The study of invisibility cloaking was sparked by the work of John Pendry et al.
1. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006). [CrossRef] [PubMed]
] who suggested a recipe for transforming the space and making objects ‘invisible’ to the incoming radiation. The experiments at microwaves [2
2. 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). [CrossRef] [PubMed]
] show that it is possible to reduce visibility of the objects, however making such cloaks for higher frequencies is quite challenging. Several other concepts were suggested to achieve similar effects. For hiding small subwavelength particles, Alu and Engheta suggested to cover them with a layer of another material, so that for a given frequency the light scattering by a nanoparticle can be made very small [3
3. A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E 72, 016623 (2005). [CrossRef]
4. A. Alu and N. Engheta, “Polarizabilities and effective parameters for collections of spherical nanoparticles formed by pairs of concentric double-negative, single-negative, andor double-positive metamaterial layers,” J. Appl. Phys. 97, 094310 (2005). [CrossRef]
]. The effect is based on the resonant cancelation of the dipole moment of the particle, and as a result it is narrowbanded. In order to increase the bandwidth of the particle invisibility, it was suggested to cover the particle with several shells of various materials [5
5. A. Alu and N. Engheta, “Multifrequency optical invisibility cloak with layered plasmonic shells,” Phys. Rev. Lett. 100, 113901 (2008). [CrossRef] [PubMed]
]. Later, this concept was demonstrated experimentally at microwaves, and it was shown that the particle cross-section can be reduced by as much as 75% [6
6. 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). [CrossRef] [PubMed]
In this paper we study the scattering properties of multi-shell plasmonic nanoparticles with a nonlinear layer. Our aim is to demonstrate that the cloaking performance of such plasmonic structures can be controlled by changing the amplitude of the incident wave, i.e. that the cloak can be made nonlinear. We consider a particle with two shells, which parameters are slightly detuned from those required for the perfect cloaking. Introducing nonlinearity, we study how it can be used for restoring the conditions of the reduced visibility of the particles.
We study a two-dimensional problem (i.e. nanowires instead of spheres), when the particle (cylindrical nanowire) of the radius R1
with dielectric constant ε1
is coated by two layers, with external radii of R2
, and dielectric constants ε2
, respectively [see the inset in Fig. 1(a)
]. Similar to the spherical case in Ref. [7
7. A. A. Zharov and N. A. Zharova, “On the electromagnetic cloaking of (Nano)particles,” Bulletin of the Russian Academy of Sciences: Physics 74, 89–92 (2010). [CrossRef]
], it can be shown that the dipole moment of this particle vanishes, and the central core is completely screened from external radiation, when ε2
= 0, and the radii of the coatings are linked by the following relation
Now we assume that the layer 2 is absorbing, and it has a nonlinear correction to the dielectric constant, so that ε2
, where ε
are real and imaginary parts of the linear part of the dielectric constant, and α
is the nonlinear coefficient. To calculate the scattering properties of such a nonlinear multi-shell plasmonic particle, we use a multipole expansion method. This method allows us to calculate not only the dipole mode excitation, but also the higher-order multipoles, and to analyze the scattering by each of the modes.
Fig. 1 (a) Normalized scattering cross-sections for several lower-order multipoles (solid lines, m=0,1,2), as well as total SCS (dashed) as functions of the dielectric constant ε2 of the second layer. Parameters are k0R1,2,3 = 0.4, 0.5, 0.866, ε1,3 = 15, 2, (ε″2) = 0.02. Inset shows geometry of the problem. (b) Dependencies of the normalized multipole and total scattering cross-sections on the dielectric constant ε3 of the external layer. Dimensions of the cylinders and dielectric permittivity of the core are the same as in (a), ε2 = −0.1+0.02i.
We assume that the incident field is TM polarized, i.e. the magnetic field is along the axis of the cylinder. The field of the incident plane wave can be represented in cylindrical coordinates
are Bessel functions of first kind, k0
. In each layer we represent the magnetic field as a sum of multipoles with azimuthal number m
are Bessel functions of second kind, ρj
is the wavenumber in the layer j. Then we write the boundary conditions at each interface for the magnetic field as well as for the azimuthal component of electric field, and we find the amplitudes of the multipoles. Azimuthal and radial components of the electric field (also contributing to the nonlinear dielectric permittivity) can be written as
To find the coefficients
we use a cylindrical transfer matrix Ŝ
, which relates the wave amplitudes in the adjacent layers
. The boundary conditions require that the field in the centre of the core is finite, and the field outside the nanoparticle is a superposition of the incident plane wave and the scattered cylindrical wave. In such a way, we can determine all coefficients
and find the total field distribution.
shows properties of the scattering cross-section per unit length (SCS) contributions by three strongest multipoles, and the total SCS is a superposition of all scattering cross-sections. We want to note, that in contrast to the spherical geometry, for the cylindrical case, we have zero order multipole. The electric field in this mode has only the azimuthal component. The dependence of the SCS on the real part of dielectric permittivity of the middle layer is shown in Fig. 1
. It clearly shows that dipole mode SCS (m
= 1) as well as the total SCS reach their minimum near our previously predicted point ε
= 0. At the same time, the total SCS remains finite, in our case Stot k0
= 0.03, where k0
. Remarkably, the SCS grows much faster towards negative values of ε2
than to the positive values.
As the next step, we verify how sensitive is the SCS to the value of the dielectric constant of the external layer, which for given sizes of the shells and for ε2
= 0 defines how well the condition of Eq. (1)
is fulfilled. However we want to demonstrate the general picture, and we consider in our calculations that ε2
. The corresponding dependencies are shown in Fig. 1(b)
. Minimum of the dipole-mode SCS is achieved close to ε3
≈ 3, which is different from the value expected from Eq. (1) ε3
= 2 because in these calculations the middle layer is absorbing.
shows distribution of the intensity of the electric field in the coated nanoparticle. Since the field in the middle layer is much larger than it is in the rest of the structure, to visualize the fields in other layers, we set the intensity to zero in the medium 2
, and plot the resulting field distribution in Fig. 2(b)
. We see that the field intensity in the epsilon-near-zero medium is enhanced by about 50 times as compared to the intensity of the incident wave. We also note that the field inside the core does not vanish, because we do not have perfect material parameters, and also because of the zero-order multipole contribution. However the field intensity in the core of the particle in our case is still 10 times weaker than that in the incident wave.
Distribution of the intensity of the electric field in (a) the whole space (b) everywhere but layer 2. Sizes of the shells are the same as in Fig. 1
= 15; 0.02 + 0.02i
; 2. The fields are normalized to the amplitude of the incident wave.
Next we consider the nanoparticle with nonlinear dielectric properties of the metal layer. Intuitively, we can expect, that the third order nonlinear correction to the dielectric constant of the metal will be able to change the cloaking conditions by adjusting incident wave intensity. We study two cases: when the dielectric constant of the metal is positive, and the nonlinear coefficient α
is negative, so that it can restore the best cloaking condition, when the dielectric constant vanishes. The second case is when ε
< 0 and α
> 0. We assume that the sizes of the shells are the same as in linear calculations, and ε1,2,3
= 15; ±0.1 + 0.02i
; 2. and nonlinear coefficient α
= ∓5 · 10−8
esu. To solve this nonlinear problem we developed a converging iterative scheme, which allows us to find the field distribution in the particle as well as radiated fields. In this scheme, we first calculate the field distribution assuming linear response of the structure. Then, we use these fields to determine the nonlinear correction to the dielectric permittivity of the screening layer. This correction is inhomogeneous, and we decompose it into the series of cos(mϕ
). For controlling convergence, we introduce parameter β
, and calculate the correction to the nonlinear dielectric permittivity as
are the nonlinear corrections to the dielectric permittivity on previous and current iterative step, respectively. Each iteration step the fields inside the screening layer have to found numerically from the Helmholtz equation
. Choice of the parameter β
has to be made empirically by looking on the convergence efficiency.
Scattering cross-sections of the nonlinear particle as a function of the incident field intensity for the two cases described above are shown in Fig. 3
. In the case of defocusing nonlinearity [see Fig. 3(a)
], the dipole SCS has a smooth profile with a minimum at approximately 9 MW/cm2
. The contribution of the monopole mode with m
= 0 first reaches minimum and then grows to a maximum value, and then slowly decreases again. At the minimum point the SCS is almost one order of magnitude smaller than in the linear case (at zero intensity on Fig. 3
), while in its maximum the nonlinear SCS is almost one order of magnitude greater than linear SCS. Figure 3(b)
shows scattering cross-sections for the focusing nonlinearity. In contrast to the previous case, the dipole-mode-induced scattering gradually decreases, however the monopole scattering has non-monotonic dependence and reaches its maximum at approximately 9 MW/cm2
. Contribution of the second and higher azimuthal modes remains small for all reasonable powers. The total scattering cross-section does not experience any dramatic changes.
Fig. 3 Normalized scattering cross-sections for several lower-order multipoles (solid lines, m = 0,1), as well as total SCS (dashed) as functions of the intensity. (a) Defocusing non-linearity, ε2 = 0.1 + 0.02i, (b) focusing nonlinearity, ε2 = −0.1 + 0.02i; α = 5 · 10−8 esu.
shows nonlinear scattering directivity for (a) defocusing and (b) focusing nonlinearity. To represent the nonlinear directivity properties, we use the following rather complicated diagram. The color represents the scattering strength, while the radial coordinate corresponds to the intensity of the incident wave, so that each circle on the figures represents scattering pattern for a given incident field intensity. In both cases we observe two threshold-like dependencies, when the direction of the scattering flips by 180 degrees. In the case (a) for low powers, the core-shell particle scatters mostly in backward direction, however with increasing power, the scattering changes to the forward direction. This transition coincides with a sharp increase of the SCS of the m
= 0 mode, shown in Fig. 3(a)
. Further increase of the intensity makes scattering more unidirectional-like until the second threshold, when the scattering again becomes mostly forward. In the case of focusing nonlinearity, shown in Fig. 4(b)
, we observe a qualitatively different behavior. The particle is initially scattering mostly in the forward direction, however above some threshold the scattering flips to be mostly in the backward direction. Further increase of the power leads to almost unidirectional scattering, and for even higher powers, the particle scatters mostly in the forward direction.
Fig. 4 Directivity of the nonlinear cloak. Color map shows scattering amplitude, while radial coordinate corresponds to the intensity of the incident wave (a) Defocusing nonlinearity, ε2 = 0.1 + 0.02i, (b) focusing nonlinearity, ε2 = −0.1 + 0.02i; α = 5 · 10−8 esu.
shows distribution of the field amplitudes for different incident power intensities in the defocusing case corresponding to Figs. 3(a)
. We observe that the overall scattering strength is dramatically changing, with a large variation of the directivity.
Fig. 5 Distribution of the scattering electric field for three different values of the incident wave intensity: (a) 2.5 · 104W/cm2; (b) 2.51 · 106W/cm2, and (c) 2.66 · 106W/cm2, α = −5 · 10−8, ε2 = 0.1 − 0.02i.
We would like to note that the requirement of the zero-index material for the shielding layer of the structure limits the frequency ranges, where such cloaking can be applied. For example, gold and silver have zero real part of the permittivity at plasma frequency, which is in ultraviolet. However, by mixing the metals with dielectrics it should be possible to create the material with plasma frequency in the visible range.
In conclusion, we have studied the wave scattering by nonlinear multi-shell nanoparticles and demonstrated that the cloaking efficiency of the particles can be controlled by varying the intensity of the incident wave. The nonlinear response of such core-shell particles is enhanced significantly in the layer with near-zero dielectric permittivity. We have revealed that the scattering direction can be abruptly changed by the incident wave due to the energy exchange between the multipole modes of the structure.