## Complex band structure of nanostructured metal-dielectric metamaterials |

Optics Express, Vol. 21, Issue 2, pp. 1593-1598 (2013)

http://dx.doi.org/10.1364/OE.21.001593

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### Abstract

We study complex eigenmodes of layered metal-dielectric metamaterials. Varying losses from weak to realistic, we analyze band structure of the metamaterial and clarify effect of lossess on its intrinsic electromagnetic properties. The structure operates in a regime with infinite numbers of eigenmodes, whereas we analyze dominant ones.

© 2013 OSA

## 1. Introduction

1. 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]

2. I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, “BW media - media with negative parameters, capable of supporting backward waves,” Microw. Opt. Technol. Lett. **31**, 129133 (2001). [CrossRef]

3. A. N. Poddubny, P. A. Belov, and Y. S. Kivshar, “Spontaneous radiation of a finite-size dipole emitter in hyperbolic media,” Phys. Rev. A **84**, 023807 (2011). [CrossRef]

4. I. Iorsh, A. Poddubny, A. Orlov, P. Belov, and Y. S. Kivshar, “Spontaneous emission enhancement in metaldi-electric metamaterials,” Phys. Lett. A **376**, 185187 (2012). [CrossRef]

1. 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]

5. J. Elser, V. A. Podolksiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. **90**, 191109 (2007). [CrossRef]

*ε*

_{1},

*ε*

_{2}and thicknesses

*d*

_{1},

*d*

_{2}(see Fig. 1, left side), we employ a local effective medium model along with the transfer matrix method [6]. The former describes the structure as an uniaxial anisotropic crystal with the effective parameters, In the case when

*ε*

_{||}and

*ε*

_{⊥}are of opposite sign, i.e.

*ε*

_{||}

*ε*

_{⊥}< 0, a material is called indefinite [7

7. D. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. **90**, 077405 (2003). [CrossRef] [PubMed]

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

9. Z. Jacob, J.-Y. Kim, G. V. Naik, A. Boltasseva, E. E. Narimanov, and V. M. Shalaev, “Engineering photonic density of states using metamaterials,” Appl. Phys. B **100**, 215–218 (2010). [CrossRef]

*k*modes. they may have very high photonic density of states [10

10. A. Poddubny, P. Belov, and Y. Kivshar, “Spontaneous radiation of a finite-size dipole emitter in hyperbolic media,” Phys. Rev. A **84**, 023807 (2011). [CrossRef]

11. O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Dipole radiation near hyperbolic metamaterials: applicability of effective-medium approximation,” Opt. Lett. **36**, 2530–2532 (2011). [CrossRef] [PubMed]

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

13. A. V. Chebykin, A. A. Orlov, C. R. Simovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B **86**, 115420 (2012). [CrossRef]

1. 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]

14. L. Shen, T.-J. Yang, and Y.-F. Chau, “Effect of internal period on the optical dispersion of indefinite-medium materials,” Phys. Rev. B **77**, 205124 (2008). [CrossRef]

15. A. P. Vinogradov, A. V. Dorofeenko, and I. A. Nechpurenko, “Analysis of plasmonic bloch waves and band structures of 1d plasmonic photonic crystals,” Metamaterials **4**, 181–200 (2010). [CrossRef]

## 2. Complex band structure

*ω*(

**k**), where

**k**= (

*k*,

_{x}*k*, 0) is the wave vector. TM polarization is assumed to take advantage of plasmonic behavior. The dispersion relation obtained by the transfer matrix method, assuming that

_{y}*i*-th layer, is given by:

*y*-direction.

*D*=

*d*

_{1}+

*d*

_{2}= 62.5nm, where

*d*

_{1}and

*d*

_{2}are the thicknesses of the layers. Dielectric permittivity of the dielectric layers is constant and equals to 4.6, while metal layers are described by dielectric function of the Drude form,

*λ*= 2

_{p}*πc/ω*= 4

_{p}*D*and the damping coefficient Γ. Layers thickness are

*d*

_{1}= 25nm and

*d*

_{2}= 37.5nm and vice versa.

*k*is considered. To find the roots of Eq. (2) in the complex plane we employ the bisection method.

_{y}*k*by the value of 8

_{y}*π/D*in our calculations.

*k*and ±

*k*

^{*}. Thus, there are forward and backward waves along with their complex conjugations, and imaginary parts of the modes are symmetric in Figs. 1(a), 2(a). It follows from the fact that if dielectric permittivities of layers are real then the right part of Eq. (2) is a holomorphic function whose restriction to the real numbers is real-valued.

16. A. Davoyan, W. Liu, A. Miroshnichenko, I. Shadrivov, Y. Kivshar, and S. Bozhevolnyi, “Mode transformation in waveguiding plasmonic structures,” Phot. Nano. Fund. Appl. **9**, 207–212 (2011). [CrossRef]

^{13}

*s*

^{−1}is supposed in our calculations. We analyze complex modes in the case of realistic losses comparing them with the lossless case to show how losses affect the behaviour of the structure. In the presence of losses, as soon as

*ε*

_{2}becomes a complex number, the right part of Eq. (2) can be complex-valued with real

*k*. Consequently, the conjugated waves with ±

_{y}*k*

^{*}are no longer supported by the structure. However, correspondence between positive and negative real and imaginary parts can be quite diverse because of the absence of symmetry in the imaginary parts of eigenmodes (see Figs. 1(b), 2(b)).

*d*

_{1}>

*d*

_{2}is shown in Fig. 1(b). Just that case corresponds to simultaneous presence of forward and backward modes in the same frequency range (

*D*/

*λ*: 0.105 − 0.108) for propagation along the layers. That is, the beam splitting phenomenon revealed in [1

**84**, 045424 (2011). [CrossRef]

*k*. In reality, however, the structure functions in a regime with infinite number of modes which imaginary parts have step-like behaviour at the resonance.

_{y}## 3. Profiles of Eigenmodes Fields

*D*/

*λ*= 0.0875. We first note that the magnetic field of the mode I averaged over each layer is equal to zero. It means that the mode I is effectively longitudinal. Such modes are well known in plasma physics as

*Langmuir*modes [17]. It has been also recently shown that such exotic modes may exist in the waveguides with metal cladding and core made of hyperbolic media [18

18. A. A. Bogdanov and R. A. Suris, “Effect of the anisotropy of a conducting layer on the dispersion law of electromagnetic waves in layered metal-dielectric structures,” JETP Lett. **96**, 49–55 (2012). [CrossRef]

*k*) is large enough, the

_{y}*k*gains large real part and the mode starts to propagate inside the structure. This leads to nonzero values of the transverse component of the Poynting vector inside the layers and to the complex shape of the field distribution inside the layers. To illustrate how the imaginary part of

_{x}*k*affects the shape of the field inside the layers we have plotted the profiles of the high order complex modes [see Fig. 4]. We can see that while we are switching to higher order modes, having the larger imaginary part of

_{y}*k*, the field profile shape becomes more and more complicated. These modes can be regarded as a specific type of coupled waveguide modes. The waveguide mode condition can be roughly estimated as

_{y}*k*=

_{i,x}d_{i}*πn*, where

*i*= 1, 2 and

*n*- is an integer. Thus, higher switching between the complex modes can be regarded as switching between different waveguide modes in the structure.

*k*).

_{y}## 4. Conclusions

## Acknowledgments

## References and links

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

2. | I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, “BW media - media with negative parameters, capable of supporting backward waves,” Microw. Opt. Technol. Lett. |

3. | A. N. Poddubny, P. A. Belov, and Y. S. Kivshar, “Spontaneous radiation of a finite-size dipole emitter in hyperbolic media,” Phys. Rev. A |

4. | I. Iorsh, A. Poddubny, A. Orlov, P. Belov, and Y. S. Kivshar, “Spontaneous emission enhancement in metaldi-electric metamaterials,” Phys. Lett. A |

5. | J. Elser, V. A. Podolksiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. |

6. | M. Born and E. Wolf, |

7. | D. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. |

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

9. | Z. Jacob, J.-Y. Kim, G. V. Naik, A. Boltasseva, E. E. Narimanov, and V. M. Shalaev, “Engineering photonic density of states using metamaterials,” Appl. Phys. B |

10. | A. Poddubny, P. Belov, and Y. Kivshar, “Spontaneous radiation of a finite-size dipole emitter in hyperbolic media,” Phys. Rev. A |

11. | O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Dipole radiation near hyperbolic metamaterials: applicability of effective-medium approximation,” Opt. Lett. |

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

13. | A. V. Chebykin, A. A. Orlov, C. R. Simovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B |

14. | L. Shen, T.-J. Yang, and Y.-F. Chau, “Effect of internal period on the optical dispersion of indefinite-medium materials,” Phys. Rev. B |

15. | A. P. Vinogradov, A. V. Dorofeenko, and I. A. Nechpurenko, “Analysis of plasmonic bloch waves and band structures of 1d plasmonic photonic crystals,” Metamaterials |

16. | A. Davoyan, W. Liu, A. Miroshnichenko, I. Shadrivov, Y. Kivshar, and S. Bozhevolnyi, “Mode transformation in waveguiding plasmonic structures,” Phot. Nano. Fund. Appl. |

17. | A. A. Andreev, A. A. Mak, and N. A. Solov’ev, |

18. | A. A. Bogdanov and R. A. Suris, “Effect of the anisotropy of a conducting layer on the dispersion law of electromagnetic waves in layered metal-dielectric structures,” JETP Lett. |

**OCIS Codes**

(260.2030) Physical optics : Dispersion

(160.3918) Materials : Metamaterials

(310.6628) Thin films : Subwavelength structures, nanostructures

**ToC Category:**

Metamaterials

**History**

Original Manuscript: October 22, 2012

Revised Manuscript: December 6, 2012

Manuscript Accepted: December 18, 2012

Published: January 15, 2013

**Citation**

Alexey Orlov, Ivan Iorsh, Pavel Belov, and Yuri Kivshar, "Complex band structure of nanostructured metal-dielectric metamaterials," Opt. Express **21**, 1593-1598 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-2-1593

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### References

- A. A. Orlov, P. M. Voroshilov, P. A. Belov, and Y. S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B84, 045424 (2011). [CrossRef]
- I. V. Lindell, S. A. Tretyakov, K. I. Nikoskinen, and S. Ilvonen, “BW media - media with negative parameters, capable of supporting backward waves,” Microw. Opt. Technol. Lett.31, 129133 (2001). [CrossRef]
- A. N. Poddubny, P. A. Belov, and Y. S. Kivshar, “Spontaneous radiation of a finite-size dipole emitter in hyperbolic media,” Phys. Rev. A84, 023807 (2011). [CrossRef]
- I. Iorsh, A. Poddubny, A. Orlov, P. Belov, and Y. S. Kivshar, “Spontaneous emission enhancement in metaldi-electric metamaterials,” Phys. Lett. A376, 185187 (2012). [CrossRef]
- J. Elser, V. A. Podolksiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett.90, 191109 (2007). [CrossRef]
- M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Oxford, Pergamon Press, 1964).
- D. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett.90, 077405 (2003). [CrossRef] [PubMed]
- M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett.94, 151105 (2009). [CrossRef]
- Z. Jacob, J.-Y. Kim, G. V. Naik, A. Boltasseva, E. E. Narimanov, and V. M. Shalaev, “Engineering photonic density of states using metamaterials,” Appl. Phys. B100, 215–218 (2010). [CrossRef]
- A. Poddubny, P. Belov, and Y. Kivshar, “Spontaneous radiation of a finite-size dipole emitter in hyperbolic media,” Phys. Rev. A84, 023807 (2011). [CrossRef]
- O. Kidwai, S. V. Zhukovsky, and J. E. Sipe, “Dipole radiation near hyperbolic metamaterials: applicability of effective-medium approximation,” Opt. Lett.36, 2530–2532 (2011). [CrossRef] [PubMed]
- A. V. Chebykin, A. A. Orlov, A. V. Vozianova, S. I. Maslovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B84, 115438 (2011). [CrossRef]
- A. V. Chebykin, A. A. Orlov, C. R. Simovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B86, 115420 (2012). [CrossRef]
- L. Shen, T.-J. Yang, and Y.-F. Chau, “Effect of internal period on the optical dispersion of indefinite-medium materials,” Phys. Rev. B77, 205124 (2008). [CrossRef]
- A. P. Vinogradov, A. V. Dorofeenko, and I. A. Nechpurenko, “Analysis of plasmonic bloch waves and band structures of 1d plasmonic photonic crystals,” Metamaterials4, 181–200 (2010). [CrossRef]
- A. Davoyan, W. Liu, A. Miroshnichenko, I. Shadrivov, Y. Kivshar, and S. Bozhevolnyi, “Mode transformation in waveguiding plasmonic structures,” Phot. Nano. Fund. Appl.9, 207–212 (2011). [CrossRef]
- A. A. Andreev, A. A. Mak, and N. A. Solov’ev, An Introduction to Hot Laser Plasma Physics (Huntington, NY: Nova Science Publishers, 2000).
- A. A. Bogdanov and R. A. Suris, “Effect of the anisotropy of a conducting layer on the dispersion law of electromagnetic waves in layered metal-dielectric structures,” JETP Lett.96, 49–55 (2012). [CrossRef]

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