Abstract

Accurate and robust characterization of metasurfaces and metamaterials in terms of effective parameters is critical to the design of novel metadevices. We compute the Cramér-Rao lower bounds on the variance of any estimator for both the electric and magnetic surface susceptibilities of metasurfaces. We show that retrieval of such effective properties is inherently difficult around resonances, most notably for low-loss metasurfaces. We also put forth a least-squares estimator to mitigate this difficulty for the normal components of susceptibility tensors, which are observed to be the most ill-behaved. The present work is relevant to the development of loss-compensated metasurfaces for which noise has to be closely considered for accurate and robust device characterization.

© 2015 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref]
  33. V. Grigoriev, G. Demésy, J. Wenger, and N. Bonod, “Singular analysis to homogenize planar metamaterials as nonlocal effective media,” Phys. Rev. B 89(24), 245102 (2014).
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    [Crossref]

2014 (2)

C. Pfeiffer and A. Grbic, “Bianisotropic metasurfaces for optimal polarization control: Analysis and synthesis,” Phys. Rev. A 2(4), 044011 (2014).
[Crossref]

V. Grigoriev, G. Demésy, J. Wenger, and N. Bonod, “Singular analysis to homogenize planar metamaterials as nonlocal effective media,” Phys. Rev. B 89(24), 245102 (2014).
[Crossref]

2013 (4)

T. Feng, F. Liu, W. Y. Tam, and J. Li, “Effective parameters retrieval for complex metamaterials with low symmetries,” EPL 102(1), 18003 (2013).
[Crossref]

M. Albooyeh, Y. Ra’di, M. Q. Adil, and C. R. Simovski, “Revised transmission line model for electromagnetic characterization of metasurfaces,” Phys. Rev. B 88(8), 085435 (2013).
[Crossref]

J. D. Caldwell, O. J. Glembocki, Y. Francescato, N. Sharac, V. Giannini, F. J. Bezares, J. P. Long, J. C. Owrutsky, I. Vurgaftman, J. G. Tischler, V. D. Wheeler, N. D. Bassim, L. M. Shirey, R. Kasica, and S. A. Maier, “Low-loss, extreme subdiffraction photon confinement via silicon carbide localized surface phonon polariton resonators,” Nano Lett. 13(8), 3690–3697 (2013).
[Crossref] [PubMed]

P.-H. Tichit, S. N. Burokur, C.-W. Qiu, and A. de Lustrac, “Experimental verification of isotropic radiation from a coherent dipole source via electric-field-driven LC resonator metamaterials,” Phys. Rev. Lett. 111(13), 133901 (2013).
[Crossref] [PubMed]

2012 (2)

C. L. Holloway, E. F. Kuester, J. A. Gordon, J. O’Hara, J. Booth, and D. R. Smith, “An overview of the theory and applications of metasurfaces: The two-dimensional equivalents of metamaterials,” IEEE Antennas Propag. Mag. 54(2), 10–35 (2012).
[Crossref]

P. Tassin, T. Koschny, and C. M. Soukoulis, “Effective material parameter retrieval for thin sheets: Theory and application to graphene, thin silver films, and single-layer metamaterials,” Physica B 407(20), 4062–4065 (2012).
[Crossref]

2011 (3)

D. Sjöberg and C. Larsson, “Cramér-Rao bounds for determination of permittivity and permeability in slabs,” IEEE Trans. Microw. Theory Tech. 59(11), 2970–2977 (2011).
[Crossref]

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: Generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

C. R. Simovski, “On electromagnetic characterization and homogenization of nanostructured metamaterials,” J. Opt. 13(1), 1–22 (2011).
[Crossref]

2010 (1)

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[Crossref] [PubMed]

2009 (3)

B. Kanté, J.-M. Lourtioz, and A. de Lustrac, “Infrared metafilms on a dielectric substrate,” Phys. Rev. B 80(20), 205120 (2009).
[Crossref]

B. Kanté, D. Germain, and A. de Lustrac, “Experimental demonstration of a nonmagnetic cloak at microwave frequencies,” Phys. Rev. B 80(20), 201104 (2009).
[Crossref]

T. Lepetit, E. Akmansoy, and J.-P. Ganne, “Experimental measurement of negative index in an all-dielectric metamaterial,” Appl. Phys. Lett. 95(12), 121101 (2009).
[Crossref]

2006 (1)

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

2005 (2)

C. L. Holloway, M. A. Mohamed, E. F. Kuester, and A. Dienstfrey, “Reflection and transmission properties of a metafilm: With an application to a controllable surface composed of resonant particles,” IEEE Trans. Electromagn. Compat. 47(4), 853–865 (2005).
[Crossref]

S. A. Ramakrishna and O. J. F. Martin, “Resolving the wave vector in negative refractive index media,” Opt. Lett. 30(19), 2626–2628 (2005).
[Crossref] [PubMed]

2004 (1)

K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, “Negative effective permeability in polaritonic photonic crystals,” Appl. Phys. Lett. 85(4), 543–545 (2004).
[Crossref]

2003 (2)

T. Koschny, P. Markoš, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(6), 065602 (2003).
[Crossref] [PubMed]

E. F. Kuester, M. A. Mohamed, M. Piket-May, and C. L. Holloway, “Averaged transition conditions for electromagnetic fields at a metafilm,” IEEE Trans. Antenn. Propag. 51(10), 2641–2651 (2003).
[Crossref]

2002 (1)

D. R. Smith, S. Schultz, P. Markoš, 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]

2000 (2)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[Crossref] [PubMed]

D. R. Smith, W. J. Padilla, D. C. 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] [PubMed]

1999 (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999).
[Crossref]

1995 (1)

A. van den Bos, “The multivariate complex normal distribution- a generalization,” IEEE Trans. Inf. Theory 41(2), 537–539 (1995).
[Crossref]

1994 (1)

A. van den Bos, “A Cramér-Rao lower bound for complex parameters,” IEEE Trans. Signal Process. 42(10), 2859 (1994).
[Crossref]

1992 (1)

J. Baker-Jarvis, R. G. Reyer, and P. D. Domich, “A nonlinear least-squares solution with causality constraints applied to transmission line permittivity and permeability determination,” IEEE Trans. Instrum. Meas. 41(5), 646–652 (1992).
[Crossref]

1974 (1)

W. B. Weir, “Automatic measurement of complex dielectric constant and permeability at microwave frequencies,” Proc. IEEE 62(1), 33–36 (1974).
[Crossref]

1970 (1)

A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time-domain techniques,” IEEE Trans. Instrum. Meas. 19(4), 377–382 (1970).
[Crossref]

Adil, M. Q.

M. Albooyeh, Y. Ra’di, M. Q. Adil, and C. R. Simovski, “Revised transmission line model for electromagnetic characterization of metasurfaces,” Phys. Rev. B 88(8), 085435 (2013).
[Crossref]

Aieta, F.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: Generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Akmansoy, E.

T. Lepetit, E. Akmansoy, and J.-P. Ganne, “Experimental measurement of negative index in an all-dielectric metamaterial,” Appl. Phys. Lett. 95(12), 121101 (2009).
[Crossref]

Albooyeh, M.

M. Albooyeh, Y. Ra’di, M. Q. Adil, and C. R. Simovski, “Revised transmission line model for electromagnetic characterization of metasurfaces,” Phys. Rev. B 88(8), 085435 (2013).
[Crossref]

Baker-Jarvis, J.

J. Baker-Jarvis, R. G. Reyer, and P. D. Domich, “A nonlinear least-squares solution with causality constraints applied to transmission line permittivity and permeability determination,” IEEE Trans. Instrum. Meas. 41(5), 646–652 (1992).
[Crossref]

Bassim, N. D.

J. D. Caldwell, O. J. Glembocki, Y. Francescato, N. Sharac, V. Giannini, F. J. Bezares, J. P. Long, J. C. Owrutsky, I. Vurgaftman, J. G. Tischler, V. D. Wheeler, N. D. Bassim, L. M. Shirey, R. Kasica, and S. A. Maier, “Low-loss, extreme subdiffraction photon confinement via silicon carbide localized surface phonon polariton resonators,” Nano Lett. 13(8), 3690–3697 (2013).
[Crossref] [PubMed]

Bezares, F. J.

J. D. Caldwell, O. J. Glembocki, Y. Francescato, N. Sharac, V. Giannini, F. J. Bezares, J. P. Long, J. C. Owrutsky, I. Vurgaftman, J. G. Tischler, V. D. Wheeler, N. D. Bassim, L. M. Shirey, R. Kasica, and S. A. Maier, “Low-loss, extreme subdiffraction photon confinement via silicon carbide localized surface phonon polariton resonators,” Nano Lett. 13(8), 3690–3697 (2013).
[Crossref] [PubMed]

Bonod, N.

V. Grigoriev, G. Demésy, J. Wenger, and N. Bonod, “Singular analysis to homogenize planar metamaterials as nonlocal effective media,” Phys. Rev. B 89(24), 245102 (2014).
[Crossref]

Booth, J.

C. L. Holloway, E. F. Kuester, J. A. Gordon, J. O’Hara, J. Booth, and D. R. Smith, “An overview of the theory and applications of metasurfaces: The two-dimensional equivalents of metamaterials,” IEEE Antennas Propag. Mag. 54(2), 10–35 (2012).
[Crossref]

Burokur, S. N.

P.-H. Tichit, S. N. Burokur, C.-W. Qiu, and A. de Lustrac, “Experimental verification of isotropic radiation from a coherent dipole source via electric-field-driven LC resonator metamaterials,” Phys. Rev. Lett. 111(13), 133901 (2013).
[Crossref] [PubMed]

Caldwell, J. D.

J. D. Caldwell, O. J. Glembocki, Y. Francescato, N. Sharac, V. Giannini, F. J. Bezares, J. P. Long, J. C. Owrutsky, I. Vurgaftman, J. G. Tischler, V. D. Wheeler, N. D. Bassim, L. M. Shirey, R. Kasica, and S. A. Maier, “Low-loss, extreme subdiffraction photon confinement via silicon carbide localized surface phonon polariton resonators,” Nano Lett. 13(8), 3690–3697 (2013).
[Crossref] [PubMed]

Capasso, F.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: Generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Chan, C. T.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[Crossref] [PubMed]

Chen, H.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[Crossref] [PubMed]

Cummer, S. A.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

de Lustrac, A.

P.-H. Tichit, S. N. Burokur, C.-W. Qiu, and A. de Lustrac, “Experimental verification of isotropic radiation from a coherent dipole source via electric-field-driven LC resonator metamaterials,” Phys. Rev. Lett. 111(13), 133901 (2013).
[Crossref] [PubMed]

B. Kanté, D. Germain, and A. de Lustrac, “Experimental demonstration of a nonmagnetic cloak at microwave frequencies,” Phys. Rev. B 80(20), 201104 (2009).
[Crossref]

B. Kanté, J.-M. Lourtioz, and A. de Lustrac, “Infrared metafilms on a dielectric substrate,” Phys. Rev. B 80(20), 205120 (2009).
[Crossref]

Demésy, G.

V. Grigoriev, G. Demésy, J. Wenger, and N. Bonod, “Singular analysis to homogenize planar metamaterials as nonlocal effective media,” Phys. Rev. B 89(24), 245102 (2014).
[Crossref]

Dienstfrey, A.

C. L. Holloway, M. A. Mohamed, E. F. Kuester, and A. Dienstfrey, “Reflection and transmission properties of a metafilm: With an application to a controllable surface composed of resonant particles,” IEEE Trans. Electromagn. Compat. 47(4), 853–865 (2005).
[Crossref]

Domich, P. D.

J. Baker-Jarvis, R. G. Reyer, and P. D. Domich, “A nonlinear least-squares solution with causality constraints applied to transmission line permittivity and permeability determination,” IEEE Trans. Instrum. Meas. 41(5), 646–652 (1992).
[Crossref]

Feng, T.

T. Feng, F. Liu, W. Y. Tam, and J. Li, “Effective parameters retrieval for complex metamaterials with low symmetries,” EPL 102(1), 18003 (2013).
[Crossref]

Francescato, Y.

J. D. Caldwell, O. J. Glembocki, Y. Francescato, N. Sharac, V. Giannini, F. J. Bezares, J. P. Long, J. C. Owrutsky, I. Vurgaftman, J. G. Tischler, V. D. Wheeler, N. D. Bassim, L. M. Shirey, R. Kasica, and S. A. Maier, “Low-loss, extreme subdiffraction photon confinement via silicon carbide localized surface phonon polariton resonators,” Nano Lett. 13(8), 3690–3697 (2013).
[Crossref] [PubMed]

Gaburro, Z.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: Generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Ganne, J.-P.

T. Lepetit, E. Akmansoy, and J.-P. Ganne, “Experimental measurement of negative index in an all-dielectric metamaterial,” Appl. Phys. Lett. 95(12), 121101 (2009).
[Crossref]

Genevet, P.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: Generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Germain, D.

B. Kanté, D. Germain, and A. de Lustrac, “Experimental demonstration of a nonmagnetic cloak at microwave frequencies,” Phys. Rev. B 80(20), 201104 (2009).
[Crossref]

Giannini, V.

J. D. Caldwell, O. J. Glembocki, Y. Francescato, N. Sharac, V. Giannini, F. J. Bezares, J. P. Long, J. C. Owrutsky, I. Vurgaftman, J. G. Tischler, V. D. Wheeler, N. D. Bassim, L. M. Shirey, R. Kasica, and S. A. Maier, “Low-loss, extreme subdiffraction photon confinement via silicon carbide localized surface phonon polariton resonators,” Nano Lett. 13(8), 3690–3697 (2013).
[Crossref] [PubMed]

Glembocki, O. J.

J. D. Caldwell, O. J. Glembocki, Y. Francescato, N. Sharac, V. Giannini, F. J. Bezares, J. P. Long, J. C. Owrutsky, I. Vurgaftman, J. G. Tischler, V. D. Wheeler, N. D. Bassim, L. M. Shirey, R. Kasica, and S. A. Maier, “Low-loss, extreme subdiffraction photon confinement via silicon carbide localized surface phonon polariton resonators,” Nano Lett. 13(8), 3690–3697 (2013).
[Crossref] [PubMed]

Gordon, J. A.

C. L. Holloway, E. F. Kuester, J. A. Gordon, J. O’Hara, J. Booth, and D. R. Smith, “An overview of the theory and applications of metasurfaces: The two-dimensional equivalents of metamaterials,” IEEE Antennas Propag. Mag. 54(2), 10–35 (2012).
[Crossref]

Grbic, A.

C. Pfeiffer and A. Grbic, “Bianisotropic metasurfaces for optimal polarization control: Analysis and synthesis,” Phys. Rev. A 2(4), 044011 (2014).
[Crossref]

Grigoriev, V.

V. Grigoriev, G. Demésy, J. Wenger, and N. Bonod, “Singular analysis to homogenize planar metamaterials as nonlocal effective media,” Phys. Rev. B 89(24), 245102 (2014).
[Crossref]

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999).
[Crossref]

Holloway, C. L.

C. L. Holloway, E. F. Kuester, J. A. Gordon, J. O’Hara, J. Booth, and D. R. Smith, “An overview of the theory and applications of metasurfaces: The two-dimensional equivalents of metamaterials,” IEEE Antennas Propag. Mag. 54(2), 10–35 (2012).
[Crossref]

C. L. Holloway, M. A. Mohamed, E. F. Kuester, and A. Dienstfrey, “Reflection and transmission properties of a metafilm: With an application to a controllable surface composed of resonant particles,” IEEE Trans. Electromagn. Compat. 47(4), 853–865 (2005).
[Crossref]

E. F. Kuester, M. A. Mohamed, M. Piket-May, and C. L. Holloway, “Averaged transition conditions for electromagnetic fields at a metafilm,” IEEE Trans. Antenn. Propag. 51(10), 2641–2651 (2003).
[Crossref]

Huang, K. C.

K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, “Negative effective permeability in polaritonic photonic crystals,” Appl. Phys. Lett. 85(4), 543–545 (2004).
[Crossref]

Joannopoulos, J. D.

K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, “Negative effective permeability in polaritonic photonic crystals,” Appl. Phys. Lett. 85(4), 543–545 (2004).
[Crossref]

Justice, B. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

Kanté, B.

B. Kanté, D. Germain, and A. de Lustrac, “Experimental demonstration of a nonmagnetic cloak at microwave frequencies,” Phys. Rev. B 80(20), 201104 (2009).
[Crossref]

B. Kanté, J.-M. Lourtioz, and A. de Lustrac, “Infrared metafilms on a dielectric substrate,” Phys. Rev. B 80(20), 205120 (2009).
[Crossref]

Kasica, R.

J. D. Caldwell, O. J. Glembocki, Y. Francescato, N. Sharac, V. Giannini, F. J. Bezares, J. P. Long, J. C. Owrutsky, I. Vurgaftman, J. G. Tischler, V. D. Wheeler, N. D. Bassim, L. M. Shirey, R. Kasica, and S. A. Maier, “Low-loss, extreme subdiffraction photon confinement via silicon carbide localized surface phonon polariton resonators,” Nano Lett. 13(8), 3690–3697 (2013).
[Crossref] [PubMed]

Kats, M. A.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: Generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Koschny, T.

P. Tassin, T. Koschny, and C. M. Soukoulis, “Effective material parameter retrieval for thin sheets: Theory and application to graphene, thin silver films, and single-layer metamaterials,” Physica B 407(20), 4062–4065 (2012).
[Crossref]

T. Koschny, P. Markoš, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(6), 065602 (2003).
[Crossref] [PubMed]

Kuester, E. F.

C. L. Holloway, E. F. Kuester, J. A. Gordon, J. O’Hara, J. Booth, and D. R. Smith, “An overview of the theory and applications of metasurfaces: The two-dimensional equivalents of metamaterials,” IEEE Antennas Propag. Mag. 54(2), 10–35 (2012).
[Crossref]

C. L. Holloway, M. A. Mohamed, E. F. Kuester, and A. Dienstfrey, “Reflection and transmission properties of a metafilm: With an application to a controllable surface composed of resonant particles,” IEEE Trans. Electromagn. Compat. 47(4), 853–865 (2005).
[Crossref]

E. F. Kuester, M. A. Mohamed, M. Piket-May, and C. L. Holloway, “Averaged transition conditions for electromagnetic fields at a metafilm,” IEEE Trans. Antenn. Propag. 51(10), 2641–2651 (2003).
[Crossref]

Larsson, C.

D. Sjöberg and C. Larsson, “Cramér-Rao bounds for determination of permittivity and permeability in slabs,” IEEE Trans. Microw. Theory Tech. 59(11), 2970–2977 (2011).
[Crossref]

Lepetit, T.

T. Lepetit, E. Akmansoy, and J.-P. Ganne, “Experimental measurement of negative index in an all-dielectric metamaterial,” Appl. Phys. Lett. 95(12), 121101 (2009).
[Crossref]

Li, J.

T. Feng, F. Liu, W. Y. Tam, and J. Li, “Effective parameters retrieval for complex metamaterials with low symmetries,” EPL 102(1), 18003 (2013).
[Crossref]

Liu, F.

T. Feng, F. Liu, W. Y. Tam, and J. Li, “Effective parameters retrieval for complex metamaterials with low symmetries,” EPL 102(1), 18003 (2013).
[Crossref]

Long, J. P.

J. D. Caldwell, O. J. Glembocki, Y. Francescato, N. Sharac, V. Giannini, F. J. Bezares, J. P. Long, J. C. Owrutsky, I. Vurgaftman, J. G. Tischler, V. D. Wheeler, N. D. Bassim, L. M. Shirey, R. Kasica, and S. A. Maier, “Low-loss, extreme subdiffraction photon confinement via silicon carbide localized surface phonon polariton resonators,” Nano Lett. 13(8), 3690–3697 (2013).
[Crossref] [PubMed]

Lourtioz, J.-M.

B. Kanté, J.-M. Lourtioz, and A. de Lustrac, “Infrared metafilms on a dielectric substrate,” Phys. Rev. B 80(20), 205120 (2009).
[Crossref]

Maier, S. A.

J. D. Caldwell, O. J. Glembocki, Y. Francescato, N. Sharac, V. Giannini, F. J. Bezares, J. P. Long, J. C. Owrutsky, I. Vurgaftman, J. G. Tischler, V. D. Wheeler, N. D. Bassim, L. M. Shirey, R. Kasica, and S. A. Maier, “Low-loss, extreme subdiffraction photon confinement via silicon carbide localized surface phonon polariton resonators,” Nano Lett. 13(8), 3690–3697 (2013).
[Crossref] [PubMed]

Markoš, P.

T. Koschny, P. Markoš, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(6), 065602 (2003).
[Crossref] [PubMed]

D. R. Smith, S. Schultz, P. Markoš, 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]

Martin, O. J. F.

Mock, J. J.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

Mohamed, M. A.

C. L. Holloway, M. A. Mohamed, E. F. Kuester, and A. Dienstfrey, “Reflection and transmission properties of a metafilm: With an application to a controllable surface composed of resonant particles,” IEEE Trans. Electromagn. Compat. 47(4), 853–865 (2005).
[Crossref]

E. F. Kuester, M. A. Mohamed, M. Piket-May, and C. L. Holloway, “Averaged transition conditions for electromagnetic fields at a metafilm,” IEEE Trans. Antenn. Propag. 51(10), 2641–2651 (2003).
[Crossref]

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. 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] [PubMed]

Nicolson, A. M.

A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time-domain techniques,” IEEE Trans. Instrum. Meas. 19(4), 377–382 (1970).
[Crossref]

O’Hara, J.

C. L. Holloway, E. F. Kuester, J. A. Gordon, J. O’Hara, J. Booth, and D. R. Smith, “An overview of the theory and applications of metasurfaces: The two-dimensional equivalents of metamaterials,” IEEE Antennas Propag. Mag. 54(2), 10–35 (2012).
[Crossref]

Owrutsky, J. C.

J. D. Caldwell, O. J. Glembocki, Y. Francescato, N. Sharac, V. Giannini, F. J. Bezares, J. P. Long, J. C. Owrutsky, I. Vurgaftman, J. G. Tischler, V. D. Wheeler, N. D. Bassim, L. M. Shirey, R. Kasica, and S. A. Maier, “Low-loss, extreme subdiffraction photon confinement via silicon carbide localized surface phonon polariton resonators,” Nano Lett. 13(8), 3690–3697 (2013).
[Crossref] [PubMed]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. 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] [PubMed]

Pendry, J. B.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[Crossref] [PubMed]

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999).
[Crossref]

Pfeiffer, C.

C. Pfeiffer and A. Grbic, “Bianisotropic metasurfaces for optimal polarization control: Analysis and synthesis,” Phys. Rev. A 2(4), 044011 (2014).
[Crossref]

Piket-May, M.

E. F. Kuester, M. A. Mohamed, M. Piket-May, and C. L. Holloway, “Averaged transition conditions for electromagnetic fields at a metafilm,” IEEE Trans. Antenn. Propag. 51(10), 2641–2651 (2003).
[Crossref]

Povinelli, M. L.

K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, “Negative effective permeability in polaritonic photonic crystals,” Appl. Phys. Lett. 85(4), 543–545 (2004).
[Crossref]

Qiu, C.-W.

P.-H. Tichit, S. N. Burokur, C.-W. Qiu, and A. de Lustrac, “Experimental verification of isotropic radiation from a coherent dipole source via electric-field-driven LC resonator metamaterials,” Phys. Rev. Lett. 111(13), 133901 (2013).
[Crossref] [PubMed]

Ra’di, Y.

M. Albooyeh, Y. Ra’di, M. Q. Adil, and C. R. Simovski, “Revised transmission line model for electromagnetic characterization of metasurfaces,” Phys. Rev. B 88(8), 085435 (2013).
[Crossref]

Ramakrishna, S. A.

Reyer, R. G.

J. Baker-Jarvis, R. G. Reyer, and P. D. Domich, “A nonlinear least-squares solution with causality constraints applied to transmission line permittivity and permeability determination,” IEEE Trans. Instrum. Meas. 41(5), 646–652 (1992).
[Crossref]

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999).
[Crossref]

Ross, G. F.

A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time-domain techniques,” IEEE Trans. Instrum. Meas. 19(4), 377–382 (1970).
[Crossref]

Schultz, S.

D. R. Smith, S. Schultz, P. Markoš, 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]

D. R. Smith, W. J. Padilla, D. C. 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] [PubMed]

Schurig, D.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

Sharac, N.

J. D. Caldwell, O. J. Glembocki, Y. Francescato, N. Sharac, V. Giannini, F. J. Bezares, J. P. Long, J. C. Owrutsky, I. Vurgaftman, J. G. Tischler, V. D. Wheeler, N. D. Bassim, L. M. Shirey, R. Kasica, and S. A. Maier, “Low-loss, extreme subdiffraction photon confinement via silicon carbide localized surface phonon polariton resonators,” Nano Lett. 13(8), 3690–3697 (2013).
[Crossref] [PubMed]

Sheng, P.

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[Crossref] [PubMed]

Shirey, L. M.

J. D. Caldwell, O. J. Glembocki, Y. Francescato, N. Sharac, V. Giannini, F. J. Bezares, J. P. Long, J. C. Owrutsky, I. Vurgaftman, J. G. Tischler, V. D. Wheeler, N. D. Bassim, L. M. Shirey, R. Kasica, and S. A. Maier, “Low-loss, extreme subdiffraction photon confinement via silicon carbide localized surface phonon polariton resonators,” Nano Lett. 13(8), 3690–3697 (2013).
[Crossref] [PubMed]

Simovski, C. R.

M. Albooyeh, Y. Ra’di, M. Q. Adil, and C. R. Simovski, “Revised transmission line model for electromagnetic characterization of metasurfaces,” Phys. Rev. B 88(8), 085435 (2013).
[Crossref]

C. R. Simovski, “On electromagnetic characterization and homogenization of nanostructured metamaterials,” J. Opt. 13(1), 1–22 (2011).
[Crossref]

Sjöberg, D.

D. Sjöberg and C. Larsson, “Cramér-Rao bounds for determination of permittivity and permeability in slabs,” IEEE Trans. Microw. Theory Tech. 59(11), 2970–2977 (2011).
[Crossref]

Smith, D. R.

C. L. Holloway, E. F. Kuester, J. A. Gordon, J. O’Hara, J. Booth, and D. R. Smith, “An overview of the theory and applications of metasurfaces: The two-dimensional equivalents of metamaterials,” IEEE Antennas Propag. Mag. 54(2), 10–35 (2012).
[Crossref]

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

T. Koschny, P. Markoš, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(6), 065602 (2003).
[Crossref] [PubMed]

D. R. Smith, S. Schultz, P. Markoš, 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]

D. R. Smith, W. J. Padilla, D. C. 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] [PubMed]

Soukoulis, C. M.

P. Tassin, T. Koschny, and C. M. Soukoulis, “Effective material parameter retrieval for thin sheets: Theory and application to graphene, thin silver films, and single-layer metamaterials,” Physica B 407(20), 4062–4065 (2012).
[Crossref]

T. Koschny, P. Markoš, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(6), 065602 (2003).
[Crossref] [PubMed]

D. R. Smith, S. Schultz, P. Markoš, 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]

Starr, A. F.

D. Schurig, J. J. Mock, B. J. Justice, S. A. Cummer, J. B. Pendry, A. F. Starr, and D. R. Smith, “Metamaterial electromagnetic cloak at microwave frequencies,” Science 314(5801), 977–980 (2006).
[Crossref] [PubMed]

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999).
[Crossref]

Tam, W. Y.

T. Feng, F. Liu, W. Y. Tam, and J. Li, “Effective parameters retrieval for complex metamaterials with low symmetries,” EPL 102(1), 18003 (2013).
[Crossref]

Tassin, P.

P. Tassin, T. Koschny, and C. M. Soukoulis, “Effective material parameter retrieval for thin sheets: Theory and application to graphene, thin silver films, and single-layer metamaterials,” Physica B 407(20), 4062–4065 (2012).
[Crossref]

Tetienne, J.-P.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: Generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Tichit, P.-H.

P.-H. Tichit, S. N. Burokur, C.-W. Qiu, and A. de Lustrac, “Experimental verification of isotropic radiation from a coherent dipole source via electric-field-driven LC resonator metamaterials,” Phys. Rev. Lett. 111(13), 133901 (2013).
[Crossref] [PubMed]

Tischler, J. G.

J. D. Caldwell, O. J. Glembocki, Y. Francescato, N. Sharac, V. Giannini, F. J. Bezares, J. P. Long, J. C. Owrutsky, I. Vurgaftman, J. G. Tischler, V. D. Wheeler, N. D. Bassim, L. M. Shirey, R. Kasica, and S. A. Maier, “Low-loss, extreme subdiffraction photon confinement via silicon carbide localized surface phonon polariton resonators,” Nano Lett. 13(8), 3690–3697 (2013).
[Crossref] [PubMed]

van den Bos, A.

A. van den Bos, “The multivariate complex normal distribution- a generalization,” IEEE Trans. Inf. Theory 41(2), 537–539 (1995).
[Crossref]

A. van den Bos, “A Cramér-Rao lower bound for complex parameters,” IEEE Trans. Signal Process. 42(10), 2859 (1994).
[Crossref]

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. 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] [PubMed]

Vurgaftman, I.

J. D. Caldwell, O. J. Glembocki, Y. Francescato, N. Sharac, V. Giannini, F. J. Bezares, J. P. Long, J. C. Owrutsky, I. Vurgaftman, J. G. Tischler, V. D. Wheeler, N. D. Bassim, L. M. Shirey, R. Kasica, and S. A. Maier, “Low-loss, extreme subdiffraction photon confinement via silicon carbide localized surface phonon polariton resonators,” Nano Lett. 13(8), 3690–3697 (2013).
[Crossref] [PubMed]

Weir, W. B.

W. B. Weir, “Automatic measurement of complex dielectric constant and permeability at microwave frequencies,” Proc. IEEE 62(1), 33–36 (1974).
[Crossref]

Wenger, J.

V. Grigoriev, G. Demésy, J. Wenger, and N. Bonod, “Singular analysis to homogenize planar metamaterials as nonlocal effective media,” Phys. Rev. B 89(24), 245102 (2014).
[Crossref]

Wheeler, V. D.

J. D. Caldwell, O. J. Glembocki, Y. Francescato, N. Sharac, V. Giannini, F. J. Bezares, J. P. Long, J. C. Owrutsky, I. Vurgaftman, J. G. Tischler, V. D. Wheeler, N. D. Bassim, L. M. Shirey, R. Kasica, and S. A. Maier, “Low-loss, extreme subdiffraction photon confinement via silicon carbide localized surface phonon polariton resonators,” Nano Lett. 13(8), 3690–3697 (2013).
[Crossref] [PubMed]

Yu, N.

N. Yu, P. Genevet, M. A. Kats, F. Aieta, J.-P. Tetienne, F. Capasso, and Z. Gaburro, “Light propagation with phase discontinuities: Generalized laws of reflection and refraction,” Science 334(6054), 333–337 (2011).
[Crossref] [PubMed]

Appl. Phys. Lett. (2)

K. C. Huang, M. L. Povinelli, and J. D. Joannopoulos, “Negative effective permeability in polaritonic photonic crystals,” Appl. Phys. Lett. 85(4), 543–545 (2004).
[Crossref]

T. Lepetit, E. Akmansoy, and J.-P. Ganne, “Experimental measurement of negative index in an all-dielectric metamaterial,” Appl. Phys. Lett. 95(12), 121101 (2009).
[Crossref]

EPL (1)

T. Feng, F. Liu, W. Y. Tam, and J. Li, “Effective parameters retrieval for complex metamaterials with low symmetries,” EPL 102(1), 18003 (2013).
[Crossref]

IEEE Antennas Propag. Mag. (1)

C. L. Holloway, E. F. Kuester, J. A. Gordon, J. O’Hara, J. Booth, and D. R. Smith, “An overview of the theory and applications of metasurfaces: The two-dimensional equivalents of metamaterials,” IEEE Antennas Propag. Mag. 54(2), 10–35 (2012).
[Crossref]

IEEE Trans. Antenn. Propag. (1)

E. F. Kuester, M. A. Mohamed, M. Piket-May, and C. L. Holloway, “Averaged transition conditions for electromagnetic fields at a metafilm,” IEEE Trans. Antenn. Propag. 51(10), 2641–2651 (2003).
[Crossref]

IEEE Trans. Electromagn. Compat. (1)

C. L. Holloway, M. A. Mohamed, E. F. Kuester, and A. Dienstfrey, “Reflection and transmission properties of a metafilm: With an application to a controllable surface composed of resonant particles,” IEEE Trans. Electromagn. Compat. 47(4), 853–865 (2005).
[Crossref]

IEEE Trans. Inf. Theory (1)

A. van den Bos, “The multivariate complex normal distribution- a generalization,” IEEE Trans. Inf. Theory 41(2), 537–539 (1995).
[Crossref]

IEEE Trans. Instrum. Meas. (2)

J. Baker-Jarvis, R. G. Reyer, and P. D. Domich, “A nonlinear least-squares solution with causality constraints applied to transmission line permittivity and permeability determination,” IEEE Trans. Instrum. Meas. 41(5), 646–652 (1992).
[Crossref]

A. M. Nicolson and G. F. Ross, “Measurement of the intrinsic properties of materials by time-domain techniques,” IEEE Trans. Instrum. Meas. 19(4), 377–382 (1970).
[Crossref]

IEEE Trans. Microw. Theory Tech. (2)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech. 47(11), 2075–2084 (1999).
[Crossref]

D. Sjöberg and C. Larsson, “Cramér-Rao bounds for determination of permittivity and permeability in slabs,” IEEE Trans. Microw. Theory Tech. 59(11), 2970–2977 (2011).
[Crossref]

IEEE Trans. Signal Process. (1)

A. van den Bos, “A Cramér-Rao lower bound for complex parameters,” IEEE Trans. Signal Process. 42(10), 2859 (1994).
[Crossref]

J. Opt. (1)

C. R. Simovski, “On electromagnetic characterization and homogenization of nanostructured metamaterials,” J. Opt. 13(1), 1–22 (2011).
[Crossref]

Nano Lett. (1)

J. D. Caldwell, O. J. Glembocki, Y. Francescato, N. Sharac, V. Giannini, F. J. Bezares, J. P. Long, J. C. Owrutsky, I. Vurgaftman, J. G. Tischler, V. D. Wheeler, N. D. Bassim, L. M. Shirey, R. Kasica, and S. A. Maier, “Low-loss, extreme subdiffraction photon confinement via silicon carbide localized surface phonon polariton resonators,” Nano Lett. 13(8), 3690–3697 (2013).
[Crossref] [PubMed]

Nat. Mater. (1)

H. Chen, C. T. Chan, and P. Sheng, “Transformation optics and metamaterials,” Nat. Mater. 9(5), 387–396 (2010).
[Crossref] [PubMed]

Opt. Lett. (1)

Phys. Rev. A (1)

C. Pfeiffer and A. Grbic, “Bianisotropic metasurfaces for optimal polarization control: Analysis and synthesis,” Phys. Rev. A 2(4), 044011 (2014).
[Crossref]

Phys. Rev. B (5)

M. Albooyeh, Y. Ra’di, M. Q. Adil, and C. R. Simovski, “Revised transmission line model for electromagnetic characterization of metasurfaces,” Phys. Rev. B 88(8), 085435 (2013).
[Crossref]

V. Grigoriev, G. Demésy, J. Wenger, and N. Bonod, “Singular analysis to homogenize planar metamaterials as nonlocal effective media,” Phys. Rev. B 89(24), 245102 (2014).
[Crossref]

D. R. Smith, S. Schultz, P. Markoš, 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]

B. Kanté, D. Germain, and A. de Lustrac, “Experimental demonstration of a nonmagnetic cloak at microwave frequencies,” Phys. Rev. B 80(20), 201104 (2009).
[Crossref]

B. Kanté, J.-M. Lourtioz, and A. de Lustrac, “Infrared metafilms on a dielectric substrate,” Phys. Rev. B 80(20), 205120 (2009).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

T. Koschny, P. Markoš, D. R. Smith, and C. M. Soukoulis, “Resonant and antiresonant frequency dependence of the effective parameters of metamaterials,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 68(6), 065602 (2003).
[Crossref] [PubMed]

Phys. Rev. Lett. (3)

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Figures (5)

Fig. 1
Fig. 1

Electric and magnetic surface susceptibilities along x and y (real and imaginary parts in dark blue-green and red-light blue, respectively), from 19 to 27 THz, for a 2D square array of sub-wavelength cubes made from Lithium Tantalate (LiTaO3). Inset shows the far-infrared metasurface along with the corresponding coordinate system.

Fig. 2
Fig. 2

Magnetic surface susceptibility along z for a 2D array of sub-wavelength LiTaO3 cubes, from 19 to 27 THz and three different collision frequencies (γ = 0.94, 1.94, 2.94 THz, in blue, green and red, respectively). (a) Real part (b) Imaginary part (c) Log-variance

Fig. 3
Fig. 3

Cramér-Rao lower bound for the electric and magnetic surface susceptibilities for a 2D array of sub-wavelength LiTaO3 cubes, from 19 to 27 THz. (a) Tangential components for three different collision frequencies (γ = 0.94, 1.94, 2.94 THz, in blue, green and red, respectively). (b) Normal component for four different oblique angles of incidence ( θ 2 =( 15,30,45,60° ) , in blue, purple, yellow, and brown, respectively).

Fig. 4
Fig. 4

Electric surface susceptibility along z for a 2D array of sub-wavelength LiTaO3 cubes, from 19 to 20.5 THz and four different estimators (Eq. (7), Eq. (24), Eq. (25), and Eq. (26), in dark blue, green, red, and light blue, respectively). No resonances are visible at higher frequencies. (a) Real part (b) Imaginary part (c) Log-variance

Fig. 5
Fig. 5

Cramér-Rao lower bound (in dark blue) for the normal electric surface susceptibility for a 2D array of sub-wavelength LiTaO3 cubes, from 10 to 35 THz and variance of the four estimators defined by Eq. (7), Eq. (24), Eq. (25), and Eq. (26) in green, red, light blue, and purple, respectively.

Equations (38)

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u z × H| z= 0 z= 0 + =+ Y ¯ ¯ s E t,av | z=0
u z × E| z= 0 z= 0 + = Z ¯ ¯ s H t,av | z=0
Y̿s=jωχ̿ES-1jωμχ̿MSzzuz×kuz×k
Z̿s=jωχ̿MS-1jωεχ̿ESzzuz×kuz×k
χ MS xx = 2j k 0 S 11 TE ( 0 ) S 21 TE ( 0 )+1 S 11 TE ( 0 ) S 21 TE ( 0 )1
χ ES yy = 2j k 0 S 11 TE ( 0 )+ S 21 TE ( 0 )1 S 11 TE ( 0 )+ S 21 TE ( 0 )+1
χ MS zz = χ ES yy si n 2 ( θ ) + 2j k 0 cos( θ ) si n 2 ( θ ) S 11 TE ( θ )+ S 21 TE ( θ )1 S 11 TE ( θ )+ S 21 TE ( θ )+1
ε r = ε ( 1+ ω L 2 ω T 2 ω T 2 ω 2 +jωγ )
z=s( ϑ )+n
z=S11TE0,S11TEθ,S21TE0,S21TEθ
ϑ= [ χ MS xx , χ ES yy , χ MS zz ] T
E[ n ]=0
E[ ( nE[ n ] ) ( nE[ n ] ) ]=R
E[ z ]=s( ϑ )
E[ ( zE[ z ] ) ( zE[ z ] ) ]=R
R est I Fisher 1
I Fisher =E[ ( lnf( z|ϑ ) ϑ ) ( lnf( z|ϑ ) ϑ ) ]=E[ 2 lnf( z|ϑ ) ϑ ϑ ]
f( z|ϑ )= 1 π 4 det( R ) exp[ ( zs ) R 1 ( zs ) ]
I Fisher =2Re[ ( s ϑ ) R 1 ( s ϑ ) ]
R mn = σ S ij ( θ k ) 2 δ m,n
I Fisher =2Re[ i,j k 1 σ S ij ( θ k ) 2 I ( ϑ ) ( θ k ; S ij ) ]
ϑ norm = k 0 [ χ MS xx , χ ES yy , χ MS zz ] T
I Fisher norm = 1 k 0 2 I Fisher
χ MS zz = χ ES yy sin 2 θ + 1 sin 2 θ S 11 TE ( θ ) j k 0 2cosθ [ 1 S 11 TE ( θ ) ] χ MS xx cos 2 θ S 11 TE ( θ ) ( k 0 2 ) 2 χ MS xx j k 0 2cosθ [ 1+ S 11 TE ( θ ) ]
χ MS zz = χ ES yy sin 2 θ 1 sin 2 θ [ 1 S 21 TE ( θ ) ] j k 0 2cosθ S 21 TE ( θ ) χ MS xx cos 2 θ [ 1+ S 21 TE ( θ ) ] ( k 0 2 ) 2 χ MS xx j k 0 2cosθ S 21 TE ( θ )
( A[ S 11 TE ( θ ), χ MS xx ] B[ S 21 TE ( θ ), χ MS xx ] ) χ MS zz =( C[ S 11 TE ( θ ), χ MS xx , χ ES yy ] D[ S 21 TE ( θ ), χ MS xx , χ ES yy ] )
I ( ϑ ) TE ( θ k ; S ij )=( | S ij ( θ k ) χ MS xx | 2 ( S ij ( θ k ) χ MS xx ) ( S ij ( θ k ) χ ES yy ) * ( S ij ( θ k ) χ MS xx ) ( S ij ( θ k ) χ MS zz ) * ( S ij ( θ k ) χ ES yy ) ( S ij ( θ k ) χ MS xx ) * | S ij ( θ k ) χ ES yy | 2 ( S ij ( θ k ) χ ES yy ) ( S ij ( θ k ) χ MS zz ) * ( S ij ( θ k ) χ MS zz ) ( S ij ( θ k ) χ MS xx ) * ( S ij ( θ k ) χ MS zz ) ( S ij ( θ k ) χ ES yy ) * | S ij ( θ k ) χ MS zz | 2 )
I ( ϑ ) TM ( θ k ; S ij )=( | S ij ( θ k ) χ ES xx | 2 ( S ij ( θ k ) χ ES xx ) ( S ij ( θ k ) χ MS yy ) * ( S ij ( θ k ) χ ES xx ) ( S ij ( θ k ) χ ES zz ) * ( S ij ( θ k ) χ MS yy ) ( S ij ( θ k ) χ ES xx ) * | S ij ( θ k ) χ MS yy | 2 ( S ij ( θ k ) χ MS yy ) ( S ij ( θ k ) χ ES zz ) * ( S ij ( θ k ) χ ES zz ) ( S ij ( θ k ) χ ES xx ) * ( S ij ( θ k ) χ ES zz ) ( S ij ( θ k ) χ MS yy ) * | S ij ( θ k ) χ ES zz | 2 )
χ ES zz = χ MS yy sin 2 θ + 1 sin 2 θ S 11 TM ( θ )+ j k 0 2cosθ [ 1+ S 11 TM ( θ ) ] χ ES xx cos 2 θ S 11 TM ( θ ) ( k 0 2 ) 2 χ ES xx + j k 0 2cosθ [ 1 S 11 TM ( θ ) ]
χ ES zz = χ MS yy sin 2 θ 1 sin 2 θ [ 1 S 21 TM ( θ ) ] j k 0 2cosθ S 21 TM ( θ ) χ ES xx cos 2 θ [ 1+ S 21 TM ( θ ) ] ( k 0 2 ) 2 χ ES xx j k 0 2cosθ S 21 TM ( θ )
A[ S 11 TE , χ MS xx ]= sin 2 θ[ S 11 TE ( k 0 2 ) 2 χ MS xx j k 0 2cosθ ( S 11 TE +1 ) ]
B[ S 21 TE , χ MS xx ]= sin 2 θ[ ( S 21 TE +1 ) ( k 0 2 ) 2 χ MS xx j k 0 2cosθ S 21 TE ]
C[ S 11 TE , χ MS xx , χ ES yy ]= S 11 TE [ 1 ( k 0 2 ) 2 χ MS xx χ ES yy ]+ j k 0 2cosθ [ χ ES yy ( S 11 TE +1 )+ χ MS xx cos 2 θ( S 11 TE 1 ) ]
D[ S 21 TE , χ MS xx , χ ES yy ]=( S 21 TE 1 )( S 21 TE +1 ) ( k 0 2 ) 2 χ MS xx χ ES yy + j k 0 2cosθ S 21 TE [ χ ES yy + χ MS xx cos 2 θ ]
A[ S 11 TM , χ ES xx ]= sin 2 θ[ S 11 TM ( k 0 2 ) 2 χ ES xx j k 0 2cosθ ( S 11 TM 1 ) ]
B[ S 21 TM , χ ES xx ]= sin 2 θ[ ( S 21 TM +1 ) ( k 0 2 ) 2 χ ES xx j k 0 2cosθ S 21 TM ]
C[ S 11 TM , χ ES xx , χ MS yy ]= S 11 TM [ 1 ( k 0 2 ) 2 χ ES xx χ MS yy ]+ j k 0 2cosθ [ χ MS yy ( S 11 TM 1 )+ χ ES xx cos 2 θ( S 11 TM +1 ) ]
D[ S 21 TM , χ ES xx , χ MS yy ]=( S 21 TM 1 )( S 21 TM +1 ) ( k 0 2 ) 2 χ ES xx χ MS yy + j k 0 2cosθ S 21 TM [ χ MS yy + χ ES xx cos 2 θ ]

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