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Theory and Applications of Toroidal Moments in Electrodynamics Nanophotonics 2018; 7(1): 93–110 Review article Nahid Talebia,*, Surong Guoa and Peter A. van Aken Theory and applications of toroidal moments in electrodynamics: their emergence, characteristics, and technological relevance https://doi.org/10.1515/nanoph-2017-0017 Received January 30, 2017; revised June 21, 2017; accepted July 9, 1 Introduction 2017 Although the history of Maxwell equations and light- Abstract: Dipole selection rules underpin much of our matter interaction started as early as 1861 [1], electromag- understanding in characterization of matter and its inter- netism has been the field of most challenging and rival action with external radiation. However, there are sev- concepts such as understanding the actual velocity of eral examples where these selection rules simply break the information transfer [2], the actual wave function of down, for which a more sophisticated knowledge of mat- photons [3–5], realization of cloaking, negative refraction, ter becomes necessary. An example, which is increasingly and transferring of the data beyond the diffraction limits becoming more fascinating, is macroscopic toroidization using plasmons and metamaterials [6–11], and the emer- (density of toroidal dipoles), which is a direct consequence gence of the Abraham-Minkowski controversy and the of retardation. In fact, dissimilar to the classical family of actual linear momentum of light [12, 13]. At the heart of electric and magnetic multipoles, which are outcomes our understanding of light-matter interaction, there exist of the Taylor expansion of the electromagnetic poten- multipole-expansion sets that tell us how to construct the tials and sources, toroidal dipoles are obtained by the extended electromagnetic sources according to the local- decomposition of the moment tensors. This review aims ized sources with well-known electromagnetic field and to discuss the fundamental and practical aspects of the radiation patterns. This concept, for example, forms the toroidal multipolar moments in electrodynamics, from its basics to understand the quasi-static dipole and quadru- emergence in the expansion set and the electromagnetic pole localized plasmon resonances in individual [14] or field associated with it, the unique characteristics of their chains of metallic nanoparticles arranged in the forms interaction with external radiations and other moments, of converter-coupler gratings [15], waveguides [16–18], or to the recent attempts to realize pronounced toroidal reso- resonators [19, 20] (the so-called longitudinal and trans- nances in smart configurations of meta-molecules. Toroi- verse resonances). The coefficients of such an expansion dal moments not only exhibit unique features in theory set known as moments were conventionally ordered as but also have promising technologically relevant appli- appearing in the expansion set according to the electric cations, such as data storage, electromagnetic-induced and magnetic multipole terms. transparency, unique magnetic responses and dichroism. In 1958, however, Zeldovich [21] reported on the possibility of elementary particles under breakdown of Keywords: toroidal moment; multipole expansion; radia- spatial parity to interact with the electromagnetic wave tion pattern; metamaterial; spin; coupling. in a peculiar form considering the interaction energy H int =−SH⋅∇× , where S is the spin and H is the mag- netic field. Considering this problem in classical elec- trodynamics, in 1967, Dubovik noticed the possibility of introducing a new class of moments being excluded from aNahid Talebi and Surong Guo: These authors contributed similarly the family of electric and magnetic moments, with differ- to this work. ent time-space symmetries but appearing in similar orders Max Planck Institute for Solid *Corresponding author: Nahid Talebi, in the expansion set to the magnetic moments [22], the so- State Research, Heisenbergstr. 1, 70569 Stuttgart, Germany, e-mail: [email protected] called toroidal moments. From those early days, discus- Surong Guo and Peter A. van Aken: Max Planck Institute for Solid sions about toroidal moment have created an impetus in State Research, Heisenbergstr. 1, 70569 Stuttgart, Germany both solid-state physics and electrodynamics. Open Access. © 2017 Nahid Talebi et al., published by De Gruyter. This work is licensed under the Creative Commons Attribution- NonCommercial-NoDerivatives 4.0 License. 94 N. Talebi et al.: Theory and applications of toroidal moments in electrodynamics What is so interesting about toroidal moments? This based on the expansion of either potentials [29–31] or question is partly entangled with the human curiosity to fields [32, 33], has several applications in classical electro- find new states of matter and ordering and partly related dynamics, such as finding solutions to inverse problems to the technological applications. Toroidal ordering in [34–36], reconstruction of images [37], and in general, solid states can open up possibilities for a new kind of decomposition of induced charges into a set of localized magnetoelectric phenomenon [23], with applications in charge distributions with well-known near-field and far- data storage and sensing. Whereas toroidal moments have field characteristics. Moreover, semianalytical methods been reviewed in Ref. [23], with a focus on static spin- based on multipole expansion sets of magnetic and elec- based toroidal moments and magnetoelectric effect (ME) tric vector potentials, such as the method of auxiliary in condensed-matter physics, we will put the emphasis of sources [38–40], the multiple multipole method [41–43], our review on different expansion sets leading to the toroi- or the generalized multipole technique [11, 18, 44–49], are dal moment and on electrodynamic toroidal moments in exponentially convergent efficient methods in numerical metamaterials and free space. Ref. [24] has provided a electromagnetics [50]. In the following, we consider the concise review in this aspect. Our review is intended to family of multipole expansions for static and dynamic provide a more complete answer to this question and to systems, considering the Taylor expansions of sources, invoke new directions in light-matter interactions involv- potentials and fields. ing toroidal moments. For this purpose, we start with the theoretical investigation of the multipole expansion sets in Section 2 leading to toroidal moments and time-space 2.1 Multipole expansions for potentials symmetry characteristics of moments, respectively. We and fields furthermore discuss in Section 3 the toroidal moments in solid-state and photonic systems, the origin of asymmetric A direct Taylor expansion of charge and current den- magnetoelectric tensors in materials, the electromagnetic sities proposed by Dubovik in 1990 already demon- interaction between external light and (meta-)materials strated the existence of a toroidal moment described as that sustain toroidal moments, and finally the electro- Tc=⋅(1/10)∫((rr Jr)2− rJ23()rd),r which originates from magnetic field and radiation pattern associated to iso- the currents flowing along the meridians of an infinitesi- lated toroidal moments. In Section 4, we further consider mal sphere as JT =∇×∇((× Trδ( ))). Jr()then continues the possibility of the excitation of materials with toroidal as the inner flow by extending from one pole to the other moments with different polarization states of light and pole of the sphere [22, 51] (see Figure 1A). Moreover, toroi- also with relativistic electrons. In Section 5, we explore the dal moment was shown to be a multipole order originated coupling of toroidal moments to other classes of moments from the transverse part of the current density J (,rt ), ⊥ like electric dipole to form radiation-free sources of elec- as the longitudinal current ((Jr , t)) is not independent tromagnetic fields and also discuss the characteristics of from the charge density, because of the charge conser- toroidal metamaterials. We continue the review to cover vation criterion ∇⋅Jr(, tr)(=−∂∂ρ , tt)/ and hence is the technological applications of toroidal moments and related to the multipole expansion of the scalar potential. finally provide a conclusion and an outlook. Here, the longitudinal current is associated with the irro- tational part of the current density vector field and points out from the charge density ρ(,rt ) toward a direction normal to any surface inclosing the charge distribution, 2 Families of multipoles whereas J⊥ (,rt ) is parallel to such a surface. Obviously, in electrodynamics a configuration of the current density distribution in the form of JT is topologically similar to the poloidal currents Mathematical expansion sets are essential for our under- flowing at the surface of a torus, hence leading to toroidal standing of the physical properties of systems and matter. magnetic fields and the so-called polar toroidal moment Not only such expansion sets help us to analytically find (see Figure 1B). Interestingly, the poloidal current distri- solutions to simple problems, but they also help us to bution leads to a diminishing of the magnetic quadrupole simplify or find approximate solutions to complicated moment and hence an increase in the detectability of toroi- systems. Among the so-called expansion sets that physi- dal moment [52]. As a simplification, a ring of either static cists routinely use are Rayleigh expansion sets [10, 25], [23, 53] or dynamic [54–57] magnetic moments is consid- Eigenmode expansion sets [11, 26, 27],
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