OSA's Digital Library

Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 6 — Mar. 17, 2008
  • pp: 3986–3992
« Show journal navigation

An all-dielectric route for terahertz cloaking

Davy P. Gaillot, Charles Croënne, and Didier Lippens  »View Author Affiliations


Optics Express, Vol. 16, Issue 6, pp. 3986-3992 (2008)
http://dx.doi.org/10.1364/OE.16.003986


View Full Text Article

Acrobat PDF (609 KB)





Browse Journals / Lookup Meetings

Browse by Journal and Year


   


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools

Share
Citations

Abstract

An original all-dielectric design that performs cloaking at 0.58 THz is demonstrated. The cloak consists of radially positioned micrometer-sized ferroelectric cylinders which exhibit under Mie theory a strong magnetic resonance. Full-wave simulations coupled with a field-summation retrieval technique were employed to adjust the rods magnetic plasma frequency; hence, the radial distribution in the permeability of the cloak. The behavior of the complete micro-structured device was simulated and results unambiguously show good reconstruction of the E-field wavefronts behind the cloak with high power transmission. This all-dielectric configuration provides an attractive route for designing cloaking devices at microwave and terahertz frequencies.

© 2008 Optical Society of America

1. Introduction

Conformal transformation of electromagnetic domains has been proposed as an exciting approach to control the flow of propagating waves [1

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

, 2

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

]. It enables the design of objects displaying unprecedented functionalities [3

3. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9803 (2006). [CrossRef] [PubMed]

] with the requirement of space gradients and anisotropy of the constitutive parameters (permittivity and permeability). Unfortunately, the fabrication of structures with such properties is challenging for the simple reason that they can not be found in Nature or synthesized in bulk. On the other hand, it is well established that strong anisotropy is obtained by structuring dielectric [4

4. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987). [CrossRef] [PubMed]

, 5

5. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]

] or metallic matter. Also, the plasma electric [6

6. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996). [CrossRef] [PubMed]

] and magnetic frequencies [7

7. J. B. Pendry, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999). [CrossRef]

] can be tailored. Originally introduced for the design of single- or double-negative media (see reviews [8

8. M. Perrin, S. Fasquel, T. Decoopman, M. X. Mélique, O. Vanbésien, E. Lheurette, and D. Lippens, “Left-handed electromagnetism obtained via nanostructured metamaterials: comparison with that from microstructured photonic crystals,” J. Opt. A, S3–S11 (2005). [CrossRef]

, 9

9. V. M. Shalaev, “Optical negative-index metamaterials,” Nature Photonics 1, 41–48 (2007). [CrossRef]

]), this artificial matter has the advantage that the desired properties can be spatially localized. This fundamental concept enables to build complex electromagnetic objects from spatially distributed discrete elements.

The magnetic cloak of invisibility demonstrated at microwave frequencies was the first experimental breakthrough under these guidelines [10

10. 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]

]. The visible spectrum is without any question the frequency range at which most impact is expected but the fabrication of nanometer-scaled elements remains a technological challenge [11

11. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with non-magnetic metamaterials,” Nature Photonics 1, 224–227 (2007). [CrossRef]

]. This constraint can be alleviated by investigating magnetic cloaking devices at THz frequencies. In 2002, O’Brien and Pendry investigated the origin of the magnetic activity in high-κ (i.e. high permittivity) ferroelectric rods for photonic band gap applications [12

12. S. O’Brien and J. B. Pendry, “Photonic band-gap effects and magnetic activity in dielectric composites,” J. Phys. 14, 4035–4044 (2002).

]. They reported that for a TE polarization (H along the rod axis), a zeroth-order Mie resonance is created from resonant displacement currents in the scatterers. Consequently, negative values in permeability were predicted within a frequency range dependent on the geometry and material parameters. A left-handed behavior (permittivity and permeability are both negative) was also observed in an ordered and random medium constructed from rectangular Mie rods [13

13. L. Peng, L. R., H. Chen, H. 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(1–4) (2007) [CrossRef] [PubMed]

].

In this work, Mie theory is applied to engineer the magnetic plasma frequency of high-κ ferroelectric rods at THz frequencies. Full-wave simulations coupled with a field-summation retrieval technique [14

14. O. Acher, J.-M. Lerat, and N. Malléjac, “Evaluation and illustration of the properties of Metamaterials using field summation,” Opt. Express 15, 1096–1106 (2007). [CrossRef] [PubMed]

, 15

15. D. R. Smith and J. B. Pendry, “Homogenization of metamaterials by field averaging (invited paper),” J. Opt. Soc. Am. B 23, 391–403 (2006). [CrossRef]

] were employed to assess the effective parameters (permittivity εeff and permeability µeff) of ferroelectric BaxSr1-xTiO3 (BST) rods excited by an H-field along the rod axis. From the results of this study, a device that performs cloaking at terahertz frequencies was carefully built from individual cylinders. Here, the idea is to superimpose the electromagnetic response of cylinders with different radii in such a way that the microstructured cloak shell displays a strong radial distribution in permeability and falls into a class of cylindrical objects derived from the conformal transformation theory. The performance of the magnetic cloak was assessed through full-wave simulations and the results are discussed in the last Section.

2. Permeability engineering in high-κ ferroelectrics rods

In this Section, we describe the electromagnetic response of micro-cut BST rods using the 3D full-wave solver HFSS (high-frequency structure simulator). The dielectric rods were excited by a plane wave along the y-axis, as the inset in Fig. 1(a) depicts. The dispersion of the investigated material was neglected and the permittivity ε′ was set to 200 with a dielectric loss tangent ε″/ε′ = 2.10−2 for all frequencies. This assumption relies on the fact that the presented micro-structured device operates, by definition, within a narrow frequency range [2

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

] where the bulk parameters can be assumed constant. Perfect magnetic conductors and perfect electric conductors were used for the lateral and top/bottom faces, respectively. Hence, the incident wave injected by the input port propagates through a unique rod along the propagation direction x and infinite set of rods in the y-z plane. Then, the scattering parameters (reflection S11 and transmission S21) were computed from the input/output ports to determine the Mie resonance frequency.

Figure 1(a) presents the frequency dependence on the reflection and transmission parameters of a plane wave impinging onto a BST cylinder with a 34 µm diameter and a height of 30 µm and for frequency values ranging from 0.36 to 0.66 THz. These frequencies are within the range where the rod electromagnetic behavior can be described by an effective medium. Here, the simulation domain was 50 µm×40 µm×50 µm. The 5 µm spacing between the top/bottom surfaces of the rod and the lateral faces insures that the incident wave does not impinge onto an infinitely long rod along the y axis. Although this aspect is not discussed in this work, note that the Mie resonance is rejected to lower frequencies with an infinite rod. For this example, the Mie resonance is predicted to occur at 0.527 THz; frequency at which the rod displays the most significant magnetic response.

Fig. 1. (a) Scattering parameters of a micro-cut BST rod computed using the finite-element solver HFSS. The inset presents the computational cell and applied boundary conditions. The rod displays a permittivity function ε BST = 200-5j (b) Top and bottom view present complex εzz and µyy, respectively, retrieved using a field-summation method. The Mie resonance occurs at 0.527 THz.

The complex effective permittivity and permeability were retrieved using a field-summation technique over the simulation domain and are presented in Fig. 1(b). The data indicate that the real part of the permeability displays a Lorentz-like behavior around the Mie resonance frequency. In contrast, the real part of the permittivity remains almost constant with a value of 1.86 over the frequency range of interest. The imaginary part of the permittivity and permeability is ~1.10−5 and ~8.10−2, respectively. Finally, the data show that the magnetic plasma frequency at which the permeability vanishes is 0.597 THz. This information is critical because it enables the design of metamaterials with permeability values ranging from 0 to 1.

In order to tailor the magnetic activity of the dielectric rods, the simplest approach would be to shift upward or downward the Mie resonance frequency by changing the dielectric function of the material. However, this solution is technologically impractical since it would require mastering the rod stoichiometry with high accuracy. Another alternative would be to increase or decrease the path length that the displacement currents undergo at the resonance. Physically, this means adjusting the radius of the rod. If one increases the radius, the resonance goes to lower frequencies and vice-versa. For instance, Fig. 2(a) presents the frequency dependence on S11 and S21 for BST rods with diameter values ranging from 34 to 40 µm by steps of 1 µm. The Mie resonance frequency is clearly red-shifted with increased radii, thus confirming our hypothesis. Consequently, one immediately grasps the advantage of geometrical modifications in high-κ rods to adjust the magnetic plasma frequency. Figure 2(b) presents the results of the retrieval technique for the investigated designs. The magnetic plasma frequency is predicted to shift from 0.597 THz to 0.543 THz. Note that we also observe a slight increase in the real part of the permittivity from ~1.9 to ~2.5.

Fig. 2. (a) Dependence of the scattering parameters on the BST rod diameter. The diameter was varied between 34 and 40 µm by steps of 1 µm. Corresponding structures are labeled from 1 to 7. (b) Top and bottom view present the real part of εzz and μyy, respectively, retrieved using a field-summation method for our 7 cases. Note that the Mie resonance shifts to lowerfrequencies with increased radii. The vertical dashed line indicates the cloaking operating frequency employed in this work.

3. Cloak design

Now that we have identified and quantified the magnetic plasma frequency in individual BST rods, we turn our attention to the design of a full cloak using Mie rods and excited under plane wave conditions. In the case of an annular cloak defined by its inner radius a and outer radius b [see Fig. 3(a)], the effective parameters of the cloak shell (permittivity and permeability) must be independently engineered to satisfy a set of equations derived from the conformal transformation theory [3

3. D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9803 (2006). [CrossRef] [PubMed]

]. On the other hand, it was shown that this design burden can be overcome by using a reduced set of equations [16

16. S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 36621–36621 (2006). [CrossRef]

]. This allows one parameter only (permittivity or permeability) to be varied with the cloak radius. In our case, a progressive variation in the permeability is obtained by positioning the Mie rods radially [11

11. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with non-magnetic metamaterials,” Nature Photonics 1, 224–227 (2007). [CrossRef]

]. Under this configuration, a transverse-electric (TE) polarized plane wave (Hr, Hθ, Ez in cylindrical coordinates) has to be employed to illuminate the magnetic cloak and the original set of equations can be reduced to:

μr(r)=(rar)2,
(1)
μθ(r)=1,
(2)
εz(r)=(bba)2.
(3)

Similarly, the first cloaking experiment was demonstrated under these conditions with metallic split-ring-resonators at 8.5 GHz. Here, we simply substitute the metallic loops with BST rods whose electromagnetic response matches closely Eqs. 1 to 3. In an arbitrarily configuration where b = 2a, the radial component in the permeability is zero at the inner surface of the cloak where r = a. In contrast, a permeability value of 0.25 is obtained at the outer surface of the cloak where r = 2b.

Here, the cloak consists of 7 levels of 30-µm-high BST rods with diameter values increasing from 34 to 40 µm by steps of 1 µm and 10 µm radial pitch separating each rod, as shown in Fig. 3(a). Finally, the inner radius of the cloak was arbitrarily set to a = 280 µm and outer radius b = 560 µm to limit the size of the computational cell.

Fig. 3. (a) Top view of the computational domain used to simulate the discretized dielectric cloak. The inset presents a 3D rendering of the device which consists of 7 concentric rings of micro-cut BST rods. An ideal copper cylinder was placed at the center to assess the cloaking efficiency. (b) Dependence of the cloak effective parameters (μr, μθ, and εz) at 0.58 THz on the normalized radius r/a where a is the cloak inner radius and for radii values ranging from a to b = 2a. The dashed line plots the theoretical distribution of the radial permeability given by Eq. 1.

Figure 3(b) presents the cloak effective parameters (μr(r), μθ(r), and εz(r)) dependence on the normalized radius r/a calculated (i) for the investigated BST rods at 0.58 THz and (ii) from Eq. 1. This operating frequency was selected by matching the discrete distribution in permeability from the rods with Eq. 1. The data also show that the permittivity value is about half of the one computed from Eq. 3. In the following Section, we will see that this permittivity mismatch does not fundamentally perturb the cloaking ability.

4. Results and discussion

In order to achieve an efficient cloaking device, key points must be satisfied. First, the device itself must not be detected. Consequently, the field pattern should not present wavefront distortions around the cloak nor shadow regions behind it. This particular point is critical because it is understandable that the electromagnetic signature of the device must not be revealed. Hence, the outer surface of the device must be well impedance matched with its surrounding environment, typically air, to minimize reflection. In addition to this impedance matching condition, the distribution of the effective parameters inside the cloak should fit the reduced set of cloak parameters given by Eqs. 1 to 3 for TE-polarized incident waves. Finally, the electromagnetic activity at the innerside of the cloak should be in principle close to zero, to minimize scattering on the hidden object.

We first studied the scattering of a plane wave at 0. 58 THz (λ = 517µm) for an uncloaked copper rod. The results of the E-field distribution achieved in this case are displayed in Fig. 4 (a). Strong distortions of the E-field are clearly observed behind and in front of the metal scatterer. Fig. 4 (b) shows the E-field z component for the metallic object now surrounded by the microstructured cloak. These results unambiguously demonstrate that cloaking is achieved at the targeted frequency. First, the E-field wavefronts are well reconstructed behind the cloak with minimal scattering. A high power transmission value of ~66% and low reflection value of ~13% were computed, thus indicating that ~20% of the total energy is either absorbed within the BST rods or reflected toward the radiation boundaries. Also few back-scattered ripples are observed at the front and side of the cloak.

Fig. 4 (c) shows a zoomed view of the cloak region. For clarity the field magnitude was magnified in order to have further insight into the field map within the cloak. In contrast with recently reported simulations of piecewise homogeneous annular cloaks, the wavefronts do not clearly follow the annular geometry of the cloak. Thus, the E-field is mostly concentrated within the first outer concentric rings of rods and does not penetrate deeply within the annular shell. This latter observation may be a consequence of the slightly negative permeability values displayed by the two inner-side set of rods. This effect, which results from the linear variation in the diameter shrinking, contributes to isolate the inner object by taking advantage of single negative inner layers.

Fig. 4. Steady-state Ez pattern calculated at 0.58 THz for a copper rod without (a) and with a microstructured cloak (b). The plane of observation is located at mid-distance between the bottom and top face of the simulation domain. The wavefronts are well reconstructed behind the cloak without noticeable backscattering. The metallic particle placed at the center of the device is nearly “invisible” to a detector located at the output port. (c) Zoomed view of the field pattern within the cloak. For clarity, the amplitude within the cloak was magnified.

5. Conclusion

Mie theory was applied to high-κ micrometer-sized BST cylinders and results showed that the ferroelectric rods exhibited a strong magnetic resonance dependent on the cylinder radii. This artificial magnetism was exploited to build an original all-dielectric design that performs cloaking at THz frequencies. Full-wave simulations coupled with a field-summation retrieval technique were employed to adjust the electromagnetic response of individual ferroelectrics rods. The rods magnetic plasma frequency was engineered such that the cloak shell displays a piece-wise variation in the radial component of the permeability; hence satisfying, for a TE polarization, the reduced equations derived from the conformal transformation theory. The cloaking performance was assessed by modeling, for the first time to our knowledge, the complete micro-structured device. Results clearly show that cloaking of any wavelengthscaled object located at the interior is achieved at 0.58 THz for the present device. In particular, the wavefronts of the electric field behind the device are well reconstructed. Additionally, a relatively high transmissivity level was obtained. These results show the potential of this all-dielectric solution for cloaking applications at microwave and terahertz frequencies.

References and links

1.

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

2.

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

3.

D. Schurig, J. B. Pendry, and D. R. Smith, “Calculation of material properties and ray tracing in transformation media,” Opt. Express 14, 9794–9803 (2006). [CrossRef] [PubMed]

4.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987). [CrossRef] [PubMed]

5.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987). [CrossRef] [PubMed]

6.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely Low Frequency Plasmons in Metallic Mesostructures,” Phys. Rev. Lett. 76, 4773–4776 (1996). [CrossRef] [PubMed]

7.

J. B. Pendry, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microwave Theory Tech. 47, 2075–2084 (1999). [CrossRef]

8.

M. Perrin, S. Fasquel, T. Decoopman, M. X. Mélique, O. Vanbésien, E. Lheurette, and D. Lippens, “Left-handed electromagnetism obtained via nanostructured metamaterials: comparison with that from microstructured photonic crystals,” J. Opt. A, S3–S11 (2005). [CrossRef]

9.

V. M. Shalaev, “Optical negative-index metamaterials,” Nature Photonics 1, 41–48 (2007). [CrossRef]

10.

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]

11.

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

12.

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

13.

L. Peng, L. R., H. Chen, H. 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(1–4) (2007) [CrossRef] [PubMed]

14.

O. Acher, J.-M. Lerat, and N. Malléjac, “Evaluation and illustration of the properties of Metamaterials using field summation,” Opt. Express 15, 1096–1106 (2007). [CrossRef] [PubMed]

15.

D. R. Smith and J. B. Pendry, “Homogenization of metamaterials by field averaging (invited paper),” J. Opt. Soc. Am. B 23, 391–403 (2006). [CrossRef]

16.

S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. Pendry, “Full-wave simulations of electromagnetic cloaking structures,” Phys. Rev. E 74, 36621–36621 (2006). [CrossRef]

OCIS Codes
(080.2710) Geometric optics : Inhomogeneous optical media
(220.2740) Optical design and fabrication : Geometric optical design
(260.1180) Physical optics : Crystal optics

ToC Category:
Physical Optics

History
Original Manuscript: October 19, 2007
Revised Manuscript: December 5, 2007
Manuscript Accepted: December 5, 2007
Published: March 11, 2008

Citation
Davy P. Gaillot, Charles Croënne, and Didier Lippens, "An all-dielectric route for terahertz cloaking," Opt. Express 16, 3986-3992 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-6-3986


Sort:  Author  |  Year  |  Journal  |  Reset  

References

  1. U. Leonhardt, "Optical conforming mapping," Science 312, 1777-1780 (2006). [CrossRef] [PubMed]
  2. J. B. Pendry, D. Shurig, and D. R. Smith, "Controlling electromagnetic fields," Science 312, 1780-1782 (2006). [CrossRef] [PubMed]
  3. D. Schurig, J. B. Pendry, and D. R. Smith, "Calculation of material properties and ray tracing in transformation media," Opt. Express 14, 9794-9803 (2006). [CrossRef] [PubMed]
  4. S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987). [CrossRef] [PubMed]
  5. E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987). [CrossRef] [PubMed]
  6. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely Low Frequency Plasmons in Metallic Mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996). [CrossRef] [PubMed]
  7. J. B. Pendry, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999). [CrossRef]
  8. M. Perrin, S. Fasquel, T. Decoopman, M. X. Mélique, O. Vanbésien, E. Lheurette, and D. Lippens, "Left-handed electromagnetism obtained via nanostructured metamaterials: comparison with that from microstructured photonic crystals," J. Opt. A, S3-S11 (2005). [CrossRef]
  9. V. M. Shalaev, "Optical negative-index metamaterials," Nat. Photonics 1, 41-48 (2007). [CrossRef]
  10. 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]
  11. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, "Optical cloaking with non-magnetic metamaterials," Nat. Photonics 1, 224 - 227 (2007). [CrossRef]
  12. S. O'Brien, and J. B. Pendry, "Photonic band-gap effects and magnetic activity in dielectric composites," J. Phys. 14, 4035-4044 (2002).
  13. L. Peng, L. Ran, H. Chen, H. 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]
  14. O. Acher, J.-M. Lerat, and N. Malléjac, "Evaluation and illustration of the properties of Metamaterials using field summation," Opt. Express 15, 1096-1106 (2007). [CrossRef] [PubMed]
  15. D. R. Smith and J. B. Pendry, "Homogenization of metamaterials by field averaging (invited paper)," J. Opt. Soc. Am. B 23, 391-403 (2006). [CrossRef]
  16. S. A. Cummer, B. I. Popa, D. Schurig, D. R. Smith, and J. Pendry, "Full-wave simulations of electromagnetic cloaking structures," Phys. Rev. E 74, 36621-36621 (2006). [CrossRef]

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

Figures

Fig. 1. Fig. 2. Fig. 3.
 
Fig. 4.
 

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited