## Nanostructured metamaterials with broadband optical properties |

Optical Materials Express, Vol. 3, Issue 2, pp. 143-156 (2013)

http://dx.doi.org/10.1364/OME.3.000143

Acrobat PDF (1004 KB)

### Abstract

We propose and develop a technique for designing a special class of nonmagnetic metamaterials possessing desired dielectric and optical properties over a broad frequency band. The technique involves the design of nanostructured metallodielectric materials (photonic crystals) with a special layered geometry where the metal content in each layer has to be determined using a fitting procedure. For illustration, we demonstrate the performance of our technique for tailoring metamaterials having epsilon-near-zero and on-demand refractive index (real or imaginary part) over a frequency band. One-, two-, as well as three-dimensional geometries have been considered. In the one-dimensional and two-dimensional cases, the results of semi-analytical calculations are validated by *ab initio* FDTD simulations.

© 2013 OSA

## 1. Introduction

2. R. Liu, Q. Cheng, J. Y. Chin, J. J. Mock, T. J. Cui, and D. R. Smith, “Broadband gradient index microwave quasi-optical elements based on non-resonant metamaterials,” Opt. Express **17**(23), 21030–21041 (2009). [CrossRef] [PubMed]

3. J. Valentine, S. Zhang, T. Zentgraf, and X. Zhang, “Development of bulk optical negative index fishnet metamaterials: achieving a low loss and broadband response through coupling,” Proc. IEEE **99**(10), 1682–1690 (2011). [CrossRef]

*x*-direction (the direction of propagation). In the

*z*-direction, the composite consists of identical unit cells of the width

*S*. Taken together, the cells form a one-dimensional (1d) photonic crystal. In turn, each unit cell may be considered as a gradient film, i.e., allowance is made for the varying permittivity

_{z}*ε*(

*z*) within each cell in the

*z*-direction.

*ε*(

*z*), is possible only in special cases [4

4. A. V. Shvartsburg, V. Kuzmiak, and G. Petite, “Optics of subwavelength gradient nanofilms,” Phys. Rep. **452**(2-3), 33–88 (2007). [CrossRef]

*ε*(

*z*) from the predetermined optical properties), which is the primary focus of our work, it presents a challenge and calls for even more sophisticated technique. Due to that, here we suggest another approach.

5. A. V. Goncharenko and K. R. Chen, “Strategy for designing epsilon-near-zero nanostructured metamaterials over a frequency range,” J. Nanophotonics **4**(1), 041530 (2010). [CrossRef]

6. L. Sun and K. W. Yu, “Strategy for designing broadband epsilon-near-zero metamaterials,” J. Opt. Soc. Am. B **29**(5), 984–989 (2012). [CrossRef]

8. L. Sun, K. W. Yu, and X. Yang, “Integrated optical devices based on broadband epsilon-near-zero meta-atoms,” Opt. Lett. **37**(15), 3096–3098 (2012). [CrossRef] [PubMed]

9. A. V. Goncharenko, V. U. Nazarov, and K. R. Chen, “Metallodielectric broadband metamaterials,” SPIE Newsroom (Feb. 6, 2012). DOI: 10.1117/2.1201201.0040207. [CrossRef]

10. A. V. Goncharenko, V. U. Nazarov, and K. R. Chen, “Development of metamaterials with desired broadband optical properties,” Appl. Phys. Lett. **101**(7), 071907 (2012). [CrossRef]

11. A. Alù, “First-principles homogenization theory for periodic metamaterials,” Phys. Rev. B **84**(7), 075153 (2011). [CrossRef]

12. C. R. Simovski, “On electromagnetic characterization and homogenization of nanostructured metamaterials,” J. Opt. **13**(1), 013001 (2011). [CrossRef]

*ab initio*finite-difference time-domain (FDTD) simulation and it is sufficient for many potential applications. More accurate technique taking into account local-field effects will be considered elsewhere.

13. K. Sakoda, N. Kawai, T. Ito, A. Chutinan, S. Noda, T. Mitsuyu, and K. Hirao, “Photonic bands of metallic systems. I. Principle of calculation and accuracy,” Phys. Rev. B **64**(4), 045116 (2001). [CrossRef]

15. A. A. Krokhin, E. Reyes, and L. Gumen, “Low-frequency index of refraction for a two-dimensional metallodielectric photonic crystal,” Phys. Rev. B **75**(4), 045131 (2007). [CrossRef]

## 2. Geometry and technique

*z*axis. Furthermore, we deal with the long-wavelength regime in which

*S*<<

_{z}*λ*/Re(

*n*) where

_{eff}*n*is the

_{eff}*effective*refractive index of the composite for the chosen orientation of the electric field and

*λ*is the vacuum wavelength of the light. So, we do not take into consideration the effect of the finite size of the unit cells (retardation effects) which can result in the non-zero phase advance across the cell [2

2. R. Liu, Q. Cheng, J. Y. Chin, J. J. Mock, T. J. Cui, and D. R. Smith, “Broadband gradient index microwave quasi-optical elements based on non-resonant metamaterials,” Opt. Express **17**(23), 21030–21041 (2009). [CrossRef] [PubMed]

*ε*ε e f f z = (

_{eff}≡*n*)

_{eff}^{2}may be written asIn particular, the profile

*ε*(

*z*) may be approximated as a step-like (piece-wise) function. In this case each unit cell is considered to be consisting of

*N*thin parallel layers (see, e.g., Fig. 1 from [5

5. A. V. Goncharenko and K. R. Chen, “Strategy for designing epsilon-near-zero nanostructured metamaterials over a frequency range,” J. Nanophotonics **4**(1), 041530 (2010). [CrossRef]

16. V. U. Nazarov, “Bulk and surface dielectric response of a superlattice with an arbitrary varying dielectric function: A general analytical solution in the local theory in the long-wave limit,” Phys. Rev. B Condens. Matter **49**(24), 17342–17350 (1994). [CrossRef] [PubMed]

*ε*(

*z*). For example, when dealing with metallodielectric composites, both the metal plasma frequency and collision frequency can be tuned by imposing a gradient of temperature in the

*z*-direction that results in the corresponding alteration of the permittivity profile along

*z*[17

17. J. P. Huang and K. W. Yu, “Optical nonlinearity enhancement of graded metallic films,” Appl. Phys. Lett. **85**(1), 94–96 (2004). [CrossRef]

*z*[18

18. L. F. Zhang, J. P. Huang, and K. W. Yu, “Gradation-controlled electric field distribution in multilayered colloidal crystals,” Appl. Phys. Lett. **92**(9), 091907 (2008). [CrossRef]

19. J. Sancho-Parramon, V. Janicki, and H. Zorc, “On the dielectric function tuning of random metal-dielectric nanocomposites for metamaterial applications,” Opt. Express **18**(26), 26915–26928 (2010). [CrossRef] [PubMed]

20. M. G. Blaber, M. D. Arnold, and M. J. Ford, “A review of the optical properties of alloys and intermetallics for plasmonics,” J. Phys. Condens. Matter **22**(14), 143201 (2010). [CrossRef] [PubMed]

*ε*(

*z*) is given in the review by Shvartsburg et

*al*[4

4. A. V. Shvartsburg, V. Kuzmiak, and G. Petite, “Optics of subwavelength gradient nanofilms,” Phys. Rep. **452**(2-3), 33–88 (2007). [CrossRef]

*almost*parallel to the metal/dielectric interface within the unit cell width. To explain how to realize such a geometry let us look at Fig. 1. Here, we sketch two metal/dielectric unit cells; each cell may be considered as consisting of 8 parallel layers of the uniform thickness. In its turn, each layer may be represented as a set of parallel segments oriented along the

*z*axis. Along the

*y*axis, the structure is considered to be homogeneous. The segments in the form of dielectric channels (pores) in a metallic host, as well as in the form of metallic channels (wires) in a dielectric host are allowed. In both cases, the width of the channels may differ in different layers. We note that all layers are considered to be of uniform thickness in contrast to our previous work [5

5. A. V. Goncharenko and K. R. Chen, “Strategy for designing epsilon-near-zero nanostructured metamaterials over a frequency range,” J. Nanophotonics **4**(1), 041530 (2010). [CrossRef]

*x*-axis as well, the composite becomes two-dimensional (2d) rather than 1d. However, if the size of the unit cell in the

*x-*direction (

*S*

_{x}) is small (

*S*

_{x}<<

*S*

_{z}), the composite may be considered as quasi one-dimensional. Furthermore, due to the continuity of the electric field which is tangential to the metal/dielectric interfaces, the permittivity of the i

*th*layer may be written asor, equivalently,where

*ε*

_{1}and

*ε*

_{2}are the permittivities of the metallic and dielectric phases,

*f*is the volume fraction of the metal phase in the i

_{i}*th*layer, and

*f(z)*is the corresponding distribution function of the metal phase. Substitution of Eq. (2a) into Eq. (1b) and Eq. (2b) into Eq. (1a) givesandrespectively, where

*s´*=

*z*/

*S*, and

_{z}*s*=

*ε*/(

_{2}*ε*–

_{1}*ε*).

_{2}*ϵ*occur at

_{eff}*f*(

*z*) = -

*s*(or

*f*= -

_{i}*s*). This feature of the effective permittivity results from a specific geometry of the photonic crystals under consideration where localized surface plasmons cannot be excited and hence resonances are not present.

*F*is an objective function, and the sought function

*f*has to be determined using a proper minimization procedure. Thus, in the present approach the unknown function is the distribution of metal content along z-coordinate, while in our previous work [5

**4**(1), 041530 (2010). [CrossRef]

## 3. Results and discussion

*ε*

_{2}= 2.5 for the permittivity of a dielectric and used the Drude representation of the form

### 3.1 Design of epsilon-near-zero MMs

21. See, e.g.,B. Edwards, A. Alu, M. G. Silveirinha, and N. Engheta, “Reflectionless sharp bends and corners in waveguides using epsilon-near-zero effects,” J. Appl. Phys. **105**(4), 044905 (2009) (and references therein). [CrossRef]

22. E. O. Liznev, A. V. Dorofeenko, L. Huizhe, A. P. Vinogradov, and S. Zouhdi, “Epsilon-near-zero material as a unique solution to three different approaches to cloaking,” Appl. Phys., A Mater. Sci. Process. **100**(2), 321–325 (2010). [CrossRef]

*broadband*ENZ MMs remains a challenging task. A possible solution has been proposed in our previous work [5

**4**(1), 041530 (2010). [CrossRef]

*z*-component of the dielectric tensor is obtained after dividing each unit cell into 8 layers and finding the metal content in each of them. Then, using Eq. (3a), the effective permittivity (see Fig. 2 ) is calculated. Close inspection shows that |Re

*ε*| does not exceed 0.05 in the actual frequency band of

_{eff}*ε*is a function oscillating about zero within the actual band where the number of its zeros corresponds to the number of layers in the unit cell. In this example, our model for Re

_{eff}*ε*yields 7 zeros within the band. It is so because in this particular case the metal contents in two of eight layers are very close. To assess the accuracy of fitting, we have also determined, and shown in Fig. 2, the root mean square of the deviation of the fit which in this case may be defined as

_{eff}*ω*running through the band [

_{i}*ω*

_{1},

*ω*

_{2}].

### 3.2 Design of MMs with on-demand refractive index

23. K. Maex, M. R. Baklanov, D. Shamiryan, F. Iacopi, S. H. Brongersma, and Z. S. Yanovitskaya, “Low dielectric constant materials for microelectronics,” J. Appl. Phys. **93**(11), 8793–8841 (2003). [CrossRef]

24. B. T. Schwartz and R. Piestun, “Total external reflection from metamaterials with ultralow refractive index,” J. Opt. Soc. Am. B **20**(12), 2448–2453 (2003). [CrossRef]

26. T. Tyc and U. Leonhardt, “Transmutation of singularities in optical instruments,” New J. Phys. **10**(11), 115038 (2008). [CrossRef]

*n*has been fitted to

_{eff}*n*= 0.5. For the case when each unit cell is divided into 16 layers, our results for two frequency bands are shown in Fig. 3 . As in the previous case, Re

_{d}*n*oscillates within the actual band. After broadening the band, the oscillation amplitude builds up, especially at low frequencies. As can be seen, at

_{eff}*N*= 16 the accuracy of fitting reaches about 1% (i.e., |Re

*n*-0.5|<0.005) over the band

_{eff}*n*; in other words, it cannot be made arbitrary close to zero. Indeed,

_{d}*ε*and Im

_{eff}*ε*are limited within the considered range, Re

_{eff}*n*is limited as well:

_{eff}### 3.3 Design of MMs with high absorption efficiency

*etc*[27

27. V. G. Kravets, F. Schedin, and A. N. Grigorenko, “Plasmonic blackbody: almost complete absorption of light in nanostructured metallic coatings,” Phys. Rev. B **78**(20), 205405 (2008). [CrossRef]

30. V. G. Kravets, S. Neubeck, A. N. Grigorenko, and A. F. Kravets, “Plasmonic blackbody: strong absorption of light by metal nanoparticles embedded in a dielectric matrix,” Phys. Rev. B **81**(16), 165401 (2010). [CrossRef]

*α*or imaginary part of the refractive index

_{c}*k*over a frequency band, the objective function may be represented as

_{c}*k*= 0.45 for two frequency bands,

_{c}*n*, oscillations of Im

_{eff}*n*occur on the high-frequency side of the band.

_{eff}*vs*the vertical coordinate for three above cases, Re

*ε*= 0 (8 layers within the unit cell), Re

_{eff}*n*= 0.5 (16 layers within the unit cell) and Im

_{eff}*n*= 0.45 (16 layers within the unit cell).

_{eff}## 4. Accuracy of the approximation used

31. M. Gaudry, J. Lerme, E. Cottancin, M. Pellarin, J.-L. Vialle, M. Broyer, B. Prevel, M. Treilleux, and P. Melinon, “Optical properties of (Au_{x}Ag_{1-x})_{n} clusters embedded in alumina: Evolution with size and stoichiometry,” Phys. Rev. B **64**(8), 085407 (2001). [CrossRef]

32. A. K. Sharma and G. J. Mohr, “On the performance of surface plasmon resonance based fibre optic sensor with different bimetallic nanoparticle alloy combinations,” J. Phys. D Appl. Phys. **41**(5), 055106 (2008). [CrossRef]

33. J. C. R. Reis, T. P. Iglesias, G. Douhéret, and M. I. Davis, “The permittivity of thermodynamically ideal liquid mixtures and the excess relative permittivity of binary dielectrics,” Phys. Chem. Chem. Phys. **11**(20), 3977–3986 (2009). [CrossRef] [PubMed]

*ab initio*numerical simulation using FDTD technique and compared the results with those obtained with the use of Eq. (3a). The simulation for the 1d case has been also performed for the comparison. To be more specific, we have found the effective permittivity with the use of the computed

*z*-component of the local electric field,where

*V*is the unit cell volume, and then compared it with that obtained with the use of Eq. (3a) for the above ENZ MM. The number of the unit cells in the

*x*direction is 16, the sizes of the unit cell are

*S*≈0.0094 λ and

_{x}*S*≈0.066 λ with respect to the band center. It should be noted that such a choice of the unit cell size satisfies the typical evaluation criteria for the correct homogenization of MM lattices [34

_{z}34. C. R. Simovski and S. A. Tretyakov, “Local constitutive parameters of metamaterials from an effective-medium perspective,” Phys. Rev. B **75**(19), 195111 (2007). [CrossRef]

12. C. R. Simovski, “On electromagnetic characterization and homogenization of nanostructured metamaterials,” J. Opt. **13**(1), 013001 (2011). [CrossRef]

*ε*are given in Fig. 6 . As can be seen, the agreement between our approximation and FDTD simulation is very good in the 1d case, but with somewhat smoothed out oscillations of Re

_{eff}*ε*curve. In the 2d case, the agreement is fairly good: it is perfect at high frequencies and certainly worse at low frequencies. Besides,some oscillations are not resolved in the actual range. Within the band

_{eff}*ε*from zero is about 0.1. The agreement is much better for the 1d geometry than in the 2d case. Obviously, this happens because Eq. (1) is exact (for the 1d case), while Eq. (3) is approximate (for the 2d case).

_{eff}*ε*nor Im

_{eff}*n*can be nulled, there are at least two ways to reduce them dealing with two-phase metamaterials. One way is to use metal having smaller values of Im

_{eff}*ε*

_{1}; for example, for silver below 3 eV, where the permittivity follows the Drude model, the dimensionless damping parameter may be estimated as

41. M. G. Blaber, M. D. Arnold, and M. J. Ford, “Designing materials for plasmonic systems: the alkali-noble intermetallics,” J. Phys. Condens. Matter **22**(9), 095501 (2010). [CrossRef] [PubMed]

*ε*is to use a dielectric phase with smaller value of its permittivity.

_{eff}26. T. Tyc and U. Leonhardt, “Transmutation of singularities in optical instruments,” New J. Phys. **10**(11), 115038 (2008). [CrossRef]

## 5. Generalization to the three-dimensional case

42. S. K. Golden and G. Papanicolaou, “Bounds for effective parameters of heterogeneous media by analytic continuation,” Commun. Math. Phys. **90**(4), 473–491 (1983) (and references therein). [CrossRef]

6. L. Sun and K. W. Yu, “Strategy for designing broadband epsilon-near-zero metamaterials,” J. Opt. Soc. Am. B **29**(5), 984–989 (2012). [CrossRef]

*G*(

*x*), containing all information about geometry of the composite, satisfies the sum rules

*p*

_{1}and

*p*

_{2}are the volume fractions of the phases 1 and 2, respectively.

43. E. Tuncer, “Extracting the spectral density function of a binary composite without a priori assumptions,” Phys. Rev. B **71**(1), 012101 (2005). [CrossRef]

46. C. Bonifasi-Lista and E. Cherkaev, “Electrical impedance spectroscopy as a potential tool for recovering bone porosity,” Phys. Med. Biol. **54**(10), 3063–3082 (2009). [CrossRef] [PubMed]

*G*(

*x*) to the real geometry of the composite. This problem is not trivial, and the above relationship is known in some particular cases only (see, e.g., [47

47. A. V. Goncharenko, V. Lozovski, and E. F. Venger, “Effective dielectric response of a shape-distributed particle system,” J. Phys. Condens. Matter **13**(35), 8217–8234 (2001). [CrossRef]

48. E. Cherkaev and M.-J. Y. Ou, “Dehomogenization: reconstruction of moments of the spectral measure of the composite,” Inv. Probl. **24**(6), 065008 (2008). [CrossRef]

49. G. A. Niklasson and C. G. Granqvist, “Optical properties and solar selectivity of coevaporated Co-Al_{2}O_{3} composite films,” J. Appl. Phys. **55**(9), 3382–3410 (1984). [CrossRef]

51. P. Mallet, C. A. Guérin, and A. Sentenac, “Maxwell-Garnett mixing rule in the presence of multiple scattering: Derivation and accuracy,” Phys. Rev. B **72**(1), 014205 (2005). [CrossRef]

*p*is the volume fractions of spherical particles of i

_{i}*th*kind (in the following all

*p*are taken to be equal to reduce the number of free parameters). Each particle is considered to be an alloy of two metals with the permittivity

_{i}*rms*. In particular, after increasing the damping parameter by the factor 1.6, the root mean square of the deviation from zero becomes five times smaller.Because

*ε*= 0 is the condition for plasmon excitation, our results show that bulk plasmons can be excited in a frequency range, as well as surface plasmons. Thus, longitudinal electromagnetic waves can propagate in such broadband MMs within a band.

## 6. Concluding remarks

*ab-initio*FDTD simulations and some limitations on its applicability have been discussed. In particular, the approach is shown to be accurate in the 1d case and fairly good in the 2d case. Finally, we have shown how our technique can be extended to the 3d case dealing with rarefied suspensions of metal alloy spheres.

53. A. N. Lagarkov and V. N. Kisel, “Losses in metamaterials: Restrictions and benefits,” Physica B **405**(14), 2925–2929 (2010). [CrossRef]

7. L. Sun and K. W. Yu, “Strategy for designing broadband epsilon-near-zero metamaterial with loss compensation by gain media,” Appl. Phys. Lett. **100**(26), 261903 (2012). [CrossRef]

54. S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature **466**(7307), 735–738 (2010). [CrossRef] [PubMed]

55. S. M. Anlage, “The physics and applications of superconducting metamaterials,” J. Opt. **13**(2), 024001 (2011). [CrossRef]

## Acknowledgments

## References and links

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41. | M. G. Blaber, M. D. Arnold, and M. J. Ford, “Designing materials for plasmonic systems: the alkali-noble intermetallics,” J. Phys. Condens. Matter |

42. | S. K. Golden and G. Papanicolaou, “Bounds for effective parameters of heterogeneous media by analytic continuation,” Commun. Math. Phys. |

43. | E. Tuncer, “Extracting the spectral density function of a binary composite without a priori assumptions,” Phys. Rev. B |

44. | E. Tuncer, “Geometrical description in binary composites and spectral density representation,” Materials |

45. | D. Zhang and E. Cherkaev, “Pade approximations for identification of air bubble volume from temperature- or frequency-dependent permittivity of a two-component mixture,” Inv. Probl. Sci. Eng. |

46. | C. Bonifasi-Lista and E. Cherkaev, “Electrical impedance spectroscopy as a potential tool for recovering bone porosity,” Phys. Med. Biol. |

47. | A. V. Goncharenko, V. Lozovski, and E. F. Venger, “Effective dielectric response of a shape-distributed particle system,” J. Phys. Condens. Matter |

48. | E. Cherkaev and M.-J. Y. Ou, “Dehomogenization: reconstruction of moments of the spectral measure of the composite,” Inv. Probl. |

49. | G. A. Niklasson and C. G. Granqvist, “Optical properties and solar selectivity of coevaporated Co-Al |

50. | S. Riikonen, I. Romero, and F. J. Garcia de Abajo, “Plasmon tunability in metallodielectric metamaterials,” Phys. Rev. B |

51. | P. Mallet, C. A. Guérin, and A. Sentenac, “Maxwell-Garnett mixing rule in the presence of multiple scattering: Derivation and accuracy,” Phys. Rev. B |

52. | C. J. F. Böttger, |

53. | A. N. Lagarkov and V. N. Kisel, “Losses in metamaterials: Restrictions and benefits,” Physica B |

54. | S. Xiao, V. P. Drachev, A. V. Kildishev, X. Ni, U. K. Chettiar, H. K. Yuan, and V. M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature |

55. | S. M. Anlage, “The physics and applications of superconducting metamaterials,” J. Opt. |

**OCIS Codes**

(160.1245) Materials : Artificially engineered materials

(260.2065) Physical optics : Effective medium theory

(160.3918) Materials : Metamaterials

(160.4236) Materials : Nanomaterials

(160.2710) Materials : Inhomogeneous optical media

**ToC Category:**

Metamaterials

**History**

Original Manuscript: December 6, 2012

Manuscript Accepted: December 7, 2012

Published: January 2, 2013

**Citation**

Anatoliy V. Goncharenko, Vladimir U. Nazarov, and Kuan-Ren Chen, "Nanostructured metamaterials with broadband optical properties," Opt. Mater. Express **3**, 143-156 (2013)

http://www.opticsinfobase.org/ome/abstract.cfm?URI=ome-3-2-143

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