## Designs for optical cloaking with high-order transformations

Optics Express, Vol. 16, Issue 8, pp. 5444-5452 (2008)

http://dx.doi.org/10.1364/OE.16.005444

Acrobat PDF (464 KB)

### Abstract

Recent advances in metamaterial research have provided us a blueprint for realistic cloaking capabilities, and it is crucial to develop practical designs to convert concepts into real-life devices. We present two structures for optical cloaking based on high-order transformations for TM and TE polarizations respectively. These designs are possible for visible and infrared wavelengths. This critical development builds upon our previous work on nonmagnetic cloak designs and high-order transformations.

© 2008 Optical Society of America

## 1. Introduction

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

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

8. Y. Benveniste and T. Miloh, “Neutral inhomogeneities in conduction phenomena,” J. Mech. Phys. Solids **47**, 1873–1892 (1999). [CrossRef]

9. A. Hendi, J. Henn, and U. Leonhardt, “Ambiguities in the scattering tomography for central potentials,” Phys. Rev. Lett. **97**, 073902 (2006). [CrossRef] [PubMed]

13. 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]

## 2. Material properties in cylindrical cloaks

*r*=

*g*(

*r*′) from (

*r*′,

*θ*′,

*z*′) to (

*r*,

*θ*,

*z*) is used to compress the region

*r*′≤

*b*into a concentric shell of

*a*≤

*r*≤

*b*, and the permittivity and permeability tensors required for an exact cloak can be determined as [15

15. W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, “Nonmagnetic cloak with minimized scattering,” Appl. Phys. Lett. **91**, 111105 (2007). [CrossRef]

*ε*,

_{z}*μ*and

_{r}*μ*(

_{θ}*μ*,

_{z}*ε*and

_{r}*ε*) enter into Maxwell’s equations. Moreover, the parameters can be further simplified to form reduced parameters which are more realistic for practical applications. Since the trajectory of the waves is determined by the cross product components of the

_{θ}*ε*and

*μ*tensors instead of the two tensors individually, the cloaking performance is sustained as long as

13. 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]

14. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics **1**, 224–227 (2007). [CrossRef]

## 3. Optical cloak with high-order transformations I: TM mode

*ε*and

_{r}*ε*using readily available materials. Apparently, a cloak cannot consist of only a single-constituent material, because a spatial variation in material properties is critical to building a cloak. To start the design, we first examine the overall flexibility we can achieve in the effective permittivity of a general two-phase composite medium. When an external field interacts with a composite consisting of two elements with permittivity of

_{θ}*ε*and

_{1}*ε*respectively, minimal screening occurs when all internal boundaries between the two constituents are parallel to the electric field, and maximal screening happens when all boundaries aligned perpendicular to the field. These two extremes are possible in an alternating layered structure, provided that the thickness of each layer is much less than the wavelength of incidence [17

_{2}17. D. E. Aspnes, “Optical-Properties of Thin-Films,” Thin Solid Films **89**, 249–262 (1982). [CrossRef]

*f*and 1-

*f*denote the volume fractions of components 1 and 2, and the subscripts ‖ and ⊥ indicate the cases with electric field polarized parallel and perpendicular to the interfaces of the layers, respectively. Such layered structures have been studied extensively in recent years for various purposes, especially in sub-diffraction imaging for both the near field and the far zone [18–23].

25. D. E. Aspnes, “Bounds on Allowed Values of the Effective Dielectric Function of 2-Component Composites at Finite Frequencies,” Phys. Rev. B **25**, 1358–1361 (1982). [CrossRef]

26. D. J. Bergman, “Exactly Solvable Microscopic Geometries and Rigorous Bounds for the Complex Dielectric-Constant of a 2-Component Composite-Material,” Phys. Rev. Lett. **44**, 1285–1287 (1980). [CrossRef]

27. G. W. Milton, “Bounds on the Complex Dielectric-Constant of a Composite-Material,” Appl. Phys. Lett. **37**, 300–302 (1980). [CrossRef]

*ε*-plane with the real and imaginary parts of

*ε*being the

*x*and

*y*axis, respectively. In this plane, the low-screening bound in Eq. (4a) corresponds to a straight line between

*ε*and

_{1}*ε*, and the high-screening bound in Eq. (4b) defines an arc which is part of the circle determined by the three points

_{2}*ε*,

_{1}*ε*and the origin.

_{2}*ε*varies from 0 at the inner boundary of the cloak (

_{r}*r*=

*a*) to 1 at the outer surface (

*r*=

*b*), while

*ε*is a function of

_{θ}*r*with varying positive value, except for the linear transformation case where ∂

*g*(

*r*′)∂

*r*′ is a constant. Now we can evaluate the possibility of fulfilling the required parameters in Eq. (3) based on alternating metal-dielectric slices whose properties are estimated by Eq. (4). With phase 1 being a metal (

*ε*=

_{1}*ε*<0) and phase 2 representing a dielectric (

_{m}*ε*=

_{2}*ε*>0), the desired material properties of the cloak are only possible when the slices are within the

_{d}*r*-

*z*plane of the cylindrical coordinates. In this case

*ε*and

_{r}*ε*correspond to

_{θ}*ε*

_{‖}and

*ε*

_{⊥}in Eq. (4), respectively. This scenario is illustrated in Fig. 1. The thick solid and dashed lines represent the two Wiener bounds

*ε*

_{‖}(

*f*) and

*ε*

_{⊥}(

*f*) respectively. The constituent materials used for the calculation are silver and silica at a green light wavelength of 532 nm. The pair of points on the bounds with the same filling fraction are connected with a straight line for clarification purposes. When

*ε*changes between 0 and 1, the value of

_{r}*ε*varies accordingly as shown by the arrow between the two thin dashed lines. Therefore, the construction of a non-magnetic cloak requires that the relationship between the two quantities

_{θ}*ε*

_{‖}and

*ε*

_{⊥}(as functions of

*f*) within the range shown in Fig. 1 fits the material properties given in Eq. (3) for a particular transformation function

*r*=

*g*(

*r*′). Another attractive feature of the proposed scheme is the minimal loss factor. As shown in Fig. 1, the loss feature described by the imaginary part of the effective permittivity is on the order of 0.01, much smaller than that of a pure metal or any resonant metal-dielectric structures. A schematic of the proposed structure consisting of interlaced metal and dielectric slices is illustrated in Fig. 2.

*a*/

*b*that fulfills the following equation:

28. P. B. Johnson and R. W. Christy, “Optical-Constants of Noble-Metals,” Phys. Rev. B **6**, 4370–4379 (1972). [CrossRef]

*a*/

*b*. When the approximate quadratic function is fixed for a given wavelength, the filling fraction function

*f*(

*r*) is determined by the following equation:

*λ*=532 nm case in Table 1. Our calculation shows that with the approximate quadratic transformation, the effective parameters

*ε*and

_{r}*ε*obtained with the Wiener bounds in Eq. (4) fit with the exact parameters required for this transformation by Eq. (2) remarkably well, with the average deviation of less than 0.5%.

_{θ}14. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics **1**, 224–227 (2007). [CrossRef]

## 4. Optical cloak with high-order transformations II: TE mode

^{-1}), which introduces a sharp Lorentz behavior in its electric permittivity. The dielectric function of SiC at mid-infrared is well described with the following model [30

30. W. G. Spitzer, D. Kleinman, and D. Walsh, “Infrared Properties of Hexagonal Silicon Carbide,” Phys. Rev. **113**, 127–132 (1959). [CrossRef]

31. D. Korobkin, Y. Urzhumov, and G. Shvets, “Enhanced near-field resolution in midinfrared using metamaterials,” J. Opt. Soc. Am. B **23**, 468–478 (2006). [CrossRef]

*∊*

_{∞}=6.5,

*ω*=972

_{L}*cm*

^{-1},

*ω*=796

_{T}*cm*

^{-1}and

*γ*=5

*cm*

^{-1}. On the high-frequency side, the dielectric function is strongly negative, which makes its optical response similar to that of metals and has been utilized in applications like a mid-infrared superlens [31

31. D. Korobkin, Y. Urzhumov, and G. Shvets, “Enhanced near-field resolution in midinfrared using metamaterials,” J. Opt. Soc. Am. B **23**, 468–478 (2006). [CrossRef]

32. T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a SiC superlens,” Science **313**, 1595–1595 (2006). [CrossRef] [PubMed]

33. J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett. **99**, 107401 (2007). [CrossRef] [PubMed]

14. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics **1**, 224–227 (2007). [CrossRef]

*ε*polaritonic material like SiC or TiO

_{2}are embedded in an IR transparent dielectric like ZnS. The non-magnetic cloak using alternating slices structure as proposed in this paper provides a more realistic design. With SiC as the negative-

*ε*material and BaF

_{2}as the positive-

*ε*slices with material properties given in [29], we can find the appropriate transformation function and shape factor that fulfills the material property requirements at a preset wavelength. The result for

*λ*=11.3 µm (CO

_{2}laser range) is shown in the last row of Table 1.

*µ*varies from 0 at the inner boundary (

_{r}*r*=

*a*) to [∂

*g*(

*r*′)/∂

*r*′]

^{2}at the outer surface (

*r*=

*b*), while the required

*ε*changes accordingly following the function [∂

_{z}*g*(

*r*′)/∂

*r*′]

^{-2}. The magnetic requirement may be accomplished using metal elements like split-ring resonators, coupled nanostrips or nanowires. However, such plasmonic structures inevitably exhibit a high loss, which is detrimental to the cloaking performance. A SiC based structure provides an all-dielectric route to a magnetic cloak for the TE mode due to the Mie resonance in a subwavelength SiC unit.

33. J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett. **99**, 107401 (2007). [CrossRef] [PubMed]

*ε*should be avoided for a minimal loss. Therefore, with the electrical field polarized along the

_{z}*z*axis of the cylindrical system, we arrange the SiC rods along the

*r*axis and form an array in the

*θ*-

*z*plane. The proposed structure is depicted in Fig. 4, where arrays of SiC wires along the radial direction are placed between the two surfaces of the cylindrical cloak.

34. S. O’Brien and J. B. Pendry, “Photonic band-gap effects and magnetic activity in dielectric composites,” J. Phys. Condens. Matter. **14**, 4035–4044 (2002). [CrossRef]

*h*and

*φ*represent the periodicities along the

*z*and

*θ*directions respectively,

*t*denotes the radius of each wire,

*k*=2π/λ

_{0}denotes the wave vector,

*a*

_{0}=[

*nJ*

_{0}(

*nkt*)

*J*

_{1}(

*kt*)-

*J*

_{0}(

*kt*)

*J*

_{1}(

*nkt*)]/[

*nJ*

_{0}(

*nkt*)

*H*

^{(1)}

_{1}(

*kt*)-

*H*

^{(1)}

_{0}(

*kt*)

*J*

_{1}(

*nkt*)] and

*c*

_{0}=[

*J*

_{0}(

*kt*)-

*a*

_{0}

*H*

^{(1)}

_{0}(

*kt*)]/

*J*

_{0}(

*nkt*) are the scattering coefficients, and the Bessel functions in the equation follow the standard notations. The permittivity along the

*z*direction is well approximated using Maxwell-Garnett method [34

34. S. O’Brien and J. B. Pendry, “Photonic band-gap effects and magnetic activity in dielectric composites,” J. Phys. Condens. Matter. **14**, 4035–4044 (2002). [CrossRef]

*μ*and

_{r}*ε*follow what is required by Eq. (2) with tolerable deviations. In Fig. 5 we plot the required and the calculated

_{z}*μ*and

_{r}*ε*for a TE cloak at

_{z}*λ*=13.5 µm. The parameters used for this calculation are

*a*=15 µm,

*a*/

*b*=0.35,

*t*=1.2 µm,

*h*=2.8 µm,

*φ*=10.6°, and the

*p*coefficient in the quadratic transformation is 0.5

*a*/

*b*

^{2}. We observe very good agreement between the required values and the calculated ones based on analytical formulae, and the imaginary part in the effective permeability is less than 0.06. This computation verifies the feasibility of the proposed cloaking system based on SiC wire arrays for the TE polarization.

## 5. Conclusions

## Acknowledgments

## References and links

1. | G. W. Milton and N. A. P. Nicorovici, “On the cloaking effects associated with anomalous localized resonance,” Proc. R. Soc. London, Ser. A |

2. | N. A. P. Nicorovici, G. W. Milton, R. C. McPhedran, and L. C. Botten, “Quasistatic cloaking of two-dimensional polarizable discrete systems by anomalous resonance,” Opt. Express |

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

4. | M. G. Silveirinha, A. Alu, and N. Engheta, “Parallel-plate metamaterials for cloaking structures,” Phys. Rev. B |

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

6. | F. J. Garcia de Abajo, G. Gomez-Santos, L. A. Blanco, A. G. Borisov, and S. V. Shabanov, “Tunneling mechanism of light transmission through metallic films,” Phys. Rev. Lett. |

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

8. | Y. Benveniste and T. Miloh, “Neutral inhomogeneities in conduction phenomena,” J. Mech. Phys. Solids |

9. | A. Hendi, J. Henn, and U. Leonhardt, “Ambiguities in the scattering tomography for central potentials,” Phys. Rev. Lett. |

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

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

12. | D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express |

13. | 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 |

14. | W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics |

15. | W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, “Nonmagnetic cloak with minimized scattering,” Appl. Phys. Lett. |

16. | R. Weder, “A rigorous analysis of high-order electromagnetic invisibility cloaks,” J. Phys. A: Math. Theor. |

17. | D. E. Aspnes, “Optical-Properties of Thin-Films,” Thin Solid Films |

18. | S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, “Imaging the near field,” J. Mod. Opt. |

19. | D. Schurig and D. R. Smith, “Sub-diffraction imaging with compensating bilayers,” New J. Phys. |

20. | P. A. Belov and Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B |

21. | S. M. Feng and J. M. Elson, “Diffraction-suppressed high-resolution imaging through metallodielectric nanofilms,” Opt. Express |

22. | Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express |

23. | A. Salandrino and N. Engheta, “Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations,” Phys. Rev. B |

24. | O. Wiener, “Die Theorie des Mischkorpers fur das Feld der stationaren Stromung,” Abh. Math.-Phys. Klasse Koniglich Sachsischen Des. Wiss. |

25. | D. E. Aspnes, “Bounds on Allowed Values of the Effective Dielectric Function of 2-Component Composites at Finite Frequencies,” Phys. Rev. B |

26. | D. J. Bergman, “Exactly Solvable Microscopic Geometries and Rigorous Bounds for the Complex Dielectric-Constant of a 2-Component Composite-Material,” Phys. Rev. Lett. |

27. | G. W. Milton, “Bounds on the Complex Dielectric-Constant of a Composite-Material,” Appl. Phys. Lett. |

28. | P. B. Johnson and R. W. Christy, “Optical-Constants of Noble-Metals,” Phys. Rev. B |

29. | E. D. Palik, |

30. | W. G. Spitzer, D. Kleinman, and D. Walsh, “Infrared Properties of Hexagonal Silicon Carbide,” Phys. Rev. |

31. | D. Korobkin, Y. Urzhumov, and G. Shvets, “Enhanced near-field resolution in midinfrared using metamaterials,” J. Opt. Soc. Am. B |

32. | T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a SiC superlens,” Science |

33. | J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett. |

34. | S. O’Brien and J. B. Pendry, “Photonic band-gap effects and magnetic activity in dielectric composites,” J. Phys. Condens. Matter. |

35. | K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, “Negative effective permeability in polaritonic photonic crystals,” Appl. Phys. Lett. |

36. | M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B |

37. | L. Peng, L. X. Ran, H. S. Chen, H. F. Zhang, J. A. Kong, and T. M. Grzegorczyk, “Experimental observation of left-handed behavior in an array of standard dielectric resonators,” Phys. Rev. Lett. |

**OCIS Codes**

(160.4760) Materials : Optical properties

(160.3918) Materials : Metamaterials

(230.3205) Optical devices : Invisibility cloaks

**ToC Category:**

Metamaterials

**History**

Original Manuscript: February 11, 2008

Revised Manuscript: March 31, 2008

Manuscript Accepted: April 1, 2008

Published: April 3, 2008

**Citation**

Wenshan Cai, Uday K. Chettiar, Alexander V. Kildishev, and Vladimir M. Shalaev, "Designs for optical cloaking with high-order transformations," Opt. Express **16**, 5444-5452 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-8-5444

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### References

- G. W. Milton, and N. A. P. Nicorovici, "On the cloaking effects associated with anomalous localized resonance," Proc. R. Soc. London, Ser. A 462, 3027-3059 (2006). [CrossRef]
- N. A. P. 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). [CrossRef] [PubMed]
- A. Alu and N. Engheta, "Achieving transparency with plasmonic and metamaterial coatings," Phys. Rev. B 72, 016623 (2005). [CrossRef]
- M. G. Silveirinha, A. Alu, and N. Engheta, "Parallel-plate metamaterials for cloaking structures," Phys. Rev. B 75, 036603 (2007). [CrossRef]
- D. A. B. Miller, "On perfect cloaking," Opt. Express 14, 12457-12466 (2006). [CrossRef] [PubMed]
- F. J. Garcia de Abajo, G. Gomez-Santos, L. A. Blanco, A. G. Borisov, and S. V. Shabanov, "Tunneling mechanism of light transmission through metallic films," Phys. Rev. Lett. 95, 067403 (2005). [CrossRef] [PubMed]
- A. Greenleaf, M. Lassas, and G. Uhlmann, "Anisotropic conductivities that cannot be detected by EIT," Physiol. Meas. 24, 413-419 (2003). [CrossRef] [PubMed]
- Y. Benveniste and T. Miloh, "Neutral inhomogeneities in conduction phenomena," J. Mech. Phys. Solids 47, 1873-1892 (1999). [CrossRef]
- A. Hendi, J. Henn, and U. Leonhardt, "Ambiguities in the scattering tomography for central potentials," Phys. Rev. Lett. 97, 073902 (2006). [CrossRef] [PubMed]
- 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. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794-9804 (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]
- W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with metamaterials," Nat. Photonics 1, 224-227 (2007). [CrossRef]
- W. Cai, U. K. Chettiar, A. V. Kildishev, V. M. Shalaev, and G. W. Milton, "Nonmagnetic cloak with minimized scattering," Appl. Phys. Lett. 91, 111105 (2007). [CrossRef]
- R. Weder, "A rigorous analysis of high-order electromagnetic invisibility cloaks," J. Phys. A: Math. Theor. 41, 065207 (2008). [CrossRef]
- D. E. Aspnes, "Optical-Properties of Thin-Films," Thin Solid Films 89, 249-262 (1982). [CrossRef]
- S. A. Ramakrishna, J. B. Pendry, M. C. K. Wiltshire, and W. J. Stewart, "Imaging the near field," J. Mod. Opt. 50, 1419-1430 (2003).
- D. Schurig, and D. R. Smith, "Sub-diffraction imaging with compensating bilayers," New J. Phys. 7, 162 (2005). [CrossRef]
- P. A. Belov, and Y. Hao, "Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime," Phys. Rev. B 73, 113110 (2006). [CrossRef]
- S. M. Feng, and J. M. Elson, "Diffraction-suppressed high-resolution imaging through metallodielectric nanofilms," Opt. Express 14, 216-221 (2006). [CrossRef] [PubMed]
- Z. Jacob, L. V. Alekseyev, and E. Narimanov, "Optical hyperlens: Far-field imaging beyond the diffraction limit," Opt. Express 14, 8247-8256 (2006). [CrossRef] [PubMed]
- A. Salandrino, and N. Engheta, "Far-field subdiffraction optical microscopy using metamaterial crystals: Theory and simulations," Phys. Rev. B 74, 075103 (2006). [CrossRef]
- O. Wiener, "Die Theorie des Mischkorpers fur das Feld der stationaren Stromung," Abh. Math.-Phys. Klasse Koniglich Sachsischen Des. Wiss. 32, 509-604 (1912).
- D. E. Aspnes, "Bounds on Allowed Values of the Effective Dielectric Function of 2-Component Composites at Finite Frequencies," Phys. Rev. B 25, 1358-1361 (1982). [CrossRef]
- D. J. Bergman, "Exactly Solvable Microscopic Geometries and Rigorous Bounds for the Complex Dielectric-Constant of a 2-Component Composite-Material," Phys. Rev. Lett. 44, 1285-1287 (1980). [CrossRef]
- G. W. Milton, "Bounds on the Complex Dielectric-Constant of a Composite-Material," Appl. Phys. Lett. 37, 300-302 (1980). [CrossRef]
- P. B. Johnson, and R. W. Christy, "Optical-Constants of Noble-Metals," Phys. Rev. B 6, 4370-4379 (1972). [CrossRef]
- E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, New York, 1997).
- W. G. Spitzer, D. Kleinman, and D. Walsh, "Infrared Properties of Hexagonal Silicon Carbide," Phys. Rev. 113, 127-132 (1959). [CrossRef]
- D. Korobkin, Y. Urzhumov, and G. Shvets, "Enhanced near-field resolution in midinfrared using metamaterials," J. Opt. Soc. Am. B 23, 468-478 (2006). [CrossRef]
- T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, "Near-field microscopy through a SiC superlens," Science 313, 1595-1595 (2006). [CrossRef] [PubMed]
- J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, "Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles," Phys. Rev. Lett. 99, 107401 (2007). [CrossRef] [PubMed]
- S. O'Brien and J. B. Pendry, "Photonic band-gap effects and magnetic activity in dielectric composites," J. Phys. Condens. Matter. 14, 4035-4044 (2002). [CrossRef]
- K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, "Negative effective permeability in polaritonic photonic crystals," Appl. Phys. Lett. 85, 543-545 (2004). [CrossRef]
- M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, "Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies," Phys. Rev. B 72, 193103 (2005). [CrossRef]
- L. Peng, L. X. Ran, H. S. Chen, H. F. Zhang, J. A. Kong, and T. M. Grzegorczyk, "Experimental observation of left-handed behavior in an array of standard dielectric resonators," Phys. Rev. Lett. 98, 157403 (2007). [CrossRef] [PubMed]

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