Zharov et al.
Vol. 31, No. 3 / March 2014 / J. Opt. Soc. Am. B
559
Liquid metacrystals Alexander A. Zharov,1 Alexander A. Zharov, Jr.,1 and Nina A. Zharova2,* 1
Institute for Physics of Microstructures, Russian Academy of Sciences, Nizhny Novgorod 603950, Russia 2 Institute of Applied Physics, Russian Academy of Sciences, Nizhny Novgorod 603950, Russia *Corresponding author:
[email protected]‑nnov.ru Received November 20, 2013; revised January 12, 2014; accepted January 20, 2014; posted January 22, 2014 (Doc. ID 201689); published February 20, 2014
We introduce a new type of metamaterial, which we call liquid metacrystals (LMCs), consisting of elongated particles (meta-atoms) suspended in viscous liquid. A constant homogeneous external electric field applied to such a material induces polarization of the meta-atoms and orients them along one axis, resulting in anisotropic electromagnetic properties of the system. The axis of anisotropy can be reoriented, and this type of tunability resembles that of liquid crystals in the nematic phase. Moreover, meta-atoms also reorient in response to the high-frequency electromagnetic waves, suggesting strong nonlinear properties of the LMCs. The artificial meta-atoms can be designed as classical oscillators to enhance their tunability. As a particular example of an electromagnetic wave control by LMCs, we study linear and nonlinear transmission of radiation through a slab of this metamaterial. © 2014 Optical Society of America OCIS codes: (160.1190) Anisotropic optical materials; (160.4760) Optical properties; (160.1245) Artificially engineered materials; (160.3918) Metamaterials. http://dx.doi.org/10.1364/JOSAB.31.000559
1. INTRODUCTION Elaboration and creation of new types of metamaterials which possess intriguing electromagnetic properties absent in natural materials is one of the most promising and very rapidly developing fields of contemporary photonics at the moment. In fact, in the last decade, metamaterials with negative refraction at frequency ranges from microwaves up to visible light have appeared [1–8]. Apart from linear left-handed metamaterials, there were created nonlinear, tunable, and elastic metamaterials with controllable electromagnetic properties [9–11]. Furthermore, other not necessarily left-handed types of metamaterials, such as plasmonic metamaterials [12–15], different nanoparticle arrays [16,17], hyperbolic artificial media (see [18,19] and references therein), etc. are currently designed for the wide range of various purposes and applications in many fields of nanophotonics. In this respect, one can mention the problems of light localization and light control at nanometer scales [20–23], nanolasing [24–26], cloaking [27,28], nanoantennas [29–31], and others. Indeed, any (even hypothetical) material expanding the potential possibilities of manipulations with electromagnetic radiation is greatly important for further detailed studies. In this work, we introduce a new type of tunable metamaterial, which we call liquid metacrystal (LMC). Such a metamaterial can be made of elongated metallic particles (meta-atoms) suspended in viscous liquid. The electric field applied to this material induces polarization of the particles, which leads to their reorientation. The homogeneous external field orients all the particles along one axis, which induces anisotropic electromagnetic properties of the material. Changing the electric field, the optical axis of such a medium can be reoriented, and this type of tunability resembles that of liquid crystals in the nematic phase [32], justifying the name liquid metacrystals. In LMC, instead of natural elongated molecules, 0740-3224/14/030559-06$15.00/0
artificial meta-atoms are suggested to be used, which can be designed to operate at different frequencies and exhibit much stronger tunability. Additionally, these meta-atoms can reorient in response to the electromagnetic waves, and this suggests that the LMC may demonstrate strong nonlinear properties. The idea of metallic dipoles dispersed in a liquid being controlled by an electric field has been mentioned in several papers, and was already experimentally demonstrated in [33]. In addition, metamaterials which reorient themselves due to the field of the incident wave, leading to a nonlinear response, were demonstrated in [34]. Several other papers have considered the influence of electromagnetic forces on meta-atoms, starting from [35]. Our work actually incorporates the orientational ability of conventional liquid crystals with resonant properties inherent in most kinds of known metamaterials.
2. SIMPLIFIED MODEL FOR ELECTROMAGNETIC RESPONSE OF LMC To describe the electromagnetic properties of LMC, one should obtain material equations connecting medium polarization with the macroscopic electric field. For this, first of all, we derive the set of equations which governs the dipole moment of individual meta-atoms in viscous liquid under a joint action of both constant and high-frequency electric fields. For convenience, we consider a specific dumbbell-like metal particle (as a meta-atom) shown in Fig. 1, enabling us to perform analytical calculations. In order to determine a dipole moment of the meta-atom induced by an external electric field, we assume that the capacitance of the particle is associated with dumbbell spheres while the inductance as well as resistance are defined by a straight dumbbell handle connecting the spheres. Thus, the considered particle of subwavelength size is equivalent to the classical LC oscillator © 2014 Optical Society of America
560
J. Opt. Soc. Am. B / Vol. 31, No. 3 / March 2014
Zharov et al.
all characteristic times in the system. So only two angular degrees of freedom described by generalized coordinates ϕ and θ remain to be considered. The Lagrange function for dumbbell-like particles and dissipation function have the forms L ml2 θ_ 2 ϕ_ 2 sin2 θ 2qlE00 E0∼ · e0 ml2 θ_ 2 ϕ_ 2 sin2 θ Fig. 1. Schematic view of meta-atom and equivalent electric circuit (inset).
schematically shown in the inset in Fig. 1. The capacitance of this oscillator is C εl a (the electrostatic unit (ESU) is used throughout this work), and the inductance and resistance are L 2lLW and R 2ρl∕πr 2 . Here, a is the radius of the dumbbell sphere, εl is the dielectric permittivity of the surrounding liquid, 2l is the length of the dumbbell handle, LW ≈ 2∕c2 logd∕r is the inductance of the direct wire of the circular section per unit length [36], ρ is the specific resistance of the metal, r is the radius of the handle, d ≈ minf2l; N −1∕3 ; λl g is some characteristic cutting distance preventing logarithmic divergence of the inductance of the straight wire, N is the p concentration of meta-atoms, and λl 2πc∕ω εl is the electromagnetic wavelength in the surrounding liquid. The p eigenfrequency pof this oscillator is ω0 2∕LC c∕ 2εl al logd∕r. The effective electromotive force acting on the particle due to external electric field is Rt Rdc Rac t 2lE00 E0∼ · e0 ;
(1)
where E00 , E0∼ are constant and high-frequency local electric fields respectively, and e0 is the unit vector along the particle axis (director). We also suppose that local fields in Eq. (1) are homogeneous on the size of meta-atoms, such that the electromotive forces induced by the gradient of the electric field can be neglected. Hence, the electric charge induced in metaatoms satisfies the equation d2 q dq R Rac t ; γ ω20 q dc dt L dt2
(2)
where γ R∕L is the damping coefficient. The orientation of the director can be characterized by spherical angles ϕ and θ, and thus its Cartesian coordinates can be expressed through these angles as e0 x cos ϕ sin θ;
e0 y sin ϕ sin θ;
e0 z cos θ: (3)
A. Lagrange Formalism and Material Equations for LMC The most convenient way to derive the equations of motion of meta-atoms is based on the Lagrange equation d ∂Lζ i ; ζ_ i ∂Lζ i ; ζ_ i ∂Dζ i ; ζ_ i − − ; _ dt ∂ζ ∂ζ i ∂ζ_ i i
(4)
where ζ i are the generalized coordinates, ζ_ i dζ i ∕dt, Lζ i ; ζ_ i is the Lagrangian, and Dζi ; ζ_ i is the so-called dissipation function which describes viscous friction [37]. Further, we disregard the effects of adhesion of the meta-atoms due to dipole–dipole interactions, which are very slow on the scale of
2qlE 0x cos ϕ sin θ E0y sin ϕ sin θ E0z cos θ; (5) D νl2 θ_ 2 ϕ_ 2 sin2 θ;
(6)
where m is the meta-atom mass, E 0i are the components of the full local electric field acting on meta-atoms including constant as well as high-frequency fields, and ν is the kinematic viscosity of the liquid. Equations (4)–(6) result in the equations of the meta-atom rotation 2 d2 θ dϕ dθ − sin θ cos θ ξ dt dt dt2 A0 qE 0x cos ϕ cos θ E0y sin ϕ cos θ − E 0z sin θ; (7) d dϕ 2 dϕ sin θ ξ sin2 θ −A0 q sin θE 0x sin ϕ − E0y cos ϕ; dt dt dt (8) which, along with the expression for the dipole charge following from Eq. (2), d2 q dq γ ω20 q B0 E0x cos ϕ sin θ E0y sin ϕ sin θ E0z cos θ; dt dt2 (9) yield the complete set of equations describing the dynamics of meta-atoms in the local field. Here, A0 3∕4πηΣ a3 l, B0 c2 ∕2 logd∕r, and ξ 3ν∕4πηΣ a3 . ηΣ is the density of particle material (metal), taking into account the density of added mass of the liquid (usually the liquid added mass is approximately half the displaced liquid mass [38], so in the considered case the correction caused by added mass to the metal density is negligibly small, except in the case of very heavy liquids; ηΣ ≈ ηM , where ηM is the metal density). If, for example, the geometry and material of meta-atoms and liquid are chosen as l 2 · 10−4 cm, a 10−4 cm, d 10−3 cm, r 5 · 10−5 cm, ηΣ ≈ ηM ≈ 10 g · cm−3 (silver), ν ≈ 1.5 · 10−3 g · s−1 (glycerin), 10−6 g · s−1 (water), p it gives p the values for resonant frequency f 0 ω0 ∕2π ≈ 8 2∕ εl THz, damping coefficient γ∕ω0 ≈ 10−3 , and the other model parameters ξ ≈ 3 · 107 s−1 (glycerin), 1.5 · 104 s−1 (water), A0 ≈ 1.2 · 1014 g−1 cm−1 , B0 ≈ 1020 cm2 s−2 . The viscous friction is quite strong, and it results in much greater times of transient processes in mechanical systems compared with the period of the electromagnetic field, which is assumed to be of the same order of inverse resonant frequency. So the director angles ϕ and θ in Eqs. (7)–(9) can be considered as the fixed values because their high-frequency vibrations near equilibrium state are negligibly small. One can also see that the material Eqs. (7)–(9) are strongly nonlinear. The nonlinearity originates due to the reorientation of meta-atoms under the joint
Zharov et al.
Vol. 31, No. 3 / March 2014 / J. Opt. Soc. Am. B
action of constant and high-frequency fields that specifies selfconsistent, intensity-dependent direction of the anisotropy axis. Within the framework of stationary interaction of the electromagnetic field with LMC, a self-consistent director orientation can be found, supposing the time-averaged right-hand parts of Eqs. (7) and (8) to be zero: hqtE0x t cos ϕ cos θ E 0y tsin ϕ cos θ − E 0z tsin θit 0; (10)
where the angular brackets denote the averaging over period of high-frequency electromagnetic field. When switching the direction or tension of the constant or high-frequency field, the transient process of director reorientation occurs with the time determined by liquid viscosity and shape and size of the meta-atoms, and this time is really much greater than the electromagnetic field period. This means that LMC is a “slow” metamaterial and it cannot be used in devices requiring a fast response. B. Effective Permittivity Tensor The set of material Eqs. (7)–(9) is written in terms of local (or acting) electric field E0 . In order to combine with Maxwell’s equations, one should rewrite them via macroscopic mean field instead of local. It can be done by means of the wellknown Lorentz–Lorenz formula [39], which establishes the connection between local E0 and macroscopic E fields in the arrays of randomly located dipoles with the interparticle distances much smaller than wavelength E0 E
4π P; 3
q ωs ω20 − ω2C ;
Fω
ω2s
3 2 ω Fω; 4π C
B 2 2 2 σˆ B @ sin ϕ cos ϕ sin θ sin ϕ sin θ
(14) 1
C sin ϕ sin θ cos θ C A: 2 cos θ
cos ϕ sin θ cos θ sin ϕ sin θ cos θ
It describes an effective uniaxial crystal with the axis characterized by spherical angles ϕ and θ. The expression for the corresponding tensor of effective dielectric permittivity of LMC follows immediately from Eq. (14): εˆ εl 4πχ m ωσϕ; ˆ θ:
(15)
3. ELECTROMAGNETIC PROPERTIES OF LMC: PARTICULAR EXAMPLES A. Linear Electromagnetic Waves in Bulk LMC An electromagnetic wave propagating in the medium with dielectric tensor [Eq. (15)] breaks up into two independent polarizations: TE with electric field orthogonal to both the wave vector k and director, and TM with the similar orientation of magnetic field. Refraction index of the TE-polarized wave does not depend on the propagation direction and frequency n2TE εl :
(16)
At the same time, the refraction index of the TM-polarized wave depends on propagation direction in the following manner: εl εl 4πχ m ω ; εl 4πχ m ωcos2 ψ
(17)
where ψ is the angle between director e0 and wavevector k: ψ e0 ∧k. In Fig. 2, the angular dependencies of refraction 3
2
3 2
1 5
1
(12)
χ m ω
sin ϕ cos ϕ sin2 θ cos ϕ sin θ cos θ
n2TM
as follows from Eq. (9). Substituting Eq. (12) into Eq. (11) and expressing the local field through the macroscopic in Eqs. (7)–(9), we come to the same equations, but with another resonant frequency, shifted due to interaction of the individual meta-atoms: qω FωB0 Eω · e0 ;
cos2 ϕ sin2 θ
(11)
where P is the medium polarization of unit volume; P 2qlNe0 for identically aligned meta-atoms. Actually, Eq. (11) gives the way of metamaterial homogenization. Then, we consider the linear problem when the dipole moments of metaatoms are directed strictly along a constant electric field E0 . This means that the high-frequency field is so weak that it does not reorient the meta-atoms. For harmonic field ∼ expiωt, the complex amplitude of the dipole oscillating charge is equal to B E0 · e qω 2 0 2ω 0 ; ω0 − ω iγω
0
nTMsinψ
hqtsin θE0x tsin ϕ − E0y tcos ϕit 0;
χˆ χ m ωσ; ˆ
561
4 0
6
−1 −2 −3
1 ; − ω2 iγω
−2
−1
n
(13)
where ω2C 8πlNB0 ∕3. The linear tensor of macroscopic susceptibility defined by the expression P χˆ E can be written as
0 cosψ
1
2
TM
Fig. 2. Angular dependence of refraction index for TM-polarized waves. Curves 1–6 correspond to the values of ω∕ωs 0.9, 0.96, 1.02, 1.08, 1.14, and 1.2. In the vicinity of resonance, for 1 < ω∕ωs < 1.135, the LMC demonstrates the properties of hyperbolic medium. Concentration of meta-atoms is N 3 · 107 cm−3 .
562
J. Opt. Soc. Am. B / Vol. 31, No. 3 / March 2014
Zharov et al.
index for TM-polarized waves are shown for different values of frequency. The isofrequency contours for waves of TE polarization are just the circles, while, for TM-polarized waves, they can take elliptic as well as hyperbolic shapes in different frequency bands. So LMC can simultaneously possess the properties of both elliptic and hyperbolic media, but at different frequencies. B. Transmission of Radiation through a Thin LMC Slab: Nonlinear Effects Simple calculations show that LMC is a potentially highly tunable resonant metamaterial, with electromagnetic properties easily changed by means of changing direction and tension of applied external electric field along with properly chosen frequency range. Below, as particular examples, we demonstrate the unique properties of LMC as a convenient tool for linear and nonlinear control of electromagnetic radiation. We consider the normal incidence of a plane electromagnetic wave of amplitude E inc onto thin plane-parallel slab of LMC mounted on the substrate with dielectric permittivity εs . We suppose that the plane of crystal is z 0, and metaatoms are oriented in this plane, i.e., θ π∕2 [see Eq. (14)]. The constant electric field is directed along the x axis, and in the linear regime, the director e0 x0 . For the incident wave of finite amplitude, the director does not keep the orientation and its direction has to be found self-consistently using Eqs. (7) and (8). Taking into account a small thickness, h, of the LMC slab in comparison with electromagnetic wavelength, the highfrequency electric field may be considered as a continuous value on the scale of the slab while the magnetic field has a jump which is proportional to the effective electric surface current flowing in the plane of LMC slab: Etr Einc Eref ; Htr − Hinc − Href
4π 4π j × z0 iωhP × z0 : c s c
(18)
The boundary conditions Eq. (18) for incident, transmitted, and reflected fields and Eq. (10) lead to the transcendental expression for the director orientation angle ϕ, tan 2ϕ
ζ 0 sin 2β ; 1 ζ 0 cos 2β
(19)
as a function of the angle β between the constant and highfrequency fields. Here, ζ 0 2Rωσ 2NL , σ NL jEinc ∕E 0 j, p E inc jEinc j, Rω ω2s ReFω∕f 1 f 2 , f 1 1 εs ik0 hεl , f 2 f 1 4πik0 hχ m ω, and “*” denotes complex conjugation. The value ζ 0 plays a role of frequency-dependent parameter of nonlinearity. It is necessary to draw attention to the fact that ζ 0 changes its sign at the resonance frequency independently of the amplitude of the incident wave. Hence, at the resonance frequency, the director is always oriented along the constant electric field. In the case of very small bias E 0 → 0 and finite high-frequency field E ∼ , the dependence of parameter σ NL on static electric field E0 becomes singular, meaning just the reorientation of the meta-atoms along the direction determined by E∼ . If the high-frequency field also decreases, the nonlinear parameter σ NL formally can be kept high enough or even singular, and obviously it is necessary to
consider other factors which lead to off-orientation of metaatoms. In conventional liquid crystals, where the molecules are densely packed up, the forces due to static friction and nearest neighbor interaction become important in the case of small bias. Here, we consider a quite rarefied medium with inter-meta-atom distances much greater than their sizes. Indeed, considering meta-atoms with size ∼2 · 10−4 cm and volume density on the order of N ∼ 106 cm−3 , the ratio of average inter-meta-atom distance (N −1∕3 ) to the meta-atom size equals 50, which is the value ≫1. Nevertheless, a thermal motion may cause the singularity disappearance. Corresponding critical static field can be evaluated through simple consideration from the equality of induced electric energy of metaatoms, 2qlE 2B0 lE2 ∕ω20 ≡ 2εl al2 E2 , and thermal energy kB T for two active degrees of freedom. This equality immediately gives an explicit estimation E2c ∼ kB T∕2εl al2 that, for considered parameters of meta-atoms and room temperature, p p equals 7 · 10−2 ∕ εl ESU or 21∕ εl V∕cm. Figure 3 shows the typical relations between the incident, reflected, and transmitted fields and also constant electric field and director orientation in the x; y plane. Actually, Eq. (19) for given ω and β has two solutions, ϕ1;2 , such as ϕ2 ϕ1 π∕2. To choose a stable one, standard analysis based upon Eqs. (8) and (9) leads to the stability criterion cos2ϕ ζ 0 cos2ϕ − 2β > 0:
(20)
Figure 4 shows a typical dependence of the stable selfconsistent angle of director, ϕ, on ω and β in the vicinity of resonance. It is interesting to note that, for stationary propagation, only the ratio of the incident wave amplitude to the value of constant field matters. The boundary conditions in Eq. (18) result in expressions for the transmission and reflection coefficients in which the angle ϕ is given in Eq. (19), cos β − 4πik0 hχ m ω cos ϕ cosϕ − β∕f 2 ; f1 sin β − 4πik0 hχ m ω sin ϕ cosϕ − β∕f 2 2 ; f1
inc 2 T ∥ Etr ∥ ∕E inc T ⊥ Etr ⊥ ∕E
inc T − cos β; R∥ Eref ∥ ∥ ∕E inc T ⊥ − sin β; R⊥ Eref ⊥ ∕E
(21)
where the signs “∥” and “⊥” indicate the components of the high-frequency electric field parallel and perpendicular to the constant field, respectively.
Eref E
inc
E
tr
e0 φ E0
Fig. 3. Schematic view of transmission of plane electromagnetic wave through a thin LMC slab placed on a substrate (left panel); typical relations between incident, reflected, and transmitted wave amplitudes, director, and constant electric field applied to the LMC (right panel).
Zharov et al.
Vol. 31, No. 3 / March 2014 / J. Opt. Soc. Am. B
563
different applications of LMC, such as polarizer, frequency filter, absorber, etc. Furthermore, the value of nonlinearity parameter ζ 0 can be changed not only by variation of frequency or incident wave amplitude, but also by variation of constant field tension. This certainly extends the capabilities of electromagnetic wave control with LMC.
4. CONCLUSION
Fig. 4. Stable self-consistent director angle, ϕ∕π, on the plane of parameters ω∕ωs and β∕π. Contour lines with the corresponding values of ϕ∕π are shown. of meta-atoms is N 106 cm−3 , pConcentration σ NL jE inc ∕E0 j 0.1.
In Fig. 5, the frequency dependencies of transmission and reflection coefficients are shown at different values of parameter σ NL and the angle β between the electric field of the linear polarized incident wave and constant electric field. One can see that quite small values of σ NL lead to strong distortion of linear curves (σ NL 0) in the vicinity of resonance. The origin of such a strong nonlinearity is due to the meta-atoms’ reorientation under the action of high-frequency fields. Since the propagation conditions of electromagnetic waves with frequency close to resonant strongly depend on the angle between the director and electric field polarization, even a small deflection of the meta-atom axis results in a large change of refraction index. Also, this nonlinearity has a distinct saturation signature because, at jσ NL j ≫ 1, all the meta-atoms align with the high-frequency field, as follows from Eq. (19). Figure 5 shows the quite complicated behavior of the transmission and reflection coefficients in the resonance region that results in divers ways of manipulations with electromagnetic waves and potentially ensures several
1
1’ 0.8
2’
3
3’
0.6
(a)
1 2
0.4 0.996
0.998
1 ω/ωs
0.35
1.004
3
0.3
2
0.25
3’
0.2
1
(b)
2’
0.15 0.1
1.002
1’ 0.996
0.998
1 ω/ωs
1.002
1.004
Fig. 5. (a) Transmission T jT ∥ j2 jT ⊥ j2 and (b) reflection R jR∥ j2 jR⊥ j2 coefficient as a function of normalized frequency to β π∕8 and solid ω∕ωs at resonance. Dashed curves correspond p ones to β 3π∕8; σ NL jEinc ∕E0 j 10−5 , 0.1, 0.1 for curves 1 (10 ), 2 (20 ), and 3 (30 ) respectively; concentration of meta-atoms is N 106 cm−3 ; film thickness is h 10−3 cm.
In conclusion, a new metamaterial called LMC was suggested, and its electromagnetic properties were discussed in this work. It was shown that LMC is a tunable metamaterial which can potentially provide light control in a wide frequency range. Operation frequency is determined by shape, size, and concentration of meta-atoms. It was also demonstrated that the same LMC may exhibit the properties of both elliptic and hyperbolic media. Due to the ability of meta-atoms to reorient in constant as well as high-frequency fields, LMC possesses incredibly strong nonlinear properties which may be also used in perspective applications.
ACKNOWLEDGMENTS The authors thank I. V. Shadrivov and V. V. Kurin for useful discussions and the RFBR for support through grants NN 14-02-00439, 13-02-97115.
REFERENCES 1. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000). 2. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001). 3. S. Linden, C. Enkrich, M. Wegener, T. Zhou, T. Kochny, and C. M. Soukoulis, “Magnetic response of metamaterials at 100 terahertz,” Science 306, 1351–1353 (2004). 4. S. Zhang, W. Fan, B. K. Minhas, A. Frauenglass, K. J. Malloy, and S. R. J. Brueck, “Midinfrared resonant magnetic nanostructures exhibiting a negative permeability,” Phys. Rev. Lett. 94, 037402 (2005). 5. G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, “Negative-index metamaterial at 780 nm wavelength,” Opt. Lett. 32, 53 (2007). 6. U. K. Chettiar, A. V. Kidishev, H. K. Yuan, W. Cai, S. Xiao, V. P. Drachev, and V. M. Shalaev, “Dual-band negative index metamaterial: double negative at 813 nm and single negative at 772 nm,” Opt. Lett. 32, 1671–1673 (2007). 7. H. J. Lezec, J. A. Dionne, and H. A. Atwater, “Negative refraction at visible frequencies,” Science 316, 430–432 (2007). 8. A. Minovich, J. Farnell, D. N. Neshev, I. McKervacher, F. Karouta, J. Tian, D. A. Powell, I. V. Shadrivov, H. H. Tan, C. Jagadish, and Y. S. Kivshar, “Liquid crystal based nonlinear fishnet metamaterials,” Appl. Phys. Lett. 100, 121113 (2012). 9. A. A. Zharov, I. V. Shadrivov, and Y. S. Kivshar, “Nonlinear properties of left-handed metamaterials,” Phys. Rev. Lett. 91, 037401 (2003). 10. M. Lapine, I. V. Shadrivov, D. A. Powell, and Y. S. Kivshar, “Magnetoelastic metamaterials,” Nat. Mater. 11, 30–33 (2012). 11. I. V. Shadrivov, D. A. Powell, and Y. S. Kivshar, “Metamaterials with conformational nonlinearity,” Sci. Rep. 1, 138 (2011). 12. W. L. Barnes, A. Dereux, and A. Ebbensen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003). 13. J. A. Fan, K. Bao, L. Sun, J. Bao, V. N. Manoharau, P. Nordlander, and F. Capasso, “Stabilized TiN nanowire arrays for highperformance and flexible supercapacitors,” Nano Lett. 12, 5376–5381 (2012). 14. A. Wiener, A. I. Fernandez-Dominguez, A. P. Horsfield, J. B. Pendry, and S. A. Maier, “Nonlocal effects in the nanofocusing
564
15. 16. 17. 18. 19. 20. 21. 22. 23. 24.
25. 26.
J. Opt. Soc. Am. B / Vol. 31, No. 3 / March 2014 performance of plasmonic tips,” Nano Lett. 12, 3308–3314 (2012). A. Boltasseva and H. A. Atwater, “Low-loss plasmonic metamaterials,” Science 331, 290–291 (2011). E. Wang, T. P. White, and K. R. Catchpole, “Resonant enhancement of dielectric and metal nanoparticle arrays for light trapping in solar cells,” Opt. Express 20, 13226–13237 (2012). J. T. Shen, P. B. Catrysse, and S. Fan, “Mechanism for designing metallic metamaterials with a high index of refraction,” Phys. Rev. Lett. 94, 197401 (2005). M. A. Noginov, Y. A. Bernakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94, 151105 (2009). C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14, 063001 (2012). V. M. Shalaev and S. Kawata, Nanophotonics with Surface Plasmons (Elsevier Science, 2007). D. K. Gramotnev and S. I. Bozhevolnyi, “Plasmonics beyond the diffraction limit,” Nat. Photonics 4, 83–91 (2010). M. I. Stockman, “Nanofocusing of optical energy in tapered plasmonic waveguides,” Phys. Rev. Lett. 93, 137404 (2004). A. R. Davoyan, I. V. Shadrivov, A. A. Zharov, D. K. Gramotnev, and Y. S. Kivshar, “Nonlinear nanofocusing in tapered plasmonic waveguides,” Phys. Rev. Lett. 105, 116804 (2010). D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. 90, 027402 (2003). I. Cao and M. L. Brongersma, “Active plasmonics: ultrafast developments,” Nat. Photonics 3, 12–13 (2009). X. Meng, U. Guler, A. V. Kidishev, K. Fujita, K. Tanaka, and V. M. Shalaev, “Unidirectional spaser in symmetry-broken plasmonic core-shell nanocavity,” Sci. Rep. 3, 1241 (2013).
Zharov et al. 27. D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314, 977–980 (2006). 28. B. Edwards, A. Alu, M. G. Silveirinha, and N. Engheta, “Experimental verification of plasmonic cloaking at microwave frequencies with metamaterials,” Phys. Rev. Lett. 103, 153901 (2009). 29. A. Alu and N. Engheta, “Tuning the scattering response of optical nanoantennas with nanocircuit loads,” Nat. Photonics 2, 307–310 (2008). 30. L. Novotny, “Optical antennas tuned to pitch,” Nature 455, 887 (2008). 31. L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics 5, 83–90 (2011). 32. P. G. deGennes and J. Prost, The Physics of Liquid Crystals, 2nd ed (Clarendon, 1995). 33. A. B. Golovin and O. D. Lavrentovich, “Electrically reconfigurable optical metamaterial based on colloidal dispersion of metal nanorods in dielectric fluid,” Appl. Phys. Lett. 95, 254104 (2009). 34. M. Liu, Y. Sun, D. A. Powell, I. V. Shadrivov, M. Lapine, R. C. McPhedran, and Y. S. Kivshar, “Nonlinear response via intrinsic rotation in metamaterials,” Phys. Rev. B 87, 235126 (2013). 35. R. Zhao, P. Tassin, T. Koschny, and C. M. Soukoulis, “Optical forces in nanowire pairs and metamaterials,” Opt. Express 18, 25665–25676 (2010). 36. J. S. Schwinger, L. L. Deraad, Jr., K. A. Milton, and W. Tsai, Classical Electrodynamics (Westview, 1998). 37. L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon, 1966). 38. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd ed. (Pergamon, 1987). 39. M. Born and E. Wolf, Principles of Optics, 2nd ed. (Pergamon, 1964).
