1. Introduction

Structural arrangement determines the magnetic properties of matter. For example, ferro- and antiferromagnetic systems can be distinguished by fully parallel or antiparallel arrangements of elementary magnetic moments [1]. The magnetic properties of solids exhibit a rich frequency-dependent behavior. The magnetic permeability µ(ω) in naturally occurring materials approaches unity for frequencies higher than several Gigahertz. This is due to the fact that the electron spins in solids can no longer follow the rapidly alternating magnetic fields. Chemists have developed new classes of magnets based on molecules [2], where the rich variety of chemical synthesis allows for the study of fundamental mechanisms. Taking this concept further, nanotechnology enables the realization of nanomagnets, where the combination of magnetic materials and structural properties of the constituent nanomagnets results in ‘designer magnetism’ [3, 4]. Nevertheless, all these concepts so far cannot be carried over to optical frequencies, owing to the fact that usually materials with given spin properties are required. Also, nature restricts the possibilities of arranging elementary magnets in space at will.

In contrast, metamaterials offer the possibility to achieve artificial magnetism [5–7] at optical frequencies without using inherently magnetic materials. By applying plasmonic metallic nanostructures with specifically tailored geometries, metamaterial dipoles can be excited and arranged in three dimensions [8, 9]. Such metamaterial nanostructures include split-ring resonators (SRRs) and cut-wire pairs [10–12]. The incident light field can excite circulating or displacement currents in these structures, in analogy to atomic orbital currents, giving rise to local magnetic dipole moments, namely, artificial “spins”. In this paper, we demonstrate longitudinal and transverse magnetic coupling in three-dimensional metamaterials at optical frequencies. In particular, by proper arrangement, magnetic dipoledipole interaction can be isolated. More specifically, we show longitudinal magnetic interaction in a 90° twisted 4-layer SRR metamaterial and transverse magnetic interaction in a 5-layer fishnet metamaterial. Our concept of arranging magnetic dipoles in three dimensions according to a designer’s plan and controlling magnetic coupling with multiple degrees of flexibility will allow the experimental study of classical magnetism in different varieties [13].

2. Design and characterization of metamaterial structures

Figure. 1(a) shows the schematic of a 4-layer twisted SRR metamaterial. Each gold SRR is twisted by 90° with respect to its vertical neighbor(s). In order to theoretically explore the resonant behavior of the structure, numerical simulations were carried out using a Finite Integration Time Domain algorithm [14]. In simulation, the structure is surrounded by air with permittivity ε=1. For excitation of the twisted SRR metamaterial, we utilize normally incident light with its polarization along the x-direction. The simulated transmission spectrum is presented in Fig. 2(a), in which four resonances (ω _L1, ω _L2, ω _L3, and ω _L4) are clearly observable.

 

Fig.1. . Structure geometry and polarization configuration. (a) 4-layer twisted SRR metamaterial. l_x=l_y=400 nm, w=100 nm, t=60 nm, d=120 nm, and the lattice constants in both x and y directions are 1200 nm. (b) 5-layer fishnet metamaterial. l_x=l_y=350 nm, t=40 nm, d=60 nm, and the lattice constants in both x and y directions are 600 nm.

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In order to clarify the origin of the spectral characteristics and to understand the physics of the magnetic response, the magnetic field intensity in dependence of the incident frequency is detected by a H_z probe, which is positioned at the center of the uppermost SRR. As shown in Fig. 2(a), large enhancement of the localized magnetic field is achieved at the four resonances. To gain insight into the coupling mechanism, the magnetic field distributions of H_z at the corresponding resonances are shown in Fig. 3(a). Each resonance is associated with the excitation of circular currents, thus giving rise to magnetic dipole moments in the individual SRRs. This elucidates that each SRR carries a magnetic moment, i.e., a classical “spin” in this analogy. Interestingly, because the electric fields in the slit gaps of neighboring SRRs are perpendicular to one another, the electric dipole-dipole interaction equals zero. In addition to the fact that the higher-order multipolar interaction is negligible in a first approximation, the electric coupling in the twisted SRR structure can thus be ignored. Therefore, we need to consider only magnetic interaction. For the twisted SRR structure, the excited four magnetic dipoles are longitudinally coupled. The close proximity of these neighboring magnetic dipoles results in strong interaction between them, giving rise to the hybridization of the magnetic response and the formation of new hybridized modes [8, 9, 15–17]. The four magnetic dipoles tend to align parallel. As a result, the fully parallel arrangement of these magnetic dipoles corresponds to the lowest frequency resonance (ω _L1). The intermediate resonances ω _L2 and ω _L3 are in terms of incomplete antiparallel arrangements of these magnetic dipoles, whereas at resonance ω _L4 the four magnetic dipoles are aligned fully antiparallel. In fact, the resonance positions in the spectrum can be easily determined by counting the increasing number of nodes of the magnetic field as shown in Fig. 3(a). A diagram depicting the interaction between these longitudinally coupled magnetic dipoles and the resulting hybridized modes is shown in Fig. 3(a).

 

Fig. 2. Optical spectra and magnetic field intensities. (a) Simulated transmission spectrum and detected H_z intensities for the 4-layer twisted SRR metamaterial (longitudinal coupling). (b) Simulated transmission spectrum and detected H_y intensities for the 5-layer fishnet metamaterial (transverse coupling).

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Through side-by-side arrangement, these magnetic dipoles can also be coupled transversely. Figure. 1(b) presents the schematic of a 5-layer fishnet metamaterial. The gold fishnet layers are surrounded by air, and the wire widths in both directions are designed to be equal for polarization independence. As has been intensively studied for the 2-layer fishnet structures, the incident light field can induce a magnetic resonance, which is correlated with the excitation of a magnetic dipole moment arising from the anti-phase current oscillation in the two metal layers [18, 19]. Intuitively, with increasing number of fishnet layers, the resonant behavior of the stacked systems would become rather complex due to the interaction between neighboring layers [20–22]. To substantiate this expectation, numerical simulations were performed for the stacked fishnet metamaterial. The simulated transmission spectrum is shown in Fig. 2(b), in which four resonances are clearly evident (ω _T1, ω _T2, ω _T3, and ω _T4). Additionally, large enhancement of the localized magnetic field at the corresponding resonances is observed by detecting the amplitudes of the magnetic field H_y via a probe placed inside the gap between the top two gold wires. In order to highlight the physical mechanism of the magnetic responses, the magnetic field distributions at the respective resonances are illustrated in Fig. 3(b). Each resonance is associated with the excitation of magnetic dipole moments inside the four gaps between each gold wire pair. The resonant behavior of the stacked fishnet system can be understood by the hybridization of magnetic response as well. The incident light excites a current loop in each wire pair and thus results in a magnetic dipole moment. These magnetic dipoles are transversely coupled and leads to the formation of four new hybridized modes, which are associated with different symmetries. More specifically, at the lowest frequency resonance ω_T1, the four magnetic dipoles tend to align fully antiparallel. The intermediate resonances ω _T2 and ω _T3 are incomplete antiparallel arrangements of the magnetic dipoles, whereas at resonance ω _T4 the four magnetic dipoles are aligned fully parallel. A diagram depicting the interaction between these transversely coupled magnetic dipoles and the resulting hybridized modes is shown in Fig. 3(b).

 

Fig. 3. (a) Resonance diagram depicting the hybridization of the modes for the 4-layer twisted SRR metamaterial and the simulated magnetic field distributions of H_z at the corresponding resonance frequencies. (b) Resonance diagram depicting the hybridization of the modes for the 5-layer fishnet metamaterial and the simulated magnetic field distributions of H_y at the corresponding resonance frequencies. Red color denotes a positive H-field, and blue denotes a negative H-field. The “spin” symbols represent the magnetic dipole moments for the respective modes. The fully parallel and antiparallel alignments of “spins” correspond to the lowest frequency resonance in (a) and (b), respectively. The number of magnetic field nodes (dotted lines) is rising and declining in (a) and (b), respectively, for increasing resonance frequencies.

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Consequently, we are going to combine the concepts of longitudinal and transverse coupling, arranging metamaterial magnetic dipoles in three dimensions. In our design, the combination of longitudinal and transverse coupling is unique in the sense that it allows for the study of purely magnetic dipole-dipole interaction in a metasolid in the lateral as well as in the vertical direction (as far as solely nearest neighbor interaction is considered). Figure. 4(a) shows a planar unit cell of such an element, consisting of four SRRs with 90° rotation angle relative to their neighbors. Longitudinal coupling as complication to the two-dimensional prototype can be introduced by stacking the planar building blocks subsequently with each layer twisted by 90° compared to its former layer. The coupling strengths of the longitudinal as well as the transverse coupling can be controlled by altering the lateral and vertical separations of constituent SRRs. Figure. 4(b) illustrates the three-dimensional arrangement of these SRRs, which constitutes a magnetic metasolid. We might term it as “photonic spin crystal”.

 

Fig. 4. (a) A unit cell of a two-dimensional magnetic metamaterial. (b) A hypothetical three-dimensional magnetic (“spin”) metamaterial. Each layer consists of SRRs that are twisted by 90° compared to its vertical neighbor layers.

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3. Conclusion

In conclusion, we have investigated longitudinal and transverse magnetic coupling in three-dimensional optical metamaterials. Depending on end-to-end or side-by-side arrangement, the metamaterial magnetic dipoles can be longitudinally or transversely coupled, giving rise to fully parallel or fully antiparallel alignment of these artificial “spins” at the lowest frequency resonance. This would constitute superparamagnetism at optical frequencies. We should remark that by using Babinet’s principle [23, 24], our concepts can be applied to complimentary metamaterial structures with the polarization of the incident light rotated by 90°. Furthermore, our concepts can also be extended to more complex metamaterial systems, in which both longitudinal and transverse magnetic interactions are involved. In the future, the challenge lies ahead to understand the rather complicated spectra of such magnetic systems and to examine the applicability of solid-state physics concepts to these novel materials.