## Nonlinear control of invisibility cloaking |

Optics Express, Vol. 20, Issue 14, pp. 14954-14959 (2012)

http://dx.doi.org/10.1364/OE.20.014954

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### Abstract

We introduce a new concept of the nonlinear control of invisibility cloaking. We study the scattering properties of multi-shell plasmonic nanoparticles with a nonlinear response of one of the shells, and demonstrate that the scattering cross-section of such particles can be controlled by a power of the incident electromagnetic radiation. More specifically, we can either increase or decrease the scattering cross-section by changing the intensity of the external field, as well as control the scattering efficiently and even reverse the radiation direction.

© 2012 OSA

*et al.*[1

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

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]

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]

5. A. Alu and N. Engheta, “Multifrequency optical invisibility cloak with layered plasmonic shells,” Phys. Rev. Lett. **100**, 113901 (2008). [CrossRef] [PubMed]

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]

*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.

*R*

_{1}with dielectric constant

*ε*

_{1}is coated by two layers, with external radii of

*R*

_{2}and

*R*

_{3}, and dielectric constants

*ε*

_{2}and

*ε*

_{3}, 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]

*ε*

_{2}= 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

*ε*

_{2}=

*ε*′

_{2}+

*iε*″

_{2}+

*α*|

*E*|

^{2}, where

*ε*′

_{2}and

*ε*″

_{2}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.

8. D. S. Filonov, A. P. Slobozhanyuk, P. A. Belov, and Yu. S. Kivshar, “Double-shell metamaterial coatings for plasmonic cloaking,” Phys. Status Solidi: Rapid Res. Lett. **6**, 46–48 (2012). [CrossRef]

*ε*′

_{2}on the cloaking performance.

*J*are Bessel functions of first kind,

_{m}*k*

_{0}=

*ω*/

*c*,

*ρ*=

*k*

_{0}

*r*. In each layer we represent the magnetic field as a sum of multipoles with azimuthal number

*m*:

*Y*are Bessel functions of second kind,

_{m}*ρ*=

_{j}*k*,

_{j}r*Ŝ*, which relates the wave amplitudes in the adjacent layers

*m*= 1) as well as the total SCS reach their minimum near our previously predicted point

*ε*′

_{2}= 0. At the same time, the total SCS remains finite, in our case

*S*

_{tot}*k*

_{0}= 0.03, where

*k*

_{0}=

*ω*/

*c*. Remarkably, the SCS grows much faster towards negative values of

*ε*

_{2}than to the positive values.

*ε*

_{2}= 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

*ε*

_{2}= −0.1+0.02

*i*. The corresponding dependencies are shown in Fig. 1(b). Minimum of the dipole-mode SCS is achieved close to

*ε*

_{3}≈ 3, which is different from the value expected from Eq. (1)

*ε*

_{3}= 2 because in these calculations the middle layer is absorbing.

*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.

*α*is negative, so that it can restore the best cloaking condition, when the dielectric constant vanishes. The second case is when

*ε*′

_{2}< 0 and

*α*> 0. We assume that the sizes of the shells are the same as in linear calculations, and

*ε*

_{1,2,3}= 15; ±0.1 + 0.02

*i*; 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

*δε*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 where

*β*has to be made empirically by looking on the convergence efficiency.

^{2}. 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/cm

^{2}. 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.

*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.

## Acknowledgments

## References and links

1. | J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science |

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 |

3. | A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E |

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. |

5. | A. Alu and N. Engheta, “Multifrequency optical invisibility cloak with layered plasmonic shells,” Phys. Rev. Lett. |

6. | B. Edwards, A. Alu, M. G. Silveirinha, and N. Engheta, “Experimental verification of plasmonic cloaking at microwave frequencies with metamaterials,” Phys. Rev. Lett. |

7. | A. A. Zharov and N. A. Zharova, “On the electromagnetic cloaking of (Nano)particles,” Bulletin of the Russian Academy of Sciences: Physics |

8. | D. S. Filonov, A. P. Slobozhanyuk, P. A. Belov, and Yu. S. Kivshar, “Double-shell metamaterial coatings for plasmonic cloaking,” Phys. Status Solidi: Rapid Res. Lett. |

**OCIS Codes**

(190.5940) Nonlinear optics : Self-action effects

(160.3918) Materials : Metamaterials

(290.5839) Scattering : Scattering, invisibility

**ToC Category:**

Metamaterials

**History**

Original Manuscript: May 10, 2012

Revised Manuscript: June 1, 2012

Manuscript Accepted: June 1, 2012

Published: June 19, 2012

**Citation**

Nina A. Zharova, Ilya V. Shadrivov, Alexander A. Zharov, and Yuri S. Kivshar, "Nonlinear control of invisibility cloaking," Opt. Express **20**, 14954-14959 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-14954

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### References

- J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science312, 1780–1782 (2006). [CrossRef] [PubMed]
- 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,” Science314, 977–980 (2006). [CrossRef] [PubMed]
- A. Alu and N. Engheta, “Achieving transparency with plasmonic and metamaterial coatings,” Phys. Rev. E72, 016623 (2005). [CrossRef]
- 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]
- A. Alu and N. Engheta, “Multifrequency optical invisibility cloak with layered plasmonic shells,” Phys. Rev. Lett.100, 113901 (2008). [CrossRef] [PubMed]
- 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]
- A. A. Zharov and N. A. Zharova, “On the electromagnetic cloaking of (Nano)particles,” Bulletin of the Russian Academy of Sciences: Physics74, 89–92 (2010). [CrossRef]
- D. S. Filonov, A. P. Slobozhanyuk, P. A. Belov, and Yu. S. Kivshar, “Double-shell metamaterial coatings for plasmonic cloaking,” Phys. Status Solidi: Rapid Res. Lett.6, 46–48 (2012). [CrossRef]

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