## Additional effective medium parameters for composite materials (excess surface currents) |

Optics Express, Vol. 19, Issue 7, pp. 6699-6704 (2011)

http://dx.doi.org/10.1364/OE.19.006699

Acrobat PDF (878 KB)

### Abstract

Modified boundary conditions for composite material are suggested. The modified *RT*-retrieval procedure yields bulk values of effective impedance and refractive index, which are independent of system size and boundary realization, whereas the conductivities of the excess surface currents depend on the property of the interface. Simultaneous treatment of all the possible realizations of the system removes the dependence. The accuracy of the latter procedure is the same as the usage of static effective parameters, namely

© 2011 OSA

4. A. P. Vinogradov and A. M. Merzlikin, “On the problem of homogenizing one-dimensional systems,” J. Exp. Theor. Phys. **94**(3), 482–488 (2002). [CrossRef]

5. A. P. Vinogradov and A. M. Merzlikin, “Electrodynamic properties of a finely layered medium,” Dokl. Phys. **46**(12), 832–834 (2001). [CrossRef]

*ε*and

*μ*in the absence of real dissipation (see e.g [6].). Thirdly, the sign of these imaginary parts may contradict to general passiveness of the system [7

7. S. O’Brien and J. B. Pendry, “Photonic band-gap effects and magnetic activity in dielectric composites,” J. Phys. Condens. Matter **14**(15), 4035–4044 (2002). [CrossRef]

9. J. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Size dependence and convergence of the retrieval parameters of metamaterials,” Photon. Nanostructures **6**(1), 96–101 (2008). [CrossRef]

7. S. O’Brien and J. B. Pendry, “Photonic band-gap effects and magnetic activity in dielectric composites,” J. Phys. Condens. Matter **14**(15), 4035–4044 (2002). [CrossRef]

9. J. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Size dependence and convergence of the retrieval parameters of metamaterials,” Photon. Nanostructures **6**(1), 96–101 (2008). [CrossRef]

10. A. Ludwig and K. J. Webb, “Accuracy of effective medium parameter extraction procedures for optical metamaterials,” Phys. Rev. B **81**(11), 113103 (2010). [CrossRef]

12. C. Fietz and G. Shvets, “Current-driven metamaterial homogenization,” Physica B **405**(14), 2930–2934 (2010). [CrossRef]

25. M. Born, *Optics* (Springer, 1933). [PubMed]

32. C. R. Simovski, “Bloch material parameters of magneto-dielectric metamaterials and the concept of Bloch lattices,” Metamaterials (Amst.) **1**(2), 62–80 (2007). [CrossRef]

27. C. R. Simovski, “Material parameters of metamaterials,” Opt. Spectrosc. **107**(5), 726–753 (2009). [CrossRef]

29. C. R. Simovski and B. Sauviac, “On the bulk averaging approach for obtaining the effective parameters of thin magnetic granular films,” Eur. Phys. J. Appl. Phys. **17**(1), 11–20 (2002). [CrossRef]

32. C. R. Simovski, “Bloch material parameters of magneto-dielectric metamaterials and the concept of Bloch lattices,” Metamaterials (Amst.) **1**(2), 62–80 (2007). [CrossRef]

34. A. P. Vinogradov, “On the form of constitutive equations in electrodynamics,” Physics-Uspekhi **45**(3), 331–338 (2002). [CrossRef]

31. 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]

32. C. R. Simovski, “Bloch material parameters of magneto-dielectric metamaterials and the concept of Bloch lattices,” Metamaterials (Amst.) **1**(2), 62–80 (2007). [CrossRef]

25. M. Born, *Optics* (Springer, 1933). [PubMed]

12. C. Fietz and G. Shvets, “Current-driven metamaterial homogenization,” Physica B **405**(14), 2930–2934 (2010). [CrossRef]

34. A. P. Vinogradov, “On the form of constitutive equations in electrodynamics,” Physics-Uspekhi **45**(3), 331–338 (2002). [CrossRef]

12. C. Fietz and G. Shvets, “Current-driven metamaterial homogenization,” Physica B **405**(14), 2930–2934 (2010). [CrossRef]

*rt*-retrieval method. Additional parameters can be retrieved from usual

*rt*-measurements caring out measurements for samples of different thickness.

4. A. P. Vinogradov and A. M. Merzlikin, “On the problem of homogenizing one-dimensional systems,” J. Exp. Theor. Phys. **94**(3), 482–488 (2002). [CrossRef]

5. A. P. Vinogradov and A. M. Merzlikin, “Electrodynamic properties of a finely layered medium,” Dokl. Phys. **46**(12), 832–834 (2001). [CrossRef]

*rt*-retrieval method to the computer simulation data [4

4. A. P. Vinogradov and A. M. Merzlikin, “On the problem of homogenizing one-dimensional systems,” J. Exp. Theor. Phys. **94**(3), 482–488 (2002). [CrossRef]

**94**(3), 482–488 (2002). [CrossRef]

5. A. P. Vinogradov and A. M. Merzlikin, “Electrodynamic properties of a finely layered medium,” Dokl. Phys. **46**(12), 832–834 (2001). [CrossRef]

*d*and permittivity

*rt*-retrieval method produces the effective impedance depending on the system thickness. This is the reflection of the fact that the input impedance of the elementary cell depends on the position of the sample surface planes.

*N*of elementary cells, cannot be adequately presented as a uniform layer (always symmetric) described by effective permittivity and permeability (or by

*L*, though the refractive index tends to the Rytov value versus

*L*(Fig. 1(a) ). Moreover, the imaginary part of

*T*-matrix of a uniform layer with refractive index

*n*, impedance

*L*is:

*T*-matrix

*T*-matrix of the elementary cell and

*T*-matrix of the layer with permittivity

*ζ*has a singularity at

*L*grows). Introduction of excess surface currents takes all these facts into account and modifies the Maxwell boundary conditions on the left and right sides of the sample as follows:

*T*-matrix

*r*and

*t*coefficients for different

*L*and extract six unknown material parameters minimizing the discrepancy

*L*. The summation in Eq. (3) is made for sample lengths from

*r*and

*t*is equal to

*t*significantly increases:

## Acknowledgement

## References and links

1. | G. W. Milton, |

2. | W. Cai, and V. Shalaev, |

3. | C. Caloz, and T. Itoh, |

4. | A. P. Vinogradov and A. M. Merzlikin, “On the problem of homogenizing one-dimensional systems,” J. Exp. Theor. Phys. |

5. | A. P. Vinogradov and A. M. Merzlikin, “Electrodynamic properties of a finely layered medium,” Dokl. Phys. |

6. | S. M. Rytov, “Electromagnetic properties of laminated medium,” Zh. Eksp. Teor. Fiz. |

7. | S. O’Brien and J. B. Pendry, “Photonic band-gap effects and magnetic activity in dielectric composites,” J. Phys. Condens. Matter |

8. | D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B |

9. | J. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Size dependence and convergence of the retrieval parameters of metamaterials,” Photon. Nanostructures |

10. | A. Ludwig and K. J. Webb, “Accuracy of effective medium parameter extraction procedures for optical metamaterials,” Phys. Rev. B |

11. | A. K. Sarychev, D. J. Bergman, and Y. Yagil, “Theory of the optical and microwave properties of metal-dielectric films,” Phys. Rev. B Condens. Matter |

12. | C. Fietz and G. Shvets, “Current-driven metamaterial homogenization,” Physica B |

13. | D. R. Smith and J. B. Pendry, “Homogenization of metamaterials by field averaging,” J. Opt. Soc. Am. B |

14. | O. Acher, J. M. Lerat, and N. Malléjac, “Evaluation and illustration of the properties of metamaterials using field summation,” Opt. Express |

15. | S. Datta, C. T. Chan, K. M. Ho, and C. M. Soukoulis, “Effective dielectric constant of periodic composite structures,” Phys. Rev. B Condens. Matter |

16. | A. P. Vinogradov and A. V. Aivazyan, “Scaling theory for homogenization of the Maxwell equations,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

17. | K. W. Whites, “Full-wave computation of constitutive parameters for lossless composite chiral materials,” IEEE Trans. Antenn. Propag. |

18. | G. Franceschetti, “A complete analysis of the reflection and transmission methods for measuring the complex permeability and permittivity of materials at microwave frequencies,” |

19. | X. Chen, T. M. Grzegorczyk, B.-I. Wu, J. Pacheco Jr, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

20. | A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time domain techniques,” IEEE Trans. Instrum. Meas. |

21. | W. B. Weir, “Automatic measurement of complex dielectric constant and permeability at microwave frequencies,” Proc. IEEE |

22. | L. A. Weinstein, |

23. | W. Śmigaj and B. Gralak, “Validity of the effective-medium approximation of photonic crystals,” Phys. Rev. B |

24. | N. A. Enkin, A. M. Merzlikin, and A. P. Vinogradov, “The difference of the refraction laws in composite materials from the Fresnel laws,” J. Commun. Technol. Electron. |

25. | M. Born, |

26. | A. P. Vinogradov and I. I. Skidanov, “Generalization of Drude’s formulas for the transition layer to chiral media,” J. Commun. Technol. Electron. |

27. | C. R. Simovski, “Material parameters of metamaterials,” Opt. Spectrosc. |

28. | A. P. Vinogradov, K. N. Rosanov, and D. P. Makhnovsky, “Effective boundary layer in composite material,” J. Commun. Technol. Electron. |

29. | C. R. Simovski and B. Sauviac, “On the bulk averaging approach for obtaining the effective parameters of thin magnetic granular films,” Eur. Phys. J. Appl. Phys. |

30. | C. R. Simovski, “Application of the Fresnel formulas for reflection and transmission of electromagnetic waves beyond the quasi-static approximation,” J. Commun. Technol. Electron. |

31. | C. R. Simovski and S. A. Tretyakov, “Local constitutive parameters of metamaterials from an effective-medium perspective,” Phys. Rev. B |

32. | C. R. Simovski, “Bloch material parameters of magneto-dielectric metamaterials and the concept of Bloch lattices,” Metamaterials (Amst.) |

33. | V. M. Agranovich, and V. L. Ginzburg, |

34. | A. P. Vinogradov, “On the form of constitutive equations in electrodynamics,” Physics-Uspekhi |

35. | T. B. A. Senior, and J. L. Volakis, |

36. | D. Bedeaux, J. Vlieger, |

37. | I. Tsukerman, “From Whitney Forms to Metamaterials: a Rigorous Homogenization Theory,” http://arxiv.org/PS_cache/arxiv/pdf/1010/1010.3632v1.pdf |

38. | S. Kim, E. F. Kuester, C. L. Holloway, A. D. Scher, and J. Baker-Jarvis, “Boundary effects on the determination of metamaterial parameters from normal incidence reflection and transmission measurements,” http://arxiv.org/abs/1009.5927 |

**OCIS Codes**

(260.2065) Physical optics : Effective medium theory

(160.5298) Materials : Photonic crystals

**ToC Category:**

Metamaterials

**History**

Original Manuscript: December 10, 2010

Revised Manuscript: January 26, 2011

Manuscript Accepted: January 31, 2011

Published: March 24, 2011

**Citation**

A. P. Vinogradov, A. I. Ignatov, A. M. Merzlikin, S. A. Tretyakov, and C. R. Simovski, "Additional effective medium parameters for composite materials (excess surface currents)," Opt. Express **19**, 6699-6704 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-7-6699

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### References

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- A. P. Vinogradov and A. M. Merzlikin, “Electrodynamic properties of a finely layered medium,” Dokl. Phys. 46(12), 832–834 (2001). [CrossRef]
- S. M. Rytov, “Electromagnetic properties of laminated medium,” Zh. Eksp. Teor. Fiz. 29, 605–616 (1955) [(Sov. Phys. - JETP. 2, 466–475 (1956)].
- S. O’Brien and J. B. Pendry, “Photonic band-gap effects and magnetic activity in dielectric composites,” J. Phys. Condens. Matter 14(15), 4035–4044 (2002). [CrossRef]
- D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002). [CrossRef]
- J. Zhou, T. Koschny, M. Kafesaki, and C. M. Soukoulis, “Size dependence and convergence of the retrieval parameters of metamaterials,” Photon. Nanostructures 6(1), 96–101 (2008). [CrossRef]
- A. Ludwig and K. J. Webb, “Accuracy of effective medium parameter extraction procedures for optical metamaterials,” Phys. Rev. B 81(11), 113103 (2010). [CrossRef]
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- K. W. Whites, “Full-wave computation of constitutive parameters for lossless composite chiral materials,” IEEE Trans. Antenn. Propag. 43(4), 376–384 (1995). [CrossRef]
- G. Franceschetti, “A complete analysis of the reflection and transmission methods for measuring the complex permeability and permittivity of materials at microwave frequencies,” 36,757–764 (1967).
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- A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time domain techniques,” IEEE Trans. Instrum. Meas. 17, 395–402 (1968). [CrossRef]
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- M. Born, Optics (Springer, 1933). [PubMed]
- A. P. Vinogradov and I. I. Skidanov, “Generalization of Drude’s formulas for the transition layer to chiral media,” J. Commun. Technol. Electron. 47, 517–520 (2002).
- C. R. Simovski, “Material parameters of metamaterials,” Opt. Spectrosc. 107(5), 726–753 (2009). [CrossRef]
- A. P. Vinogradov, K. N. Rosanov, and D. P. Makhnovsky, “Effective boundary layer in composite material,” J. Commun. Technol. Electron. 44, 317–322 (1999).
- C. R. Simovski and B. Sauviac, “On the bulk averaging approach for obtaining the effective parameters of thin magnetic granular films,” Eur. Phys. J. Appl. Phys. 17(1), 11–20 (2002). [CrossRef]
- C. R. Simovski, “Application of the Fresnel formulas for reflection and transmission of electromagnetic waves beyond the quasi-static approximation,” J. Commun. Technol. Electron. 52(9), 953–971 (2007). [CrossRef]
- 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]
- C. R. Simovski, “Bloch material parameters of magneto-dielectric metamaterials and the concept of Bloch lattices,” Metamaterials (Amst.) 1(2), 62–80 (2007). [CrossRef]
- V. M. Agranovich, and V. L. Ginzburg, Spatial Dispersion in Crystal Optics and the Theory of Excitons, Monographs and Texts in Physics and Astronomy, Vol. XIII (Interscience, 1966).
- A. P. Vinogradov, “On the form of constitutive equations in electrodynamics,” Physics-Uspekhi 45(3), 331–338 (2002). [CrossRef]
- T. B. A. Senior, and J. L. Volakis, Approximate Boundary Conditions in Electrodynamics (The Institute of Electrical Engineers, 1995).
- D. Bedeaux, J. Vlieger, Optical Properties of Surfaces, 2 ed. (Imperial College Press, 2004).
- I. Tsukerman, “From Whitney Forms to Metamaterials: a Rigorous Homogenization Theory,” http://arxiv.org/PS_cache/arxiv/pdf/1010/1010.3632v1.pdf
- S. Kim, E. F. Kuester, C. L. Holloway, A. D. Scher, and J. Baker-Jarvis, “Boundary effects on the determination of metamaterial parameters from normal incidence reflection and transmission measurements,” http://arxiv.org/abs/1009.5927

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