Electromagnetic toroidal excitations in matter and free space

Journal name:
Nature Materials
Year published:
Published online


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


  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.


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  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


All authors made equal contributions to the review and edited the text and figures.

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