Abstract

While the effective medium treatment of unbounded metamaterials appears to be well established and firmly proven, related phenomena in finite structures have not received sufficient attention. We report on mesoscopic effects associated with the boundaries of finite discrete metamaterial samples, which can invalidate an effective medium description. We show how to avoid such effects by proper choice of boundary configuration. As all metamaterial implementations are naturally finite, we are confident that our findings are crucial for future metamaterial research.

© 2012 OSA

1. Introduction

Metamaterials — artificial media engineered with various structural units which play the role of atoms — have been in the focus of active research for more than a decade. One of the milestones in understanding metamaterial properties is the effective medium theory (EMT), which aims to describe metamaterials in terms of effective parameters: permittivity and permeability [1]. In particular, EMT has been essential for the lattices of split-ring-resonators (SRRs) —the key structure for obtaining an artificial magnetic response [2]. Approaches for determining the effective permeability of bulk SRR-metamaterials were attempted prior to metamaterials outburst [3]. Subsequently, fruitful theoretical procedures were developed to account for mutual interaction [4], bianisotropy [5], magnetoinductive waves [6] and their interaction with the electromagnetic spectrum [7, 8], spatial dispersion [810], mutual interaction between the electric and magnetic subsystems [11], specificity of boundary conditions [12, 13], arbitrary geometry of elements [14], multipole expansion [15], disorder [16,17] and noise [18].

However, effective medium theories, in principle, consider unbounded lattices, whereas all practical metamaterials are of finite size. For conventional materials, it is well known [13, 19, 20] that at the boundaries a transition layer is formed, with properties different from those of the bulk (see the historical introduction in [13] for a detailed background). For large samples, however, transition layer does not significantly affect the overall macroscopic properties, and only manifests itself under specific conditions such as strongly oblique incidence, with surface modes not affecting the bulk considerably. But metamaterials are often constructed with a much fewer number of elements than reasonable pieces of conventional materials, so a comparison to atomic clusters might be more appropriate. At the same time, internal structure of metamaterials is often more complex than that of natural materials, in a sense that metamaterial elements may be quite densely arranged, be of different types or form geometrically independent sub-lattices [21]. This raises an issue of pronounced surface and spatial resonances which may appear in finite structures even in a quasi-static regime [22,23].

For metamaterial slabs, having a finite size in one dimension (and the same applies to the small samples placed inside a waveguide, which is equivalent to a system infinite in transverse direction), bulk effective parameters can be, with certain precautions, extracted from transmission and reflection characteristics [24, 25] (see e.g. [26, 27] for a recent update on the methodology). On this way, it is known that the effective properties extracted for the slabs of finite thickness quickly converge to the EMT predictions when the number of unit cells across the slab increases, and for most structures with low spatial dispersion a thickness of a few unit cells is sufficient to yield the bulk properties.

In this paper, we address the most general case of practical relevance: metamaterials with finite size in all three dimensions, observed in free space. We will analyse whether such metamaterials can be described with EMT parameters, and what are the requirements for such a description to be valid. It turns out that while for a slab of metamaterial a few unit cells across the slab are sufficient for EMT to work well, for the finite size in 3D it may be not sufficient to take several unit cells in each direction, and that a dramatic effect is caused by the ambiguity in the boundary structure (see Fig. 1).

 

Fig. 1 Scheme of a symmetric finite cube with discrete structure, showing two options of surface configuration (for the same unit cell in the bulk): “flat” geometry (a) and “ragged” geometry (b). Note that the actual number of elements is not necessarily reflected here.

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Indeed, a unit cell of the isotropic system shown in Fig. 1 contains three resonant loops non-symmetrically positioned with respect to the geometrical centre of the cell: three sides of the cube carry resonators, and the other three do not (as those resonators belong to the adjacent unit cells). This makes no difference in the middle of the structure. However, at the boundaries, overall symmetry of the sample can be achieved either by adding extra surface resonators on the three sides, Fig. 1(a), or, less trivially, by removing the unnecessary resonators on the opposite sides, Fig. 1(b). We will further refer to these methods of assembly as “flat” and “ragged”.

We should note that early analysis [4] confirmed that anisotropic stacks of uniformly oriented SRRs (all having parallel axes), with about one thousand elements can be successfully described with EMT; however that structure has no ambiguity of the kind described above.

2. Results and discussion

We will now analyse metamaterial cubes with isotropic cubic unit cells, and having different surface geometry as discussed above, and compare their characteristics to those of homogeneous pieces of a bulk material with the corresponding effective parameters [8]. In order to avoid any complications related to spatial dispersion and fit well into the EMT domain, we study a system with deeply subwavelength and strongly interacting resonant current loops: kresa ≈ 0.02 and a/r0 ≈ 3.1 (where kres is the value of the free space wavenumber k0 at resonance, a is the lattice constant and r0 is the mean radius of the loop). A high quality factor of the resonators, Q ≈ 500, has been chosen in order to present more illustrative results.

The finite size in all three directions does not permit a consistent study in terms of transmission characteristics, but there is a convenient macroscopic characteristic that can be attributed to finite samples: normalised polarisability, i.e. the total magnetic moment of the sample per unit external magnetic field and per unit volume. For subwavelength objects made of a homogeneous isotropic medium, this quantity is a scalar independent of the size. To find the polarisability of our discrete cubes, we solved a system of circuit equations for coupled resonators [6,28], assuming a uniform external magnetic field perpendicular to the face of the cube; note that taking all the mutual interactions into account is essential for such analysis. The total magnetic moment is then computed by summing the individual magnetic moments of each ring.

In Fig. 2, we compare the normalised polarisabilities of several metamaterial samples with various number of layers in each direction, calculated for either “flat” or “ragged” boundary. The normalised polarisability of the discrete cubes changes remarkably as the number of elements grows (unlike what we would expect for a homogeneous material), and it does not show a trend towards similarity between “flat” and “ragged” versions. However, the results obtained for the “ragged” geometry show more uniformity than those obtained for “flat” cubes.

 

Fig. 2 Real part of the normalised polarisability (arbitrary units) of discrete cubes with 5, 8 or 11 layers of resonators in each direction, having either a “flat” geometry (a) or a “ragged” geometry (b).

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Further distinctions are revealed (Fig. 3) by comparing discrete cubes with an equivalent cube made of a homogeneous medium, with the permeability given by Eq. (13) of [8],

μ=1+γ(kresa)2/(k0a)212ϰa4ϰcγ/3,
where γ = 0.162 is a coefficient related to the polarisability of a ring (determined by its geometry), ϰa = 0.0195 and ϰc = −0.0173 are inductive coupling coefficients to the axial and coplanar nearest neighbours; the numerical values specified here, correspond to the example used in our illustrations. Note that, although it is generally necessary to take more remote neighbours into account as well [4], for an unbounded cubic lattice taking only the nearest neighbours explicitly and then using Lorentz correction for farther neighbours [8] provides a sufficiently good accuracy. The polarisability of the homogeneous cube was found using the CST “Microwave Studio” commercial package. In order to avoid the appearance of multiple non-physical resonances, the edges and corners of the homogeneous cube must be rounded [29]. We have used a rounding radius corresponding to one half of the unit cell size of the discrete cubes. Such rounding leads to a minor difference between the simulation results for cubes of various absolute sizes, but this difference is very small in comparison with discrete cubes, and is not of a qualitative nature. Figure 3 shows that the polarisability of the “ragged” cube is significantly closer to that of the homogeneous cube, than the polarisability of the “flat” cube.

 

Fig. 3 Real (a) and imaginary (b) parts of the normalised polarisability (arbitrary units) of discrete cubes with 11 layers in each direction, having either a “flat” geometry (dashed lines) or “ragged” geometry (dotted lines), in comparison with the polarizability of a homogeneous cube (solid lines).

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These results can be understood in view of the essential role of surface elements [13, 20], since they do not have the same surroundings as the elements in the bulk and form transition layers with different properties. Then the difference between a “flat” structure and a “ragged” structure can be qualitatively assessed by noting that the elements of a “flat” surface do not have 5 out of 14 nearest neighbours, as compared to fully immersed rings, while in “ragged” surfaces, only 3 of the nearest neighbours are absent. This suggests that the behaviour of the “ragged” cube should be closer to the behaviour of an unbounded piece.

Although the considered metamaterial samples are formed by hundreds or even thousands of elements, they still behave like mesoscopic systems. The reason for this is that the number of periods is relatively small, so the surface excitations do not sufficiently dissipate within the sample, making a consistent macroscopic field averaging unreliable. However, we can attempt to extract the effective permeability directly from the polarisability of our cubic samples, as these quantities must be related. For this purpose, we assume that the functional form of the polarisability of a cube is similar to that of a sphere αo = 3(μ − 1)/(μ + 2). So, we formally express the polarisability of a cube as

α=A(μ1)/(μ+C),
where A and C are real frequency-dependent coefficients (see Fig. 4 for an example of their functional form). But for a homogeneous cube we know both the permeability function μ given by Eq. (1), and the polarisability α (which was computed numerically and shown with solid lines in Fig. 3). Thus, we can uniquely calculate A and C (shown in Fig. 4) as
C=|μ|2ReμReαImαImμ1Reμ+ReαImαImμ,A=(Reμ+C)2+(Imμ)2Imμ(C+1)Imα.
Now, these A and C coefficients can be used to evaluate the effective permeability of discrete cubes from their polarisability α# which is available from our analysis (shown with the dashed and dotted lines in Fig. 3), using an inverse of equation (2):
μ#=(A+Cα#)/(Aα#).
The result of this procedure is shown in Fig. 5: the curve corresponding to the “ragged” cube is very similar to the actual permeability of a homogeneous cube, unlike the curve of the “flat” cube. This result can be regarded as a strong argument in favour of the “ragged” configuration, in a sense that its effective permeability retrieved through the above procedure, matches the EMT prediction, whereas for the “flat” configuration such retrieval fails.

 

Fig. 4 Frequency dependence of the A and C coefficients calculated with Eq. (3) based on the effective permeability μ of a homogeneous cube Eq. (1) and its polarisability α, obtained from numerical simulations.

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Fig. 5 A comparison between the permeability μ of a homogeneous cube (solid line) and the effective permeability μ# obtained for 11-layer discrete cubes with a “flat” geometry (dashed lines) or a “ragged” geometry (dotted lines); real (a) and imaginary (b) parts.

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3. Conclusions

We have shown that the behaviour of finite metamaterial samples can deviate significantly from continuous medium expectations. We conclude that for isotropic SRR-metamaterials with up to several thousands of elements, the “ragged” boundary structure should be used for practical applications to permit a description in terms of the effective parameters of an unbounded metamaterial. We are confident that, for the size range in which most practical metamaterials fit, our results provide valuable information and design guidelines, allowing for to control the properties of finite metamaterial structures. Our results can be also of interest for the general theory of mesoscopic electromagnetic systems.

Acknowledgment

We are grateful to C.R. Simovski, M.V. Gorkunov and R.C. McPhedran for helpful discussions. This work has been supported by the Spanish Ministry of Science and Innovation ( CSD2008-00066), European Union FEDER fund ( TEC2010-16948), Czech Grant Agency ( 102/09/0314), Czech Technical University in Prague ( SGS10/271/OHK3/3T/13) and Australian Research Council ( DP110105484 and CE110001018).

References and links

1. C. R. Simovski, “Material parameters of metamaterials,” Opt. Spectrosc. 107, 726–753 (2009). [CrossRef]  

2. S. A. Schelkunoff and H. T. Friis, Antennas Theory and Practice (Wiley, 1966).

3. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47, 2075–2084 (1999). [CrossRef]  

4. M. Gorkunov, M. Lapine, E. Shamonina, and K. H. Ringhofer, “Effective magnetic properties of a composite material with circular conductive elements,” Eur. Phys. J. B 28, 263–269 (2002). [CrossRef]  

5. R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65, 144440 (2002). [CrossRef]  

6. E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magnetoinductive waves in one, two, and three dimensions,” J. Appl. Phys. 92, 6252–6261 (2002). [CrossRef]  

7. R. R. A. Syms, E. Shamonina, V. Kalinin, and L. Solymar, “A theory of metamaterials based on periodically loaded transmission lines: interaction between magnetoinductive and electromagnetic waves,” J. Appl. Phys. 97, 064909 (2005). [CrossRef]  

8. J. D. Baena, L. Jelinek, R. Marqués, and M. Silveirinha, “Unified homogenization theory for magnetoinductive and electromagnetic waves in split-ring metamaterials,” Phys. Rev. A 78, 013842 (2008). [CrossRef]  

9. M. Silveirinha, J. Baena, L. Jelinek, and R. Marques, “Nonlocal homogenization of an array of cubic particles made of resonant rings,” Metamaterials 3, 115–128 (2009). [CrossRef]  

10. V. M. Agranovich and Y. N. Gartstein, “Electrodynamics of metamaterials and the Landau–Lifshitz approach to the magnetic permeability,” Metamaterials 3, 1–9 (2009). [CrossRef]  

11. C. R. Simovski, “Analytical modelling of double-negative composites,” Metamaterials 2, 169–185 (2008). [CrossRef]  

12. P. A. Belov and C. R. Simovski, “Boundary conditions for interfaces of electromagnetic crystals and the generalized Ewald-Oseen extinction principle,” Phys. Rev. B 73, 045102 (2006). [CrossRef]  

13. M. Albooyeh, D. Morits, and C. R. Simovski, “Electromagnetic characterization of substrated metasurfaces,” Metamaterials 5, 178–205 (2011). [CrossRef]  

14. M. Silveirinha, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters,” Phys. Rev. B 75, 115104 (2007). [CrossRef]  

15. A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B 84, 075153 (2011). [CrossRef]  

16. M. Gorkunov, S. A. Gredeskul, I. V. Shadrivov, and Y. S. Kivshar, “Effect of microscopic disorder on magnetic properties of metamaterials,” Phys. Rev. E 73, 056605 (2006). [CrossRef]  

17. J. Rico-García, J. López-Alonso, and A. Aradian, “Toy model to describe the effect of positional blocklike disorder in metamaterials composites,” J. Opt. Soc. Am. B 29, 53–67 (2012). [CrossRef]  

18. R. R. A. Syms, O. Sydoruk, and L. Solymar, “Lossy metamaterials: no effective medium properties without noise,” Phys. Rev. B 84, 235150 (2011). [CrossRef]  

19. A. P. Vinogradov, D. P. Makhnovskii, and K. N. Rozanov, “Effective boundary layer in composite materials,” J. Communication Technol. Electron. 44, 317–322 (1999).

20. O. N. Gadomskii and S. V. Sukhov, “Microscopic theory of a transition layer on the ideal surface of semiinfinite dielectric media and the near-field effect,” Opt. Spectrosc. 89, 261–266 (2000). [CrossRef]  

21. M. Lapine and S. Tretyakov, “Contemporary notes on metamaterials,” IET Microw. Antenn. Propag. 1, 3–11 (2007). [CrossRef]  

22. O. Zhuromskyy, E. Shamonina, and L. Solymar, “2D metamaterials with hexagonal structure: spatial resonances and near field imaging,” Opt. Express 13, 9299–9309 (2005). [CrossRef]   [PubMed]  

23. R. Marqués, L. Jelinek, M. Freire, J. Baena, and M. Lapine, “Bulk metamaterials made of resonant rings,” Proc. IEEE 99, 1660–1668 (2011). [CrossRef]  

24. A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time-domain techniques,” IEEE Trans. Instrum. Meas. IM–19, 377–382 (1970). [CrossRef]  

25. W. B. Weir, “Automatic measurement of complex dielectric constant and permeability at microwave frequencies,” Proc. IEEE 62, 33–36 (1974). [CrossRef]  

26. X.-X. Liu, D. A. Powell, and A. Alù, “Correcting the Fabry-Perot artifacts in metamaterial retrieval procedures,” Phys. Rev. B 84, 235106 (2011). [CrossRef]  

27. O. Luukkonen, S. I. Maslovski, and S. A. Tretyakov, “A stepwise Nicolson–Ross–Weir–based material parameter extraction method,” IEEE Antenn. Wireless Propag. Lett. 10, 1295–1298 (2011). [CrossRef]  

28. M. Lapine, L. Jelinek, R. Marqués, and M. Freire, “Exact modelling method for discrete finite metamaterial lens,” IET Microw. Antenn. Propag. 4, 1132–1139 (2010). [CrossRef]  

29. H. Wallén, H. Kettunen, and A. Sihvola, “Surface modes of negative-parameter interfaces and the importance of rounding sharp corners,” Metamaterials 2, 113–121 (2008). [CrossRef]  

References

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  1. C. R. Simovski, “Material parameters of metamaterials,” Opt. Spectrosc. 107, 726–753 (2009).
    [Crossref]
  2. S. A. Schelkunoff and H. T. Friis, Antennas Theory and Practice (Wiley, 1966).
  3. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47, 2075–2084 (1999).
    [Crossref]
  4. M. Gorkunov, M. Lapine, E. Shamonina, and K. H. Ringhofer, “Effective magnetic properties of a composite material with circular conductive elements,” Eur. Phys. J. B 28, 263–269 (2002).
    [Crossref]
  5. R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65, 144440 (2002).
    [Crossref]
  6. E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magnetoinductive waves in one, two, and three dimensions,” J. Appl. Phys. 92, 6252–6261 (2002).
    [Crossref]
  7. R. R. A. Syms, E. Shamonina, V. Kalinin, and L. Solymar, “A theory of metamaterials based on periodically loaded transmission lines: interaction between magnetoinductive and electromagnetic waves,” J. Appl. Phys. 97, 064909 (2005).
    [Crossref]
  8. J. D. Baena, L. Jelinek, R. Marqués, and M. Silveirinha, “Unified homogenization theory for magnetoinductive and electromagnetic waves in split-ring metamaterials,” Phys. Rev. A 78, 013842 (2008).
    [Crossref]
  9. M. Silveirinha, J. Baena, L. Jelinek, and R. Marques, “Nonlocal homogenization of an array of cubic particles made of resonant rings,” Metamaterials 3, 115–128 (2009).
    [Crossref]
  10. V. M. Agranovich and Y. N. Gartstein, “Electrodynamics of metamaterials and the Landau–Lifshitz approach to the magnetic permeability,” Metamaterials 3, 1–9 (2009).
    [Crossref]
  11. C. R. Simovski, “Analytical modelling of double-negative composites,” Metamaterials 2, 169–185 (2008).
    [Crossref]
  12. P. A. Belov and C. R. Simovski, “Boundary conditions for interfaces of electromagnetic crystals and the generalized Ewald-Oseen extinction principle,” Phys. Rev. B 73, 045102 (2006).
    [Crossref]
  13. M. Albooyeh, D. Morits, and C. R. Simovski, “Electromagnetic characterization of substrated metasurfaces,” Metamaterials 5, 178–205 (2011).
    [Crossref]
  14. M. Silveirinha, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters,” Phys. Rev. B 75, 115104 (2007).
    [Crossref]
  15. A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B 84, 075153 (2011).
    [Crossref]
  16. M. Gorkunov, S. A. Gredeskul, I. V. Shadrivov, and Y. S. Kivshar, “Effect of microscopic disorder on magnetic properties of metamaterials,” Phys. Rev. E 73, 056605 (2006).
    [Crossref]
  17. J. Rico-García, J. López-Alonso, and A. Aradian, “Toy model to describe the effect of positional blocklike disorder in metamaterials composites,” J. Opt. Soc. Am. B 29, 53–67 (2012).
    [Crossref]
  18. R. R. A. Syms, O. Sydoruk, and L. Solymar, “Lossy metamaterials: no effective medium properties without noise,” Phys. Rev. B 84, 235150 (2011).
    [Crossref]
  19. A. P. Vinogradov, D. P. Makhnovskii, and K. N. Rozanov, “Effective boundary layer in composite materials,” J. Communication Technol. Electron. 44, 317–322 (1999).
  20. O. N. Gadomskii and S. V. Sukhov, “Microscopic theory of a transition layer on the ideal surface of semiinfinite dielectric media and the near-field effect,” Opt. Spectrosc. 89, 261–266 (2000).
    [Crossref]
  21. M. Lapine and S. Tretyakov, “Contemporary notes on metamaterials,” IET Microw. Antenn. Propag. 1, 3–11 (2007).
    [Crossref]
  22. O. Zhuromskyy, E. Shamonina, and L. Solymar, “2D metamaterials with hexagonal structure: spatial resonances and near field imaging,” Opt. Express 13, 9299–9309 (2005).
    [Crossref] [PubMed]
  23. R. Marqués, L. Jelinek, M. Freire, J. Baena, and M. Lapine, “Bulk metamaterials made of resonant rings,” Proc. IEEE 99, 1660–1668 (2011).
    [Crossref]
  24. A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time-domain techniques,” IEEE Trans. Instrum. Meas. IM–19, 377–382 (1970).
    [Crossref]
  25. W. B. Weir, “Automatic measurement of complex dielectric constant and permeability at microwave frequencies,” Proc. IEEE 62, 33–36 (1974).
    [Crossref]
  26. X.-X. Liu, D. A. Powell, and A. Alù, “Correcting the Fabry-Perot artifacts in metamaterial retrieval procedures,” Phys. Rev. B 84, 235106 (2011).
    [Crossref]
  27. O. Luukkonen, S. I. Maslovski, and S. A. Tretyakov, “A stepwise Nicolson–Ross–Weir–based material parameter extraction method,” IEEE Antenn. Wireless Propag. Lett. 10, 1295–1298 (2011).
    [Crossref]
  28. M. Lapine, L. Jelinek, R. Marqués, and M. Freire, “Exact modelling method for discrete finite metamaterial lens,” IET Microw. Antenn. Propag. 4, 1132–1139 (2010).
    [Crossref]
  29. H. Wallén, H. Kettunen, and A. Sihvola, “Surface modes of negative-parameter interfaces and the importance of rounding sharp corners,” Metamaterials 2, 113–121 (2008).
    [Crossref]

2012 (1)

2011 (6)

R. R. A. Syms, O. Sydoruk, and L. Solymar, “Lossy metamaterials: no effective medium properties without noise,” Phys. Rev. B 84, 235150 (2011).
[Crossref]

A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B 84, 075153 (2011).
[Crossref]

M. Albooyeh, D. Morits, and C. R. Simovski, “Electromagnetic characterization of substrated metasurfaces,” Metamaterials 5, 178–205 (2011).
[Crossref]

R. Marqués, L. Jelinek, M. Freire, J. Baena, and M. Lapine, “Bulk metamaterials made of resonant rings,” Proc. IEEE 99, 1660–1668 (2011).
[Crossref]

X.-X. Liu, D. A. Powell, and A. Alù, “Correcting the Fabry-Perot artifacts in metamaterial retrieval procedures,” Phys. Rev. B 84, 235106 (2011).
[Crossref]

O. Luukkonen, S. I. Maslovski, and S. A. Tretyakov, “A stepwise Nicolson–Ross–Weir–based material parameter extraction method,” IEEE Antenn. Wireless Propag. Lett. 10, 1295–1298 (2011).
[Crossref]

2010 (1)

M. Lapine, L. Jelinek, R. Marqués, and M. Freire, “Exact modelling method for discrete finite metamaterial lens,” IET Microw. Antenn. Propag. 4, 1132–1139 (2010).
[Crossref]

2009 (3)

C. R. Simovski, “Material parameters of metamaterials,” Opt. Spectrosc. 107, 726–753 (2009).
[Crossref]

M. Silveirinha, J. Baena, L. Jelinek, and R. Marques, “Nonlocal homogenization of an array of cubic particles made of resonant rings,” Metamaterials 3, 115–128 (2009).
[Crossref]

V. M. Agranovich and Y. N. Gartstein, “Electrodynamics of metamaterials and the Landau–Lifshitz approach to the magnetic permeability,” Metamaterials 3, 1–9 (2009).
[Crossref]

2008 (3)

C. R. Simovski, “Analytical modelling of double-negative composites,” Metamaterials 2, 169–185 (2008).
[Crossref]

J. D. Baena, L. Jelinek, R. Marqués, and M. Silveirinha, “Unified homogenization theory for magnetoinductive and electromagnetic waves in split-ring metamaterials,” Phys. Rev. A 78, 013842 (2008).
[Crossref]

H. Wallén, H. Kettunen, and A. Sihvola, “Surface modes of negative-parameter interfaces and the importance of rounding sharp corners,” Metamaterials 2, 113–121 (2008).
[Crossref]

2007 (2)

M. Lapine and S. Tretyakov, “Contemporary notes on metamaterials,” IET Microw. Antenn. Propag. 1, 3–11 (2007).
[Crossref]

M. Silveirinha, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters,” Phys. Rev. B 75, 115104 (2007).
[Crossref]

2006 (2)

M. Gorkunov, S. A. Gredeskul, I. V. Shadrivov, and Y. S. Kivshar, “Effect of microscopic disorder on magnetic properties of metamaterials,” Phys. Rev. E 73, 056605 (2006).
[Crossref]

P. A. Belov and C. R. Simovski, “Boundary conditions for interfaces of electromagnetic crystals and the generalized Ewald-Oseen extinction principle,” Phys. Rev. B 73, 045102 (2006).
[Crossref]

2005 (2)

R. R. A. Syms, E. Shamonina, V. Kalinin, and L. Solymar, “A theory of metamaterials based on periodically loaded transmission lines: interaction between magnetoinductive and electromagnetic waves,” J. Appl. Phys. 97, 064909 (2005).
[Crossref]

O. Zhuromskyy, E. Shamonina, and L. Solymar, “2D metamaterials with hexagonal structure: spatial resonances and near field imaging,” Opt. Express 13, 9299–9309 (2005).
[Crossref] [PubMed]

2002 (3)

M. Gorkunov, M. Lapine, E. Shamonina, and K. H. Ringhofer, “Effective magnetic properties of a composite material with circular conductive elements,” Eur. Phys. J. B 28, 263–269 (2002).
[Crossref]

R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65, 144440 (2002).
[Crossref]

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magnetoinductive waves in one, two, and three dimensions,” J. Appl. Phys. 92, 6252–6261 (2002).
[Crossref]

2000 (1)

O. N. Gadomskii and S. V. Sukhov, “Microscopic theory of a transition layer on the ideal surface of semiinfinite dielectric media and the near-field effect,” Opt. Spectrosc. 89, 261–266 (2000).
[Crossref]

1999 (2)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47, 2075–2084 (1999).
[Crossref]

A. P. Vinogradov, D. P. Makhnovskii, and K. N. Rozanov, “Effective boundary layer in composite materials,” J. Communication Technol. Electron. 44, 317–322 (1999).

1974 (1)

W. B. Weir, “Automatic measurement of complex dielectric constant and permeability at microwave frequencies,” Proc. IEEE 62, 33–36 (1974).
[Crossref]

1970 (1)

A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time-domain techniques,” IEEE Trans. Instrum. Meas. IM–19, 377–382 (1970).
[Crossref]

Agranovich, V. M.

V. M. Agranovich and Y. N. Gartstein, “Electrodynamics of metamaterials and the Landau–Lifshitz approach to the magnetic permeability,” Metamaterials 3, 1–9 (2009).
[Crossref]

Albooyeh, M.

M. Albooyeh, D. Morits, and C. R. Simovski, “Electromagnetic characterization of substrated metasurfaces,” Metamaterials 5, 178–205 (2011).
[Crossref]

Alù, A.

A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B 84, 075153 (2011).
[Crossref]

X.-X. Liu, D. A. Powell, and A. Alù, “Correcting the Fabry-Perot artifacts in metamaterial retrieval procedures,” Phys. Rev. B 84, 235106 (2011).
[Crossref]

Aradian, A.

Baena, J.

R. Marqués, L. Jelinek, M. Freire, J. Baena, and M. Lapine, “Bulk metamaterials made of resonant rings,” Proc. IEEE 99, 1660–1668 (2011).
[Crossref]

M. Silveirinha, J. Baena, L. Jelinek, and R. Marques, “Nonlocal homogenization of an array of cubic particles made of resonant rings,” Metamaterials 3, 115–128 (2009).
[Crossref]

Baena, J. D.

J. D. Baena, L. Jelinek, R. Marqués, and M. Silveirinha, “Unified homogenization theory for magnetoinductive and electromagnetic waves in split-ring metamaterials,” Phys. Rev. A 78, 013842 (2008).
[Crossref]

Belov, P. A.

P. A. Belov and C. R. Simovski, “Boundary conditions for interfaces of electromagnetic crystals and the generalized Ewald-Oseen extinction principle,” Phys. Rev. B 73, 045102 (2006).
[Crossref]

Freire, M.

R. Marqués, L. Jelinek, M. Freire, J. Baena, and M. Lapine, “Bulk metamaterials made of resonant rings,” Proc. IEEE 99, 1660–1668 (2011).
[Crossref]

M. Lapine, L. Jelinek, R. Marqués, and M. Freire, “Exact modelling method for discrete finite metamaterial lens,” IET Microw. Antenn. Propag. 4, 1132–1139 (2010).
[Crossref]

Friis, H. T.

S. A. Schelkunoff and H. T. Friis, Antennas Theory and Practice (Wiley, 1966).

Gadomskii, O. N.

O. N. Gadomskii and S. V. Sukhov, “Microscopic theory of a transition layer on the ideal surface of semiinfinite dielectric media and the near-field effect,” Opt. Spectrosc. 89, 261–266 (2000).
[Crossref]

Gartstein, Y. N.

V. M. Agranovich and Y. N. Gartstein, “Electrodynamics of metamaterials and the Landau–Lifshitz approach to the magnetic permeability,” Metamaterials 3, 1–9 (2009).
[Crossref]

Gorkunov, M.

M. Gorkunov, S. A. Gredeskul, I. V. Shadrivov, and Y. S. Kivshar, “Effect of microscopic disorder on magnetic properties of metamaterials,” Phys. Rev. E 73, 056605 (2006).
[Crossref]

M. Gorkunov, M. Lapine, E. Shamonina, and K. H. Ringhofer, “Effective magnetic properties of a composite material with circular conductive elements,” Eur. Phys. J. B 28, 263–269 (2002).
[Crossref]

Gredeskul, S. A.

M. Gorkunov, S. A. Gredeskul, I. V. Shadrivov, and Y. S. Kivshar, “Effect of microscopic disorder on magnetic properties of metamaterials,” Phys. Rev. E 73, 056605 (2006).
[Crossref]

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47, 2075–2084 (1999).
[Crossref]

Jelinek, L.

R. Marqués, L. Jelinek, M. Freire, J. Baena, and M. Lapine, “Bulk metamaterials made of resonant rings,” Proc. IEEE 99, 1660–1668 (2011).
[Crossref]

M. Lapine, L. Jelinek, R. Marqués, and M. Freire, “Exact modelling method for discrete finite metamaterial lens,” IET Microw. Antenn. Propag. 4, 1132–1139 (2010).
[Crossref]

M. Silveirinha, J. Baena, L. Jelinek, and R. Marques, “Nonlocal homogenization of an array of cubic particles made of resonant rings,” Metamaterials 3, 115–128 (2009).
[Crossref]

J. D. Baena, L. Jelinek, R. Marqués, and M. Silveirinha, “Unified homogenization theory for magnetoinductive and electromagnetic waves in split-ring metamaterials,” Phys. Rev. A 78, 013842 (2008).
[Crossref]

Kalinin, V.

R. R. A. Syms, E. Shamonina, V. Kalinin, and L. Solymar, “A theory of metamaterials based on periodically loaded transmission lines: interaction between magnetoinductive and electromagnetic waves,” J. Appl. Phys. 97, 064909 (2005).
[Crossref]

Kalinin, V. A.

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magnetoinductive waves in one, two, and three dimensions,” J. Appl. Phys. 92, 6252–6261 (2002).
[Crossref]

Kettunen, H.

H. Wallén, H. Kettunen, and A. Sihvola, “Surface modes of negative-parameter interfaces and the importance of rounding sharp corners,” Metamaterials 2, 113–121 (2008).
[Crossref]

Kivshar, Y. S.

M. Gorkunov, S. A. Gredeskul, I. V. Shadrivov, and Y. S. Kivshar, “Effect of microscopic disorder on magnetic properties of metamaterials,” Phys. Rev. E 73, 056605 (2006).
[Crossref]

Lapine, M.

R. Marqués, L. Jelinek, M. Freire, J. Baena, and M. Lapine, “Bulk metamaterials made of resonant rings,” Proc. IEEE 99, 1660–1668 (2011).
[Crossref]

M. Lapine, L. Jelinek, R. Marqués, and M. Freire, “Exact modelling method for discrete finite metamaterial lens,” IET Microw. Antenn. Propag. 4, 1132–1139 (2010).
[Crossref]

M. Lapine and S. Tretyakov, “Contemporary notes on metamaterials,” IET Microw. Antenn. Propag. 1, 3–11 (2007).
[Crossref]

M. Gorkunov, M. Lapine, E. Shamonina, and K. H. Ringhofer, “Effective magnetic properties of a composite material with circular conductive elements,” Eur. Phys. J. B 28, 263–269 (2002).
[Crossref]

Liu, X.-X.

X.-X. Liu, D. A. Powell, and A. Alù, “Correcting the Fabry-Perot artifacts in metamaterial retrieval procedures,” Phys. Rev. B 84, 235106 (2011).
[Crossref]

López-Alonso, J.

Luukkonen, O.

O. Luukkonen, S. I. Maslovski, and S. A. Tretyakov, “A stepwise Nicolson–Ross–Weir–based material parameter extraction method,” IEEE Antenn. Wireless Propag. Lett. 10, 1295–1298 (2011).
[Crossref]

Makhnovskii, D. P.

A. P. Vinogradov, D. P. Makhnovskii, and K. N. Rozanov, “Effective boundary layer in composite materials,” J. Communication Technol. Electron. 44, 317–322 (1999).

Marques, R.

M. Silveirinha, J. Baena, L. Jelinek, and R. Marques, “Nonlocal homogenization of an array of cubic particles made of resonant rings,” Metamaterials 3, 115–128 (2009).
[Crossref]

Marqués, R.

R. Marqués, L. Jelinek, M. Freire, J. Baena, and M. Lapine, “Bulk metamaterials made of resonant rings,” Proc. IEEE 99, 1660–1668 (2011).
[Crossref]

M. Lapine, L. Jelinek, R. Marqués, and M. Freire, “Exact modelling method for discrete finite metamaterial lens,” IET Microw. Antenn. Propag. 4, 1132–1139 (2010).
[Crossref]

J. D. Baena, L. Jelinek, R. Marqués, and M. Silveirinha, “Unified homogenization theory for magnetoinductive and electromagnetic waves in split-ring metamaterials,” Phys. Rev. A 78, 013842 (2008).
[Crossref]

R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65, 144440 (2002).
[Crossref]

Maslovski, S. I.

O. Luukkonen, S. I. Maslovski, and S. A. Tretyakov, “A stepwise Nicolson–Ross–Weir–based material parameter extraction method,” IEEE Antenn. Wireless Propag. Lett. 10, 1295–1298 (2011).
[Crossref]

Medina, F.

R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65, 144440 (2002).
[Crossref]

Morits, D.

M. Albooyeh, D. Morits, and C. R. Simovski, “Electromagnetic characterization of substrated metasurfaces,” Metamaterials 5, 178–205 (2011).
[Crossref]

Nicolson, A. M.

A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time-domain techniques,” IEEE Trans. Instrum. Meas. IM–19, 377–382 (1970).
[Crossref]

Pendry, J. B.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47, 2075–2084 (1999).
[Crossref]

Powell, D. A.

X.-X. Liu, D. A. Powell, and A. Alù, “Correcting the Fabry-Perot artifacts in metamaterial retrieval procedures,” Phys. Rev. B 84, 235106 (2011).
[Crossref]

Rafii-El-Idrissi, R.

R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65, 144440 (2002).
[Crossref]

Rico-García, J.

Ringhofer, K. H.

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magnetoinductive waves in one, two, and three dimensions,” J. Appl. Phys. 92, 6252–6261 (2002).
[Crossref]

M. Gorkunov, M. Lapine, E. Shamonina, and K. H. Ringhofer, “Effective magnetic properties of a composite material with circular conductive elements,” Eur. Phys. J. B 28, 263–269 (2002).
[Crossref]

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47, 2075–2084 (1999).
[Crossref]

Ross, G. F.

A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time-domain techniques,” IEEE Trans. Instrum. Meas. IM–19, 377–382 (1970).
[Crossref]

Rozanov, K. N.

A. P. Vinogradov, D. P. Makhnovskii, and K. N. Rozanov, “Effective boundary layer in composite materials,” J. Communication Technol. Electron. 44, 317–322 (1999).

Schelkunoff, S. A.

S. A. Schelkunoff and H. T. Friis, Antennas Theory and Practice (Wiley, 1966).

Shadrivov, I. V.

M. Gorkunov, S. A. Gredeskul, I. V. Shadrivov, and Y. S. Kivshar, “Effect of microscopic disorder on magnetic properties of metamaterials,” Phys. Rev. E 73, 056605 (2006).
[Crossref]

Shamonina, E.

O. Zhuromskyy, E. Shamonina, and L. Solymar, “2D metamaterials with hexagonal structure: spatial resonances and near field imaging,” Opt. Express 13, 9299–9309 (2005).
[Crossref] [PubMed]

R. R. A. Syms, E. Shamonina, V. Kalinin, and L. Solymar, “A theory of metamaterials based on periodically loaded transmission lines: interaction between magnetoinductive and electromagnetic waves,” J. Appl. Phys. 97, 064909 (2005).
[Crossref]

M. Gorkunov, M. Lapine, E. Shamonina, and K. H. Ringhofer, “Effective magnetic properties of a composite material with circular conductive elements,” Eur. Phys. J. B 28, 263–269 (2002).
[Crossref]

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magnetoinductive waves in one, two, and three dimensions,” J. Appl. Phys. 92, 6252–6261 (2002).
[Crossref]

Sihvola, A.

H. Wallén, H. Kettunen, and A. Sihvola, “Surface modes of negative-parameter interfaces and the importance of rounding sharp corners,” Metamaterials 2, 113–121 (2008).
[Crossref]

Silveirinha, M.

M. Silveirinha, J. Baena, L. Jelinek, and R. Marques, “Nonlocal homogenization of an array of cubic particles made of resonant rings,” Metamaterials 3, 115–128 (2009).
[Crossref]

J. D. Baena, L. Jelinek, R. Marqués, and M. Silveirinha, “Unified homogenization theory for magnetoinductive and electromagnetic waves in split-ring metamaterials,” Phys. Rev. A 78, 013842 (2008).
[Crossref]

M. Silveirinha, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters,” Phys. Rev. B 75, 115104 (2007).
[Crossref]

Simovski, C. R.

M. Albooyeh, D. Morits, and C. R. Simovski, “Electromagnetic characterization of substrated metasurfaces,” Metamaterials 5, 178–205 (2011).
[Crossref]

C. R. Simovski, “Material parameters of metamaterials,” Opt. Spectrosc. 107, 726–753 (2009).
[Crossref]

C. R. Simovski, “Analytical modelling of double-negative composites,” Metamaterials 2, 169–185 (2008).
[Crossref]

P. A. Belov and C. R. Simovski, “Boundary conditions for interfaces of electromagnetic crystals and the generalized Ewald-Oseen extinction principle,” Phys. Rev. B 73, 045102 (2006).
[Crossref]

Solymar, L.

R. R. A. Syms, O. Sydoruk, and L. Solymar, “Lossy metamaterials: no effective medium properties without noise,” Phys. Rev. B 84, 235150 (2011).
[Crossref]

O. Zhuromskyy, E. Shamonina, and L. Solymar, “2D metamaterials with hexagonal structure: spatial resonances and near field imaging,” Opt. Express 13, 9299–9309 (2005).
[Crossref] [PubMed]

R. R. A. Syms, E. Shamonina, V. Kalinin, and L. Solymar, “A theory of metamaterials based on periodically loaded transmission lines: interaction between magnetoinductive and electromagnetic waves,” J. Appl. Phys. 97, 064909 (2005).
[Crossref]

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magnetoinductive waves in one, two, and three dimensions,” J. Appl. Phys. 92, 6252–6261 (2002).
[Crossref]

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47, 2075–2084 (1999).
[Crossref]

Sukhov, S. V.

O. N. Gadomskii and S. V. Sukhov, “Microscopic theory of a transition layer on the ideal surface of semiinfinite dielectric media and the near-field effect,” Opt. Spectrosc. 89, 261–266 (2000).
[Crossref]

Sydoruk, O.

R. R. A. Syms, O. Sydoruk, and L. Solymar, “Lossy metamaterials: no effective medium properties without noise,” Phys. Rev. B 84, 235150 (2011).
[Crossref]

Syms, R. R. A.

R. R. A. Syms, O. Sydoruk, and L. Solymar, “Lossy metamaterials: no effective medium properties without noise,” Phys. Rev. B 84, 235150 (2011).
[Crossref]

R. R. A. Syms, E. Shamonina, V. Kalinin, and L. Solymar, “A theory of metamaterials based on periodically loaded transmission lines: interaction between magnetoinductive and electromagnetic waves,” J. Appl. Phys. 97, 064909 (2005).
[Crossref]

Tretyakov, S.

M. Lapine and S. Tretyakov, “Contemporary notes on metamaterials,” IET Microw. Antenn. Propag. 1, 3–11 (2007).
[Crossref]

Tretyakov, S. A.

O. Luukkonen, S. I. Maslovski, and S. A. Tretyakov, “A stepwise Nicolson–Ross–Weir–based material parameter extraction method,” IEEE Antenn. Wireless Propag. Lett. 10, 1295–1298 (2011).
[Crossref]

Vinogradov, A. P.

A. P. Vinogradov, D. P. Makhnovskii, and K. N. Rozanov, “Effective boundary layer in composite materials,” J. Communication Technol. Electron. 44, 317–322 (1999).

Wallén, H.

H. Wallén, H. Kettunen, and A. Sihvola, “Surface modes of negative-parameter interfaces and the importance of rounding sharp corners,” Metamaterials 2, 113–121 (2008).
[Crossref]

Weir, W. B.

W. B. Weir, “Automatic measurement of complex dielectric constant and permeability at microwave frequencies,” Proc. IEEE 62, 33–36 (1974).
[Crossref]

Zhuromskyy, O.

Eur. Phys. J. B (1)

M. Gorkunov, M. Lapine, E. Shamonina, and K. H. Ringhofer, “Effective magnetic properties of a composite material with circular conductive elements,” Eur. Phys. J. B 28, 263–269 (2002).
[Crossref]

IEEE Antenn. Wireless Propag. Lett. (1)

O. Luukkonen, S. I. Maslovski, and S. A. Tretyakov, “A stepwise Nicolson–Ross–Weir–based material parameter extraction method,” IEEE Antenn. Wireless Propag. Lett. 10, 1295–1298 (2011).
[Crossref]

IEEE Trans. Instrum. Meas. (1)

A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time-domain techniques,” IEEE Trans. Instrum. Meas. IM–19, 377–382 (1970).
[Crossref]

IEEE Trans. Microw. Theory Tech. (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47, 2075–2084 (1999).
[Crossref]

IET Microw. Antenn. Propag. (2)

M. Lapine and S. Tretyakov, “Contemporary notes on metamaterials,” IET Microw. Antenn. Propag. 1, 3–11 (2007).
[Crossref]

M. Lapine, L. Jelinek, R. Marqués, and M. Freire, “Exact modelling method for discrete finite metamaterial lens,” IET Microw. Antenn. Propag. 4, 1132–1139 (2010).
[Crossref]

J. Appl. Phys. (2)

E. Shamonina, V. A. Kalinin, K. H. Ringhofer, and L. Solymar, “Magnetoinductive waves in one, two, and three dimensions,” J. Appl. Phys. 92, 6252–6261 (2002).
[Crossref]

R. R. A. Syms, E. Shamonina, V. Kalinin, and L. Solymar, “A theory of metamaterials based on periodically loaded transmission lines: interaction between magnetoinductive and electromagnetic waves,” J. Appl. Phys. 97, 064909 (2005).
[Crossref]

J. Communication Technol. Electron. (1)

A. P. Vinogradov, D. P. Makhnovskii, and K. N. Rozanov, “Effective boundary layer in composite materials,” J. Communication Technol. Electron. 44, 317–322 (1999).

J. Opt. Soc. Am. B (1)

Metamaterials (5)

M. Albooyeh, D. Morits, and C. R. Simovski, “Electromagnetic characterization of substrated metasurfaces,” Metamaterials 5, 178–205 (2011).
[Crossref]

M. Silveirinha, J. Baena, L. Jelinek, and R. Marques, “Nonlocal homogenization of an array of cubic particles made of resonant rings,” Metamaterials 3, 115–128 (2009).
[Crossref]

V. M. Agranovich and Y. N. Gartstein, “Electrodynamics of metamaterials and the Landau–Lifshitz approach to the magnetic permeability,” Metamaterials 3, 1–9 (2009).
[Crossref]

C. R. Simovski, “Analytical modelling of double-negative composites,” Metamaterials 2, 169–185 (2008).
[Crossref]

H. Wallén, H. Kettunen, and A. Sihvola, “Surface modes of negative-parameter interfaces and the importance of rounding sharp corners,” Metamaterials 2, 113–121 (2008).
[Crossref]

Opt. Express (1)

Opt. Spectrosc. (2)

O. N. Gadomskii and S. V. Sukhov, “Microscopic theory of a transition layer on the ideal surface of semiinfinite dielectric media and the near-field effect,” Opt. Spectrosc. 89, 261–266 (2000).
[Crossref]

C. R. Simovski, “Material parameters of metamaterials,” Opt. Spectrosc. 107, 726–753 (2009).
[Crossref]

Phys. Rev. A (1)

J. D. Baena, L. Jelinek, R. Marqués, and M. Silveirinha, “Unified homogenization theory for magnetoinductive and electromagnetic waves in split-ring metamaterials,” Phys. Rev. A 78, 013842 (2008).
[Crossref]

Phys. Rev. B (6)

R. R. A. Syms, O. Sydoruk, and L. Solymar, “Lossy metamaterials: no effective medium properties without noise,” Phys. Rev. B 84, 235150 (2011).
[Crossref]

M. Silveirinha, “Metamaterial homogenization approach with application to the characterization of microstructured composites with negative parameters,” Phys. Rev. B 75, 115104 (2007).
[Crossref]

A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B 84, 075153 (2011).
[Crossref]

R. Marqués, F. Medina, and R. Rafii-El-Idrissi, “Role of bianisotropy in negative permeability and left-handed metamaterials,” Phys. Rev. B 65, 144440 (2002).
[Crossref]

P. A. Belov and C. R. Simovski, “Boundary conditions for interfaces of electromagnetic crystals and the generalized Ewald-Oseen extinction principle,” Phys. Rev. B 73, 045102 (2006).
[Crossref]

X.-X. Liu, D. A. Powell, and A. Alù, “Correcting the Fabry-Perot artifacts in metamaterial retrieval procedures,” Phys. Rev. B 84, 235106 (2011).
[Crossref]

Phys. Rev. E (1)

M. Gorkunov, S. A. Gredeskul, I. V. Shadrivov, and Y. S. Kivshar, “Effect of microscopic disorder on magnetic properties of metamaterials,” Phys. Rev. E 73, 056605 (2006).
[Crossref]

Proc. IEEE (2)

W. B. Weir, “Automatic measurement of complex dielectric constant and permeability at microwave frequencies,” Proc. IEEE 62, 33–36 (1974).
[Crossref]

R. Marqués, L. Jelinek, M. Freire, J. Baena, and M. Lapine, “Bulk metamaterials made of resonant rings,” Proc. IEEE 99, 1660–1668 (2011).
[Crossref]

Other (1)

S. A. Schelkunoff and H. T. Friis, Antennas Theory and Practice (Wiley, 1966).

Cited By

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Figures (5)

Fig. 1
Fig. 1

Scheme of a symmetric finite cube with discrete structure, showing two options of surface configuration (for the same unit cell in the bulk): “flat” geometry (a) and “ragged” geometry (b). Note that the actual number of elements is not necessarily reflected here.

Fig. 2
Fig. 2

Real part of the normalised polarisability (arbitrary units) of discrete cubes with 5, 8 or 11 layers of resonators in each direction, having either a “flat” geometry (a) or a “ragged” geometry (b).

Fig. 3
Fig. 3

Real (a) and imaginary (b) parts of the normalised polarisability (arbitrary units) of discrete cubes with 11 layers in each direction, having either a “flat” geometry (dashed lines) or “ragged” geometry (dotted lines), in comparison with the polarizability of a homogeneous cube (solid lines).

Fig. 4
Fig. 4

Frequency dependence of the A and C coefficients calculated with Eq. (3) based on the effective permeability μ of a homogeneous cube Eq. (1) and its polarisability α, obtained from numerical simulations.

Fig. 5
Fig. 5

A comparison between the permeability μ of a homogeneous cube (solid line) and the effective permeability μ# obtained for 11-layer discrete cubes with a “flat” geometry (dashed lines) or a “ragged” geometry (dotted lines); real (a) and imaginary (b) parts.

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

μ = 1 + γ ( k res a ) 2 / ( k 0 a ) 2 1 2 ϰ a 4 ϰ c γ / 3 ,
α = A ( μ 1 ) / ( μ + C ) ,
C = | μ | 2 Re μ Re α Im α Im μ 1 Re μ + Re α Im α Im μ , A = ( Re μ + C ) 2 + ( Im μ ) 2 Im μ ( C + 1 ) Im α .
μ # = ( A + C α # ) / ( A α # ) .

Metrics