## Homogenization of quasi-1d metamaterials and the problem of extended bandwidth |

Optics Express, Vol. 22, Issue 3, pp. 2429-2442 (2014)

http://dx.doi.org/10.1364/OE.22.002429

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### Abstract

We derive approximate analytical expressions for the effective permittivity tensor of two-phase metamaterials whose geometry is close to one-dimensional (quasi-one-dimensional metamaterials). Specifically, we consider the metamaterial made of parallel slabs with width given by a linear or parabolic function. Using our approach, the design of epsilon-near-zero, ultra-low and high refractive index metallodielectric metamaterials with extended bandwidth has been demonstrated. In addition, generalizations to the three-dimensional case and some limitations of the presented technique are briefly considered.

© 2014 Optical Society of America

## 1. Introduction

1. G. W. Milton, *The Theory of Composites* (Cambridge University, 2002). [CrossRef]

3. A. Reuss, “Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle,” Z. Angew. Math. Mech. **9**, 49–58 (1929). [CrossRef]

1. G. W. Milton, *The Theory of Composites* (Cambridge University, 2002). [CrossRef]

4. W.T. Perrins and R.C. McPedran, “Metamaterials and the homogenization of composite materials,” Metamaterials **4**, 24–31 (2010). [CrossRef]

*et al.*[5

5. A.A. Krokhin, P. Halevi, and J. Arriaga, “Long-wavelength limit (homogenization) for two-dimensional photonic crystals,” Phys. Rev. B **65**, 115208 (2002). [CrossRef]

7. A.V. Goncharenko, “Limiting geometries and dielectric tensor of superlattices,” Tech. Phys. Lett. **26**(7), 594–596 (2000). [CrossRef]

8. M. Scalora, M.J. Bloemer, A.S. Pethel, J.P. Dowling, C.M. Bowden, and A.S. Manka, “Transparent, metallodielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. **83**, 2377–2383 (1998). [CrossRef]

9. H. Rauh, G.I. Yampolskaya, and S.V. Yampolskii, “Optical transmittance of photonic structures with linearly graded dielectric constituents,” New J. Phys. **12**, 073033 (2010). [CrossRef]

10. L.V. Alekseyev, E.E. Narimarov, T. Tumkur, H. Li, Yu. A. Barnakov, and M.A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. **97**, 131107 (2010). [CrossRef]

11. A.V. Kabashin, P. Evans, S. Pastkovsky, W. Hendren, G.A. Wurtz, R. Atkinson, R. Pollard, V.A. Podolskiy, and A.V. Zayats, “Plasmonic nanorod metamaterials for biosensing,” Nature Mater. **8**, 867–871 (2009). [CrossRef]

12. P. Ginzburg, F. J. Rodriguez Fortuno, G.A. Wurtz, W. Dickson, A. Murphy, F. Morgan, R.J. Pollard, I. Iorsh, A. Atrashchenko, P.A. Belov, Y.S. Kivshar, A. Nevet, G. Ankonina, M. Orenstein, and A.V. Zayats, “Manipulating polarization of light with ultrathin epsilon-near-zero metamaterials,” Opt. Express **21**, 14907–14917 (2013). [CrossRef] [PubMed]

13. P. 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]

15. C.R. Simovski, P.A. Belov, A.A. Atrashchenko, and Y.S. Kivshar, “Wire metamaterials: Physics and applications,” Adv. Mater. **24**, 4229–4248 (2012). [CrossRef] [PubMed]

14. Z. Jacob, L.V. Alekseev, and E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express **14**, 8247–8256 (2006). [CrossRef] [PubMed]

16. D.R. Smith, P. Kolinko, and D. Schurig, “Negative refraction in indefinite media,” J. Opt. Soc. Am. B **21**, 1032–1043 (2004). [CrossRef]

18. M.A. Noginov, Yu.A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E.E. Narimanov, “Bulk photonic metamaterials with hyperbolic dispersion,” Appl. Phys. Lett. **94**, 151105 (2009). [CrossRef]

19. A.V. Goncharenko, V.U. Nazarov, and K.R. Chen, “Development of metamaterials with desired broadband optical properties,” Appl. Phys. Lett. **101**, 071907 (2012). [CrossRef]

20. A.V. Goncharenko and K.R. Chen, “Strategy for designing epsilon-near-zero nanostructured metamaterials over a frequency range,” J. Nanophoton. **4**, 041530 (2010). [CrossRef]

21. A.V. Goncharenko, V.U. Nazarov, and K.R. Chen, “Nanostructured metamaterials with broadband optical properties,” Opt. Mater. Express **3**, 143–156 (2013). [CrossRef]

19. A.V. Goncharenko, V.U. Nazarov, and K.R. Chen, “Development of metamaterials with desired broadband optical properties,” Appl. Phys. Lett. **101**, 071907 (2012). [CrossRef]

21. A.V. Goncharenko, V.U. Nazarov, and K.R. Chen, “Nanostructured metamaterials with broadband optical properties,” Opt. Mater. Express **3**, 143–156 (2013). [CrossRef]

## 2. Basic formalism

*y*-direction, and homogeneous in the

*xz*-plane. So, the

*y*-axis is the direction along which the local permittivity can change. To validate the efeective permittivity formalism, the period

*S*is considered to be much less than the wavelength (quasistatic approximation). Then, the effective permittivity can be represented as a tensor of the form

_{y}*ε̃*=

*diag*(

*ε*,

_{x}*ε*) with where the components

_{y}*ε*and

_{x}*ε*of the tensor correspond to the orientation of the electric field along the

_{y}*x*- and

*y*- axes, respectively.

*ε*

_{1}and

*ε*

_{2}. In the following, the phases 1 and 2 are called the slab and the host, respectively. Therefore, the above MM is a periodic array of parallel slabs of varied width. Then, the local width of the slab normalized to the size of the unit cell along the

*x*–axis

*S*can be defined as the local filling factor of the phase 1,

_{x}*f*(

*y*). Obviously, the total filling factor of the phase 1 is

*f*

_{2}= 1 −

*f*

_{1}. Next we break the unit cell down into thin layers of the thickness d

*y*(one such a layer is shown in Fig. 1). The assumption, used in the following analysis, is that the interface between two phases remains

*almost parallel*to the

*y*–axis within each layer. To meet this condition, the function

*f*(

*y*) should, obviously, be smooth and satisfy the inequality where

*f*(

*y*) =

_{max}*max f*(

*y*) and

*f*(

*y*) =

_{min}*min f*(

*y*). The electric field, being oriented along the

*y*–axis, remains almost parallel to the interfaces within the layer, while being oriented along the

*x*–axis, becomes almost perpendicular to those. This allows one to write down for the corresponding layer permittivities

*ε*

_{||}(

*y*) and

*ε*

_{⊥}(

*y*) After substituting

*ε*(

*y*) =

*ε*

_{⊥}(

*y*) into Eq. (1) and

*ε*(

*y*) =

*ε*

_{||}(

*y*) into Eq. (2), one has where

*y′*=

*y/S*. As is easy to check, the principal components of the tensor

_{y}*ε̃*given by Eqs. (6) and (7) satisfy the Keller’s theorem (duality relation) [22

22. J.B. Keller, “A theorem on the conductivity of a composite medium,” J. Math. Phys. **5**, 548–549 (1964). [CrossRef]

20. A.V. Goncharenko and K.R. Chen, “Strategy for designing epsilon-near-zero nanostructured metamaterials over a frequency range,” J. Nanophoton. **4**, 041530 (2010). [CrossRef]

21. A.V. Goncharenko, V.U. Nazarov, and K.R. Chen, “Nanostructured metamaterials with broadband optical properties,” Opt. Mater. Express **3**, 143–156 (2013). [CrossRef]

**3**, 143–156 (2013). [CrossRef]

19. A.V. Goncharenko, V.U. Nazarov, and K.R. Chen, “Development of metamaterials with desired broadband optical properties,” Appl. Phys. Lett. **101**, 071907 (2012). [CrossRef]

*ε*, we solved an inverse problem and found the piece-wise function

_{y}*f*. Such an approach can, however, involve computational difficulties. Here, we suggest to parametrize the distribution function

*f*(

*y*) using only a few parameters that considerably simplifies the practical use of Eqs. (6) and (7).

## 3. Particular cases of the distribution function *f*(*y′*)

*y′*

_{0},

*f*, and

_{min}*f*. The linear profile can be analytically represented as and the parabolic profile as

_{max}*s*=

*ε*

_{2}/(

*ε*

_{1}−

*ε*

_{2}) and

*s′*=

*ε*

_{1}/(

*ε*

_{2}−

*ε*

_{1}).

## 4. Examples of application: Numerical results

23. A. Vial and T. Laroche, “Comparison of gold and silver dispersion laws suitable for FDTD simulations,” Appl. Phys. B **93**, 139–143 (2008). [CrossRef]

### 4.1. ENZ metamaterials

*ε*fitting to zero for the bands 570 – 670 nm, 570 – 720 nm, and 570 – 770 nm (here we use the parabolic profile for the distribution function; the use of the linear profile yields worse fitting). The root mean square (

_{y}*rms*) of the deviation of the fit from zero for each band is 8.8 · 10

^{−4}, 0.0015, and 0.0022, respectively. It should be also noted that the shift of the actual band to the long wavelength side allows one to considerably improve the accuracy of fitting. For example,

*rms*= 1.3 · 10

^{−4}for the band 700 – 800 nm (see Fig. 4, blue solid curve). In this figure, two solutions for Re

*ε*, fitted to zero over the band 700 – 800 nm, as well as the corresponding imaginary parts of the effective permittivity are shown.

_{y}### 4.2. ULRI metamaterials

*n*

^{*}is similar to that for ENZ MMs [19

**101**, 071907 (2012). [CrossRef]

*Re*{

*ε*[

_{y}*ε*

_{1}(

*ω*),

*ε*

_{2},

*f*(

*y′*

_{0},

*f*,

_{min}*f*)]}

_{max}^{1/2}−

*n*

^{*}|| on a frequency band [

*ω*

_{1},

*ω*

_{2}] [19

**101**, 071907 (2012). [CrossRef]

**3**, 143–156 (2013). [CrossRef]

*y′*

_{0},

*f*, and

_{min}*f*, multiple solution branches can coexist for the same

_{max}*n*

^{*}. As an example, in Fig. 5 we show the effective refractive index of the designed MM fitted to

*n*

^{*}= 0.25 over the band

*λ*= 620 – 720 nm (the red edge of the visible spectrum). As before, we have used the parabolic profile for the distribution function, although the use of the linear profile also yields satisfactory results. The four curves (1–4) correspond to four sets of the parameters

*y′*

_{0},

*f*, and

_{min}*f*which allow one to obtain local minima of the above norm. The root mean square of the deviation of the fit from

_{max}*n*

^{*}for each solution branch is shown in Fig. 6. As one can see, only one solution branch (curve 2) exists for

*n*

^{*}< 0.1475, while another branch (curve 4) provides appropriate solution (with small

*rms*) as

*n*

^{*}→ 1. At

*n*

^{*}= 0.16, the minimal

*rms*can be as small as 0.0045. At the same time, the root mean square rapidly rises as

*n*

^{*}becomes smaller than 0.14.

### 4.3. HRI metamaterials

*Re*{

*ε*[

_{x}*ε*

_{1}(

*ω*),

*ε*

_{2},

*f*(

*y′*

_{0},

*f*,

_{min}*f*)]}

_{max}^{1/2}−

*n*

^{*}|| on a frequency band [

*ω*

_{1},

*ω*

_{2}]. The effective refractive index for two possible solutions (both are for the parabolic distribution function) for

*n*

^{*}= 4.25 are shown in Fig. 7. We note that although both curves have been obtained for the same actual band 620 – 720 nm, one of the solutions (solid curve) provides appropriate fitting over a broader band. Furthermore, as can be seen from Fig. 8, three solution branches can coexist below

*n*

^{*}= 3 and two branches above that. One of them (curve 1) provides lower

*rms*below

*n*

^{*}= 4.25, while another provides lower

*rms*above it. For example, at

*n*

^{*}= 4.5 the best fit provides

*rms*≃ 0.02 and then the root mean square rapidly rises with

*n*

^{*}. It should be also noted that shifting the actual band to the long wavelength side allows one to considerably extend the bandwidth providing high values of the effective refractive index and keeping

*rms*relatively small. For example, fitting

*Re*(

*ε*)

_{x}^{1/2}to

*n*

^{*}= 5 over the band of 660 – 860 nm yields

*rms*= 0.006, while fitting to

*n*

^{*}= 5.5 over the band of 700 – 900 nm yields

*rms*= 0.0091 (see Fig. 9). Fitting to

*n*

^{*}= 6 over the band of 770 – 1020 nm (not shown here) yields the root mean square as small as 0.0116.

## 5. Discussion

*ε*, Im

*ε*) which is bounded by the straight line, connecting

*ε*

_{1}and

*ε*

_{2},

*ε*=

_{u}*f*

_{1}

*ε*

_{1}+

*f*

_{2}

*ε*

_{2}, and by the arc, connecting the origin,

*ε*

_{1}, and

*ε*

_{2},

*ε*= (

_{l}*f*

_{1}/

*ε*

_{1}+

*f*

_{2}/

*ε*

_{2})

^{−1}(the above bounds are known as upper and lower Wiener’s bound, respectively, in terms of the effective conductivity). In turn, the allowable values of the effective refractive index must lie within a region in the complex plane (Re

*n*, Im

*n*), bounded by the curves

*n*≤Re

_{u}*n*, and no allowed value of

_{l}*ε*exists such that

_{eff}*ε′*

_{1}−

*ε*

_{2}, has one minimum. Upon differentiating Eq. (15) and equating the result to zero, it is easy to find that this minimum is achieved at Then, the minimum value

*n*of the real part of the effective refractive index can be obtained after substitution of Eq. (16) into Eq. (15),

_{min}*n*can be achieved by reducing

_{min}*ε*

_{2}. Besides, if the metal permittivity is described by Drude’s model, at low frequencies

*n*is large:

_{min}*ω*→ 0 (here

*γ*is the damping coefficient or collision frequency). On the other hand, it is also large at high frequencies, because the dielectric constrast Δ is relatively small in this case. Thus, at some frequency,

*n*takes its minimum value.

_{min}*n*≤Re

_{u}*n*, the point with the maximum value of the effective refractive index lies on the lower Wiener’s bound. That is why just Eq. (6) (which, as we remind, corresponds to electric field oriented normally to the wires) is convenient to realize the HRI regime. Upon differentiating Re

_{l}*n*(

_{l}*f*) and equating the result to zero, it is possible to find

*ω*≡ 2

*πc/λ*.

*n*(

_{min}*λ*) and

*n*(

_{max}*λ*) are shown in Fig. 10. So,

*n*is large at high frequencies, reduces with reducing the frequency, and takes its smallest value (0.078) at about

_{min}*λ*= 860 nm. Then, it slowly rises with

*λ*, i.e., with a decrease in the frequency. The results of calculations performed with the use of exact and approximate formulas are almost indistinguishable at wavelengths above

*λ*= 560 nm. At the same time,

*n*monotonically rises with

_{max}*λ*.

*n*and

_{min}*n*have been obtained for a fixed frequency (the narrowband case). Obviously,

_{max}*n*cannot be reduced, while

_{min}*n*cannot be enlarged when extending the bandwidth.

_{max}*rms*is minimal. However, this is not always so. Let us rewrite the condition of quasi-one-dimensionality, In eq. (3), in the form Then, if

*y′*

_{0}is close 1/2, and/or

*f*is close to 1, and

_{max}*f*is close to 0, the condition

_{min}*S*≪

_{x}*S*must be satisfied for In eq. (20) to be valid. However, becase

_{y}*S*cannot be too large (

_{y}*S*≪

_{y}*λ*), this imposes a severe upper limit on

*S*which can be difficult to attain using existing fabrication techniques. For example, in Fig. 5 we show four solutions for which

_{x}*rms*= 0.0035 (curve 1), 0.0089 (curve 2), and 0.021 (curves 3 and 4) for

*n*

^{*}= 0.25. So, after substituting specific parameters

*y′*

_{0},

*f*, and

_{min}*f*, obtained after fitting, into In eq. (20), we have got 96.1(

_{max}*S*) ≪ 1, 453(

_{x}/S_{y}*S*) ≪ 1, 1.55(

_{x}/S_{y}*S*) ≪ 1, and 0.94(

_{x}/S_{y}*S*) ≪ 1, for curves 1,2,3, and 4, respectively. It is obvious that, althogh

_{x}/S_{y}*rms*is very small for the curves 1 and 2, real experiment with such geometrical parameters seems to be hardly feasible, because

*S*must be too small to satisfy the condition of quasi-one-dimensionality. Otherwise, Eqs. (6) and (7) become invalid, and the calculations within the framewotk of our approach lose their accuracy.

_{x}26. A.K. Popov and S.A. Myslivets, “Transformable broad-band transparency and amplification in negative-index films,” Appl. Phys. Lett. **93**, 191117 (2008). [CrossRef]

27. A.N. Lagarkov, V.N. Kisel, and A.K. Sarychev, “Loss and gain in metamaterials,” J. Opt. Soc. Am. B **27**, 648–659 (2010). [CrossRef]

*n*,Im

*n*) plane for three wavelengths, 620, 670, and 720 nm. We note that in the ULRI regime, which is realized near the upper Wiener bound, the imaginary part of the effective refractive index distinctly rises as

*n*

^{*}→

*n*. At the same time, it can be small at moderate values on

_{min}*n*

^{*}and very small when

*n*

^{*}is about unity or slightly above it (this is because the ULRI, as well as ENZ regime are nonresonant in character [19

**101**, 071907 (2012). [CrossRef]

20. A.V. Goncharenko and K.R. Chen, “Strategy for designing epsilon-near-zero nanostructured metamaterials over a frequency range,” J. Nanophoton. **4**, 041530 (2010). [CrossRef]

## 6. Generalizations and limitations

*ε̃*=

*diag*(

*ε*,

_{x}*ε*,

_{z}*ε*) where the

_{y}*ε*component is of the same form as in Eq. (2), with

_{y}*ε*

_{||}(

*y*) given by Eq. (4). It should be noted that both

*ε*

_{||}and

*ε*depend neither on the shape of the wires’ cross-section nor the lattice type. For the

_{y}*ε*and

_{x}*ε*components we may write where both

_{z}*f*(

*y*) << 1, they may be evaluated using generalized Maxwell-Garnett technique [28

28. J. Elser, R. Wangberg, V.A. Podolskiy, and E.E. Narimanov, “Nanowire metamaterials with extreme optical anisotropy,” Appl. Phys. Lett. **89**, 261102 (2006). [CrossRef]

29. W. Yan, N.A. Mortensen, and M. Wubs, “Hypebolic metamaterial lens with hydrodynamic nonlocal response,” Opt. Express **21**, 15026–15036 (2013). [CrossRef] [PubMed]

31. X.X. Liu and A. Alu, “Limitations and potential of metamaterial lenses,” J. Nanophoton. **5**, 053509 (2011). [CrossRef]

32. C.R. Simovski, “On electromagnetic characterization and homogenization of nanostructured metamaterials,” J. Opt. **13**, 013001 (2011). [CrossRef]

33. A.A. Orlov, P.M. Voroshilov, P.A. Belov, and Y.S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B **84**, 045424 (2011). [CrossRef]

34. A.V. Chebykin, A.A. Orlov, A.V. Vozianova, S.I. Maslovski, Yu.S. Kivshar, and P.A. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B **84**, 115438 (2011). [CrossRef]

35. P.A. Belov, R. Marques, S.I. Maslovski, I.S. Nefedov, M. Silveirinha, C.R. Simovski, and S.A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B **67**, 113103 (2003). [CrossRef]

15. C.R. Simovski, P.A. Belov, A.A. Atrashchenko, and Y.S. Kivshar, “Wire metamaterials: Physics and applications,” Adv. Mater. **24**, 4229–4248 (2012). [CrossRef] [PubMed]

11. A.V. Kabashin, P. Evans, S. Pastkovsky, W. Hendren, G.A. Wurtz, R. Atkinson, R. Pollard, V.A. Podolskiy, and A.V. Zayats, “Plasmonic nanorod metamaterials for biosensing,” Nature Mater. **8**, 867–871 (2009). [CrossRef]

36. S.I. Maslovski and M.G. Silveirinha, “Nonlocal permittivity from a quasistatic model for a class of wire media,” Phys. Rev. B **80**, 245101 (2009). [CrossRef]

*k*= 0); in addition, its strength can be controlled by changing a ratio between the thicknesses of metal and dielectric layers [33

_{y}33. A.A. Orlov, P.M. Voroshilov, P.A. Belov, and Y.S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B **84**, 045424 (2011). [CrossRef]

37. N. Dubrovina, L.O. Le Cunff, N. Burokur, R. Ghasemi, A. Degiron, A. De Lustrac, A. Vial, G. Leronded, and A. Lupu, “Single metafilm effective medium behavior in optical domain: Maxwell-Garnett approximation and beyond,” Appl. Phys. A **109**, 901–906 (2012). [CrossRef]

38. E.A. Gibson, I.R. Gabitov, A.I. Maimistov, and N.M. Litchinitser, “Transition metamaterials with spatially separated zeros,” Opt. Lett. **36**, 3624–3626 (2011). [CrossRef] [PubMed]

*γ*should be corrected to take into account the interfacial scattering effects, and may be written as

*γ*=

*γ*+

_{b}*Av*where

_{F}/t*γ*is the bulk damping constant,

_{b}*A*is a constant which depends on the surrface scattering mechanism,

*v*is the Fermi velocity of electrons, and

_{F}*t*is the characteristic thickness of the layer. However, as we noticed earlier [20

**4**, 041530 (2010). [CrossRef]

*ε*component of the dielectric tensor. At the same time, this effect can take place as the electric field is perpendicular to the layers which, in turn, are thin enough. In this case the metal permittivity

_{y}*ε*

_{1}can change, that has to be taken into account in practical calculations.

## 7. Conclusion

## Acknowledgments

## References and links

1. | G. W. Milton, |

2. | W. Voight, |

3. | A. Reuss, “Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle,” Z. Angew. Math. Mech. |

4. | W.T. Perrins and R.C. McPedran, “Metamaterials and the homogenization of composite materials,” Metamaterials |

5. | A.A. Krokhin, P. Halevi, and J. Arriaga, “Long-wavelength limit (homogenization) for two-dimensional photonic crystals,” Phys. Rev. B |

6. | M. Sahimi, |

7. | A.V. Goncharenko, “Limiting geometries and dielectric tensor of superlattices,” Tech. Phys. Lett. |

8. | M. Scalora, M.J. Bloemer, A.S. Pethel, J.P. Dowling, C.M. Bowden, and A.S. Manka, “Transparent, metallodielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. |

9. | H. Rauh, G.I. Yampolskaya, and S.V. Yampolskii, “Optical transmittance of photonic structures with linearly graded dielectric constituents,” New J. Phys. |

10. | L.V. Alekseyev, E.E. Narimarov, T. Tumkur, H. Li, Yu. A. Barnakov, and M.A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. |

11. | A.V. Kabashin, P. Evans, S. Pastkovsky, W. Hendren, G.A. Wurtz, R. Atkinson, R. Pollard, V.A. Podolskiy, and A.V. Zayats, “Plasmonic nanorod metamaterials for biosensing,” Nature Mater. |

12. | P. Ginzburg, F. J. Rodriguez Fortuno, G.A. Wurtz, W. Dickson, A. Murphy, F. Morgan, R.J. Pollard, I. Iorsh, A. Atrashchenko, P.A. Belov, Y.S. Kivshar, A. Nevet, G. Ankonina, M. Orenstein, and A.V. Zayats, “Manipulating polarization of light with ultrathin epsilon-near-zero metamaterials,” Opt. Express |

13. | P. 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 |

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

15. | C.R. Simovski, P.A. Belov, A.A. Atrashchenko, and Y.S. Kivshar, “Wire metamaterials: Physics and applications,” Adv. Mater. |

16. | D.R. Smith, P. Kolinko, and D. Schurig, “Negative refraction in indefinite media,” J. Opt. Soc. Am. B |

17. | J. Yao, Z Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A.M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science |

18. | M.A. Noginov, Yu.A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E.E. Narimanov, “Bulk photonic metamaterials with hyperbolic dispersion,” Appl. Phys. Lett. |

19. | A.V. Goncharenko, V.U. Nazarov, and K.R. Chen, “Development of metamaterials with desired broadband optical properties,” Appl. Phys. Lett. |

20. | A.V. Goncharenko and K.R. Chen, “Strategy for designing epsilon-near-zero nanostructured metamaterials over a frequency range,” J. Nanophoton. |

21. | A.V. Goncharenko, V.U. Nazarov, and K.R. Chen, “Nanostructured metamaterials with broadband optical properties,” Opt. Mater. Express |

22. | J.B. Keller, “A theorem on the conductivity of a composite medium,” J. Math. Phys. |

23. | A. Vial and T. Laroche, “Comparison of gold and silver dispersion laws suitable for FDTD simulations,” Appl. Phys. B |

24. | L. Sun and K.W. Yu, “Strategy for designing broadband epsilon-near-zero metamaterials,” J. Opt. Soc. Am. B |

25. | L. Sun, K.W. Yu, and X. Yang, “Integrated optical devices based on broadband epsilon-near-zero meta-atoms,” Opt. Lett. |

26. | A.K. Popov and S.A. Myslivets, “Transformable broad-band transparency and amplification in negative-index films,” Appl. Phys. Lett. |

27. | A.N. Lagarkov, V.N. Kisel, and A.K. Sarychev, “Loss and gain in metamaterials,” J. Opt. Soc. Am. B |

28. | J. Elser, R. Wangberg, V.A. Podolskiy, and E.E. Narimanov, “Nanowire metamaterials with extreme optical anisotropy,” Appl. Phys. Lett. |

29. | W. Yan, N.A. Mortensen, and M. Wubs, “Hypebolic metamaterial lens with hydrodynamic nonlocal response,” Opt. Express |

30. | C. David, N.A. Mortensen, and J. Christensen, “Perfect imaging, epsilon-near zero phenomena and waveguiding in the scope of nonlocal effects,” Sci. Rept. |

31. | X.X. Liu and A. Alu, “Limitations and potential of metamaterial lenses,” J. Nanophoton. |

32. | C.R. Simovski, “On electromagnetic characterization and homogenization of nanostructured metamaterials,” J. Opt. |

33. | A.A. Orlov, P.M. Voroshilov, P.A. Belov, and Y.S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B |

34. | A.V. Chebykin, A.A. Orlov, A.V. Vozianova, S.I. Maslovski, Yu.S. Kivshar, and P.A. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B |

35. | P.A. Belov, R. Marques, S.I. Maslovski, I.S. Nefedov, M. Silveirinha, C.R. Simovski, and S.A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B |

36. | S.I. Maslovski and M.G. Silveirinha, “Nonlocal permittivity from a quasistatic model for a class of wire media,” Phys. Rev. B |

37. | N. Dubrovina, L.O. Le Cunff, N. Burokur, R. Ghasemi, A. Degiron, A. De Lustrac, A. Vial, G. Leronded, and A. Lupu, “Single metafilm effective medium behavior in optical domain: Maxwell-Garnett approximation and beyond,” Appl. Phys. A |

38. | E.A. Gibson, I.R. Gabitov, A.I. Maimistov, and N.M. Litchinitser, “Transition metamaterials with spatially separated zeros,” Opt. Lett. |

**OCIS Codes**

(260.2065) Physical optics : Effective medium theory

(160.3918) Materials : Metamaterials

**ToC Category:**

Metamaterials

**History**

Original Manuscript: November 21, 2013

Manuscript Accepted: January 14, 2014

Published: January 28, 2014

**Citation**

A. V. Goncharenko, E. F. Venger, and A. O. Pinchuk, "Homogenization of quasi-1d metamaterials and the problem of extended bandwidth," Opt. Express **22**, 2429-2442 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-3-2429

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### References

- G. W. Milton, The Theory of Composites (Cambridge University, 2002). [CrossRef]
- W. Voight, Lehrbuch der Kristallphysik (Teubner-Verlag, 1928).
- A. Reuss, “Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle,” Z. Angew. Math. Mech. 9, 49–58 (1929). [CrossRef]
- W.T. Perrins, R.C. McPedran, “Metamaterials and the homogenization of composite materials,” Metamaterials 4, 24–31 (2010). [CrossRef]
- A.A. Krokhin, P. Halevi, J. Arriaga, “Long-wavelength limit (homogenization) for two-dimensional photonic crystals,” Phys. Rev. B 65, 115208 (2002). [CrossRef]
- M. Sahimi, Heterogeneous Materials. I. Linear Transport and Optical Properties (Springer, 2003).
- A.V. Goncharenko, “Limiting geometries and dielectric tensor of superlattices,” Tech. Phys. Lett. 26(7), 594–596 (2000). [CrossRef]
- M. Scalora, M.J. Bloemer, A.S. Pethel, J.P. Dowling, C.M. Bowden, A.S. Manka, “Transparent, metallodielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377–2383 (1998). [CrossRef]
- H. Rauh, G.I. Yampolskaya, S.V. Yampolskii, “Optical transmittance of photonic structures with linearly graded dielectric constituents,” New J. Phys. 12, 073033 (2010). [CrossRef]
- L.V. Alekseyev, E.E. Narimarov, T. Tumkur, H. Li, Yu. A. Barnakov, M.A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. 97, 131107 (2010). [CrossRef]
- A.V. Kabashin, P. Evans, S. Pastkovsky, W. Hendren, G.A. Wurtz, R. Atkinson, R. Pollard, V.A. Podolskiy, A.V. Zayats, “Plasmonic nanorod metamaterials for biosensing,” Nature Mater. 8, 867–871 (2009). [CrossRef]
- P. Ginzburg, F. J. Rodriguez Fortuno, G.A. Wurtz, W. Dickson, A. Murphy, F. Morgan, R.J. Pollard, I. Iorsh, A. Atrashchenko, P.A. Belov, Y.S. Kivshar, A. Nevet, G. Ankonina, M. Orenstein, A.V. Zayats, “Manipulating polarization of light with ultrathin epsilon-near-zero metamaterials,” Opt. Express 21, 14907–14917 (2013). [CrossRef] [PubMed]
- P. Belov, 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]
- Z. Jacob, L.V. Alekseev, E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14, 8247–8256 (2006). [CrossRef] [PubMed]
- C.R. Simovski, P.A. Belov, A.A. Atrashchenko, Y.S. Kivshar, “Wire metamaterials: Physics and applications,” Adv. Mater. 24, 4229–4248 (2012). [CrossRef] [PubMed]
- D.R. Smith, P. Kolinko, D. Schurig, “Negative refraction in indefinite media,” J. Opt. Soc. Am. B 21, 1032–1043 (2004). [CrossRef]
- J. Yao, Z Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A.M. Stacy, X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008). [CrossRef] [PubMed]
- M.A. Noginov, Yu.A. Barnakov, G. Zhu, T. Tumkur, H. Li, E.E. Narimanov, “Bulk photonic metamaterials with hyperbolic dispersion,” Appl. Phys. Lett. 94, 151105 (2009). [CrossRef]
- A.V. Goncharenko, V.U. Nazarov, K.R. Chen, “Development of metamaterials with desired broadband optical properties,” Appl. Phys. Lett. 101, 071907 (2012). [CrossRef]
- A.V. Goncharenko, K.R. Chen, “Strategy for designing epsilon-near-zero nanostructured metamaterials over a frequency range,” J. Nanophoton. 4, 041530 (2010). [CrossRef]
- A.V. Goncharenko, V.U. Nazarov, K.R. Chen, “Nanostructured metamaterials with broadband optical properties,” Opt. Mater. Express 3, 143–156 (2013). [CrossRef]
- J.B. Keller, “A theorem on the conductivity of a composite medium,” J. Math. Phys. 5, 548–549 (1964). [CrossRef]
- A. Vial, T. Laroche, “Comparison of gold and silver dispersion laws suitable for FDTD simulations,” Appl. Phys. B 93, 139–143 (2008). [CrossRef]
- L. Sun, K.W. Yu, “Strategy for designing broadband epsilon-near-zero metamaterials,” J. Opt. Soc. Am. B 29, 984–989 (2012). [CrossRef]
- L. Sun, K.W. Yu, X. Yang, “Integrated optical devices based on broadband epsilon-near-zero meta-atoms,” Opt. Lett. 37, 3096–3098 (2012). [CrossRef] [PubMed]
- A.K. Popov, S.A. Myslivets, “Transformable broad-band transparency and amplification in negative-index films,” Appl. Phys. Lett. 93, 191117 (2008). [CrossRef]
- A.N. Lagarkov, V.N. Kisel, A.K. Sarychev, “Loss and gain in metamaterials,” J. Opt. Soc. Am. B 27, 648–659 (2010). [CrossRef]
- J. Elser, R. Wangberg, V.A. Podolskiy, E.E. Narimanov, “Nanowire metamaterials with extreme optical anisotropy,” Appl. Phys. Lett. 89, 261102 (2006). [CrossRef]
- W. Yan, N.A. Mortensen, M. Wubs, “Hypebolic metamaterial lens with hydrodynamic nonlocal response,” Opt. Express 21, 15026–15036 (2013). [CrossRef] [PubMed]
- C. David, N.A. Mortensen, J. Christensen, “Perfect imaging, epsilon-near zero phenomena and waveguiding in the scope of nonlocal effects,” Sci. Rept. 3, 02526 (2013).
- X.X. Liu, A. Alu, “Limitations and potential of metamaterial lenses,” J. Nanophoton. 5, 053509 (2011). [CrossRef]
- C.R. Simovski, “On electromagnetic characterization and homogenization of nanostructured metamaterials,” J. Opt. 13, 013001 (2011). [CrossRef]
- A.A. Orlov, P.M. Voroshilov, P.A. Belov, Y.S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84, 045424 (2011). [CrossRef]
- A.V. Chebykin, A.A. Orlov, A.V. Vozianova, S.I. Maslovski, Yu.S. Kivshar, P.A. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B 84, 115438 (2011). [CrossRef]
- P.A. Belov, R. Marques, S.I. Maslovski, I.S. Nefedov, M. Silveirinha, C.R. Simovski, S.A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B 67, 113103 (2003). [CrossRef]
- S.I. Maslovski, M.G. Silveirinha, “Nonlocal permittivity from a quasistatic model for a class of wire media,” Phys. Rev. B 80, 245101 (2009). [CrossRef]
- N. Dubrovina, L.O. Le Cunff, N. Burokur, R. Ghasemi, A. Degiron, A. De Lustrac, A. Vial, G. Leronded, A. Lupu, “Single metafilm effective medium behavior in optical domain: Maxwell-Garnett approximation and beyond,” Appl. Phys. A 109, 901–906 (2012). [CrossRef]
- E.A. Gibson, I.R. Gabitov, A.I. Maimistov, N.M. Litchinitser, “Transition metamaterials with spatially separated zeros,” Opt. Lett. 36, 3624–3626 (2011). [CrossRef] [PubMed]

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