currents, and an electric dipole, represented by a pair of opposite charges, oscillating on the same frequency as the currents. With appropriate phase difference and oscillation amplitudes, destructive interference takes place: the combined source does not radiate electromagnetic fields. However, the scalar (ϕ) and vector (A) potentials associated with radiation of these dipoles do not cancel, but instead propagate to the far field. Hence, a non-radiating configuration acts as a source of electromagnetic potentials (but not electromagnetic fields). The physical significance and detectability of these potentials are not established and are being actively discussed in the literature.

  5. Focused doughnut pulses.
    Figure 5: Focused doughnut pulses.

    Artistic representation of a TM focused doughnut pulse propagating from right to left. Here, the magnetic field (H) is azimuthally polarized and confined in a torus-shaped region, and the electric field (E) is winding along the meridians of the torus resulting in a longitudinal component at the centre of the pulse. Focused doughnut pulses have broad spectrums and are characterized by two parameters, q1, which represents an effective wavelength, and q2, which quantifies the focal depth and is analogous to the Rayleigh range of conventional beam optics. The projected cross-section demonstrates the confinement of the pulse energy in two adjacent toroidal regions, and the white arrow indicates the propagation direction.

References

  1. Raab, R. E. & de Lange, O. L. Multipole Theory in Electromagnetism (Oxford University Press, 2004).
  2. Dubovik, V. M., Tosunyan, L. A. & Tugushev, V. V. Axial toroidal moments in electrodynamics and solid-state physics. Zh. Eksp.Teor. Fiz. 90, 590605 (1986); (English translation in Sov. Phys. JETP 63, 344351).
  3. Dubovik, V. M. & Tugushev, V. V. Toroid moments in electrodynamics and solid-state physics. Phys. Rep. 187, 145202 (1990).
  4. Vrejoiu, C. Electromagnetic multipoles in Cartesian coordinates. J. Phys. A. Math. Gen. 35, 99119922 (2002).
  5. Gongora, A. T. & Ley-Koo, E. Complete electromagnetic multipole expansion including toroidal moments. Rev. Mex. Fís 52, 188197 (2006).
  6. Zel'Dovich, Ia. B. Electromagnetic interaction with parity violation. J. Exp. Theor. Phys. 33, 15311533 (1957).
  7. Flambaum, V. V. & Murray, D. W. Anapole moment and nucleon weak interactions. Phys. Rev. C 56, 16411644 (1997).
  8. Flambaum, V. V. & Khriplovich, I. B. P-odd nuclear forces — a source of parity violation in atoms. Zh. Eksp.Teor. Fiz 79, 16561663 (1980); (English translation in Sov. Phys. JETP 52, 835839).
  9. Ceulemans, A. & Chibotaru, L. F. Molecular anapole moments. Phys. Rev. Lett. 80, 18611864 (1998).
  10. Afanasiev, G. N. Simplest sources of electromagnetic fields as a tool for testing the reciprocity-like theorems. J. Phys. D. Appl. Phys. 34, 539 (2001).
  11. Afanasiev, G. N. & Dubovik, V. M. Some remarkable charge–current configurations. Phys. Part. Nuclei 29, 366391 (1998).
  12. Dubovik, V. M. & Cheshkov, A. A. Multipole expansion in classic and quantum field theory and radiation. Sov. J. Particles. Nucl. 5, 318337 (1974).
  13. Afanasiev, G. N. The electromagnetic field of solenoids with time-dependent currents. J. Phys. A. Math. Gen. 23, 57555764 (1990).
  14. Afanasiev, G. N. & Stepanovsky, Y. P. The electromagnetic field of elementary time-dependent toroidal sources. J. Phys. A. Math. Gen. 28, 45654580 (1995).
  15. Radescu, E. E. & Vlad, D. H. Angular momentum loss by a radiating toroidal dipole. Phys. Rev. E 57, 60306037 (1998).
  16. Radescu, E. E. & Vaman, G. Exact calculation of the angular momentum loss, recoil force, and radiation intensity for an arbitrary source in terms of electric, magnetic, and toroid multipoles. Phys. Rev. E 65, 046609 (2002).
  17. Spaldin, N. A., Fiebig, M. & Mostovoy, M. The toroidal moment in condensed-matter physics and its relation to the magnetoelectric effect. J. Phys. Condens. Matter 20, 434203 (2008).
  18. Kittel, C. Theory of the structure of ferromagnetic domains in films and small particles. Phys. Rev. 70, 965971 (1946).
  19. Zheludev, I. S., Perekalina, T. M., Smirnovskaya, E. M., Fonton, S. S. & Yarmukhamedov, Y. N. Magnetic properties of nickel-boracite iodide. JETP Lett. 20, 129130 (1974).
  20. Ginzburg, V. L., Gorbatsevich, A. A., Kopayev, Y. V. & Volkov, B. A. On the problem of superdiamagnetism. Solid State Commun. 50, 339343 (1984).
  21. Sannikov, D. G. & Zheludev, I. S. On the possibility of phase transitions with spontaneous toroidal moment formation in nickel boracites. Sov. Phys. Solid State 27, 826828 (1985).
  22. Crone J. C. & Chung, P. W. Modeling of Toroidal Ordering in Ferroelectric Nanodots (Army Research Laboratory, 2007).
  23. Dubovik, V. M. Material equations for electromagnetism with toroidal polarizations. Phys. Rev. E 61, 70877097 (2000).
  24. Ederer, C. & Spaldin, N. A. Towards a microscopic theory of toroidal moments in bulk periodic crystals. Phys. Rev. B 76, 214404 (2007).
  25. Van Aken, B. B., Rivera, J.-P., Schmid, H. & Fiebig, M. Observation of ferrotoroidic domains. Nature 449, 702705 (2007).
  26. Naumov, I. I., Bellaiche, L. & Fu, H. Unusual phase transitions in ferroelectric nanodisks and nanorods. Nature 432, 737740 (2004).
  27. Planes, A., Castán, T. & Saxena, A. Recent progress in the thermodynamics of ferrotoroidic materials. Multiferroic Mater. 1, 922 (2015).
  28. Khomskii, D. Classifying multiferroics: mechanisms and effects. Physics 2, 20 (2009).
  29. Pyatakov, A. P. & Zvezdin, A. K. Magnetoelectric and multiferroic media. Phys. Usp. 55, 557581 (2012).
  30. Tolstoi, N. A. & Spartakov, A. A. Aromagnetism: a new type of magnetism. JETP Lett. 52, 161164 (1990).
  31. Fedotov, V. A., Marinov, K., Boardman, A. D. & Zheludev, N. I. On the aromagnetism and anapole moment of anthracene nanocrystals. New J. Phys. 9, 95 (2007).
  32. Martsenyuk, M. A. & Martsenyuk, N. M. Origin of aromagnetism. JETP Lett. 53, 243246 (1991).
  33. Toledano, P., Khalyavin, D. D. & Chapon, L. C. Spontaneous toroidal moment and field-induced magnetotoroidic effects in Ba2CoGe2O7. Phys. Rev. B 84, 094421 (2011).
  34. Tokura, Y. Multiferroics — toward strong coupling between magnetization and polarization in a solid. J. Magn. Magn. Mater. 310, 11451150 (2007).
  35. Sannikov, D. G. Phenomenological theory of the magnetoelectric effect in some boracites. Zh. Eksp. Teor. Fiz. 111, 536546 (1997); (English translation in J. Exp. Theor. Phys. 84, 293299).
  36. Mettout, B., Tolédano, P. & Fiebig, M. Symmetry replication and toroidic effects in the multiferroic pyroxene NaFeSi2O6. Phys. Rev. B 81, 214417 (2010).
  37. Feng, H.-J. & Liu, F.-M. Ab initio prediction on ferrotoroidic and electronic properties of olivine Li4 MnFeCoNiP4O16. Chinese Phys. B 18, 24812486 (2009).
  38. Hayami, S., Kusunose, H. & Motome, Y. Toroidal order in metals without local inversion symmetry. Phys. Rev. B 90, 024432 (2014).
  39. Yamaguchi, Y. & Kimura, T. Magnetoelectric control of frozen state in a toroidal glass. Nature Commun. 4, 2063 (2013).
  40. Lin, S.-Y. et al. Coupling Dy3 triangles to maximize the toroidal moment. Angew. Chem. Int. Ed. 51, 1276712771 (2012).
  41. Berger, R. J. F. Prediction of a cyclic helical oligoacetylene showing anapolar ring currents in the magnetic field. Z. Naturforsch. B 67b, 11271131 (2012).
  42. Popov, Y. F. et al. Magnetoelectric effect and toroidal ordering in Ga2−xFexO3. Zh. Eksp. Teor. Fiz. 114, 263272 (1998); (English translation in J. Exp. Theor. Phys. 87, 146151).
  43. Ressouche, E. et al. Magnetoelectric MnPS3 as a candidate for ferrotoroidicity. Phys. Rev. B 82, 100408 (2010).
  44. Zimmermann, A. S., Meier, D. & Fiebig, M. Ferroic nature of magnetic toroidal order. Nature Commun. 5, 4796 (2014).
  45. Grahn, P., Shevchenko, A. & Kaivola, M. Electromagnetic multipole theory for optical nanomaterials. New J. Phys. 14, 093033 (2012).
  46. Arango, F. B. & Koenderink, A. F. Polarizability tensor retrieval for magnetic and plasmonic antenna design. New J. Phys. 15, 073023 (2013).
  47. Cho, K. Microscopic Expression of Chiral Susceptibilities. Metamaterials '2011: The Fifth International Congress on Advanced Electromagnetic Materials in Microwaves and Optics 672674 (2011); http://go.nature.com/PNQmvj
  48. Miroshnichenko, A. E. et al. Nonradiating anapole modes in dielectric nanoparticles. Nature Commun. 6, 8069 (2015).
  49. Zhang, X.-L., Wang, S. B., Lin, Z., Sun, H.-B. & Chan, C. T. Optical force on toroidal nanostructures: toroidal dipole versus renormalized electric dipole. Phys. Rev. A 92, 043804 (2015).
  50. Savinov, V., Fedotov, V. A. & Zheludev, N. I. Toroidal dipolar excitation and macroscopic electromagnetic properties of metamaterials. Phys. Rev. B 89, 205112 (2014).
  51. Kaelberer, T., Fedotov, V. A., Papasimakis, N., Tsai, D. P. & Zheludev, N. I. Toroidal dipolar response in a metamaterial. Science 330, 15101512 (2010).
  52. Aggarwal K. M. Keenan, F. P. Radiative rates for E1, E2, M1 and M2 transitions in Fe X. Astron. Astrophys. 427, 763767 (2004).
  53. Marinov, K., Boardman, A. D., Fedotov, V. A. & Zheludev, N. I. Toroidal metamaterial. New J. Phys. 9, 324 (2007).
  54. Papasimakis, N., Fedotov, V. A., Marinov, K. & Zheludev, N. I. Gyrotropy of a metamolecule: wire on a torus. Phys. Rev. Lett. 103, 093901 (2009).
  55. Dong, Z.-G., Ni, P., Zhu, J., Yin, X. & Zhang, X. Toroidal dipole response in a multifold double-ring metamaterial. Opt. Express 20, 1306513070 (2012).
  56. Ye, Q. W. et al. The magnetic toroidal dipole in steric metamaterial for permittivity sensor application. Phys. Scripta 88, 055002 (2013).
  57. Fan, Y., Wei, Z., Li, H., Chen, H. & Soukoulis, C. M. Low-loss and high-Q planar metamaterial with toroidal moment. Phys. Rev. B 87, 115417 (2013).
  58. Savinov, V., Delfanazari, K., Fedotov, V. A. & Zheludev, N. I. Planar superconducting toroidal metamaterial: a source for oscillating vector-potential? 2014 Conference on Lasers and Electro-Optics (CLEO) FTu1C.1 (2014).
  59. Ding, C. et al. Stable terahertz toroidal dipolar resonance in a planar metamaterial. Phys. Status Solidi 252, 13881393 (2015).
  60. Huang, Y. W. et al. Design of plasmonic toroidal metamaterials at optical frequencies. Opt. Express 20, 17601768 (2012).
  61. Wu, P. C. et al. Three-dimensional metamaterials: from split ring resonator to toroidal metamolecule. Proc. SPIE 9163 (2014).
  62. Dong, Z.-G. et al. Optical toroidal dipolar response by an asymmetric double-bar metamaterial. Appl. Phys. Lett. 101, 144105 (2012).
  63. Dong, Z.-G. et al. All-optical Hall effect by the dynamic toroidal moment in a cavity-based metamaterial. Phys. Rev. B 87, 245429 (2013).
  64. Zhang, Q., Xiao, J. J. & Wang, S. L. Optical characteristics associated with magnetic resonance in toroidal metamaterials of vertically coupled plasmonic nanodisks. J. Opt. Soc. Am. B 31, 11031108 (2014).
  65. Liu, W., Zhang, J. & Miroshnichenko, A. E. Toroidal dipole induced transparency in core-shell nanoparticles. Laser Photon. Rev. 9, 564570 (2015).
  66. Kim, S.-H. et al. Subwavelength localization and toroidal dipole moment of spoof surface plasmon polaritons. Phys. Rev. B 91, 035116 (2015).
  67. Li, J. et al. Excitation of plasmon toroidal mode at optical frequencies by angle-resolved reflection. Opt. Lett. 39, 66836686 (2014).
  68. Ögüt, B., Talebi, N., Vogelgesang, R., Sigle, W. & van Aken, P. A. Toroidal plasmonic eigenmodes in oligomer nanocavities for the visible. Nano Lett. 12, 52395244 (2012).
  69. Huang, Y. W. et al. Toroidal lasing spaser. Sci. Rep. 3, 1237 (2013).
  70. Basharin, A. A. et al. Dielectric metamaterials with toroidal dipolar response. Phys. Rev. X 5, 011036 (2015).
  71. Vinogradov, A. P. & Aivazyan, A. V. Scaling theory for homogenization of the Maxwell equations. Phys. Rev. E 60, 987993 (1999).
  72. Fernandez-Corbaton, I., Nanz, S. & Rockstuhl, C. On the dynamic toroidal multipoles. Preprint at http://arxiv.org/abs/1507.00755 (2015).
  73. Fedotov, V. A., Rogacheva, A. V., Savinov, V., Tsai, D. P. & Zheludev, N. I. Resonant transparency and non-trivial non-radiating excitations in toroidal metamaterials. Sci. Rep. 3, 2967 (2013).
  74. Liu, W., Zhang, J., Lei, B., Hu, H. & Miroshnichenko, A. E. Invisible nanowires with interfering electric and toroidal dipoles. Opt. Lett. 40, 22932296 (2015).
  75. Boardman, A. D. & Marinov, K., Zheludev, N. I. & Fedotov, V. A. Dispersion properties of nonradiating configurations: finite-difference time-domain modeling. Phys. Rev. E 72, 036603 (2005).
  76. Marengo, E. A. & Ziolkowski, R. W. Nonradiating sources, the Aharonov-Bohm effect, and the question of measurability of electromagnetic potentials. Radio Sci. 37, 1019 (2002).
  77. Zagoskin, A. M., Chipouline, A., Il'ichev, E., Johansson, J. R. & Nori, F. Toroidal qubits: naturally-decoupled quiet artificial atoms. Preprint at: http://arxiv.org/abs/1406.7678 (2014).
  78. Raybould, T. A. et al. Toroidal optical activity. Preprint at: http://arxiv.org/abs/1508.06192 (2015).
  79. Hellwarth, R. W. & Nouchi, P. Focused one-cycle electromagnetic pulses. Phys. Rev. E 54, 889 (1996).
  80. Zheludev, N. I., Fedotov, V., Papasimakis, N., Savinov, V. & Raybould, T. Propagating and localized toroidal excitations in free space and metamaterials. Proc. SPIE 9544 (2015).
  81. Raybould, T. A., Fedotov, V. A., Papasimakis, N., Youngs, I. J. & Zheludev, N. I. Focused electromagnetic doughnut pulses and their interaction with interfaces and nanostructures. Opt. Express 24, 31503161 (2016).
  82. Ziolkowski, R. W. Localized transmission of electromagnetic energy. Phys. Rev. A 39, 20052033 (1989).
  83. Ziolkowski, R. W. Properties of electromagnetic beams generated by ultra-wide bandwidth pulse-driven arrays. IEEE Trans. Antennas Propag. 40, 888905 (1992).
  84. Lemak, S. et al. Toroidal structure and DNA cleavage by the CRISPR-associated [4Fe-4S]-cluster containing Cas4 nuclease SSO0001 from Sulfolobus solfataricus. J. Am. Chem. Soc. 135, 1747617487 (2013).
  85. Watson, D. W., Jenkins, S. D., Ruostekoski, J., Fedotov, V. A. & Zheludev, N. I. Toroidal dipole excitations in metamolecules formed by interacting plasmonic nanorods. Preprint at: http://arxiv.org/abs/1510.05609 (2015).
  86. Bao, Y., Zhu, X. & Fang, Z. Plasmonic toroidal dipolar response under radially polarized excitation. Sci. Rep. 5, 11793 (2015).
  87. Thorner, G., Kiat, J.-M., Bogicevic, C. & Kornev, I. Axial hypertoroidal moment in a ferroelectric nanotorus: a way to switch local polarization. Phys. Rev. B 89, 220103 (2014).
  88. Savinov, V. Novel toroidal and superconducting metamaterials PhD thesis, Univ. Southampton (2014).
  89. Leroy, B. How to convert the equations of electromagnetism from Gaussian to SI units in less than no time. Am. J. Phys. 53, 589590 (1985).
  90. Bohren, C. F. & Huffman, D. R. Absorption and scattering of light by small particles. (Wiley, 1983).

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Affiliations

  1. Optoelectronics Research Centre & Centre for Photonic Metamaterials, University of Southampton, Highfield SO17 1BJ, UK

    • N. Papasimakis,
    • V. A. Fedotov,
    • V. Savinov,
    • T. A. Raybould &
    • N. I. Zheludev
  2. TPI and Centre for Disruptive Photonic Technologies, Nanyang Technological University, Singapore 637378, Singapore

    • N. I. Zheludev

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