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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 22, Iss. 7 — Apr. 7, 2014
  • pp: 8640–8653
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Magnetic dipole super-resonances and their impact on mechanical forces at optical frequencies

Iñigo Liberal, Iñigo Ederra, Ramón Gonzalo, and Richard W. Ziolkowski  »View Author Affiliations


Optics Express, Vol. 22, Issue 7, pp. 8640-8653 (2014)
http://dx.doi.org/10.1364/OE.22.008640


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Abstract

Artificial magnetism enables various transformative optical phenomena, including negative refraction, Fano resonances, and unconventional nanoantennas, beamshapers, polarization transformers and perfect absorbers, and enriches the collection of electromagnetic field control mechanisms at optical frequencies. We demonstrate that it is possible to excite a magnetic dipole super-resonance at optical frequencies by coating a silicon nanoparticle with a shell impregnated with active material. The resulting response is several orders of magnitude stronger than that generated by bare silicon nanoparticles and is comparable to electric dipole super-resonances excited in spaser-based nanolasers. Furthermore, this configuration enables an exceptional control over the optical forces exerted on the nanoparticle. It expedites huge pushing or pulling actions, as well as a total suppression of the force in both far-field and near-field scenarios. These effects empower advanced paradigms in electromagnetic manipulation and microscopy.

© 2014 Optical Society of America

1. Introduction

Because of the inherent weakness of the magnetic response of matter at optical frequencies [1

1. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).

, 2

2. R. Merlin, “Metamaterials and the Landau-Lifshitz permeability argument: large permittivity begets high-frequency magnetism,” Proc. Natl Acad. Sci. USA 106, 1693–1698 (2009). [CrossRef] [PubMed]

], there has been a great deal of excitement in the recent development of artificial magnetic properties based on metamaterial-inspired concepts [3

3. V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photon. 1, 41–48 (2007). [CrossRef]

6

6. A. Alù and N. Engheta, “The quest for magnetic plasmons at optical frequencies,” Opt. Express 17, 5723–5730 (2009). [CrossRef] [PubMed]

]. This ability to tailor magnetic, as well as electric optical responses has facilitated the pursuit of negative refraction [3

3. V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photon. 1, 41–48 (2007). [CrossRef]

], cloaking [7

7. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006). [CrossRef] [PubMed]

, 8

8. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]

] and perfect lensing [9

9. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000). [CrossRef] [PubMed]

]. It has also stimulated the current investigations of Kerker-inspired Huygen’s sources [10

10. M. Kerker, D. S. Wang, and C. L. Giles, “Electromagnetic scattering by magnetic spheres,” J. Opt. Soc. Am. A 73, 765–767 (1983). [CrossRef]

] at optical and infrared frequencies, aimed at the advancement of highly directive and efficient nanoantennas [11

11. Y. H. Fu, A. I. Kuznetsov, A. E. Miroshnichenko, Y. F. Yu, and B. Lukyanchuk, “Directional visible light scattering by silicon nanoparticles,” Nature Commun. 4, 1527 (2013). [CrossRef]

, 12

12. S. Person, M. Jain, Z. Lapin, J. J. Sáenz, G. Wicks, and L. Novotny, “Demonstration of zero optical backscattering from single nanoparticles,” Nano Lett. 13, 1806–1809 (2013). [PubMed]

] and the control of thermal emissions [13

13. S. D. Campbell and R. W. Ziolkowski, “Simultaneous excitation of electric and magnetic dipole modes in a resonant core-shell particle at infrared frequencies to achieve minimal backscattering,” J. Sel. Top. Quantum Electron. 19, 4700209 (2013). [CrossRef]

]. Magnetic-based Fano resonances have also attracted great interest; they feature huge field enhancements and sharp spectral features with consequent applications in the field of sensing [14

14. F. Shafiei, F. Monticone, K. Q. Le, X.-X. Liu, T. Hartsfield, A. Alù, and X. Li, “A subwavelength plasmonic metamolecule exhibiting magnetic-based optical Fano resonance,” Nature Nanotech. 8, 95–99 (2013). [CrossRef]

, 15

15. S. N. Sheikholeslami, A. García-Etxarri, and J. A. Dionne, “Controlling the interplay of electric and magnetic modes via Fano-like plasmon resonances,” Nano Lett. 11, 3927–3934 (2011). [CrossRef] [PubMed]

]. Moreover, a strong magnetic response is essential to the ultimate design of thin (one-particle thickness) beam-shapers [16

16. C. Pfeiffer and A. Grbic, “Metamaterial Huygens surfaces: tailoring wave fronts with reflectionless sheets,” Phys. Rev. Lett. 110, 7401–7405 (2013). [CrossRef]

], polarization transformers [17

17. T. Niemi, A. O. Karilainen, and S. A. Tretyakov, “Synthesis of polarization transformers,” IEEE Trans. Antennas Propag. 61, 3102–3111 (2013). [CrossRef]

] and perfect electromagnetic absorbers [18

18. Y. Ra’di, V. S. Asadchy, and S. A. Tretyakov, “Total absorption of electromagnetic waves in ultimately thin layers,” IEEE Trans. Antennas Propag. 61, 4606–4614 (2013). [CrossRef]

].

As predicted theoretically [19

19. A. García-Etxarri, R. Gómez-Medina, L. S. Froufe-Pérez, C. López, L. Chantada, F. Scheffold, J. Aizpurua, M. Nieto-Vesperinas, and J. J. Sáenz, “Strong magnetic response of submicron silicon particles in the infrared,” Opt. Express 19, 4815–4826 (2011). [CrossRef] [PubMed]

], the relatively high refractive index of silicon (Si) n ∼ 3.5 (see, e.g., [20

20. H. H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 9, 561 (1980). [CrossRef]

]) enables the isotropic and low-loss excitation of magnetic dipole modes at optical frequencies. The existence of these magnetic resonances has been experimentally verified recently [11

11. Y. H. Fu, A. I. Kuznetsov, A. E. Miroshnichenko, Y. F. Yu, and B. Lukyanchuk, “Directional visible light scattering by silicon nanoparticles,” Nature Commun. 4, 1527 (2013). [CrossRef]

, 14

14. F. Shafiei, F. Monticone, K. Q. Le, X.-X. Liu, T. Hartsfield, A. Alù, and X. Li, “A subwavelength plasmonic metamolecule exhibiting magnetic-based optical Fano resonance,” Nature Nanotech. 8, 95–99 (2013). [CrossRef]

, 21

21. A. I. Kuznetsov, A. E. Miroshnichenko, Y. H. Fu, J. Zhang, and B. Lukyanchuk, “Magnetic light,” Nature Sci. Reports 2, 492 (2012).

, 22

22. A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region,” Nano Lett. 12, 3749–3755 (2012). [CrossRef] [PubMed]

]. In this article, we demonstrate that it is possible to boost the magnetic dipole resonance response of a Si nanosphere by several orders of magnitude by coating it with a shell impregnated with an active material. Moreover, although the nanoparticle retains a dominant magnetic dipolar response, this enhancement is comparable to those achieved with spaser-based nano-lasers [23

23. J. A. Gordon and R. W. Ziolkowski, “The design and simulated performance of a coated nano-particle laser,” Opt. Express 15, 2622–2653 (2007). [CrossRef] [PubMed]

26

26. S. Arslanagić and R. W. Ziolkowski, “Active coated nano-particle excited by an arbitrarily located electric Hertzian dipoleresonance and transparency effects” J. Opt. 12, 024014 (2010). [CrossRef]

]. Thus, this configuration attains unprecedented levels of magnetic activity at optical frequencies.

2. Far-field analysis

Fig. 1 (a) Scattering efficiency spectrum Qscat, and the contributions to it from the electric and magnetic dipoles, Qscate and Qscatm, respectively, for the passive case, κ = 0. Inset: Sketch of the coated Si particle illuminated by a plane-wave, and the resulting dipolar excitations. (b) Qscat at the magnetic dipole resonance as a function of the imaginary part of the index of refraction, κ, as well as the mechanical force exerted on the nanoparticle, normalized to the incident electromagnetic power projected onto its physical area, Fnorm=F/(SRπa22). Insets: Top-right: Zero-force region. Bottom-right: Geometry.

Figure 1(a) shows the scattering efficiency Qscat, defined as the total scattering cross section normalized to its cross sectional area πa22 (see, e.g., [37

37. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 2008).

] p. 72), for the passive case, i.e., when κ = 0. As found for bare, passive Si nanospheres [22

22. A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region,” Nano Lett. 12, 3749–3755 (2012). [CrossRef] [PubMed]

], the Qscat behavior of the core-shell configuration is dominated by the superposition of the responses arising from the electric and magnetic dipolar resonances. Figure 1(b) depicts Qscat at the magnetic dipolar resonance as a function of the imaginary part of the shell refractive index, κ. In order to construct Fig. 1(b), a Qscat spectrum is calculated for each κ value as in Fig. 1(a). Next, its maximum is identified and included in Fig. 1(b). Therefore, each point in Fig. 1(b) corresponds to a slightly different wavelength. Nevertheless, all of these wavelengths are located in the immediate neighborhood of 725 nm. The figure demonstrates that, although Qscat monotonically increases for small κ values, there is a optimal value: κ ∼ 0.275, for which a super-resonant state of the coated nanoparticle is excited and Qscat is increased by several orders of magnitude. While this effect is somewhat analogous to super-resonant states excited in spaser-based nano-lasers [23

23. J. A. Gordon and R. W. Ziolkowski, “The design and simulated performance of a coated nano-particle laser,” Opt. Express 15, 2622–2653 (2007). [CrossRef] [PubMed]

26

26. S. Arslanagić and R. W. Ziolkowski, “Active coated nano-particle excited by an arbitrarily located electric Hertzian dipoleresonance and transparency effects” J. Opt. 12, 024014 (2010). [CrossRef]

], a collection of scattering directivity patterns, scattering efficiency spectra and field plots provided in Fig. 2 for different gain values demonstrates that the magnetic dipolar resonance is dominant at such a super-resonant state. Therefore, it can be concluded that the active coated Si nanosphere is able to provide an unprecedented magnetic response at optical frequencies. In theory, the scattering efficiency at the super-resonance is unbounded for this canonical model. In practice however, it will be limited by fabrication tolerances and the difficulty of dealing with an increasingly narrow bandwidth as the core-shell system begins to lase, as well as the saturation of the gain media and other non-linear effects. The extent of these effects will be confined to dampening the super-resonance within a small interval of κ values centered on the super-resonance, so that the net enhancement at the super-resonance will be limited to a few of orders of magnitude as a function of the fabrication tolerances and gain medium properties.

Fig. 2 First column: Scattering efficiency spectrum Qscat, and the contributions to it from the electric and magnetic dipoles, Qscate and Qscatm. Second column: Scattering directivity patterns, Dscat, in the XZ- and YZ-planes. Dscat = 4πr2 ( · Sscat/Pscat), where Sscat=12Re[Escat×(Hscat)*] represents the time-averaged Poynting vector field associated with the scattered field, and Pscat=SSscatr^dS is the time-averaged total scattered power. Third column: Colormap and quiver (arrow) plots of the electric field at the maximum of Qscat. Each row corresponds to a different gain value (a) κ = 0.1, (d) κ = 0.2, (c) κ = 0.275, and (d) κ = 0.365.

To emphasize the field mechanism that leads to this super-resonant state, the third column of Fig. 2 gives the combined colormap and quiver (arrow) plots of the electric field at the maximum of the Qscat spectrum. It can be concluded that as the particle is tuned close to the super-resonance (see Fig. 2(c)), its response is dominated by a strong circulating electric field, similar to that excited at the magnetic dipole resonance of a high-permittivity sphere [21

21. A. I. Kuznetsov, A. E. Miroshnichenko, Y. H. Fu, J. Zhang, and B. Lukyanchuk, “Magnetic light,” Nature Sci. Reports 2, 492 (2012).

]. The circulating field is concentrated in the outer part of the Si core and at its interface with the active shell. In this manner, and in analogy with spaser-based nano-lasers [23

23. J. A. Gordon and R. W. Ziolkowski, “The design and simulated performance of a coated nano-particle laser,” Opt. Express 15, 2622–2653 (2007). [CrossRef] [PubMed]

26

26. S. Arslanagić and R. W. Ziolkowski, “Active coated nano-particle excited by an arbitrarily located electric Hertzian dipoleresonance and transparency effects” J. Opt. 12, 024014 (2010). [CrossRef]

], the proposed configuration benefits from the concentration of the fields into its nanometer-sized core-shell form, which provides the feedback mechanism that leads to the super-resonance. This fact also enables the excitation of the super-resonance with realistic κ values. In particular, we note that commercially available [35] core-shell CdSe/ZnS are characterized by optical extinction coefficients ranging from 0.01 cm−1 to 5.9 cm−1. Following available models for QDs in the strong confinement regime [38

38. P. Holmström, L. Thylén, and A. Bratkovsky, “Dielectric function of quantum dots in the strong confinement regime,” J. Appl. Phys. 107, 4307–4313 (2010). [CrossRef]

], it can be found that such values correspond to κ values ranging from 0.01 to 3 [36

36. S. D. Campbell and R. W. Ziolkowski, “The performance of active coated nanoparticles based on quantum-dot gain media,” Adv. OptoElectronics 36, 8786–8791 (2012).

]. Even higher values κ ∼ 4 have been experimentally retrieved for PbS QDs [39

39. I. Moreels, D. Kruschke, P. Glas, and J. W. Tomm, “The dielectric function of PbS quantum dots in a glass matrix,” Opt. Mater. Express 2, 496–500 (2012). [CrossRef]

]. Therefore, it can be concluded that it is possible to excite a super-resonant state in coated Si nanospheres with realistic material parameters.

This active configuration also provides new and exciting opportunities in electromagnetic manipulation and microscopy. To illustrate this fact, Fig. 1(b) gives the mechanical force exerted on the particle when it is illuminated by a plane-wave of electric field magnitude E0 = 1V/m. The force exerted on the particle has been computed by using the analytical formalism introduced in [27

27. P. C. Chaumet and A. Rahmani, “Electromagnetic force and torque on magnetic and negative-index scatterers,” Opt. Express 17, 2224–2234 (2009). [CrossRef] [PubMed]

, 28

28. M. Nieto-Vesperinas, J. J. Sáenz, R. Gómez-Medina, and L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express 18, 428–443 (2010). [CrossRef]

], later summarized in this article as Eq. (1). Moreover, since the force is independent of its position for this plane-wave excitation, the particle is centered at the origin of the coordinates for the sake of simplicity. It is apparent from Fig. 1(b) that the mechanical force exerted on the particle is enhanced by several orders of magnitude at its super-resonant state. More strikingly, it can be positive (pushing) or negative (pulling). As suggested in [40

40. T. Kudo and H. Ishihara, “Proposed nonlinear resonance laser technique for manipulating nanoparticles,” Phys. Rev. Lett. 109, 7402–7406 (2012). [CrossRef]

, 41

41. A. Mizrahi and Y. Fainman, “Negative radiation pressure on gain medium structures,” Opt. Lett. 35, 3405–3407 (2010). [CrossRef] [PubMed]

], the latter dragging forces correspond to those cases in which the kinetic momentum carried by the incident plane-wave is increased by the stimulated emission being generated by the nanoparticle. Following these studies [40

40. T. Kudo and H. Ishihara, “Proposed nonlinear resonance laser technique for manipulating nanoparticles,” Phys. Rev. Lett. 109, 7402–7406 (2012). [CrossRef]

, 41

41. A. Mizrahi and Y. Fainman, “Negative radiation pressure on gain medium structures,” Opt. Lett. 35, 3405–3407 (2010). [CrossRef] [PubMed]

], here we assume a symmetric pumping orthogonal to the direction of propagation (see Fig. 1(b)), so that any force action in the nanoparticle mediated by the pumping can be neglected. However, the aforementioned acceleration and/or dragging effects enhanced by means of stimulated emission entail large scattered fields, possibly resulting in a large and undesired interaction with the environment. In this perspective, the proposed approach of a magnetic dipole super-resonance is advantageous, since the reactive scattered fields excited by the coated Si nanosphere are dominated by the magnetic rather than electric fields, and thus the coupling with the immediate environment is minimized with respect to electric resonances. Moreover, it can be concluded from Fig. 1(b) that the force is totally suppressed at κ ≃ 0.365. This fact allow us to achieve a high visibility nanoparticle: Qscat ≃ 48 at κ ≃ 0.365 (see Fig. 2(d)), with a zero-force effect. This configuration could be exploited to develop recoilless optical microscopy techniques. Furthermore, because the electric polarizability itself is not necessary small and is tunable, zero-force effects can be obtained with a certain degree of freedom in the scattering directivity pattern. For example, at the present zero-force configuration (κ = 0.365), the scattered power is mostly directed against the direction of propagation of the incident field, with a maximum directivity of Dscat ≃ 2.1 (see Fig. 2(d)).

3. Near-field analysis

Despite this fact, the symmetry between the contributions to the force associated with the electric and magnetic fields in Eq. (1) suggest that, in theory, it should be possible to suppress the force by opposing those contributions. To achieve this, not only must there be a proper balance between the electric and magnetic responses of the particle, but there also must be a certain degree of symmetry between the electric and magnetic fields illuminating it. In order to construct this equilibrium configuration, it is important to note that, taking advantage of the coated Si nanoparticle, it is possible to find designs in which αee=αmm/η02. This equality applies to both the real and imaginary parts of the polarizabilities, which can only be achieved naturally with an active particle that has a substantial magnetic response, e.g., the proposed active coated Si nanoparticle. Specifically, it has been found that the αee=αmm/η02 condition is satisfied for the particular geometry considered in this article when the active shell has κ = 0.568 at 708nm.

Fig. 3 (a) Sketch of the geometry: Coated Si particle illuminated by two aligned electric and magnetic dipoles with balanced magnitudes Iel = Iml/η0 = 10mA · nm. Scattering directivity patterns in the XZ and XY-planes when the nanoparticle is located on the Z-axis. (b) Mechanical force F and (c) scattered power Pscat spectra (normalized to the power radiated by the source in free-space P0) when the nanoparticle is positioned on the −Z-axis at the distances d = 2a2, d = 4a2 and d = 20a2 from the source location. Colormaps of the (d) force magnitude and (e) scattered power (in dB scale, normalized, respectively, to 1 pN and P0) as functions of the particle location on the XZ-plane at λ = 708nm. The grey areas indicate those locations which are not physically accessible to the nanoparticle.

Interestingly, both residual terms vanish along the z-axis (θ = 0, π). Therefore, a particle placed along this axis, as schematically depicted on Fig. 3(a), will feel no mechanical force. This effect is evidenced in Fig. 3(b), which illustrates the force exerted on the nanoparticle for different separation distances from the source, r = d. It can be concluded from Fig. 3(b) that the force is always zero at λ = 708nm, no matter what the separation distance from the source is. We believe that this effect can be effectively exploited in particle sorting. For example, small dielectric particles (e.g., those arising as impurities in nano-fabrication processes), are essentially described by an electric polarizability, and therefore are attracted towards the source region due to gradient forces, which are in fact maximized on the z-axis [42

42. I. Liberal, I. Ederra, R. Gonzalo, and R. W. Ziolkowski, “Near-field electromagnetic trapping through curl-spin forces,” Phys. Rev. A 87, 3807–3816 (2013). [CrossRef]

]. Therefore, the acceleration exerted by the sources on the coated Si particle, even though it has a much larger scattering cross-section, will be smaller than that of small dielectric particles, allowing to remove those impurities from diluted mixtures of particles with strong magnetic activity.

Moreover, since both the electric and magnetic polarizabilities are meaningful, the nanoparticle is scattering a significant amount of power; and, hence, it is visible to external observers. This effect is confirmed by Fig. 3(c), which shows the scattered power spectrum for different distances of separation from the source. This figure also manifests the fact that the particle is not only scattering a significant amount of power, but this occurs at a local maximum. In this configuration and, as illustrated in Fig. 3(a), the electric and magnetic dipoles excited in the nanoparticle are parallel and directed along the z-axis. Consequently, the scattering directivity pattern is characterized by a sin2θ angular variation, no matter what the relative magnitude and phase are between the electric and magnetic dipoles. We believe that this powerful combination of mechanical force suppression and high visibility might trigger innovative paradigms in near-field microscopy.

To further assess these concepts, Figs. 3(d) and 3(e) are, respectively, colormaps of the force magnitude and scattered power as a function of the particle location (center of the particle) on the XZ-plane, at λ = 708nm. On the one hand, Fig. 3(d) illustrates how the force is suppressed along the z-axis, and how it is weighted by an sinθ factor for other locations, as it was predicted by Eq. (2). On the other hand, Fig. 3(e) confirms that the particle scatters a significant amount of power when it is placed along the (zero-force) z-axis. In fact, although there is a null on the source radiation pattern, the reactive fields are maximized along this axis [42

42. I. Liberal, I. Ederra, R. Gonzalo, and R. W. Ziolkowski, “Near-field electromagnetic trapping through curl-spin forces,” Phys. Rev. A 87, 3807–3816 (2013). [CrossRef]

]. Consequently, the scattered power is maximized in this axis when the particle is located in the near-field of the sources.

To finalize the discussion, one might wonder what the forces would be in this near field scenario at the magnetic dipole super-resonance (i.e., for κ values closer to the threshold value 0.275). In such a case, the nanoparticle response is dominated by the magnetic dipole resonance, and the force field will be analogous to the force field produced by a localized source acting on a resonant electric dipolar nanoparticle, as studied in [42

42. I. Liberal, I. Ederra, R. Gonzalo, and R. W. Ziolkowski, “Near-field electromagnetic trapping through curl-spin forces,” Phys. Rev. A 87, 3807–3816 (2013). [CrossRef]

]. In this manner, the force field will consist of a balance of the gradient, radiation pressure, and curl-spin force components [42

42. I. Liberal, I. Ederra, R. Gonzalo, and R. W. Ziolkowski, “Near-field electromagnetic trapping through curl-spin forces,” Phys. Rev. A 87, 3807–3816 (2013). [CrossRef]

], whose magnitudes will all be enhanced by the magnetic dipole super-resonance.

4. Conclusion

In summary, we have demonstrated that it is possible to boost the magnetic dipole resonances present in Si nanoparticles by covering them with a shell impregnated with an active material. This configuration benefits from the concentration of a circulating electric field in the interface between the Si core and the active shell, and thus the magnetic dipole response is several orders of magnitude larger than those excited in bare silicon nanoparticles, and comparable to electric dipole super-resonances excited in spaser-based nanolasers. We believe that such extraordinarily strong magnetic response can be exploited in a wide range of technological applications. Furthermore, this configuration enables a great control on the optical forces exerted on the nanoparticle in both far-field and near-field scenarios. Specifically, colossal pushing and pulling forces are feasible close to the nanoparticle super-resonance. Moreover, it is possible to suppress the force exerted on the nanoparticle even when it is scattering a large amount of power. These effects open up advanced paradigms in electromagnetic manipulation and microscopy. Future efforts might also include the manipulation of the environment surrounding the active nanoparticle. These could include, among others, displacing and rotating the source to control its beam/pattern, optical binding of active/passive nanoparticles to a localized source, and/or compressing the medium in which the nanoparticle is immersed [45

45. I. Liberal, I. Ederra, R. Gonzalo, and R. W. Ziolkowski, “Electromagnetic force density in electrically and magnetically polarizable media,” Phys. Rev. A 88, 053808 (2013). [CrossRef]

].

Appendix A

This appendix provides explicit expressions for the electric and magnetic polarizabilities of a core-shell nanoparticle. To this end, equating the scattered fields predicted by Mie theory to the fields radiated by equivalent electric and magnetic dipoles, it is possible to derive the following relationships between the n = 1 TM and TE scattered field coefficients, and the electric and magnetic polarizabilities of a dipolar particle [46

46. A. Alù and N. Engheta, “Polarizabilities and effective parameters for collections of spherical nanoparticles formed by pairs of concentric double-negative, single-negative, and or double-positive metamaterial layers,” J. Appl. Phys. 97, 094310 (2005). [CrossRef]

]:
αee=j6πε0k03b1TM,αmm=j6πμ0k03b1TE
(3)

The above formulation can be applied to any object, provided that the b1TM and b1TE scatttered field coefficients are known. For a core-shell structure, these coefficients can be explicitly written as [37

37. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 2008).

]
b1TM=ηsJ^1(k0a2)[J^1(ksa2)+d1TMH^1(2)(ksa2)]η0J1^(k0a2)[J^1(ksa2)+d1TMH^1(2)(ksa2)]η0H^1(2)(k0a2)[J^1(ksa2)+d1TMH^1(2)(ksa2)]ηsH^1(2)(k0a2)[J1^(ksa2)+d1TMH^1(2)(ksa2)]
(4)
with
d1TM=ηcJ1^(kca1)J^1(ksa1)ηsJ^1(kca1)J1^(ksa1)ηsJ^1(kca1)H^1(2)(ksa1)ηcJn^(kca1)H^1(2)(ksa1)
(5)
and
b1TE=ηsJ^1(k0a2)[J^1(ksa2)+d1TMH^1(2)(ksa2)]η0J^1(k0a2)[J^1(ksa2)+d1TMH^1(2)(ksa2)]η0H^1(2)(k0a2)[J1^(ksa2)+d1TMH^1(2)(ksa2)]ηsH^1(2)(k0a2)[J^1(ksa2)+d1TMH^1(2)(ksa2)]
(6)
with
d1TE=ηcJ^1(kca1)J1^(ksa1)ηsJ^1(kca1)J^1(ksa1)ηsJ^1(kca1)H^1(2)(ksa1)ηcJ^n(kca1)H^1(2)(ksa1)
(7)
where a2 and a1 are, respectively, the external and internal radii of the core-shell structure, which is characterized by the propagation constants kc and ks, and medium impedances ηc and ηs, in the core and shell layers, respectively. The terms Ĵn(−) and H^n(2)() are the Schelkunoff forms of the spherical Bessel functions of the first kind and spherical Hankel functions of the second kind, respectively.

To finalize, it is worth remarking that the polarizabilities have been formulated by examining the plane-wave scattering problem [37

37. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 2008).

, 46

46. A. Alù and N. Engheta, “Polarizabilities and effective parameters for collections of spherical nanoparticles formed by pairs of concentric double-negative, single-negative, and or double-positive metamaterial layers,” J. Appl. Phys. 97, 094310 (2005). [CrossRef]

]. However, since the response of a spherical particle is isotropic, the derived polarizabilities can be applied for arbitrary incident fields.

Appendix B

Aligned electric and magnetic Hertzian dipole fields

Consider an electric Hertzian dipole of current moment Iel located at the origin of the coordinates and oriented along the + direction. Assuming that the dipole is driven with a sinusoid at the angular frequency ω, and adopting the ejωt time convention, the components of the time-harmonic electromagnetic fields produced by the electric Hertzian dipole: Ee=Erer^+Eθeθ^, He=Hϕeϕ^, are given by the closed form analytical expressions [44

44. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1961).

]
Ere=η0k024πIel2cosθ[1(k0r)2+j(k0r)3]ejk0r
(8)
Eθe=η0k024πIelsinθ[jk0r+1(k0r)2+j(k0r)3]ejk0r
(9)
Hϕe=k024πIelsinθ[jk0r+1(k0r)2]ejk0r
(10)
where η0=μ0/ε0 and k0=ωε0μ0 are the impedance and propagation constant, respectively, in free-space. By duality, the fields produced by the magnetic Hertzian dipole of magnetic dipole moment Iml = η0Iel, also located at the origin of the coordinates and oriented along +, can be written as [44

44. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1961).

]
Em=η0He
(11)
Hm=Eeη0
(12)

The total electromagnetic field produced by a source composed of these two elementary dipoles is given by the superposition of their fields, i.e.,
E=Ee+Em=Ee+η0He
(13)
H=He+Hm=HeEeη0
(14)
Since Ee · Em = 0 and He · Hm = 0, the electric and magnetic field intensities can be written as a superposition of the intensities produced by the electric and magnetic Hertzian dipoles alone:
|E|2=|Ee|2+|Em|2=|Ee|2+η02|He|2
(15)
|H|2=|He|2+|Hm|2=|He|2+|Ee|2η02
(16)
It is interesting to notice that the electric and magnetic fields are proportional, with proportionality constant η02, i.e.,
|E|2=η02|H|2
(17)
This balance between the electric and magnetic field intensities also imposes a zero reactive power result, i.e.,
Preac=ω2V(μ0|H|2ε0|E|2)dV=0
(18)
In fact, the density of reactive power ω2(μ0|H|2ε0|E|2) is cancelled out identically at all points when this balanced condition is satisfied.

Equation (18) can also be tested by studying the complex Poynting vector field S=SR+jSI=12E×H*. To this end, note, in virtue of Poynting’s theorem, that the reactive power excited by a localized source can be computed via the flux integral of SI through a surface that completely encloses the sources: [44

44. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1961).

]
Preac=limr0SSIn^dS
(19)
The complex Poynting vector field and its real and imaginary parts are given by the superposition of the corresponding Poynting vector fields associated with the electric and magnetic dipole fields, as well as their cross terms, which represent the interference effects. In particular, one has:
S=Se+Sm+Scross
(20)
SR=SRe+SRm+SRcross
(21)
SI=SIe+SIm+SIcross
(22)

Since the electric and magnetic dipoles are aligned for our configuration and since they radiate with the same angular power distribution, it can be readily checked that SRm=SRe. Moreover, since Re[A × A*] ≡ 0, the contribution of the cross product terms to the radiated power is zero: SRcross=0. Thus, the total Poynting vector field can be written simply as twice the one associated with the electric Hertzian dipole:
SR=2SRe
(23)
As a consequence, the power radiated into free-space by the combination source, P0, is simply two times the power radiated by the electric Hertzian dipole or its dual into free-space:
P0=2P0e=η0k026π|Iel|2
(24)
On the other hand, the symmetry of the fields lead us to SIm=SIe and SIcross=2ωcLSEe for the imaginary parts of the complex Poynting vector field. This means
SI=SIcross=2ωcLSEe
(25)
In other words, the electric and magnetic contributions to SI cancel out. The remaining part is due to the cross-terms only; it is proportional to the density of the spin angular momentum of the electric dipole. This implies · SI = 0; and, in view of Eq. (19), this result is consistent with the zero-reactive power property.

Finally, the electric and magnetic densities of the spin angular momentum:
LSE=ε04jω(E)*×E
(26)
LSH=μ04jω(H)*×H
(27)
can also be subdivided according to the electric dipole, magnetic dipole and interference terms as
LSE=LSEe+LSEm+LSEcross
(28)
LSH=LSHe+LSHm+LSHcross
(29)
Note that since He and Em are linearly polarized, their corresponding densities of the spin angular momentum are zero, i.e., LSEm=0 and LSHe=0. In addition, due to the duality of the sources, it can also be readily checked that
LSEe=LSMm
(30)
LSEcross=LSMcross=SIeωc
(31)

Mechanical forces exerted on a magnetoelectric particle

According to available analytical techniques, the time-averaged force, F, exerted on a particle that is characterized by electric, αee, and magnetic, αmm, polarizabilities and is illuminated by an arbitrary electromagnetic field (E, H), can be written as [27

27. P. C. Chaumet and A. Rahmani, “Electromagnetic force and torque on magnetic and negative-index scatterers,” Opt. Express 17, 2224–2234 (2009). [CrossRef] [PubMed]

, 28

28. M. Nieto-Vesperinas, J. J. Sáenz, R. Gómez-Medina, and L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express 18, 428–443 (2010). [CrossRef]

]
F=Fe+Fm+Fem
(32)
where
Fe=12Re{αeeE()E*}=αee4|E|2+αee[η0k0SRωε0×LSE]
(33)
Fm=12Re{αmmH()H*}=αmm4|H|2+αmm[k0η0SRωμ0×LSH]
(34)
Fem=η0k0412πμ0Re{p×m*}=η0k046πμ0{Re[αeeαmm*]SRIm[αeeαmm*]SI}
(35)
Reorganizing these terms, the force can also be written as the combination:
F=Fgrad+Frp+Fcurl+Fint
(36)
where the gradient, radiation pressure, curl-spin and electric-magnetic interference force components are, respectively, given by [27

27. P. C. Chaumet and A. Rahmani, “Electromagnetic force and torque on magnetic and negative-index scatterers,” Opt. Express 17, 2224–2234 (2009). [CrossRef] [PubMed]

, 28

28. M. Nieto-Vesperinas, J. J. Sáenz, R. Gómez-Medina, and L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express 18, 428–443 (2010). [CrossRef]

]
Fgrad=14(αee|E|2+αmm|H|2)
(37)
Frp=η0k0(αee+αmmη02k036πμ0Re[αeeαmm*])SR
(38)
Fcurl=ω[αeeε0×LSE+αmmμ0×LSH]
(39)
Fint=η0k046πμ0Im[αeeαmm*]SI
(40)

For our particular choice of a source consisting of two aligned electric and magnetic Hertzian dipoles, these force expressions can be conveniently rewritten as
Fgrad=14(αee+αmmη02)|E|2
(41)
Frp=η0k0(αee+αmmη02k036πμ0Re[αeeαmm*])SR
(42)
Fcurl=ωε0[(αee+αmmη02)×LSEe+(αeeαmmη02)×SIeωc]
(43)
Fint=2ωcη0k046πμ0Im[αeeαmm*]LSEe
(44)
Explicit expressions of these force components for Hertzian dipole fields can be found by introducing those fields Eqs. (8)(10) into Eqs. (41)(44).

This exercise leads to the following relations:
r^Fgrad=Caux(αee+αmmη02)[sin2θ(k0r)3+4cos2θ(k0r)5+32cos2θ+sin2θ(k0r)7]
(45)
θ^Fgrad=Caux(αee+αmmη02)[1(k0r)32(k0r)51(k0r)7]sinθcosθ
(46)
ϕ^Fgrad=0
(47)
r^Frp=Caux2(αee+αmmη02k036πμ0Re[αeeαmm*])sin2θ(k0r)2
(48)
θ^Frp=0
(49)
ϕ^Frp=0
(50)
r^Fcurl=Caux(αee+αmmη02)2cos2θsin2θ(k0r)4
(51)
θ^Fcurl=Caux2(αee+αmmη02)sinθcosθ(k0r)4
(52)
ϕ^Fcurl=Caux(αeeαmmη02){2(k0r)4+3(k0r)6}sinθcosθ
(53)
r^Fint=0
(54)
ϕ^Fint=Caux2k036πμ0Im[αeeαmm*]sinθcosθ(k0r)3
(55)
θ^Fint=0
(56)
where Caux ∈ ℝ+ is an auxiliary constant defined as
Caux=k0|η0k024πIel|2
(57)
Then the total force on a balanced active particle, i.e., an active particle for which
αee=αmmη02
(58)
reduces simply to Eq. (2):
F=2k043|αee|2SR+2αee×SIeωc=2η02k05|Iel4π(k0r)|2{r^2k033η0|αee|2sinθϕ^αeeε0ω[2(k0r)2+3(k0r)4]cosθ}sinθ
(59)

Acknowledgments

This work was supported in part by the Spanish Ministry of Science and Innovation, Projects No. TEC2009-11995 and No. CSD2008-00066 and by the NSF Contract No. ECCS-1126572.

References and links

1.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).

2.

R. Merlin, “Metamaterials and the Landau-Lifshitz permeability argument: large permittivity begets high-frequency magnetism,” Proc. Natl Acad. Sci. USA 106, 1693–1698 (2009). [CrossRef] [PubMed]

3.

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photon. 1, 41–48 (2007). [CrossRef]

4.

N. Liu, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Plasmonic building blocks for magnetic molecules in three-dimensional optical metamaterials,” Adv. Mater. 20, 3859–3865 (2008). [CrossRef]

5.

H.-K. Yuan, U. K. Chettiar, W. Cai, A. V. Kildishev, A. Boltasseva, V. P. Drachev, and V. M. Shalaev, “A negative permeability material at red light,” Opt. Express 15, 1076–1083 (2007). [CrossRef] [PubMed]

6.

A. Alù and N. Engheta, “The quest for magnetic plasmons at optical frequencies,” Opt. Express 17, 5723–5730 (2009). [CrossRef] [PubMed]

7.

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006). [CrossRef] [PubMed]

8.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]

9.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000). [CrossRef] [PubMed]

10.

M. Kerker, D. S. Wang, and C. L. Giles, “Electromagnetic scattering by magnetic spheres,” J. Opt. Soc. Am. A 73, 765–767 (1983). [CrossRef]

11.

Y. H. Fu, A. I. Kuznetsov, A. E. Miroshnichenko, Y. F. Yu, and B. Lukyanchuk, “Directional visible light scattering by silicon nanoparticles,” Nature Commun. 4, 1527 (2013). [CrossRef]

12.

S. Person, M. Jain, Z. Lapin, J. J. Sáenz, G. Wicks, and L. Novotny, “Demonstration of zero optical backscattering from single nanoparticles,” Nano Lett. 13, 1806–1809 (2013). [PubMed]

13.

S. D. Campbell and R. W. Ziolkowski, “Simultaneous excitation of electric and magnetic dipole modes in a resonant core-shell particle at infrared frequencies to achieve minimal backscattering,” J. Sel. Top. Quantum Electron. 19, 4700209 (2013). [CrossRef]

14.

F. Shafiei, F. Monticone, K. Q. Le, X.-X. Liu, T. Hartsfield, A. Alù, and X. Li, “A subwavelength plasmonic metamolecule exhibiting magnetic-based optical Fano resonance,” Nature Nanotech. 8, 95–99 (2013). [CrossRef]

15.

S. N. Sheikholeslami, A. García-Etxarri, and J. A. Dionne, “Controlling the interplay of electric and magnetic modes via Fano-like plasmon resonances,” Nano Lett. 11, 3927–3934 (2011). [CrossRef] [PubMed]

16.

C. Pfeiffer and A. Grbic, “Metamaterial Huygens surfaces: tailoring wave fronts with reflectionless sheets,” Phys. Rev. Lett. 110, 7401–7405 (2013). [CrossRef]

17.

T. Niemi, A. O. Karilainen, and S. A. Tretyakov, “Synthesis of polarization transformers,” IEEE Trans. Antennas Propag. 61, 3102–3111 (2013). [CrossRef]

18.

Y. Ra’di, V. S. Asadchy, and S. A. Tretyakov, “Total absorption of electromagnetic waves in ultimately thin layers,” IEEE Trans. Antennas Propag. 61, 4606–4614 (2013). [CrossRef]

19.

A. García-Etxarri, R. Gómez-Medina, L. S. Froufe-Pérez, C. López, L. Chantada, F. Scheffold, J. Aizpurua, M. Nieto-Vesperinas, and J. J. Sáenz, “Strong magnetic response of submicron silicon particles in the infrared,” Opt. Express 19, 4815–4826 (2011). [CrossRef] [PubMed]

20.

H. H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 9, 561 (1980). [CrossRef]

21.

A. I. Kuznetsov, A. E. Miroshnichenko, Y. H. Fu, J. Zhang, and B. Lukyanchuk, “Magnetic light,” Nature Sci. Reports 2, 492 (2012).

22.

A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region,” Nano Lett. 12, 3749–3755 (2012). [CrossRef] [PubMed]

23.

J. A. Gordon and R. W. Ziolkowski, “The design and simulated performance of a coated nano-particle laser,” Opt. Express 15, 2622–2653 (2007). [CrossRef] [PubMed]

24.

M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, and U. Wiesner, “Demonstration of a spaser-based nanolaser”. Nature Photon. 460, 1110–1113 (2009).

25.

N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nature Photon. 2, 351–354 (2008). [CrossRef]

26.

S. Arslanagić and R. W. Ziolkowski, “Active coated nano-particle excited by an arbitrarily located electric Hertzian dipoleresonance and transparency effects” J. Opt. 12, 024014 (2010). [CrossRef]

27.

P. C. Chaumet and A. Rahmani, “Electromagnetic force and torque on magnetic and negative-index scatterers,” Opt. Express 17, 2224–2234 (2009). [CrossRef] [PubMed]

28.

M. Nieto-Vesperinas, J. J. Sáenz, R. Gómez-Medina, and L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express 18, 428–443 (2010). [CrossRef]

29.

M. Nieto-Vesperinas, R. Gómez-Medina, and J. J. Sáenz, “Angle-suppressed scattering and optical forces on submicrometer dielectric particles,” J. Opt. Soc. Am. A 28, 54–60 (2011). [CrossRef]

30.

R. Gómez-Medina, B. García-Cámara, I. Suárez-Lacalle, F. González, F. Moreno, M. Nieto-Vesperinas, and J. J. Sáenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophoton. 5, 3512 (2011). [CrossRef]

31.

S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature (London) 466, 735–738 (2010). [CrossRef]

32.

N. Meinzer, M. Ruther, S. Linden, C. M. Soukoulis, G. Khitrova, J. Hendrickson, J. D. Olitsky, H. M. Gibbs, and M. Wegener, “Arrays of Ag split-ring resonators coupled to InGaAs single-quantum-well gain,” Opt. Express 18, 24140–24151 (2010). [CrossRef] [PubMed]

33.

N. Meinzer, M. Konig, M. Ruther, S. Linden, G. Khitrova, H. M. Gibbs, K. Busch, and M. Wegener, “Distance-dependence of the coupling between split-ring resonators and single-quantum-well gain,” Appl. Phys. Lett. 99, 111104 (2011). [CrossRef]

34.

M. Wegener, J. L. García-Pomar, C. M. Soukoulis, N. Meinzer, M. Ruther, and S. Linden, “Toy model for plasmonic metamaterial resonances coupled to two-level system gain,” Opt. Express 16, 19785–19798 (2008). [CrossRef] [PubMed]

35.

Sigma-Aldrich Corporation, url: http://www.sigmaaldrich.com/materials-science/nanomaterials/lumidots.html (2013).

36.

S. D. Campbell and R. W. Ziolkowski, “The performance of active coated nanoparticles based on quantum-dot gain media,” Adv. OptoElectronics 36, 8786–8791 (2012).

37.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 2008).

38.

P. Holmström, L. Thylén, and A. Bratkovsky, “Dielectric function of quantum dots in the strong confinement regime,” J. Appl. Phys. 107, 4307–4313 (2010). [CrossRef]

39.

I. Moreels, D. Kruschke, P. Glas, and J. W. Tomm, “The dielectric function of PbS quantum dots in a glass matrix,” Opt. Mater. Express 2, 496–500 (2012). [CrossRef]

40.

T. Kudo and H. Ishihara, “Proposed nonlinear resonance laser technique for manipulating nanoparticles,” Phys. Rev. Lett. 109, 7402–7406 (2012). [CrossRef]

41.

A. Mizrahi and Y. Fainman, “Negative radiation pressure on gain medium structures,” Opt. Lett. 35, 3405–3407 (2010). [CrossRef] [PubMed]

42.

I. Liberal, I. Ederra, R. Gonzalo, and R. W. Ziolkowski, “Near-field electromagnetic trapping through curl-spin forces,” Phys. Rev. A 87, 3807–3816 (2013). [CrossRef]

43.

S. Tricarico, F. Bilotti, and L. Vegni, “Reduction of optical forces exerted on nanoparticles covered by scattering cancellation based plasmonic cloaks,” Phys. Rev. B 82, 5109–5117 (2010). [CrossRef]

44.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, 1961).

45.

I. Liberal, I. Ederra, R. Gonzalo, and R. W. Ziolkowski, “Electromagnetic force density in electrically and magnetically polarizable media,” Phys. Rev. A 88, 053808 (2013). [CrossRef]

46.

A. Alù and N. Engheta, “Polarizabilities and effective parameters for collections of spherical nanoparticles formed by pairs of concentric double-negative, single-negative, and or double-positive metamaterial layers,” J. Appl. Phys. 97, 094310 (2005). [CrossRef]

OCIS Codes
(290.0290) Scattering : Scattering
(290.4020) Scattering : Mie theory
(160.3918) Materials : Metamaterials
(250.5403) Optoelectronics : Plasmonics

ToC Category:
Metamaterials

History
Original Manuscript: January 30, 2014
Revised Manuscript: March 19, 2014
Manuscript Accepted: March 24, 2014
Published: April 3, 2014

Citation
Iñigo Liberal, Iñigo Ederra, Ramón Gonzalo, and Richard W. Ziolkowski, "Magnetic dipole super-resonances and their impact on mechanical forces at optical frequencies," Opt. Express 22, 8640-8653 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-7-8640


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References

  1. L. D. Landau, E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).
  2. R. Merlin, “Metamaterials and the Landau-Lifshitz permeability argument: large permittivity begets high-frequency magnetism,” Proc. Natl Acad. Sci. USA 106, 1693–1698 (2009). [CrossRef] [PubMed]
  3. V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photon. 1, 41–48 (2007). [CrossRef]
  4. N. Liu, L. Fu, S. Kaiser, H. Schweizer, H. Giessen, “Plasmonic building blocks for magnetic molecules in three-dimensional optical metamaterials,” Adv. Mater. 20, 3859–3865 (2008). [CrossRef]
  5. H.-K. Yuan, U. K. Chettiar, W. Cai, A. V. Kildishev, A. Boltasseva, V. P. Drachev, V. M. Shalaev, “A negative permeability material at red light,” Opt. Express 15, 1076–1083 (2007). [CrossRef] [PubMed]
  6. A. Alù, N. Engheta, “The quest for magnetic plasmons at optical frequencies,” Opt. Express 17, 5723–5730 (2009). [CrossRef] [PubMed]
  7. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006). [CrossRef] [PubMed]
  8. J. B. Pendry, D. Schurig, D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]
  9. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000). [CrossRef] [PubMed]
  10. M. Kerker, D. S. Wang, C. L. Giles, “Electromagnetic scattering by magnetic spheres,” J. Opt. Soc. Am. A 73, 765–767 (1983). [CrossRef]
  11. Y. H. Fu, A. I. Kuznetsov, A. E. Miroshnichenko, Y. F. Yu, B. Lukyanchuk, “Directional visible light scattering by silicon nanoparticles,” Nature Commun. 4, 1527 (2013). [CrossRef]
  12. S. Person, M. Jain, Z. Lapin, J. J. Sáenz, G. Wicks, L. Novotny, “Demonstration of zero optical backscattering from single nanoparticles,” Nano Lett. 13, 1806–1809 (2013). [PubMed]
  13. S. D. Campbell, R. W. Ziolkowski, “Simultaneous excitation of electric and magnetic dipole modes in a resonant core-shell particle at infrared frequencies to achieve minimal backscattering,” J. Sel. Top. Quantum Electron. 19, 4700209 (2013). [CrossRef]
  14. F. Shafiei, F. Monticone, K. Q. Le, X.-X. Liu, T. Hartsfield, A. Alù, X. Li, “A subwavelength plasmonic metamolecule exhibiting magnetic-based optical Fano resonance,” Nature Nanotech. 8, 95–99 (2013). [CrossRef]
  15. S. N. Sheikholeslami, A. García-Etxarri, J. A. Dionne, “Controlling the interplay of electric and magnetic modes via Fano-like plasmon resonances,” Nano Lett. 11, 3927–3934 (2011). [CrossRef] [PubMed]
  16. C. Pfeiffer, A. Grbic, “Metamaterial Huygens surfaces: tailoring wave fronts with reflectionless sheets,” Phys. Rev. Lett. 110, 7401–7405 (2013). [CrossRef]
  17. T. Niemi, A. O. Karilainen, S. A. Tretyakov, “Synthesis of polarization transformers,” IEEE Trans. Antennas Propag. 61, 3102–3111 (2013). [CrossRef]
  18. Y. Ra’di, V. S. Asadchy, S. A. Tretyakov, “Total absorption of electromagnetic waves in ultimately thin layers,” IEEE Trans. Antennas Propag. 61, 4606–4614 (2013). [CrossRef]
  19. A. García-Etxarri, R. Gómez-Medina, L. S. Froufe-Pérez, C. López, L. Chantada, F. Scheffold, J. Aizpurua, M. Nieto-Vesperinas, J. J. Sáenz, “Strong magnetic response of submicron silicon particles in the infrared,” Opt. Express 19, 4815–4826 (2011). [CrossRef] [PubMed]
  20. H. H. Li, “Refractive index of silicon and germanium and its wavelength and temperature derivatives,” J. Phys. Chem. Ref. Data 9, 561 (1980). [CrossRef]
  21. A. I. Kuznetsov, A. E. Miroshnichenko, Y. H. Fu, J. Zhang, B. Lukyanchuk, “Magnetic light,” Nature Sci. Reports 2, 492 (2012).
  22. A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, B. N. Chichkov, “Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region,” Nano Lett. 12, 3749–3755 (2012). [CrossRef] [PubMed]
  23. J. A. Gordon, R. W. Ziolkowski, “The design and simulated performance of a coated nano-particle laser,” Opt. Express 15, 2622–2653 (2007). [CrossRef] [PubMed]
  24. M. A. Noginov, G. Zhu, A. M. Belgrave, R. Bakker, V. M. Shalaev, E. E. Narimanov, S. Stout, E. Herz, T. Suteewong, U. Wiesner, “Demonstration of a spaser-based nanolaser”. Nature Photon. 460, 1110–1113 (2009).
  25. N. I. Zheludev, S. L. Prosvirnin, N. Papasimakis, V. A. Fedotov, “Lasing spaser,” Nature Photon. 2, 351–354 (2008). [CrossRef]
  26. S. Arslanagić, R. W. Ziolkowski, “Active coated nano-particle excited by an arbitrarily located electric Hertzian dipoleresonance and transparency effects” J. Opt. 12, 024014 (2010). [CrossRef]
  27. P. C. Chaumet, A. Rahmani, “Electromagnetic force and torque on magnetic and negative-index scatterers,” Opt. Express 17, 2224–2234 (2009). [CrossRef] [PubMed]
  28. M. Nieto-Vesperinas, J. J. Sáenz, R. Gómez-Medina, L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express 18, 428–443 (2010). [CrossRef]
  29. M. Nieto-Vesperinas, R. Gómez-Medina, J. J. Sáenz, “Angle-suppressed scattering and optical forces on submicrometer dielectric particles,” J. Opt. Soc. Am. A 28, 54–60 (2011). [CrossRef]
  30. R. Gómez-Medina, B. García-Cámara, I. Suárez-Lacalle, F. González, F. Moreno, M. Nieto-Vesperinas, J. J. Sáenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophoton. 5, 3512 (2011). [CrossRef]
  31. S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H.-K. Yuan, V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature (London) 466, 735–738 (2010). [CrossRef]
  32. N. Meinzer, M. Ruther, S. Linden, C. M. Soukoulis, G. Khitrova, J. Hendrickson, J. D. Olitsky, H. M. Gibbs, M. Wegener, “Arrays of Ag split-ring resonators coupled to InGaAs single-quantum-well gain,” Opt. Express 18, 24140–24151 (2010). [CrossRef] [PubMed]
  33. N. Meinzer, M. Konig, M. Ruther, S. Linden, G. Khitrova, H. M. Gibbs, K. Busch, M. Wegener, “Distance-dependence of the coupling between split-ring resonators and single-quantum-well gain,” Appl. Phys. Lett. 99, 111104 (2011). [CrossRef]
  34. M. Wegener, J. L. García-Pomar, C. M. Soukoulis, N. Meinzer, M. Ruther, S. Linden, “Toy model for plasmonic metamaterial resonances coupled to two-level system gain,” Opt. Express 16, 19785–19798 (2008). [CrossRef] [PubMed]
  35. Sigma-Aldrich Corporation, url: http://www.sigmaaldrich.com/materials-science/nanomaterials/lumidots.html (2013).
  36. S. D. Campbell, R. W. Ziolkowski, “The performance of active coated nanoparticles based on quantum-dot gain media,” Adv. OptoElectronics 36, 8786–8791 (2012).
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