Abstract

In this work, we present a simple method for the direct retrieval of the effective permittivity and permeability of a bulk semi-infinite metamaterial from variable-angle spectroscopic ellipsometry measurements. Starting from the well-known Fresnel equations, we derive an analytical expression in which unknown coefficients are fitted to the experimental data using a linear regression model. The effective permittivity and permeability are then determined by solving a simple system and the correct solution is selected based on physical criteria. As an example, the method is applied to the case of a self-assembled metamaterial exhibiting strong isotropic optical magnetism.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (John Wiley & Sons, 2007).
  2. D. R. Smith, W. J. Padilla, D. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
    [Crossref]
  3. D. Smith, S. Schultz, P. Markoš, and C. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).
    [Crossref]
  4. C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics 5(9), 523–530 (2011).
    [Crossref]
  5. A. Baron, A. Aradian, V. Ponsinet, and P. Barois, “Self-assembled optical metamaterials,” Opt. Laser Technol. 82, 94–100 (2016).
    [Crossref]
  6. S. A. Tretyakov, “A personal view on the origins and developments of the metamaterial concept,” J. Opt. 19(1), 013002 (2017).
    [Crossref]
  7. A. Baron, A. Iazzolino, K. Ehrhardt, J.-B. Salmon, A. Aradian, V. Kravets, A. N. Grigorenko, J. Leng, A. Le Beulze, M. Tréguer-Delapierre, M. A. Correa-Duarte, and P. Barois, “Bulk optical metamaterials assembled by microfluidic evaporation,” Opt. Mater. Express 3(11), 1792 (2013).
    [Crossref]
  8. S. Gomez-Graña, A. Le Beulze, M. Treguer-Delapierre, S. Mornet, E. Duguet, E. Grana, E. Cloutet, G. Hadziioannou, J. Leng, J.-B. Salmon, V. G. Kravets, A. N. Grigorenko, N. A. Peyyety, V. Ponsinet, P. Richetti, A. Baron, D. Torrent, and P. Barois, “Hierarchical self-assembly of a bulk metamaterial enables isotropic magnetic permeability at optical frequencies,” Mater. Horiz. 3(6), 596–601 (2016).
    [Crossref]
  9. D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71(3), 036617 (2005).
    [Crossref]
  10. V. G. Kravets, F. Schedin, S. Taylor, D. Viita, and A. N. Grigorenko, “Plasmonic resonances in optomagnetic metamaterials based on double dot arrays,” Opt. Express 18(10), 9780–9790 (2010).
    [Crossref]
  11. C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77(19), 195328 (2008).
    [Crossref]
  12. E. Hecht, Optics, 5th ed. (Pearson Education Limited, 2017),
  13. A. N. Grigorenko, “Negative refractive index in artificial metamaterials,” Opt. Lett. 31(16), 2483 (2006).
    [Crossref]

2017 (1)

S. A. Tretyakov, “A personal view on the origins and developments of the metamaterial concept,” J. Opt. 19(1), 013002 (2017).
[Crossref]

2016 (2)

S. Gomez-Graña, A. Le Beulze, M. Treguer-Delapierre, S. Mornet, E. Duguet, E. Grana, E. Cloutet, G. Hadziioannou, J. Leng, J.-B. Salmon, V. G. Kravets, A. N. Grigorenko, N. A. Peyyety, V. Ponsinet, P. Richetti, A. Baron, D. Torrent, and P. Barois, “Hierarchical self-assembly of a bulk metamaterial enables isotropic magnetic permeability at optical frequencies,” Mater. Horiz. 3(6), 596–601 (2016).
[Crossref]

A. Baron, A. Aradian, V. Ponsinet, and P. Barois, “Self-assembled optical metamaterials,” Opt. Laser Technol. 82, 94–100 (2016).
[Crossref]

2013 (1)

2011 (1)

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics 5(9), 523–530 (2011).
[Crossref]

2010 (1)

2008 (1)

C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77(19), 195328 (2008).
[Crossref]

2006 (1)

2005 (1)

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71(3), 036617 (2005).
[Crossref]

2002 (1)

D. Smith, S. Schultz, P. Markoš, and C. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).
[Crossref]

2000 (1)

D. R. Smith, W. J. Padilla, D. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[Crossref]

Aradian, A.

Barois, P.

S. Gomez-Graña, A. Le Beulze, M. Treguer-Delapierre, S. Mornet, E. Duguet, E. Grana, E. Cloutet, G. Hadziioannou, J. Leng, J.-B. Salmon, V. G. Kravets, A. N. Grigorenko, N. A. Peyyety, V. Ponsinet, P. Richetti, A. Baron, D. Torrent, and P. Barois, “Hierarchical self-assembly of a bulk metamaterial enables isotropic magnetic permeability at optical frequencies,” Mater. Horiz. 3(6), 596–601 (2016).
[Crossref]

A. Baron, A. Aradian, V. Ponsinet, and P. Barois, “Self-assembled optical metamaterials,” Opt. Laser Technol. 82, 94–100 (2016).
[Crossref]

A. Baron, A. Iazzolino, K. Ehrhardt, J.-B. Salmon, A. Aradian, V. Kravets, A. N. Grigorenko, J. Leng, A. Le Beulze, M. Tréguer-Delapierre, M. A. Correa-Duarte, and P. Barois, “Bulk optical metamaterials assembled by microfluidic evaporation,” Opt. Mater. Express 3(11), 1792 (2013).
[Crossref]

Baron, A.

A. Baron, A. Aradian, V. Ponsinet, and P. Barois, “Self-assembled optical metamaterials,” Opt. Laser Technol. 82, 94–100 (2016).
[Crossref]

S. Gomez-Graña, A. Le Beulze, M. Treguer-Delapierre, S. Mornet, E. Duguet, E. Grana, E. Cloutet, G. Hadziioannou, J. Leng, J.-B. Salmon, V. G. Kravets, A. N. Grigorenko, N. A. Peyyety, V. Ponsinet, P. Richetti, A. Baron, D. Torrent, and P. Barois, “Hierarchical self-assembly of a bulk metamaterial enables isotropic magnetic permeability at optical frequencies,” Mater. Horiz. 3(6), 596–601 (2016).
[Crossref]

A. Baron, A. Iazzolino, K. Ehrhardt, J.-B. Salmon, A. Aradian, V. Kravets, A. N. Grigorenko, J. Leng, A. Le Beulze, M. Tréguer-Delapierre, M. A. Correa-Duarte, and P. Barois, “Bulk optical metamaterials assembled by microfluidic evaporation,” Opt. Mater. Express 3(11), 1792 (2013).
[Crossref]

Cloutet, E.

S. Gomez-Graña, A. Le Beulze, M. Treguer-Delapierre, S. Mornet, E. Duguet, E. Grana, E. Cloutet, G. Hadziioannou, J. Leng, J.-B. Salmon, V. G. Kravets, A. N. Grigorenko, N. A. Peyyety, V. Ponsinet, P. Richetti, A. Baron, D. Torrent, and P. Barois, “Hierarchical self-assembly of a bulk metamaterial enables isotropic magnetic permeability at optical frequencies,” Mater. Horiz. 3(6), 596–601 (2016).
[Crossref]

Correa-Duarte, M. A.

Duguet, E.

S. Gomez-Graña, A. Le Beulze, M. Treguer-Delapierre, S. Mornet, E. Duguet, E. Grana, E. Cloutet, G. Hadziioannou, J. Leng, J.-B. Salmon, V. G. Kravets, A. N. Grigorenko, N. A. Peyyety, V. Ponsinet, P. Richetti, A. Baron, D. Torrent, and P. Barois, “Hierarchical self-assembly of a bulk metamaterial enables isotropic magnetic permeability at optical frequencies,” Mater. Horiz. 3(6), 596–601 (2016).
[Crossref]

Ehrhardt, K.

Fujiwara, H.

H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (John Wiley & Sons, 2007).

Gomez-Graña, S.

S. Gomez-Graña, A. Le Beulze, M. Treguer-Delapierre, S. Mornet, E. Duguet, E. Grana, E. Cloutet, G. Hadziioannou, J. Leng, J.-B. Salmon, V. G. Kravets, A. N. Grigorenko, N. A. Peyyety, V. Ponsinet, P. Richetti, A. Baron, D. Torrent, and P. Barois, “Hierarchical self-assembly of a bulk metamaterial enables isotropic magnetic permeability at optical frequencies,” Mater. Horiz. 3(6), 596–601 (2016).
[Crossref]

Grana, E.

S. Gomez-Graña, A. Le Beulze, M. Treguer-Delapierre, S. Mornet, E. Duguet, E. Grana, E. Cloutet, G. Hadziioannou, J. Leng, J.-B. Salmon, V. G. Kravets, A. N. Grigorenko, N. A. Peyyety, V. Ponsinet, P. Richetti, A. Baron, D. Torrent, and P. Barois, “Hierarchical self-assembly of a bulk metamaterial enables isotropic magnetic permeability at optical frequencies,” Mater. Horiz. 3(6), 596–601 (2016).
[Crossref]

Grigorenko, A. N.

Hadziioannou, G.

S. Gomez-Graña, A. Le Beulze, M. Treguer-Delapierre, S. Mornet, E. Duguet, E. Grana, E. Cloutet, G. Hadziioannou, J. Leng, J.-B. Salmon, V. G. Kravets, A. N. Grigorenko, N. A. Peyyety, V. Ponsinet, P. Richetti, A. Baron, D. Torrent, and P. Barois, “Hierarchical self-assembly of a bulk metamaterial enables isotropic magnetic permeability at optical frequencies,” Mater. Horiz. 3(6), 596–601 (2016).
[Crossref]

Hecht, E.

E. Hecht, Optics, 5th ed. (Pearson Education Limited, 2017),

Iazzolino, A.

Koschny, T.

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71(3), 036617 (2005).
[Crossref]

Kravets, V.

Kravets, V. G.

S. Gomez-Graña, A. Le Beulze, M. Treguer-Delapierre, S. Mornet, E. Duguet, E. Grana, E. Cloutet, G. Hadziioannou, J. Leng, J.-B. Salmon, V. G. Kravets, A. N. Grigorenko, N. A. Peyyety, V. Ponsinet, P. Richetti, A. Baron, D. Torrent, and P. Barois, “Hierarchical self-assembly of a bulk metamaterial enables isotropic magnetic permeability at optical frequencies,” Mater. Horiz. 3(6), 596–601 (2016).
[Crossref]

V. G. Kravets, F. Schedin, S. Taylor, D. Viita, and A. N. Grigorenko, “Plasmonic resonances in optomagnetic metamaterials based on double dot arrays,” Opt. Express 18(10), 9780–9790 (2010).
[Crossref]

Le Beulze, A.

S. Gomez-Graña, A. Le Beulze, M. Treguer-Delapierre, S. Mornet, E. Duguet, E. Grana, E. Cloutet, G. Hadziioannou, J. Leng, J.-B. Salmon, V. G. Kravets, A. N. Grigorenko, N. A. Peyyety, V. Ponsinet, P. Richetti, A. Baron, D. Torrent, and P. Barois, “Hierarchical self-assembly of a bulk metamaterial enables isotropic magnetic permeability at optical frequencies,” Mater. Horiz. 3(6), 596–601 (2016).
[Crossref]

A. Baron, A. Iazzolino, K. Ehrhardt, J.-B. Salmon, A. Aradian, V. Kravets, A. N. Grigorenko, J. Leng, A. Le Beulze, M. Tréguer-Delapierre, M. A. Correa-Duarte, and P. Barois, “Bulk optical metamaterials assembled by microfluidic evaporation,” Opt. Mater. Express 3(11), 1792 (2013).
[Crossref]

Lederer, F.

C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77(19), 195328 (2008).
[Crossref]

Leng, J.

S. Gomez-Graña, A. Le Beulze, M. Treguer-Delapierre, S. Mornet, E. Duguet, E. Grana, E. Cloutet, G. Hadziioannou, J. Leng, J.-B. Salmon, V. G. Kravets, A. N. Grigorenko, N. A. Peyyety, V. Ponsinet, P. Richetti, A. Baron, D. Torrent, and P. Barois, “Hierarchical self-assembly of a bulk metamaterial enables isotropic magnetic permeability at optical frequencies,” Mater. Horiz. 3(6), 596–601 (2016).
[Crossref]

A. Baron, A. Iazzolino, K. Ehrhardt, J.-B. Salmon, A. Aradian, V. Kravets, A. N. Grigorenko, J. Leng, A. Le Beulze, M. Tréguer-Delapierre, M. A. Correa-Duarte, and P. Barois, “Bulk optical metamaterials assembled by microfluidic evaporation,” Opt. Mater. Express 3(11), 1792 (2013).
[Crossref]

Markoš, P.

D. Smith, S. Schultz, P. Markoš, and C. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).
[Crossref]

Menzel, C.

C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77(19), 195328 (2008).
[Crossref]

Mornet, S.

S. Gomez-Graña, A. Le Beulze, M. Treguer-Delapierre, S. Mornet, E. Duguet, E. Grana, E. Cloutet, G. Hadziioannou, J. Leng, J.-B. Salmon, V. G. Kravets, A. N. Grigorenko, N. A. Peyyety, V. Ponsinet, P. Richetti, A. Baron, D. Torrent, and P. Barois, “Hierarchical self-assembly of a bulk metamaterial enables isotropic magnetic permeability at optical frequencies,” Mater. Horiz. 3(6), 596–601 (2016).
[Crossref]

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[Crossref]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[Crossref]

Paul, T.

C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77(19), 195328 (2008).
[Crossref]

Pertsch, T.

C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77(19), 195328 (2008).
[Crossref]

Peyyety, N. A.

S. Gomez-Graña, A. Le Beulze, M. Treguer-Delapierre, S. Mornet, E. Duguet, E. Grana, E. Cloutet, G. Hadziioannou, J. Leng, J.-B. Salmon, V. G. Kravets, A. N. Grigorenko, N. A. Peyyety, V. Ponsinet, P. Richetti, A. Baron, D. Torrent, and P. Barois, “Hierarchical self-assembly of a bulk metamaterial enables isotropic magnetic permeability at optical frequencies,” Mater. Horiz. 3(6), 596–601 (2016).
[Crossref]

Ponsinet, V.

S. Gomez-Graña, A. Le Beulze, M. Treguer-Delapierre, S. Mornet, E. Duguet, E. Grana, E. Cloutet, G. Hadziioannou, J. Leng, J.-B. Salmon, V. G. Kravets, A. N. Grigorenko, N. A. Peyyety, V. Ponsinet, P. Richetti, A. Baron, D. Torrent, and P. Barois, “Hierarchical self-assembly of a bulk metamaterial enables isotropic magnetic permeability at optical frequencies,” Mater. Horiz. 3(6), 596–601 (2016).
[Crossref]

A. Baron, A. Aradian, V. Ponsinet, and P. Barois, “Self-assembled optical metamaterials,” Opt. Laser Technol. 82, 94–100 (2016).
[Crossref]

Richetti, P.

S. Gomez-Graña, A. Le Beulze, M. Treguer-Delapierre, S. Mornet, E. Duguet, E. Grana, E. Cloutet, G. Hadziioannou, J. Leng, J.-B. Salmon, V. G. Kravets, A. N. Grigorenko, N. A. Peyyety, V. Ponsinet, P. Richetti, A. Baron, D. Torrent, and P. Barois, “Hierarchical self-assembly of a bulk metamaterial enables isotropic magnetic permeability at optical frequencies,” Mater. Horiz. 3(6), 596–601 (2016).
[Crossref]

Rockstuhl, C.

C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77(19), 195328 (2008).
[Crossref]

Salmon, J.-B.

S. Gomez-Graña, A. Le Beulze, M. Treguer-Delapierre, S. Mornet, E. Duguet, E. Grana, E. Cloutet, G. Hadziioannou, J. Leng, J.-B. Salmon, V. G. Kravets, A. N. Grigorenko, N. A. Peyyety, V. Ponsinet, P. Richetti, A. Baron, D. Torrent, and P. Barois, “Hierarchical self-assembly of a bulk metamaterial enables isotropic magnetic permeability at optical frequencies,” Mater. Horiz. 3(6), 596–601 (2016).
[Crossref]

A. Baron, A. Iazzolino, K. Ehrhardt, J.-B. Salmon, A. Aradian, V. Kravets, A. N. Grigorenko, J. Leng, A. Le Beulze, M. Tréguer-Delapierre, M. A. Correa-Duarte, and P. Barois, “Bulk optical metamaterials assembled by microfluidic evaporation,” Opt. Mater. Express 3(11), 1792 (2013).
[Crossref]

Schedin, F.

Schultz, S.

D. Smith, S. Schultz, P. Markoš, and C. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).
[Crossref]

D. R. Smith, W. J. Padilla, D. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[Crossref]

Smith, D.

D. Smith, S. Schultz, P. Markoš, and C. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).
[Crossref]

Smith, D. R.

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71(3), 036617 (2005).
[Crossref]

D. R. Smith, W. J. Padilla, D. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[Crossref]

Soukoulis, C.

D. Smith, S. Schultz, P. Markoš, and C. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).
[Crossref]

Soukoulis, C. M.

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics 5(9), 523–530 (2011).
[Crossref]

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71(3), 036617 (2005).
[Crossref]

Taylor, S.

Torrent, D.

S. Gomez-Graña, A. Le Beulze, M. Treguer-Delapierre, S. Mornet, E. Duguet, E. Grana, E. Cloutet, G. Hadziioannou, J. Leng, J.-B. Salmon, V. G. Kravets, A. N. Grigorenko, N. A. Peyyety, V. Ponsinet, P. Richetti, A. Baron, D. Torrent, and P. Barois, “Hierarchical self-assembly of a bulk metamaterial enables isotropic magnetic permeability at optical frequencies,” Mater. Horiz. 3(6), 596–601 (2016).
[Crossref]

Treguer-Delapierre, M.

S. Gomez-Graña, A. Le Beulze, M. Treguer-Delapierre, S. Mornet, E. Duguet, E. Grana, E. Cloutet, G. Hadziioannou, J. Leng, J.-B. Salmon, V. G. Kravets, A. N. Grigorenko, N. A. Peyyety, V. Ponsinet, P. Richetti, A. Baron, D. Torrent, and P. Barois, “Hierarchical self-assembly of a bulk metamaterial enables isotropic magnetic permeability at optical frequencies,” Mater. Horiz. 3(6), 596–601 (2016).
[Crossref]

Tréguer-Delapierre, M.

Tretyakov, S. A.

S. A. Tretyakov, “A personal view on the origins and developments of the metamaterial concept,” J. Opt. 19(1), 013002 (2017).
[Crossref]

Vier, D.

D. R. Smith, W. J. Padilla, D. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[Crossref]

Vier, D. C.

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71(3), 036617 (2005).
[Crossref]

Viita, D.

Wegener, M.

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics 5(9), 523–530 (2011).
[Crossref]

J. Opt. (1)

S. A. Tretyakov, “A personal view on the origins and developments of the metamaterial concept,” J. Opt. 19(1), 013002 (2017).
[Crossref]

Mater. Horiz. (1)

S. Gomez-Graña, A. Le Beulze, M. Treguer-Delapierre, S. Mornet, E. Duguet, E. Grana, E. Cloutet, G. Hadziioannou, J. Leng, J.-B. Salmon, V. G. Kravets, A. N. Grigorenko, N. A. Peyyety, V. Ponsinet, P. Richetti, A. Baron, D. Torrent, and P. Barois, “Hierarchical self-assembly of a bulk metamaterial enables isotropic magnetic permeability at optical frequencies,” Mater. Horiz. 3(6), 596–601 (2016).
[Crossref]

Nat. Photonics (1)

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics 5(9), 523–530 (2011).
[Crossref]

Opt. Express (1)

Opt. Laser Technol. (1)

A. Baron, A. Aradian, V. Ponsinet, and P. Barois, “Self-assembled optical metamaterials,” Opt. Laser Technol. 82, 94–100 (2016).
[Crossref]

Opt. Lett. (1)

Opt. Mater. Express (1)

Phys. Rev. B (2)

D. Smith, S. Schultz, P. Markoš, and C. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65(19), 195104 (2002).
[Crossref]

C. Menzel, C. Rockstuhl, T. Paul, F. Lederer, and T. Pertsch, “Retrieving effective parameters for metamaterials at oblique incidence,” Phys. Rev. B 77(19), 195328 (2008).
[Crossref]

Phys. Rev. E (1)

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, “Electromagnetic parameter retrieval from inhomogeneous metamaterials,” Phys. Rev. E 71(3), 036617 (2005).
[Crossref]

Phys. Rev. Lett. (1)

D. R. Smith, W. J. Padilla, D. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[Crossref]

Other (2)

H. Fujiwara, Spectroscopic Ellipsometry: Principles and Applications (John Wiley & Sons, 2007).

E. Hecht, Optics, 5th ed. (Pearson Education Limited, 2017),

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Figures (5)

Fig. 1.
Fig. 1. Electric field $E$, magnetic induction $B$ and wavevector $k$ corresponding to the incident, reflected and transmitted wave for a) $p-$polarization and b) $s-$polarization. $B$ in a) and $E$ in b) are perpendicular to the plane of the paper and are pointing to the reader. NB: the case represented here is that for which $\theta _2$ is positive, which corresponds to a positive refractive index, $n_2$. However the retrieval method presented in this paper is equally valid for a negative refractive index.
Fig. 2.
Fig. 2. Examples showing the line of best fit (red line) for experimental data from [8] (blue crosses) based on the linear regression model described in Eqs. (10)–(16) for three different wavelengths: a), d) $\lambda = 350 \, \mathrm {nm}$; b), e) $\lambda = 401\, \mathrm {nm}$ ; c), f) $\lambda = 427 \, \mathrm {nm}$. The experimental data was collected at five angles of incidence $\theta _1$ (50, 55, 60, 65 and 70$^{\circ }$). Graphs a), b), c) represent the real part of the experimental variable Y, while graphs d), e) and f) represent the imaginary part. The linear model fits well Re(Y) at $\lambda = 350 \, \mathrm {nm}$ but poorly at $\lambda = 401 \, \mathrm {nm}$ and $\lambda = 427 \, \mathrm {nm}$. The linear model fits reasonably well Im(Y) in the three cases.
Fig. 3.
Fig. 3. Four possible solutions for the values of the real and imaginary parts of the permittivity $\varepsilon$ and the permeability $\mu$ obtained using the retrieval method presented in this manuscript on ellipsometric data from [8]. The graphs correspond respectively to the solution: a) $\varepsilon _a^+$ (Eq. (19)), b) $\varepsilon _b^+$ (Eq. (20)), c) $\varepsilon _a^-$ (Eq. (21)) and d) $\varepsilon _b^-$ (Eq. (22)). The thick blue sections of the curves correspond to wavelengths for which both coefficients of determination $R^2$ of the linear regressions (Eq. (15) and Eq. (16)) are higher than 0.9, indicating that the model provides a satisfactory description of the data.
Fig. 4.
Fig. 4. Four possible solutions for the values of the real and imaginary parts of the refractive index $n$ and the impedance $Z$ obtained using the retrieval method presented in this manuscript on ellipsometric data from [8]. The graphs correspond respectively to the solution: a) $\varepsilon _a^+$ (Eq. (19)), b) $\varepsilon _b^+$ (Eq. (20)), c) $\varepsilon _a^-$ (Eq. (21)) and d) $\varepsilon _b^-$ (Eq. (22)). The thick blue sections of the curves correspond to wavelengths for which both coefficients of determination $R^2$ of the linear regressions (Eq. (15) and Eq. (16)) are higher than 0.9, indicating that the model provides a satisfactory description of the data.
Fig. 5.
Fig. 5. a) Coefficient of determination for the linear regression on the real part (Eq. (15)) and the imaginary part (Eq. (16)) of Y as a function of wavelength, indicating the goodness of the fit for the experimental data from [8]. b) Experimental range for the real and the imaginary part of the variable Y as a function of wavelength. c) 1/e penetration depth calculated as $\delta _p = \lambda / 4 \pi n"$ as a function of wavelength. The thick blue sections of the curve corresponds to wavelengths for which both coefficients of determination $R^2$ of the linear regressions (Eq. (15) and Eq. (16)) are higher than 0.9, indicating that the model provides a satisfactory description of the data.

Equations (24)

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ρ = r p r s = tan ψ e i Δ
n = n b a c k g r o u n d sin θ 1 [ 1 + ( 1 ρ 1 + ρ ) 2 tan 2 θ 1 ] 1 / 2
r p = Z 1 cos θ 1 Z 2 cos θ 2 Z 1 cos θ 1 + Z 2 cos θ 2
r s = Z 2 cos θ 1 Z 1 cos θ 2 Z 2 cos θ 1 + Z 1 cos θ 2
ρ = ( Z 1 cos θ 1 Z 2 cos θ 2 ) ( Z 2 cos θ 1 + Z 1 cos θ 2 ) ( Z 1 cos θ 1 + Z 2 cos θ 2 ) ( Z 2 cos θ 1 Z 1 cos θ 2 )
1 ρ 1 + ρ = cos θ 1 cos θ 2 cos 2 θ 1 cos 2 θ 2 Z 2 2 Z 1 2 Z 1 Z 2
( 1 ρ 1 + ρ ) 2 sin 4 θ 1 cos 2 θ 1 = ( 1 n 1 2 n 2 2 sin 2 θ 1 ) ( n 2 2 n 1 2 n 2 2 ) 2 ( Z 2 2 Z 1 2 Z 1 Z 2 ) 2
( 1 ρ 1 + ρ ) 2 sin 4 θ 1 cos 2 θ 1 = ( ε 2 μ 1 ε 1 μ 2 ε 1 μ 1 ε 2 μ 2 ) 2 ( ε 2 μ 2 ε 1 μ 1 sin 2 θ 1 )
( 1 ρ 1 + ρ ) 2 sin 4 θ 1 cos 2 θ 1 = ( ε μ 1 ε μ ) 2 ( ε μ sin 2 θ 1 )
Y = A X + B
X = sin 2 θ 1
Y = ( 1 ρ 1 + ρ ) 2 sin 4 θ 1 cos 2 θ 1
A = ( ε μ 1 ε μ ) 2
B = ( ε μ 1 ε μ ) 2 ε μ
R e ( Y ) = R e ( A ) X + R e ( B )
I m ( Y ) = I m ( A ) X + I m ( B )
ε μ = n 2 = A B
ε 2 ± ε A ( 1 B / A ) B / A = 0
ε a + = A ( 1 B A ) + Δ 2
ε b + = A ( 1 B A ) Δ 2
ε a = A ( 1 B A ) + Δ 2
ε b = A ( 1 B A ) Δ 2
Δ = ( A B ) 2 + 4 B A
n = | ε | | μ | exp [ i ( δ ϵ + δ μ 2 ) ] s g n [ cos ( δ ϵ δ μ 2 ) ]

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