Abstract

Electromagnetic or acoustic metamaterials can be described in terms of equivalent effective, in general anisotropic, media and several techniques exist to determine the effective permeability and permittivity (or effective mass density and bulk modulus in the context of acoustics). Among these techniques, retrieval methods use the measured reflection and transmission coefficients (or scattering coefficients) for waves incident on a metamaterial slab containing few unit cells. Until now, anisotropic effective slabs have been considered in the literature but they are limited to the case where one of the axes of anisotropy is aligned with the slab interface. We propose an extension to arbitrary orientations of the principal axes of anisotropy and oblique incidence. The retrieval method is illustrated in the electromagnetic case for layered media, and in the acoustic case for array of tilted elliptical particles.

© 2014 Optical Society of America

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References

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2014 (1)

A Maurel, J.-F. Mercier, and S. Félix, “Wave propagation through penetrable scatterers in a waveguide and though a penetrable grating,” J. Acoust. Soc. Am. 135, 165–174 (2014).
[Crossref] [PubMed]

2013 (5)

S. Arslanagić, T. V. Hansen, N. A. Mortensen, A. H. Gregersen, O. Sigmund, R. W. Ziolkowski, and O. Breinbjerg, “A review of the scattering-parameter extraction method with clarification of ambiguity issues in relation to metamaterial homogenization,” IEEE Antennas and Propagation Magazine 55, 91–106 (2013).
[Crossref]

A. Maurel, S. Félix, and J.-F. Mercier, “Enhanced transmission through gratings: Structural and geometrical effects,” Phys. Rev. B 88, 115416 (2013).
[Crossref]

C.P. Berraquero, A. Maurel, P. Petitjeans, and V. Pagneux, “Experimental realization of a water-wave metamaterial shifter,” Phys. Rev. E 88, 051002(R) (2013).
[Crossref]

T. Antonakakis, R. V. Craster, and S. Guenneau, “High-frequency homogenization of zero-frequency stop band photonic and phononic crystals,” New J. Phys. 15, 103014 (2013).
[Crossref]

Y. Xie, B.-I. Popa, L. Zigoneanu, and S. A. Cummer, “Measurement of a broadband negative index with space-coiling acoustic metamaterials,” Phys. Rev. Lett. 110, 175501 (2013).
[Crossref] [PubMed]

2012 (3)

J. Sánchez-Dehesa, D. Torrent, and J. Carbonell, “Anisotropic metamaterials as sensing devices in acoustics and electromagnetism,” Proc. of SPIE 8346, 834606 (2012).
[Crossref]

Z. Liang and J. Li, “Extreme acoustic metamaterial by coiling up space,” Phys. Rev. Lett. 108, 114301 (2012).
[Crossref] [PubMed]

F.-J. Hsieh and W.-C. Wang, “Full extraction methods to retrieve effective refractive index and parameters of a bianisotropic metamaterial based on material dispersion models,” J. Appl. Phys. 112, 064907 (2012)
[Crossref]

2011 (5)

Z. H. Jiang, J. A. Bossard, X. Wang, and D. H. Werner, “Synthesizing metamaterials with angularly independent effective medium properties based on an anisotropic parameter retrieval technique coupled with a genetic algorithm,” J. Appl. Phys. 109, 013515 (2011)
[Crossref]

A. I. Căbuz, A. Nicolet, F. Zolla, D. Felbacq, and G. Bouchitté, “Homogenization of nonlocal wire metamaterial via a renormalization approach,” J. Opt. Soc. Am. B, 28, 1275–1282 (2011).
[Crossref]

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

L. Zigoneanu, B.-I. Popa, and S. A. Cummer, “Design and measurements of a broadband two-dimensional acoustic lens,” Phys. Rev. B 84, 024305 (2011).
[Crossref]

L. Zigoneanu, B.-I. Popa, A.F. Starr, and S. A. Cummer, “Design and measurements of a broadband two-dimensional acoustic metamaterial with anisotropic effective mass density,” J. Appl. Phys. 109, 054906 (2011).
[Crossref]

2010 (3)

C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, “Validity of effective material parameters for optical fishnet metamaterials,” Phys. Rev B, 81, 035320 (2010).
[Crossref]

S. B. Raghunathan and N. V. Budko, “Effective permittivity of finite inhomogeneous objects,” Phys. Rev. B 81, 054206 (2010).
[Crossref]

C. R. Simovski and S. A. Tretyakov, “On effective electromagnetic parameters of artificial nanostructured magnetic materials,” Photonics and Nanostructures - Fundamentals and Applications 8, 254–263 (2010).
[Crossref]

2009 (3)

2008 (2)

Y. Yuan, L. Shen, L. Ran, T. Jiang, J. Huangfu, and J. A. Kong, “Directive emission based on anisotropic meta-materials,” Phys. Rev. A 77, 053821 (2008).
[Crossref]

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

2007 (1)

V. Fokin, M. Ambati, C. Sun, and X. Zhang, “Method for retrieving effective properties of locally resonant acoustic metamaterials,” Phys. Rev. B 76, 144302 (2007).
[Crossref]

2006 (3)

H. Chen, J. Zhang, Y. Bai, Y. Luo, L. Ran, Q. Jiang, and J. A. Kong, “Experimental retrieval of the effective parameters of metamaterials based on a waveguide method,” Opt. Express 14, 12944–12949 (2006).
[Crossref] [PubMed]

B. L. Wu, W. J. Wang, J. Pacheco, X. Chen, J. Lu, T. M. Grzegorczyk, J. A. Kong, P. Kao, P. A. Theophelakes, and M. J. Hogan, “Anisotropic metamaterials as antenna substrate to enhance directivity,” Microw. and Opt. Techn. Lett. 48, 680–683 (2006).
[Crossref]

D. Torrent and J. Sánchez-Dehesa, “Effective parameters of clusters of cylinders embedded in a nonviscous fluid or gas,” Phys. Rev. B 74, 224305 (2006).
[Crossref]

2005 (2)

T. M. Grzegorczyk, Z. M. Thomas, and J. A. Kong, “Inversion of critical angle and Brewster angle in anisotropic left-handed metamaterials,” Appl. Phys. Lett. 86, 251909 (2005).
[Crossref]

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

2004 (1)

X. Chen, T. M. Grzegorczyk, B. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E., 70, 016608 (2004).
[Crossref]

2003 (1)

2002 (1)

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, 195104 (2002).
[Crossref]

2000 (1)

1996 (1)

N. A. Nicorovici and R. C. McPhedran, “Transport properties of arrays of elliptical cylinders,” Phys. Rev. E 54(2), 1945–1957 (1996).
[Crossref]

Ambati, M.

V. Fokin, M. Ambati, C. Sun, and X. Zhang, “Method for retrieving effective properties of locally resonant acoustic metamaterials,” Phys. Rev. B 76, 144302 (2007).
[Crossref]

Antonakakis, T.

T. Antonakakis, R. V. Craster, and S. Guenneau, “High-frequency homogenization of zero-frequency stop band photonic and phononic crystals,” New J. Phys. 15, 103014 (2013).
[Crossref]

Arslanagic, S.

S. Arslanagić, T. V. Hansen, N. A. Mortensen, A. H. Gregersen, O. Sigmund, R. W. Ziolkowski, and O. Breinbjerg, “A review of the scattering-parameter extraction method with clarification of ambiguity issues in relation to metamaterial homogenization,” IEEE Antennas and Propagation Magazine 55, 91–106 (2013).
[Crossref]

Bai, Y.

Berraquero, C.P.

C.P. Berraquero, A. Maurel, P. Petitjeans, and V. Pagneux, “Experimental realization of a water-wave metamaterial shifter,” Phys. Rev. E 88, 051002(R) (2013).
[Crossref]

Bossard, J. A.

Z. H. Jiang, J. A. Bossard, X. Wang, and D. H. Werner, “Synthesizing metamaterials with angularly independent effective medium properties based on an anisotropic parameter retrieval technique coupled with a genetic algorithm,” J. Appl. Phys. 109, 013515 (2011)
[Crossref]

Bouchitté, G.

A. I. Căbuz, A. Nicolet, F. Zolla, D. Felbacq, and G. Bouchitté, “Homogenization of nonlocal wire metamaterial via a renormalization approach,” J. Opt. Soc. Am. B, 28, 1275–1282 (2011).
[Crossref]

Breinbjerg, O.

S. Arslanagić, T. V. Hansen, N. A. Mortensen, A. H. Gregersen, O. Sigmund, R. W. Ziolkowski, and O. Breinbjerg, “A review of the scattering-parameter extraction method with clarification of ambiguity issues in relation to metamaterial homogenization,” IEEE Antennas and Propagation Magazine 55, 91–106 (2013).
[Crossref]

Budko, N. V.

S. B. Raghunathan and N. V. Budko, “Effective permittivity of finite inhomogeneous objects,” Phys. Rev. B 81, 054206 (2010).
[Crossref]

Cabuz, A. I.

A. I. Căbuz, A. Nicolet, F. Zolla, D. Felbacq, and G. Bouchitté, “Homogenization of nonlocal wire metamaterial via a renormalization approach,” J. Opt. Soc. Am. B, 28, 1275–1282 (2011).
[Crossref]

Carbonell, J.

J. Sánchez-Dehesa, D. Torrent, and J. Carbonell, “Anisotropic metamaterials as sensing devices in acoustics and electromagnetism,” Proc. of SPIE 8346, 834606 (2012).
[Crossref]

Chen, H.

Chen, X.

B. L. Wu, W. J. Wang, J. Pacheco, X. Chen, J. Lu, T. M. Grzegorczyk, J. A. Kong, P. Kao, P. A. Theophelakes, and M. J. Hogan, “Anisotropic metamaterials as antenna substrate to enhance directivity,” Microw. and Opt. Techn. Lett. 48, 680–683 (2006).
[Crossref]

X. Chen, T. M. Grzegorczyk, B. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E., 70, 016608 (2004).
[Crossref]

Craster, R. V.

T. Antonakakis, R. V. Craster, and S. Guenneau, “High-frequency homogenization of zero-frequency stop band photonic and phononic crystals,” New J. Phys. 15, 103014 (2013).
[Crossref]

Cui, W.

Cummer, S. A.

Y. Xie, B.-I. Popa, L. Zigoneanu, and S. A. Cummer, “Measurement of a broadband negative index with space-coiling acoustic metamaterials,” Phys. Rev. Lett. 110, 175501 (2013).
[Crossref] [PubMed]

L. Zigoneanu, B.-I. Popa, and S. A. Cummer, “Design and measurements of a broadband two-dimensional acoustic lens,” Phys. Rev. B 84, 024305 (2011).
[Crossref]

L. Zigoneanu, B.-I. Popa, A.F. Starr, and S. A. Cummer, “Design and measurements of a broadband two-dimensional acoustic metamaterial with anisotropic effective mass density,” J. Appl. Phys. 109, 054906 (2011).
[Crossref]

B.-I. Popa and S. A. Cummer, “Design and characterization of broadband acoustic composite metamaterials,” Phys. Rev. B 80, 174303 (2009).
[Crossref]

Felbacq, D.

A. I. Căbuz, A. Nicolet, F. Zolla, D. Felbacq, and G. Bouchitté, “Homogenization of nonlocal wire metamaterial via a renormalization approach,” J. Opt. Soc. Am. B, 28, 1275–1282 (2011).
[Crossref]

Félix, S.

A Maurel, J.-F. Mercier, and S. Félix, “Wave propagation through penetrable scatterers in a waveguide and though a penetrable grating,” J. Acoust. Soc. Am. 135, 165–174 (2014).
[Crossref] [PubMed]

A. Maurel, S. Félix, and J.-F. Mercier, “Enhanced transmission through gratings: Structural and geometrical effects,” Phys. Rev. B 88, 115416 (2013).
[Crossref]

Fokin, V.

V. Fokin, M. Ambati, C. Sun, and X. Zhang, “Method for retrieving effective properties of locally resonant acoustic metamaterials,” Phys. Rev. B 76, 144302 (2007).
[Crossref]

Gregersen, A. H.

S. Arslanagić, T. V. Hansen, N. A. Mortensen, A. H. Gregersen, O. Sigmund, R. W. Ziolkowski, and O. Breinbjerg, “A review of the scattering-parameter extraction method with clarification of ambiguity issues in relation to metamaterial homogenization,” IEEE Antennas and Propagation Magazine 55, 91–106 (2013).
[Crossref]

Grzegorczyk, T. M.

B. L. Wu, W. J. Wang, J. Pacheco, X. Chen, J. Lu, T. M. Grzegorczyk, J. A. Kong, P. Kao, P. A. Theophelakes, and M. J. Hogan, “Anisotropic metamaterials as antenna substrate to enhance directivity,” Microw. and Opt. Techn. Lett. 48, 680–683 (2006).
[Crossref]

T. M. Grzegorczyk, Z. M. Thomas, and J. A. Kong, “Inversion of critical angle and Brewster angle in anisotropic left-handed metamaterials,” Appl. Phys. Lett. 86, 251909 (2005).
[Crossref]

X. Chen, T. M. Grzegorczyk, B. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E., 70, 016608 (2004).
[Crossref]

Guenneau, S.

T. Antonakakis, R. V. Craster, and S. Guenneau, “High-frequency homogenization of zero-frequency stop band photonic and phononic crystals,” New J. Phys. 15, 103014 (2013).
[Crossref]

Hansen, T. V.

S. Arslanagić, T. V. Hansen, N. A. Mortensen, A. H. Gregersen, O. Sigmund, R. W. Ziolkowski, and O. Breinbjerg, “A review of the scattering-parameter extraction method with clarification of ambiguity issues in relation to metamaterial homogenization,” IEEE Antennas and Propagation Magazine 55, 91–106 (2013).
[Crossref]

Hao, Y.

Y. Hao and R. Mittra, FDTD Modeling of Metamaterials: Theory and Applications, Chap. 7, (Artech House, Boston, 2009).

Hogan, M. J.

B. L. Wu, W. J. Wang, J. Pacheco, X. Chen, J. Lu, T. M. Grzegorczyk, J. A. Kong, P. Kao, P. A. Theophelakes, and M. J. Hogan, “Anisotropic metamaterials as antenna substrate to enhance directivity,” Microw. and Opt. Techn. Lett. 48, 680–683 (2006).
[Crossref]

Hsieh, F.-J.

F.-J. Hsieh and W.-C. Wang, “Full extraction methods to retrieve effective refractive index and parameters of a bianisotropic metamaterial based on material dispersion models,” J. Appl. Phys. 112, 064907 (2012)
[Crossref]

Huangfu, J.

Jiang, Q.

Jiang, T.

Jiang, Z. H.

Z. H. Jiang, J. A. Bossard, X. Wang, and D. H. Werner, “Synthesizing metamaterials with angularly independent effective medium properties based on an anisotropic parameter retrieval technique coupled with a genetic algorithm,” J. Appl. Phys. 109, 013515 (2011)
[Crossref]

Kao, P.

B. L. Wu, W. J. Wang, J. Pacheco, X. Chen, J. Lu, T. M. Grzegorczyk, J. A. Kong, P. Kao, P. A. Theophelakes, and M. J. Hogan, “Anisotropic metamaterials as antenna substrate to enhance directivity,” Microw. and Opt. Techn. Lett. 48, 680–683 (2006).
[Crossref]

Kong, J. A.

Y. Yuan, L. Shen, L. Ran, T. Jiang, J. Huangfu, and J. A. Kong, “Directive emission based on anisotropic meta-materials,” Phys. Rev. A 77, 053821 (2008).
[Crossref]

B. L. Wu, W. J. Wang, J. Pacheco, X. Chen, J. Lu, T. M. Grzegorczyk, J. A. Kong, P. Kao, P. A. Theophelakes, and M. J. Hogan, “Anisotropic metamaterials as antenna substrate to enhance directivity,” Microw. and Opt. Techn. Lett. 48, 680–683 (2006).
[Crossref]

H. Chen, J. Zhang, Y. Bai, Y. Luo, L. Ran, Q. Jiang, and J. A. Kong, “Experimental retrieval of the effective parameters of metamaterials based on a waveguide method,” Opt. Express 14, 12944–12949 (2006).
[Crossref] [PubMed]

T. M. Grzegorczyk, Z. M. Thomas, and J. A. Kong, “Inversion of critical angle and Brewster angle in anisotropic left-handed metamaterials,” Appl. Phys. Lett. 86, 251909 (2005).
[Crossref]

X. Chen, T. M. Grzegorczyk, B. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E., 70, 016608 (2004).
[Crossref]

Koschny, T.

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

Lederer, F.

C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, “Validity of effective material parameters for optical fishnet metamaterials,” Phys. Rev B, 81, 035320 (2010).
[Crossref]

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

Li, J.

Z. Liang and J. Li, “Extreme acoustic metamaterial by coiling up space,” Phys. Rev. Lett. 108, 114301 (2012).
[Crossref] [PubMed]

Liang, Z.

Z. Liang and J. Li, “Extreme acoustic metamaterial by coiling up space,” Phys. Rev. Lett. 108, 114301 (2012).
[Crossref] [PubMed]

Lu, J.

B. L. Wu, W. J. Wang, J. Pacheco, X. Chen, J. Lu, T. M. Grzegorczyk, J. A. Kong, P. Kao, P. A. Theophelakes, and M. J. Hogan, “Anisotropic metamaterials as antenna substrate to enhance directivity,” Microw. and Opt. Techn. Lett. 48, 680–683 (2006).
[Crossref]

Luo, Y.

Ma, W.

Markos, P.

P. Markos and C. M. Soukoulis, “Transmission properties and effective electromagnetic parameters of double negative metamaterials,” Opt. Express 11, 649–661 (2003).
[Crossref] [PubMed]

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, 195104 (2002).
[Crossref]

Maurel, A

A Maurel, J.-F. Mercier, and S. Félix, “Wave propagation through penetrable scatterers in a waveguide and though a penetrable grating,” J. Acoust. Soc. Am. 135, 165–174 (2014).
[Crossref] [PubMed]

Maurel, A.

A. Maurel, S. Félix, and J.-F. Mercier, “Enhanced transmission through gratings: Structural and geometrical effects,” Phys. Rev. B 88, 115416 (2013).
[Crossref]

C.P. Berraquero, A. Maurel, P. Petitjeans, and V. Pagneux, “Experimental realization of a water-wave metamaterial shifter,” Phys. Rev. E 88, 051002(R) (2013).
[Crossref]

McPhedran, R. C.

N. A. Nicorovici and R. C. McPhedran, “Transport properties of arrays of elliptical cylinders,” Phys. Rev. E 54(2), 1945–1957 (1996).
[Crossref]

Menzel, C.

C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, “Validity of effective material parameters for optical fishnet metamaterials,” Phys. Rev B, 81, 035320 (2010).
[Crossref]

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

Mercier, J.-F.

A Maurel, J.-F. Mercier, and S. Félix, “Wave propagation through penetrable scatterers in a waveguide and though a penetrable grating,” J. Acoust. Soc. Am. 135, 165–174 (2014).
[Crossref] [PubMed]

A. Maurel, S. Félix, and J.-F. Mercier, “Enhanced transmission through gratings: Structural and geometrical effects,” Phys. Rev. B 88, 115416 (2013).
[Crossref]

Mittra, R.

Y. Hao and R. Mittra, FDTD Modeling of Metamaterials: Theory and Applications, Chap. 7, (Artech House, Boston, 2009).

Mortensen, N. A.

S. Arslanagić, T. V. Hansen, N. A. Mortensen, A. H. Gregersen, O. Sigmund, R. W. Ziolkowski, and O. Breinbjerg, “A review of the scattering-parameter extraction method with clarification of ambiguity issues in relation to metamaterial homogenization,” IEEE Antennas and Propagation Magazine 55, 91–106 (2013).
[Crossref]

Nevière, M.

Nicolet, A.

A. I. Căbuz, A. Nicolet, F. Zolla, D. Felbacq, and G. Bouchitté, “Homogenization of nonlocal wire metamaterial via a renormalization approach,” J. Opt. Soc. Am. B, 28, 1275–1282 (2011).
[Crossref]

Nicorovici, N. A.

N. A. Nicorovici and R. C. McPhedran, “Transport properties of arrays of elliptical cylinders,” Phys. Rev. E 54(2), 1945–1957 (1996).
[Crossref]

Oleinik, O. A.

O. A. Oleinik, A. S. Shamaev, and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization (North Holland, Amsterdam, 1992).

Pacheco, J.

B. L. Wu, W. J. Wang, J. Pacheco, X. Chen, J. Lu, T. M. Grzegorczyk, J. A. Kong, P. Kao, P. A. Theophelakes, and M. J. Hogan, “Anisotropic metamaterials as antenna substrate to enhance directivity,” Microw. and Opt. Techn. Lett. 48, 680–683 (2006).
[Crossref]

X. Chen, T. M. Grzegorczyk, B. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E., 70, 016608 (2004).
[Crossref]

Pagneux, V.

C.P. Berraquero, A. Maurel, P. Petitjeans, and V. Pagneux, “Experimental realization of a water-wave metamaterial shifter,” Phys. Rev. E 88, 051002(R) (2013).
[Crossref]

Paul, T.

C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, “Validity of effective material parameters for optical fishnet metamaterials,” Phys. Rev B, 81, 035320 (2010).
[Crossref]

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

Peng, L.

Pertsch, T.

C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, “Validity of effective material parameters for optical fishnet metamaterials,” Phys. Rev B, 81, 035320 (2010).
[Crossref]

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

Petitjeans, P.

C.P. Berraquero, A. Maurel, P. Petitjeans, and V. Pagneux, “Experimental realization of a water-wave metamaterial shifter,” Phys. Rev. E 88, 051002(R) (2013).
[Crossref]

Popa, B.-I.

Y. Xie, B.-I. Popa, L. Zigoneanu, and S. A. Cummer, “Measurement of a broadband negative index with space-coiling acoustic metamaterials,” Phys. Rev. Lett. 110, 175501 (2013).
[Crossref] [PubMed]

L. Zigoneanu, B.-I. Popa, and S. A. Cummer, “Design and measurements of a broadband two-dimensional acoustic lens,” Phys. Rev. B 84, 024305 (2011).
[Crossref]

L. Zigoneanu, B.-I. Popa, A.F. Starr, and S. A. Cummer, “Design and measurements of a broadband two-dimensional acoustic metamaterial with anisotropic effective mass density,” J. Appl. Phys. 109, 054906 (2011).
[Crossref]

B.-I. Popa and S. A. Cummer, “Design and characterization of broadband acoustic composite metamaterials,” Phys. Rev. B 80, 174303 (2009).
[Crossref]

Popov, E.

Raghunathan, S. B.

S. B. Raghunathan and N. V. Budko, “Effective permittivity of finite inhomogeneous objects,” Phys. Rev. B 81, 054206 (2010).
[Crossref]

Ran, L.

Rockstuhl, C.

C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, “Validity of effective material parameters for optical fishnet metamaterials,” Phys. Rev B, 81, 035320 (2010).
[Crossref]

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

Sánchez-Dehesa, J.

J. Sánchez-Dehesa, D. Torrent, and J. Carbonell, “Anisotropic metamaterials as sensing devices in acoustics and electromagnetism,” Proc. of SPIE 8346, 834606 (2012).
[Crossref]

D. Torrent and J. Sánchez-Dehesa, “Effective parameters of clusters of cylinders embedded in a nonviscous fluid or gas,” Phys. Rev. B 74, 224305 (2006).
[Crossref]

Schultz, S.

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, 195104 (2002).
[Crossref]

Shamaev, A. S.

O. A. Oleinik, A. S. Shamaev, and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization (North Holland, Amsterdam, 1992).

Shamonina, E.

L. Solymar and E. Shamonina, Waves in Metamaterials (Oxford University, Oxford, 2009).

Shen, L.

Y. Yuan, L. Shen, L. Ran, T. Jiang, J. Huangfu, and J. A. Kong, “Directive emission based on anisotropic meta-materials,” Phys. Rev. A 77, 053821 (2008).
[Crossref]

Sigmund, O.

S. Arslanagić, T. V. Hansen, N. A. Mortensen, A. H. Gregersen, O. Sigmund, R. W. Ziolkowski, and O. Breinbjerg, “A review of the scattering-parameter extraction method with clarification of ambiguity issues in relation to metamaterial homogenization,” IEEE Antennas and Propagation Magazine 55, 91–106 (2013).
[Crossref]

Simovski, C. R.

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

C. R. Simovski and S. A. Tretyakov, “On effective electromagnetic parameters of artificial nanostructured magnetic materials,” Photonics and Nanostructures - Fundamentals and Applications 8, 254–263 (2010).
[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, 036617 (2005).
[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, 195104 (2002).
[Crossref]

Solymar, L.

L. Solymar and E. Shamonina, Waves in Metamaterials (Oxford University, Oxford, 2009).

Soukoulis, C. M.

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

P. Markos and C. M. Soukoulis, “Transmission properties and effective electromagnetic parameters of double negative metamaterials,” Opt. Express 11, 649–661 (2003).
[Crossref] [PubMed]

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, 195104 (2002).
[Crossref]

Starr, A.F.

L. Zigoneanu, B.-I. Popa, A.F. Starr, and S. A. Cummer, “Design and measurements of a broadband two-dimensional acoustic metamaterial with anisotropic effective mass density,” J. Appl. Phys. 109, 054906 (2011).
[Crossref]

Sun, C.

V. Fokin, M. Ambati, C. Sun, and X. Zhang, “Method for retrieving effective properties of locally resonant acoustic metamaterials,” Phys. Rev. B 76, 144302 (2007).
[Crossref]

Theophelakes, P. A.

B. L. Wu, W. J. Wang, J. Pacheco, X. Chen, J. Lu, T. M. Grzegorczyk, J. A. Kong, P. Kao, P. A. Theophelakes, and M. J. Hogan, “Anisotropic metamaterials as antenna substrate to enhance directivity,” Microw. and Opt. Techn. Lett. 48, 680–683 (2006).
[Crossref]

Thomas, Z. M.

T. M. Grzegorczyk, Z. M. Thomas, and J. A. Kong, “Inversion of critical angle and Brewster angle in anisotropic left-handed metamaterials,” Appl. Phys. Lett. 86, 251909 (2005).
[Crossref]

Torrent, D.

J. Sánchez-Dehesa, D. Torrent, and J. Carbonell, “Anisotropic metamaterials as sensing devices in acoustics and electromagnetism,” Proc. of SPIE 8346, 834606 (2012).
[Crossref]

D. Torrent and J. Sánchez-Dehesa, “Effective parameters of clusters of cylinders embedded in a nonviscous fluid or gas,” Phys. Rev. B 74, 224305 (2006).
[Crossref]

Tretyakov, S.

C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, “Validity of effective material parameters for optical fishnet metamaterials,” Phys. Rev B, 81, 035320 (2010).
[Crossref]

Tretyakov, S. A.

C. R. Simovski and S. A. Tretyakov, “On effective electromagnetic parameters of artificial nanostructured magnetic materials,” Photonics and Nanostructures - Fundamentals and Applications 8, 254–263 (2010).
[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, 036617 (2005).
[Crossref]

Wang, W. J.

B. L. Wu, W. J. Wang, J. Pacheco, X. Chen, J. Lu, T. M. Grzegorczyk, J. A. Kong, P. Kao, P. A. Theophelakes, and M. J. Hogan, “Anisotropic metamaterials as antenna substrate to enhance directivity,” Microw. and Opt. Techn. Lett. 48, 680–683 (2006).
[Crossref]

Wang, W.-C.

F.-J. Hsieh and W.-C. Wang, “Full extraction methods to retrieve effective refractive index and parameters of a bianisotropic metamaterial based on material dispersion models,” J. Appl. Phys. 112, 064907 (2012)
[Crossref]

Wang, X.

Z. H. Jiang, J. A. Bossard, X. Wang, and D. H. Werner, “Synthesizing metamaterials with angularly independent effective medium properties based on an anisotropic parameter retrieval technique coupled with a genetic algorithm,” J. Appl. Phys. 109, 013515 (2011)
[Crossref]

Wang, Z.

Werner, D. H.

Z. H. Jiang, J. A. Bossard, X. Wang, and D. H. Werner, “Synthesizing metamaterials with angularly independent effective medium properties based on an anisotropic parameter retrieval technique coupled with a genetic algorithm,” J. Appl. Phys. 109, 013515 (2011)
[Crossref]

Wu, B.

X. Chen, T. M. Grzegorczyk, B. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E., 70, 016608 (2004).
[Crossref]

Wu, B. L.

B. L. Wu, W. J. Wang, J. Pacheco, X. Chen, J. Lu, T. M. Grzegorczyk, J. A. Kong, P. Kao, P. A. Theophelakes, and M. J. Hogan, “Anisotropic metamaterials as antenna substrate to enhance directivity,” Microw. and Opt. Techn. Lett. 48, 680–683 (2006).
[Crossref]

Xie, Y.

Y. Xie, B.-I. Popa, L. Zigoneanu, and S. A. Cummer, “Measurement of a broadband negative index with space-coiling acoustic metamaterials,” Phys. Rev. Lett. 110, 175501 (2013).
[Crossref] [PubMed]

Yang, T.-C.

Yang, Y.-H.

Yen, T.-J.

Yosifian, G. A.

O. A. Oleinik, A. S. Shamaev, and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization (North Holland, Amsterdam, 1992).

Yuan, Y.

Y. Yuan, L. Shen, L. Ran, T. Jiang, J. Huangfu, and J. A. Kong, “Directive emission based on anisotropic meta-materials,” Phys. Rev. A 77, 053821 (2008).
[Crossref]

Zhang, J.

Zhang, X.

V. Fokin, M. Ambati, C. Sun, and X. Zhang, “Method for retrieving effective properties of locally resonant acoustic metamaterials,” Phys. Rev. B 76, 144302 (2007).
[Crossref]

Zigoneanu, L.

Y. Xie, B.-I. Popa, L. Zigoneanu, and S. A. Cummer, “Measurement of a broadband negative index with space-coiling acoustic metamaterials,” Phys. Rev. Lett. 110, 175501 (2013).
[Crossref] [PubMed]

L. Zigoneanu, B.-I. Popa, A.F. Starr, and S. A. Cummer, “Design and measurements of a broadband two-dimensional acoustic metamaterial with anisotropic effective mass density,” J. Appl. Phys. 109, 054906 (2011).
[Crossref]

L. Zigoneanu, B.-I. Popa, and S. A. Cummer, “Design and measurements of a broadband two-dimensional acoustic lens,” Phys. Rev. B 84, 024305 (2011).
[Crossref]

Ziolkowski, R. W.

S. Arslanagić, T. V. Hansen, N. A. Mortensen, A. H. Gregersen, O. Sigmund, R. W. Ziolkowski, and O. Breinbjerg, “A review of the scattering-parameter extraction method with clarification of ambiguity issues in relation to metamaterial homogenization,” IEEE Antennas and Propagation Magazine 55, 91–106 (2013).
[Crossref]

Zolla, F.

A. I. Căbuz, A. Nicolet, F. Zolla, D. Felbacq, and G. Bouchitté, “Homogenization of nonlocal wire metamaterial via a renormalization approach,” J. Opt. Soc. Am. B, 28, 1275–1282 (2011).
[Crossref]

Appl. Phys. Lett. (1)

T. M. Grzegorczyk, Z. M. Thomas, and J. A. Kong, “Inversion of critical angle and Brewster angle in anisotropic left-handed metamaterials,” Appl. Phys. Lett. 86, 251909 (2005).
[Crossref]

IEEE Antennas and Propagation Magazine (1)

S. Arslanagić, T. V. Hansen, N. A. Mortensen, A. H. Gregersen, O. Sigmund, R. W. Ziolkowski, and O. Breinbjerg, “A review of the scattering-parameter extraction method with clarification of ambiguity issues in relation to metamaterial homogenization,” IEEE Antennas and Propagation Magazine 55, 91–106 (2013).
[Crossref]

J. Acoust. Soc. Am. (1)

A Maurel, J.-F. Mercier, and S. Félix, “Wave propagation through penetrable scatterers in a waveguide and though a penetrable grating,” J. Acoust. Soc. Am. 135, 165–174 (2014).
[Crossref] [PubMed]

J. Appl. Phys. (3)

F.-J. Hsieh and W.-C. Wang, “Full extraction methods to retrieve effective refractive index and parameters of a bianisotropic metamaterial based on material dispersion models,” J. Appl. Phys. 112, 064907 (2012)
[Crossref]

Z. H. Jiang, J. A. Bossard, X. Wang, and D. H. Werner, “Synthesizing metamaterials with angularly independent effective medium properties based on an anisotropic parameter retrieval technique coupled with a genetic algorithm,” J. Appl. Phys. 109, 013515 (2011)
[Crossref]

L. Zigoneanu, B.-I. Popa, A.F. Starr, and S. A. Cummer, “Design and measurements of a broadband two-dimensional acoustic metamaterial with anisotropic effective mass density,” J. Appl. Phys. 109, 054906 (2011).
[Crossref]

J. Opt. (1)

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

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B, (1)

A. I. Căbuz, A. Nicolet, F. Zolla, D. Felbacq, and G. Bouchitté, “Homogenization of nonlocal wire metamaterial via a renormalization approach,” J. Opt. Soc. Am. B, 28, 1275–1282 (2011).
[Crossref]

Microw. and Opt. Techn. Lett. (1)

B. L. Wu, W. J. Wang, J. Pacheco, X. Chen, J. Lu, T. M. Grzegorczyk, J. A. Kong, P. Kao, P. A. Theophelakes, and M. J. Hogan, “Anisotropic metamaterials as antenna substrate to enhance directivity,” Microw. and Opt. Techn. Lett. 48, 680–683 (2006).
[Crossref]

New J. Phys. (1)

T. Antonakakis, R. V. Craster, and S. Guenneau, “High-frequency homogenization of zero-frequency stop band photonic and phononic crystals,” New J. Phys. 15, 103014 (2013).
[Crossref]

Opt. Express (4)

Photonics and Nanostructures - Fundamentals and Applications (1)

C. R. Simovski and S. A. Tretyakov, “On effective electromagnetic parameters of artificial nanostructured magnetic materials,” Photonics and Nanostructures - Fundamentals and Applications 8, 254–263 (2010).
[Crossref]

Phys. Rev B, (1)

C. Menzel, T. Paul, C. Rockstuhl, T. Pertsch, S. Tretyakov, and F. Lederer, “Validity of effective material parameters for optical fishnet metamaterials,” Phys. Rev B, 81, 035320 (2010).
[Crossref]

Phys. Rev. A (1)

Y. Yuan, L. Shen, L. Ran, T. Jiang, J. Huangfu, and J. A. Kong, “Directive emission based on anisotropic meta-materials,” Phys. Rev. A 77, 053821 (2008).
[Crossref]

Phys. Rev. B (8)

D. Torrent and J. Sánchez-Dehesa, “Effective parameters of clusters of cylinders embedded in a nonviscous fluid or gas,” Phys. Rev. B 74, 224305 (2006).
[Crossref]

S. B. Raghunathan and N. V. Budko, “Effective permittivity of finite inhomogeneous objects,” Phys. Rev. B 81, 054206 (2010).
[Crossref]

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

V. Fokin, M. Ambati, C. Sun, and X. Zhang, “Method for retrieving effective properties of locally resonant acoustic metamaterials,” Phys. Rev. B 76, 144302 (2007).
[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, 195104 (2002).
[Crossref]

B.-I. Popa and S. A. Cummer, “Design and characterization of broadband acoustic composite metamaterials,” Phys. Rev. B 80, 174303 (2009).
[Crossref]

L. Zigoneanu, B.-I. Popa, and S. A. Cummer, “Design and measurements of a broadband two-dimensional acoustic lens,” Phys. Rev. B 84, 024305 (2011).
[Crossref]

A. Maurel, S. Félix, and J.-F. Mercier, “Enhanced transmission through gratings: Structural and geometrical effects,” Phys. Rev. B 88, 115416 (2013).
[Crossref]

Phys. Rev. E (3)

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

C.P. Berraquero, A. Maurel, P. Petitjeans, and V. Pagneux, “Experimental realization of a water-wave metamaterial shifter,” Phys. Rev. E 88, 051002(R) (2013).
[Crossref]

N. A. Nicorovici and R. C. McPhedran, “Transport properties of arrays of elliptical cylinders,” Phys. Rev. E 54(2), 1945–1957 (1996).
[Crossref]

Phys. Rev. E., (1)

X. Chen, T. M. Grzegorczyk, B. Wu, J. Pacheco, and J. A. Kong, “Robust method to retrieve the constitutive effective parameters of metamaterials,” Phys. Rev. E., 70, 016608 (2004).
[Crossref]

Phys. Rev. Lett. (2)

Z. Liang and J. Li, “Extreme acoustic metamaterial by coiling up space,” Phys. Rev. Lett. 108, 114301 (2012).
[Crossref] [PubMed]

Y. Xie, B.-I. Popa, L. Zigoneanu, and S. A. Cummer, “Measurement of a broadband negative index with space-coiling acoustic metamaterials,” Phys. Rev. Lett. 110, 175501 (2013).
[Crossref] [PubMed]

Proc. of SPIE (1)

J. Sánchez-Dehesa, D. Torrent, and J. Carbonell, “Anisotropic metamaterials as sensing devices in acoustics and electromagnetism,” Proc. of SPIE 8346, 834606 (2012).
[Crossref]

Other (5)

Y. Hao and R. Mittra, FDTD Modeling of Metamaterials: Theory and Applications, Chap. 7, (Artech House, Boston, 2009).

O. A. Oleinik, A. S. Shamaev, and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization (North Holland, Amsterdam, 1992).

N. Engheta and R. W. Ziolkowski, eds. Metamaterials: Physics and Engineering Explorations (John Wiley & Sons, 2006).
[Crossref]

L. Solymar and E. Shamonina, Waves in Metamaterials (Oxford University, Oxford, 2009).

R. V. Craster and S. Guenneau, eds. Acoustic metamaterials: Negative Refraction, Imaging, Lensing and Cloaking, Vol. 166 (Springer, 2012).

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Figures (15)

Fig. 1
Fig. 1 (a) Typical configuration of the real medium, the reflection R and transmission T through a slab containing few cells (otherwise x-periodic in the x-direction) are calculated for an obliquely incident plane wave with angle θ and wavenumber k. (b) Effective anisotropic medium leading to the same scattering properties (R and T).
Fig. 2
Fig. 2 Configuration of the study. The slab contains slanted layers, made of alternating air and a material with parameters relative to those of the air ε1 = 10 and μ1 = 0.2; the wave is incident with angle θ and we measure R and T.
Fig. 3
Fig. 3 (a) Propagation of an incident plane wave (with θ = 45°) through a slab with kL = 5, containing a layered structure with kd = 0.5 and α = 0 (the layers are represented only on half of the slab to make visible the wavefield inside the slab). (b) Propagation through the slab (kL = 5) filled with the effective anisotropic medium defined by the retrieved parameters (α ≃ 0, εX = 1.880, εY = 4.573 and μZ = 0.612).
Fig. 4
Fig. 4 Same representation as in Fig. 3 for (a) slanted layers in a slab with length kL = 2.5, with kd = 0.25 and α = 45°. (b) The effective parameters derived from the inversion method are α = 44.62°, εX = 1.848, εY = 5.014 and μZ = 0.605.
Fig. 5
Fig. 5 Illustration of the preliminary step of the retrieval method: Calculation of the compensated transmission coefficient Tc, Eq. (21). Here T (θ) and T (−θ) have been calculated for slanted layers with α = 45° and kd = 0.5.
Fig. 6
Fig. 6 Step 1 of the retrieval method (main step): the impedance ratio ξ (θ) and the effective index (along the x-axis) n(θ) are retrieved, Eq. (12); for kd = 0.01 (○, blue), 0.1 (•, red) and 0.5 (⋄, green).
Fig. 7
Fig. 7 Step 2: εy is deduced from ξ and n; the final corresponding εX values are indicated. Same color and symbol conventions as in Fig. 6 are used.
Fig. 8
Fig. 8 Step 3: Determination of μZ and of the product εX εY by fitting the quantity n2/εy = μZ − sin2 θ εyX εY. Symbols correspond to the retrieved values (with the same color and symbol conventions as in Fig. 6), and plain lines to the fit.
Fig. 9
Fig. 9 Step 4: T (θ)/T (−θ) is fitted to find εxy, Eq. (20).
Fig. 10
Fig. 10 Variation of the retrieved parameters with α, for kd = 0.01 (○, blue), kd = 0.1 (•, red) and kd = 0.5 (⋄, green). Gray dotted lines correspond to the homogenized values, Eqs. (25) and (14).
Fig. 11
Fig. 11 Variation of the retrieved parameters (as defined in Eq. (6)) as a function of kd, for α = 0 (○, blue) and α = 45° (⋄, green). Gray dotted lines correspond to the homogenized values, Eqs. (25).
Fig. 12
Fig. 12 Square lattices of tilted elliptical particles; the unit cell is square x × x and (a, b) are the major and minor diameters. Note that tilting the ellipse inside the unit cell produces an array (b) with arrangement different from tilting the whole array (a). The surrounding medium has normalized mass density and bulk modulus; the elliptical particles have relative mass density ρ1 and relative bulk modulus B1.
Fig. 13
Fig. 13 Top panel: Propagation of an incident wave (with θ = 45°) through a slab with length L = 50x containing tilted elliptical particles in a square lattice with unit cell of size kℓx = 0.25 (the ellipse is given by a = 0.8x, b = 0.2x). The particles are materialized with black line only on the lower part of the slab; the insets show a single particle in the unit cell. Bottom : Propagation through the slab of same length filled with the effective anisotropic medium defined using the retrieved parameters (ρX = 1.171, ρY = 1.785, B = 1.138), and (a) α = 0, (b) α = 45°, α = 90°.
Fig. 14
Fig. 14 Variation of the retrieved parameters with α for N = 5 (○, blue), N = 15 (•, red) and N = 30 (⋄, green). Dotted gray lines are the theoretical predictions, Eqs. (27) and (28).
Fig. 15
Fig. 15 Variation of the retrieved parameters with the array size N (y = Nℓx) for α = 0 (○, blue) and α = 90° (⋄, green). The dotted gray lines indicate the theoretical predictions, Eqs. (27) and (28).).

Tables (1)

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Table 1 Coefficients in the 2D wave equation for anisotropic media in electromagnetism and acoustics.

Equations (28)

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ε diag = ( ε X 0 0 0 ε Y 0 0 0 ε Z ) , μ diag = ( μ X 0 0 0 μ Y 0 0 0 μ Z ) , in electromagnetism , ρ diag = ( ρ X 0 0 0 ρ Y 0 0 0 ρ Z ) , in acoustics .
R = ( cos α sin α 0 sin α cos α 0 0 0 1 )
in electromagnetism { × E = i k ( R t μ diag R ) H , × H = i k ( R t μ diag R ) E , in acoustics { p = i k ( R t ρ diag R ) u , B u = i k p ,
[ ( 1 / ε x 1 / ε x y 1 / ε x y 1 / ε y ) H ] + μ Z k 2 H = 0 , TM waves , [ ( 1 / μ x 1 / μ x y 1 / μ x y 1 / μ y ) E ] + ε Z k 2 E = 0 , TE waves .
[ ( 1 / ρ x 1 / ρ x y 1 / ρ x 1 / ρ y ) p ] + k 2 B p = 0 ,
[ ( 1 / ε Y 0 0 1 / ε X ) H ] + μ Z k 2 H = 0 ,
{ H ( y 0 ) = e i k sin θ x [ e i k cos θ y + R e i k cos θ y ] , H ( 0 < y y ) = e i k sin θ x [ a e i n k y + b e i n k y ] , H ( y y ) = e i k sin θ x T e i k cos θ ( y y )
R = ( 1 ξ 2 ) ( e i n k y e i n k y ) ( 1 + ξ ) 2 e i n k y ( 1 ξ ) 2 e i n k y , T = 4 ξ ( 1 + ξ ) 2 e i n k y ( 1 ξ ) 2 e i n k y
n k Y k = μ Z ε X ε X ε Y sin 2 θ , ξ = ε X cos θ n .
n = ε Z μ X μ X μ Y sin 2 θ , and ξ = μ X cos θ n ,
n = ρ Y B ρ Y ρ X sin 2 θ , and ξ ρ Y cos θ n .
{ n = i k y [ log ( 1 R 2 + T 2 + ξ ˜ ) 2 T + 2 i m π ] , with ξ ˜ ± ( 1 R 2 + T 2 ) 2 4 T 2 , ξ = ξ ˜ ( 1 R ) 2 T 2 ,
[ ( 1 / ε y 1 / ε x y 1 / ε x y 1 / ε x ) H ] + μ z k 2 H = 0 ,
{ 1 ε x = cos 2 α ε Y + sin 2 α ε X , 1 ε y = sin 2 α ε Y + cos 2 α ε X , 1 ε x y = sin α cos α ( 1 ε X 1 ε Y ) .
H ( 0 < y y ) = e i k sin θ x [ a e i k + y + b e i k y ] .
k ± 2 / ε y + 2 k ± k sin θ / ε x y + k 2 ( sin 2 θ / ε x μ Z ) = 0 ,
k ± = ε y ε x y k sin θ ± n k , with n μ Z ε y ε y 2 ε X ε Y sin 2 θ ,
R = ( 1 ξ 2 ) ( e i n k y e i n k y ) ( 1 + ξ ) 2 e i n k y ( 1 ξ ) 2 e i n k y , T = 4 ξ e i k y sin θ ε y / ε x y ( 1 + ξ ) 2 e i n k y ( 1 ξ ) 2 e i n k y
n μ Z ε y ε y 2 ε X ε Y sin 2 θ , ξ ε y cos θ n .
e 2 i k y sin θ ε y / ε x y = T ( θ ) T ( θ ) ,
T c ( θ ) T ( θ ) T ( θ ) = 4 ξ ( 1 + ξ ) 2 e i n k y ( 1 ξ ) 2 e i n k y ,
1 ε x = ε y ε X ε Y + ε y ε x y 2 .
{ 1 ε X = 1 2 [ 1 ε x + 1 ε y + ( 1 ε x 1 ε y ) 2 + 4 ε x y 2 ] , 1 ε Y = 1 2 [ 1 ε x + 1 ε y ( 1 ε x 1 ε y ) 2 + 4 ε x y 2 ] ,
{ α = sin 1 [ 1 / ε Y 1 / ε x 1 / ε Y 1 / ε X ] , if ε x y 0 α = 180 ° sin 1 [ 1 / ε Y 1 / ε x 1 / ε Y 1 / ε X ] , if ε x y > 0 ,
{ ε X homog = ( 1 φ + φ / ε 1 ) 1 = 1.818 , ε Y homog = 1 φ + φ ε 1 = 5.500 , μ homog = 1 φ + φ μ 1 = 0.6 ,
ρ X = 1.171 , ρ Y = 1.785 , B = 1.138 .
B th = [ φ + B 1 + 1 φ ] 1 = 1.143 ,
{ ρ X th = 2 r X φ + ρ 1 ( r X + φ ) 2 r X + φ + ρ 1 ( r X φ ) 1.168 , ρ Y th = 2 r Y φ + ρ 1 ( r Y + φ ) 2 r Y + φ + ρ 1 ( r Y φ ) 1.853 ,

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