## Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance

Optics Express, Vol. 15, Issue 10, pp. 6314-6323 (2007)

http://dx.doi.org/10.1364/OE.15.006314

Acrobat PDF (4059 KB)

### Abstract

Discrete systems of infinitely long polarizable line dipoles are considered in the quasistatic limit, interacting with a two-dimensional cloaking system consisting of a hollow plasmonic cylindrical shell. A numerical procedure is described for accurately calculating electromagnetic fields arising in the quasistatic limit, for the case when the relative permittivity of the cloaking shell has a very small imaginary part. Animations are given which illustrate cloaking of discrete systems, both for the case of induced dipoles and induced quadrupoles on the interacting particles. The simulations clarify the physical mechanism for the cloaking.

© 2007 Optical Society of America

## 1. Introduction

*et al*. [1

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

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

3. 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]

*et al*. [4

4. A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that
cannot be detected by EIT,” Physiol.
Meas. **24**413–419 (2003). [CrossRef] [PubMed]

5. A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Full-wave invisibility of active devices at all frequencies,” http://arxiv.org/abs/math.AP/0611185.

*et al*. [6

6. G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and
physical equations with a transformation invariant
form,” New. J. Phys. **8**, 248–267
(2006). [CrossRef]

*et al*. [7

7. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical Cloaking with Non-Magnetic Metamaterials,” http://arxiv.org/pdf/physics/0611242.

15. M. Kerker, “Invisible
bodies,” J. Opt. Soc. Am. **65**, 376–379
(1975). [CrossRef]

16. A. Alu and N. Engheta, “Pairing an epsilon-negative slab
with a mu-negative slab: resonance, tunneling and
transparency,” IEEE Trans. Antennas Propag. **51**, 2558–2571
(2003). [CrossRef]

17. A. Alu and N. Engheta, “Achieving transparency with
plasmonic and metamaterial coatings,”
Phys. Rev. E **72**, 016623 (2005). [CrossRef]

18. A. G. Ramm, “Invisible
obstacles,” Ann. Polon. Math. **90**, 145–148
(2007). [CrossRef]

19. D. A. B. Miller, “On perfect
cloaking,” Opt. Express **14**, 12457–12466
(2006). [CrossRef] [PubMed]

20. P. Sheng, “Waves on the
horizon,” Science **313**, 1399–1400
(2006). [CrossRef] [PubMed]

21. P. Weiss, “Out of Sight: Physicists get serious
about invisibility shields,” Science News **170**, 42–44
(2006). [CrossRef]

13. G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated
with anomalous localized resonance,”
Proc. Roy. Soc. A **462**, 3027–3059
(2006). [CrossRef]

## 2. Theoretical considerations

*N*polarizable line dipoles placed at the points

*P*

_{1},

*P*

_{2},…

*P*,…

_{j}*P*,

_{N}*P*= (

_{j}*x*,

_{j}*y*), outside the cylinder. The coated cylinder has a core of radius

_{j}*r*and a relative permittivity

_{c}*ε*= 1, and a shell of radius

_{c}*r*and an almost lossless relative permittivity

_{s}*ε*close to -1. The medium outside the cylinder has a relative permittivity

_{s}*ε*= 1. An external electric field is applied to this system (see Fig. 1). Note that this is an electrostatic treatment of a quasistatic problem. By quasistatic we mean that the diameter of the coated cylinder is sufficiently small, and hence that the gradients of the fields are sufficiently large, that the spatial parts of the fields may be derived from the solution of the Laplace equation rather than the Helmholtz equation.

_{m}*D*of the matrix (exterior to the shell and extending as far as the nearest source), the complex potential is the solution of the Laplace equation. For example, in region

*D*(see Fig. 1), it has the form [12

12. G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, and V. A. Podolskiy, “A proof of superlensing in the
quasistatic regime, and limitations of superlenses in this regime due to
anomalous localized resonance,” Proc. R.
Soc. Lond. A **461**, 3999–4034
(2005). [CrossRef]

13. G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated
with anomalous localized resonance,”
Proc. Roy. Soc. A **462**, 3027–3059
(2006). [CrossRef]

22. Supporting Online Material, http://www.physics.usyd.edu.au/cudos/research/plasmon.html.

*θ*is measured from the x-axis. The superscripts ‘e’ and ‘o’ denote functions which are, respectively, symmetric and antisymmetric under the transformation

*θ*→-

*θ*. Note that because

*ε*, is complex, the coefficients in (1) are also complex.

_{S}*B*in Eq. (1) may be expressed in terms of

^{e/o}_{l}*A*

^{e}_{0},

*A*, and

^{e}_{l}*A*.

^{o}_{l}*in the matrix and outside the cylindrical lens, the total complex potential has the form*

**z***V*

_{appl}(

*) is the potential of the external electric field,*

**z***V*(

_{n}*) is the potential of the induced dipole moment of particle*

**z***n*,

*V*̃

_{appl}(

*) represents the response of the cylindrical lens to the external field, and*

**z***V*̃

*n*(

*) represents the response of the cylindrical lens to the electric field of the*

**z***n*

^{th}induced dipole moment. Note that when writing equation (2) we understand that, in fact, each term is expressed in terms of analytic functions of

*and*

**z***̄, as shown in [22*

**z**22. Supporting Online Material, http://www.physics.usyd.edu.au/cudos/research/plasmon.html.

*j*is characterized by its moment

**d**

*= (*

_{j}*d*

_{x}^{(j)},

*d*

_{y}^{(j)}) which is given by the equation

*α*is the polarizability of the dipole, and

_{j}*=*

**z***is due to the applied field, the response to it from the cylinder, fields coming from all other dipoles, and the response of the cylinder to all dipoles. Equations of the form (3) for all*

**z**_{j}*j*= 1,2,…,

*N*constitute a system of linear equations that we solve for the dipole moments [22

22. Supporting Online Material, http://www.physics.usyd.edu.au/cudos/research/plasmon.html.

*ε*). This cannot be set equal to zero, since the problem of the interaction between the applied field, the polarizable line dipoles and the coated cylinder then has no convergent solution. However, it should be kept as small as possible, since otherwise the cloaking is much less distinct. In the results given here, we have chosen it to be typically 10

_{s}^{-12}.

*α*) with their configuration such that a strong resonance occurs, which can mask the main features of the equipotential lines, and also the interaction with the cylindrical lens. (For an isolated pair of polarizable line dipoles this resonance occurs when

*α*is the square of the interparticle distance). This difficulty is overcome by adjusting a to an appropriate value. (In [movies 3] and [4] the value of

*α*has been chosen anomalously large to make the cloaking effects more evident.)

*Mathematica*which, for small polarizable systems, is able to produce results of very high accuracy, even close to singular points. However, the computational efficiency was very poor and so we rewrote the code using the utility

*MathLink*so as to perform all of the demanding numerical calculations in Fortran, employing

*Mathematica*only to control the flow of operations and produce the final equipotential contour plots. Using this method the time required for simulations was decreased by up to a factor of approximately 100, making possible the animations of quite large systems of polarizable line dipoles discussed below. Using an Intel Pentium M processor of 1.86 GHz with Windows XP, the timing to obtain the contour plot in Fig. 6 (right panel), with 50,000 plot points for 26 line dipoles was 296 seconds.

*V*exp(-

*iωt*)) this means that we illustrate ”snap-shots” of the physical potential at a subset of those times

*t*such that

_{m}*ωt*= 2

_{m}*mπ*, for integer

*m*.

## 3. Numerical simulations of cloaking

**d**

^{(j)}= (

*d*

_{x}^{(j)},

*d*

_{y}^{(j)}) is the dipole moment of the dipole at

*=*

**z**_{j}*r*exp(i

_{j}*θ*).

_{j}*r*

_{#}= √(

*r*

_{s}^{3}/

*r*) defined in Refs.[12

_{c}12. G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, and V. A. Podolskiy, “A proof of superlensing in the
quasistatic regime, and limitations of superlenses in this regime due to
anomalous localized resonance,” Proc. R.
Soc. Lond. A **461**, 3999–4034
(2005). [CrossRef]

13. G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated
with anomalous localized resonance,”
Proc. Roy. Soc. A **462**, 3027–3059
(2006). [CrossRef]

*r*

_{#}, we first see core-shell resonances, and then shell-matrix resonances, which rapidly quench the dipole moment of the particle (the resonant regions of alternating blue and red bands are where the potential is oscillatory and enormous, and has been truncated to avoid crowding of equipotential lines). Within the cloaking region

*r*<

*r*

_{#}, the resonances diminish, while the equipotential lines outside

*r*

_{#}are accurately those of the applied field alone. In Figs 3 and 4 we show the equipotential lines together with the electric field lines (Fig. 3) and the direction of the electric field (Fig. 4). Note the reversal of direction of electric field lines in Fig. 4 at the shell boundaries, a consequence of its negative relative permittivity, and also that the electric field lines in Fig. 3 are essentially equivalent to electric displacement field lines.

*x*axis. For

*x*≥ 0 one polarizable dipole is always further to the left than the other.

*r*

_{#}. Note that when the leftmost particle is quenched, the other particle no longer interacts with it, but only with the cloaking cylinder, which however is effectively invisible to the other particle beyond the radius

*r*

_{#}.

*r*

_{#}are quite close to those of the applied field when the cluster is deepest into the cloaking region (halfway through the movie).

*r*

_{#}, and that the regions of most evident distortion are (not surprisingly) associated with regions of tightest curvature on the silhouette.

## 4. Conclusion

12. G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, and V. A. Podolskiy, “A proof of superlensing in the
quasistatic regime, and limitations of superlenses in this regime due to
anomalous localized resonance,” Proc. R.
Soc. Lond. A **461**, 3999–4034
(2005). [CrossRef]

## References and links

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

2. | U. Leonhardt, “Optical conformal
mapping,” Science |

3. | 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 |

4. | A. Greenleaf, M. Lassas, and G. Uhlmann, “Anisotropic conductivities that
cannot be detected by EIT,” Physiol.
Meas. |

5. | A. Greenleaf, Y. Kurylev, M. Lassas, and G. Uhlmann, “Full-wave invisibility of active devices at all frequencies,” http://arxiv.org/abs/math.AP/0611185. |

6. | G. W. Milton, M. Briane, and J. R. Willis, “On cloaking for elasticity and
physical equations with a transformation invariant
form,” New. J. Phys. |

7. | W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical Cloaking with Non-Magnetic Metamaterials,” http://arxiv.org/pdf/physics/0611242. |

8. | N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, “Optical and dielectric properties of
partially resonant composites,” Phys.
Rev. B |

9. | T. J. Cui, Q. Cheng, W. B. Lu, Q. Jiang, and J. A. Kong, “Localization of electromagnetic
energy using a left-handed-medium slab,”
Phy. Rev. B |

10. | A. D. Boardman and K. Marinov, “Non-radiating and radiating
configurations driven by left-handed
metamate-rials,” J. Opt. Soc. Am. B |

11. | V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, and N. I. Zheludev, “Planar electromagnetic metamaterial
with a fish scale structure,” Phys. Rev.
E |

12. | G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, and V. A. Podolskiy, “A proof of superlensing in the
quasistatic regime, and limitations of superlenses in this regime due to
anomalous localized resonance,” Proc. R.
Soc. Lond. A |

13. | G. W. Milton and N.-A. P. Nicorovici, “On the cloaking effects associated
with anomalous localized resonance,”
Proc. Roy. Soc. A |

14. | G. W. Milton, N.-A. P. Nicorovici, and R. C. McPhedran, “Opaque perfect
lenses,” Physica B |

15. | M. Kerker, “Invisible
bodies,” J. Opt. Soc. Am. |

16. | A. Alu and N. Engheta, “Pairing an epsilon-negative slab
with a mu-negative slab: resonance, tunneling and
transparency,” IEEE Trans. Antennas Propag. |

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

18. | A. G. Ramm, “Invisible
obstacles,” Ann. Polon. Math. |

19. | D. A. B. Miller, “On perfect
cloaking,” Opt. Express |

20. | P. Sheng, “Waves on the
horizon,” Science |

21. | P. Weiss, “Out of Sight: Physicists get serious
about invisibility shields,” Science News |

22. | Supporting Online Material, http://www.physics.usyd.edu.au/cudos/research/plasmon.html. |

23. | O. P. Bruno and S. Lintner, “Superlens-cloaking of small dielectric bodies in the quasistatic regime,” submitted. |

**OCIS Codes**

(160.4760) Materials : Optical properties

(260.2110) Physical optics : Electromagnetic optics

(260.5740) Physical optics : Resonance

(350.7420) Other areas of optics : Waves

**ToC Category:**

Physical Optics

**History**

Original Manuscript: March 22, 2007

Revised Manuscript: May 3, 2007

Manuscript Accepted: May 4, 2007

Published: May 7, 2007

**Citation**

N. A. Nicorovici, G. W. Milton, R. C. McPhedran, and L. C. Botten, "Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance," Opt. Express **15**, 6314-6323 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-10-6314

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### References

- J. B. Pendry, D. Schurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006). [CrossRef] [PubMed]
- U. Leonhardt, "Optical conformal mapping," Science 312, 1777-1780 (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," Science 314, 977-980 (2006). [CrossRef] [PubMed]
- A. Greenleaf, M. Lassas, and G. Uhlmann, "Anisotropic conductivities that cannot be detected by EIT," Physiol. Meas. 24413-419 (2003). [CrossRef] [PubMed]
- A. Greenleaf, Y. Kurylev,M. Lassas, and G. Uhlmann, "Full-wave invisibility of active devices at all frequencies," http://arxiv.org/abs/math.AP/0611185.
- G.W. Milton,M. Briane, and J. R. Willis, "On cloaking for elasticity and physical equations with a transformation invariant form," New Journal of Physics 8, 248-267 (2006). [CrossRef]
- W. Cai, U. K. Chettiar, A. V. Kildishev, and V.M. Shalaev, "Optical Cloaking with Non-MagneticMetamaterials," http://arxiv.org/pdf/physics/0611242.
- N. A. Nicorovici, R. C. McPhedran, and G. W. Milton, "Optical and dielectric properties of partially resonant composites," Phys. Rev. B 490, 8479-8482 (1994). [CrossRef]
- T. J. Cui, Q. Cheng,W. B. Lu, Q. Jiang, J. A. Kong, "Localization of electromagnetic energy using a left-handedmedium slab," Phy. Rev. B 71, 045114 (2005). [CrossRef]
- A. D. Boardman and K. Marinov, "Non-radiating and radiating configurations driven by left-handed metamaterials," J. Opt. Soc. Am. B 23, 543-552 (2006). [CrossRef]
- V. A. Fedotov, P. L. Mladyonov, S. L. Prosvirnin, and N. I. Zheludev, "Planar electromagnetic metamaterial with a fish scale structure," Phys. Rev. E 72, 056613 (2005). [CrossRef]
- G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, and V. A. Podolskiy, "A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance," Proc. R. Soc. Lond. A 461, 3999-4034 (2005). [CrossRef]
- G. W. Milton and N.-A. P. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. Roy. Soc. A 462, 3027-3059 (2006). [CrossRef]
- G. W. Milton, N.-A. P. Nicorovici, and R. C. McPhedran, "Opaque perfect lenses," Physica B, in press, http://www.arxiv.org/abs/physics/0608225. [CrossRef]
- M. Kerker, "Invisible bodies," J. Opt. Soc. Am. 65, 376-379 (1975). [CrossRef]
- A. Alu and N. Engheta, "Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency," IEEE Trans. Antennas Propag. 51, 2558-2571 (2003). [CrossRef]
- A. Alu and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. E 72, 016623 (2005). [CrossRef]
- A. G. Ramm, "Invisible obstacles," Ann. Polon. Math. 90, 145-148 (2007). [CrossRef]
- D. A. B. Miller, "On perfect cloaking," Opt. Express 14, 12457-12466 (2006). [CrossRef] [PubMed]
- P. Sheng, "Waves on the horizon," Science 313, 1399-1400 (2006). [CrossRef] [PubMed]
- P. Weiss, "Out of Sight: Physicists get serious about invisibility shields," Science News 170, 42-44 (2006). [CrossRef]
- Supporting Online Material, http://www.physics.usyd.edu.au/cudos/research/plasmon.html.
- O. P. Bruno and S. Lintner, "Superlens-cloaking of small dielectric bodies in the quasistatic regime," submitted.

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