Magnetic effects are at the basis of several relevant microwave applications, e.g., imaging, computer memory modules, magneto-inductive waveguides and metamaterials. Commonly designed at low frequencies, purely natural magnetic molecules are not readily available in the visible, due to intrinsic natural limitations of optical materials. Here, using the anomalous wave interaction of electric-plasmonic nanoparticles, we consider a basic geometry that may constitute a lumped isotropic magneto-plasmonic “molecule” at optical frequencies, with applications for cloaking, imaging and optical communications.
©2009 Optical Society of America
Surface plasmons and plasmonic nanoresonances are at the basis of several exciting optical phenomena, from the sparkling colors of ancient stained glasses and art masterpieces [1-2] to recent applications for sub-wavelength imaging , sub-diffractive communications , optical lumped nanocircuits [5-6] and scattering-cancellation-based plasmonic cloaking . Plasmonic effects are provided by the collective resonant excitation of conduction electrons in metals , supported by the local optical electric field. Although the source-free Maxwell equations are elegantly symmetric in electric and magnetic fields, Nature lacks symmetry in charges, manifested in abundance of electric monopoles, but absence of a magnetic monopole. This is reflected in the fact that the analogous of electric-based plasmonic properties are not available under a magnetic excitation. This intriguing lack of symmetry of natural materials is associated with the low value of the fine-structure constant α = 1/137, which explains absence of magnetic monopoles in nature and the associated lack of magnetic effects in optical materials . Some renowned electromagnetic textbooks, e.g., , have even questioned the same meaning of magnetic permeability and the notion of magnetic polarization above the far infrared, as it was noticed in recent papers on the topic [10-11].
However, recent theoretical and experimental advancement in the physics and engineering of metamaterials have encouraged many researchers to look into the possibility of inducing magnetic effects in artificial optical materials. This has led to the synthesis of negative-index optical metamaterials, with many breakthrough applications in several fields of optics . Several approaches have been proposed in this direction, like metallic particles with the shape of split loop resonators [13-16], electrostatic resonances [17-18], metal nanoclusters [19-22], high permittivity inclusions [11, 23], pairs of plasmonic nanoparticles [24-26], plasmonic waveguides  and fishnet structures [28-29]. All these setups aim to provide a noticeable magnetic response to the optical wave, overcoming the intrinsic saturation problems  associated with the limited conducting properties of metals at optical frequencies. However, many of these geometries exhibit relatively noticeable levels of losses, strong anisotropy and polarization dependence, due to inherent asymmetries in their geometries, Moreover, their size is often required to be comparable to, or a good fraction of, the operating wavelength in order to provide a non-negligible response, and their magnetic response is strongly affected and influenced by an intrinsic quadrupolar contribution, which introduces additional radiation losses and impurity of radiation and polarization . The quest for an isotropic, polarization-independent magnetic-based plasmonic effect in the form of magnetic “lumped” molecule at optical frequencies is therefore still open  and its realization may indeed constitute an important milestone in the fields of nano-optics and plasmonics.
2. An optical magnetic molecule
It is known that a simple nanosphere with permittivity ε ≃ -2εb [Fig. 1(a)] embedded in a background of permittivity εb, exhibits a strong electric-based plasmonic resonance and it may be regarded as the fundamental optical electric resonator at the nanoscale, providing a strong electric response, isotropic and independent of the electric field polarization. Its resonant features, available even when the size of the nanosphere is much smaller than the operating wavelength, are associated with its plasmonic properties and they have been exploited since centuries in many applications [1-7]. When combined in arrays or lattices, its electric interaction may be tailored at will [30-34] for several interesting optical applications. Would it be possible to envision a dual scenario, i.e., a deeply subwavelength optical element that may somehow provide a magnetic-based plasmonic resonant response, isotropic and independent of the magnetic polarization?
In [10, 35] we have suggested that a circular ring of electric-based plasmonic nanoparticles may create a strong magnetic response to the impinging light. At its resonance, such nanoring has been shown to provide a strong magnetic polarization, while avoiding quadrupolar radiation and achieving less Ohmic and radiation losses  when compared to the other available techniques [12-29]. It was brought to our attention that an analogous idea to use a planar loop of plasmonic nanoparticles to produce optical magnetism was envisioned by Shalaev some time ago, but unpublished , although in his suggestion the loop was comparable in size with the wavelength of the impinging wave, whereas in [10, 35] and here the overall size of the magnetic molecule is much smaller than the operating wavelength -- a relevant point in using its magnetic response in an optical metamaterial. Still, this geometry, as well as the others available in the literature, is highly anisotropic and polarization-dependent.
Here, we consider and introduce the concept of a fully symmetric 3D isotropic spherical collection of six nanoparticles, effectively playing the role of a molecule consisting of the equivalent of six split-ring resonators, but operating at optical frequencies, each placed at the face of a cube , that may constitute the fundamental basic optical magnetic molecule, providing magnetic-based plasmonic effects at optical frequencies, i.e., the electromagnetic dual of an electric-based plasmonic nanosphere. An extension of our previous planar work to a 3D cluster of core-shell nanoparticles surrounding a dielectric sphere has also been recently envisioned by Simovski and Tretyakov in . Our geometry for magnetic plasmons proposed here may resemble such structure, however, it is smaller in scale and simpler to manufacture, without a need for the central dielectric core. Preliminary results for the optical wave interaction with our 3D isotropic molecule were first presented at two conferences [39-40].
Consider the geometry of Fig. 1(b), i.e., a properly arranged collection of six electric-plasmonic nanoparticles of permittivity ε , equally spaced, tightly packed (but not touching) with their centers on a mathematical spherical surface (or equivalently on the faces of a cube, effectively resembling six split-ring resonators  at optical frequencies). In the limit for which this structure in its entirety is much smaller than the wavelength of operation, its response is effectively isotropic and independent of the specific polarization and direction of the electromagnetic wave. Moreover, consistent with our findings in [10, 35], the loop of four spheres lying on the arbitrary plane orthogonal to the impinging magnetic field may support, under proper conditions, a resonant circulating electric displacement current, which supports a dominant magnetic response from the structure in its entirety. This is evident when we compare the resonant interaction with the local electric field in the field plots of Fig. 1, which show the electric field distributions on the E-plane for an individual silver nanosphere of radius R + a = 33.5 nm [Fig. 1(a)] and for a structure of the same size that is composed of six silver nanospheres, each of radius a = 13.5nm, placed symmetrically over a mathematical spherical surface of radius R = 20nm. Both geometries are embedded in a glass background (εb = 2.2 ε 0) and our numerical simulations consider realistic loss and the Drude frequency dispersion of the involved materials. The field plots report full-wave numerical simulations using commercial software at the electric-based plasmon [Fig. 1(a)] and magnetic-based plasmon [Fig. 1(b)] resonance for the two geometries, arising at λ 0=476nm and λ 0 = 502 nm, respectively.
In Fig. 1(a), the isolated electric-plasmonic nanosphere is dominated by a uniform electric field, providing a strong electric response; in Fig. 1(b), dually, the electric field rotates in the plane orthogonal to the impinging magnetic field, providing a resonant magnetic response and a negligible quadrupolar response . Indeed, the “molecule” of Fig. 1(b) may, by all means, be regarded as a lumped isotropic, polarization-independent magnetic element with a magnetic-based plasmonic resonance at optical frequencies.
It is interesting to note that each of the plasmonic nanoparticles composing the molecule of Fig. 1(b) individually interacts only with the local electric field, since they are composed of the plasmonic material that is non-responsive to magnetic excitation. However, the specific geometry in which they are arranged forces the electric field to circulate in the plane orthogonal to the impinging magnetic field, isotropically and independent of its specific orientation, inducing an overall magnetic resonance. In many senses, this nanoscale structure may be regarded as the optical analogue of six artificial magnetic inclusions at microwave frequencies, like the split-ring resonator [36, 41], whose geometry forces the conduction current to circulate in a similar fashion. Here at optical frequencies, however, the displacement current -iωεE, with E being the local electric field and ω the radian frequency of operation, takes the more dominant role and thus ensures a strong magnetic response from this structure. Moreover, the overall symmetry of this specific geometry ensures its response to be isotropic and polarization independent.
Figure 2 compares the magnitude of the normalized electric and magnetic polarizabilities for the two geometries of Fig. 1. These quantities reveal the capabilities of the “molecule” to induce an electric/magnetic dipole response to an electric/magnetic excitation. In this figure, their values are normalized to the maximum theoretically ideal values achievable by a lossless passive resonant particle. The individual silver nanoparticle, as expected, provides a relatively broad electric-based plasmonic resonant response centered at λ 0 = 476 nm, and a very weak magnetic response. In comparison, the molecule of Fig. 1(b) ensures a strong magnetic resonance at λ 0 = 502 nm, slightly red-shifted when compared to the individual electric resonance supported by each of the nanoparticles composing the molecule, consistent with our general findings in [10, 35].
In addition, the magnetic molecule also supports a resonant electric response at a shorter wavelength, due to the collective electric resonance of the six nanoparticles. The proper superposition of these two resonances, which relates to orthogonal field component with respect to each other and radiate in different planes of polarization, may pave the way to interesting applications for negative-index metamaterial technology in the visible, extending the findings of [10, 35] to 3D isotropic arrangements. In this regard, it is noticed that although here we are focusing on the electromagnetic response of this single collection of plasmonic nanoparticles, the use of this “pure” magnetic molecule in optical metamaterials may indeed be very promising. Our preliminary results in this matter show how such an artificial molecule, when used as the basic inclusion of an optical nanomaterial, may produce an effective negative permittivity and permeability similar to those found in [10, 35], but with purely isotropic response. We will report a detailed study of these properties in a future paper.
Returning to the individual molecule of Fig. 1(b), despite the presence of realistic material losses and frequency dispersion, Fig. 2 predicts that such molecule may indeed provide a strong magnetic response in the visible, supporting the interpretation of this nanostructure as a lumped isotropic magnetic molecule. Its bandwidth of operation, although clearly narrower than that of the electric resonance of Fig. 2(a), is still quite promising (it is noticed that here the geometries of Fig. 1(a) and 1(b) have been chosen to have the same overall radius, explaining the difference in operational bandwidth in the two geometries). It is worth noting that the relative robustness of this isotropic magnetic molecule, despite the small electrical size of each of its constituents, is provided by the strong coupling among the six nanoparticles composing the structure, consistent with the general bandwidth broadening and robustness to losses provided by closely spaced nanoresonators [33-34].
It should be mentioned that other geometries involving plasmonic nanoclusters have been proposed in the recent literature to induce artificial magnetism in the visible [17-22]. However, as we have shown in details in  the specific number of inclusions chosen here on each plane of polarization and the specific shape and geometry of the 6-particle collection ensures a “pure” magnetic response, with no residual unwanted quadrupolar radiation, and a more isotropic interaction with light ╌ properties that make the molecule of Fig. 1(b) especially interesting for applications in optical metamaterials.
Another relevant aspect to underline here refers to the special magnetic resonance that this molecule supports, a resonance that does not depend on the overall size of the molecule , but rather on the specific plasmonic properties of the individual nanoparticles composing the molecule, which in turn support the circulating displacement current giving rise to its overall magnetic response. This is a relevant and important exception to the general theory reported in , which predicted the possibility of artificial optical magnetism only for large values of Re[ε] and/or Im[ε] , limiting the range of realization of optical metamaterials. We notice instead that here the nanoparticles composing the cluster of Fig. 1b operate near their individual electric plasmonic resonance, for which their permittivity is close to -2εb. It is evident that in this case the resonant scattering required in  to achieve an overall magnetic response is achieved by means of a plasmonic effect, rather than a large index of refraction of the materials involved. The specific geometry and disposition of the molecule allows one to use this resonant response in order to create a magnetic effect, without relying on large permittivity materials. This geometry indeed opens up interesting possibilities for development of artificial optical magnetism.
3. Magnetic plasmons in the visible
It follows from the above discussion that envisioning an isotropic lumped magnetic molecule at optical frequencies may pave the way to several novel optical applications based on magnetism. In Fig. 3, following this analogy, we compare a guided electric-based plasmon wave [Fig. 3(a)] traveling along a chain of silver nanoparticles, each of the geometry of Fig. 1a, with a magnetic-based plasmon wave [Fig. 3(b)] traveling along a chain of our magnetic molecules of Fig. 1(b). This may be considered as the optical-plasmonic analog of magneto-inductive transmission-lines analyzed and realized in recent papers [42-45].
Both panels in Fig. 3 plot the electric field distribution (snapshot in time) on the E plane of polarization and the chains are embedded in a glass background. The drastic difference between the two scenarios is clearly seen: in Fig. 3(a) the nanoparticles are all polarized by a transverse electric field exciting a transverse plasmon wave that travels along the chain of particles, consistent with the results in [30-33]; in Fig. 3(b), instead, the magnetic field (normal to the figure) excites a rotating electric field on each molecule, that supports a magnetic plasmon wave traveling along the chain of magnetic molecules.
In Fig. 3(b), a residual vertical electric polarization of the molecules is also noticed, consistent with the results of Fig. 2(b), which predicted a non-negligible electric polarizability (in addition to the magnetic-based polarizability) at the magnetic resonance. Such electric response is not necessarily undesirable, since it may support negative-index propagation, as suggested in [10, 35]. To the best of our knowledge, this is the first deeply subwavelength structure for which a purely magnetic-based plasmon wave is envisioned in the visible domain.
The design of the lumped isotropic magnetic molecule that we have introduced here may lead to different possible applications of magnetism in optics, spanning imaging, cloaking, nanomaterials and optical communications. Future experimental realization of such geometry seems feasible and realistic. The quest for achieving a sensible, isotropic, polarization-independent, magnetic response from a lumped molecule at visible frequencies, which nature seems to have an aversion for, may soon reach a conclusion with the use of nanotechnology and plasmonic interaction of nanoparticles.
This work is supported in part by the U.S. Air Force Office of Scientific Research (AFOSR) grant number FA9550-08-1-0220.
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