Abstract

We derive approximate analytical expressions for the effective permittivity tensor of two-phase metamaterials whose geometry is close to one-dimensional (quasi-one-dimensional metamaterials). Specifically, we consider the metamaterial made of parallel slabs with width given by a linear or parabolic function. Using our approach, the design of epsilon-near-zero, ultra-low and high refractive index metallodielectric metamaterials with extended bandwidth has been demonstrated. In addition, generalizations to the three-dimensional case and some limitations of the presented technique are briefly considered.

© 2014 Optical Society of America

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    [CrossRef]
  34. A.V. Chebykin, A.A. Orlov, A.V. Vozianova, S.I. Maslovski, Yu.S. Kivshar, P.A. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B 84, 115438 (2011).
    [CrossRef]
  35. P.A. Belov, R. Marques, S.I. Maslovski, I.S. Nefedov, M. Silveirinha, C.R. Simovski, S.A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B 67, 113103 (2003).
    [CrossRef]
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    [CrossRef]
  37. N. Dubrovina, L.O. Le Cunff, N. Burokur, R. Ghasemi, A. Degiron, A. De Lustrac, A. Vial, G. Leronded, A. Lupu, “Single metafilm effective medium behavior in optical domain: Maxwell-Garnett approximation and beyond,” Appl. Phys. A 109, 901–906 (2012).
    [CrossRef]
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    [CrossRef] [PubMed]

2013 (4)

2012 (5)

L. Sun, K.W. Yu, “Strategy for designing broadband epsilon-near-zero metamaterials,” J. Opt. Soc. Am. B 29, 984–989 (2012).
[CrossRef]

L. Sun, K.W. Yu, X. Yang, “Integrated optical devices based on broadband epsilon-near-zero meta-atoms,” Opt. Lett. 37, 3096–3098 (2012).
[CrossRef] [PubMed]

N. Dubrovina, L.O. Le Cunff, N. Burokur, R. Ghasemi, A. Degiron, A. De Lustrac, A. Vial, G. Leronded, A. Lupu, “Single metafilm effective medium behavior in optical domain: Maxwell-Garnett approximation and beyond,” Appl. Phys. A 109, 901–906 (2012).
[CrossRef]

A.V. Goncharenko, V.U. Nazarov, K.R. Chen, “Development of metamaterials with desired broadband optical properties,” Appl. Phys. Lett. 101, 071907 (2012).
[CrossRef]

C.R. Simovski, P.A. Belov, A.A. Atrashchenko, Y.S. Kivshar, “Wire metamaterials: Physics and applications,” Adv. Mater. 24, 4229–4248 (2012).
[CrossRef] [PubMed]

2011 (5)

E.A. Gibson, I.R. Gabitov, A.I. Maimistov, N.M. Litchinitser, “Transition metamaterials with spatially separated zeros,” Opt. Lett. 36, 3624–3626 (2011).
[CrossRef] [PubMed]

X.X. Liu, A. Alu, “Limitations and potential of metamaterial lenses,” J. Nanophoton. 5, 053509 (2011).
[CrossRef]

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

A.A. Orlov, P.M. Voroshilov, P.A. Belov, Y.S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84, 045424 (2011).
[CrossRef]

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

2010 (5)

A.N. Lagarkov, V.N. Kisel, A.K. Sarychev, “Loss and gain in metamaterials,” J. Opt. Soc. Am. B 27, 648–659 (2010).
[CrossRef]

A.V. Goncharenko, K.R. Chen, “Strategy for designing epsilon-near-zero nanostructured metamaterials over a frequency range,” J. Nanophoton. 4, 041530 (2010).
[CrossRef]

H. Rauh, G.I. Yampolskaya, S.V. Yampolskii, “Optical transmittance of photonic structures with linearly graded dielectric constituents,” New J. Phys. 12, 073033 (2010).
[CrossRef]

L.V. Alekseyev, E.E. Narimarov, T. Tumkur, H. Li, Yu. A. Barnakov, M.A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. 97, 131107 (2010).
[CrossRef]

W.T. Perrins, R.C. McPedran, “Metamaterials and the homogenization of composite materials,” Metamaterials 4, 24–31 (2010).
[CrossRef]

2009 (3)

A.V. Kabashin, P. Evans, S. Pastkovsky, W. Hendren, G.A. Wurtz, R. Atkinson, R. Pollard, V.A. Podolskiy, A.V. Zayats, “Plasmonic nanorod metamaterials for biosensing,” Nature Mater. 8, 867–871 (2009).
[CrossRef]

S.I. Maslovski, M.G. Silveirinha, “Nonlocal permittivity from a quasistatic model for a class of wire media,” Phys. Rev. B 80, 245101 (2009).
[CrossRef]

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

2008 (3)

A. Vial, T. Laroche, “Comparison of gold and silver dispersion laws suitable for FDTD simulations,” Appl. Phys. B 93, 139–143 (2008).
[CrossRef]

A.K. Popov, S.A. Myslivets, “Transformable broad-band transparency and amplification in negative-index films,” Appl. Phys. Lett. 93, 191117 (2008).
[CrossRef]

J. Yao, Z Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A.M. Stacy, X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

2006 (3)

P. Belov, Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B 73, 113110 (2006).
[CrossRef]

Z. Jacob, L.V. Alekseev, E. Narimanov, “Optical hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14, 8247–8256 (2006).
[CrossRef] [PubMed]

J. Elser, R. Wangberg, V.A. Podolskiy, E.E. Narimanov, “Nanowire metamaterials with extreme optical anisotropy,” Appl. Phys. Lett. 89, 261102 (2006).
[CrossRef]

2004 (1)

2003 (1)

P.A. Belov, R. Marques, S.I. Maslovski, I.S. Nefedov, M. Silveirinha, C.R. Simovski, S.A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B 67, 113103 (2003).
[CrossRef]

2002 (1)

A.A. Krokhin, P. Halevi, J. Arriaga, “Long-wavelength limit (homogenization) for two-dimensional photonic crystals,” Phys. Rev. B 65, 115208 (2002).
[CrossRef]

2000 (1)

A.V. Goncharenko, “Limiting geometries and dielectric tensor of superlattices,” Tech. Phys. Lett. 26(7), 594–596 (2000).
[CrossRef]

1998 (1)

M. Scalora, M.J. Bloemer, A.S. Pethel, J.P. Dowling, C.M. Bowden, A.S. Manka, “Transparent, metallodielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377–2383 (1998).
[CrossRef]

1964 (1)

J.B. Keller, “A theorem on the conductivity of a composite medium,” J. Math. Phys. 5, 548–549 (1964).
[CrossRef]

1929 (1)

A. Reuss, “Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle,” Z. Angew. Math. Mech. 9, 49–58 (1929).
[CrossRef]

Alekseev, L.V.

Alekseyev, L.V.

L.V. Alekseyev, E.E. Narimarov, T. Tumkur, H. Li, Yu. A. Barnakov, M.A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. 97, 131107 (2010).
[CrossRef]

Alu, A.

X.X. Liu, A. Alu, “Limitations and potential of metamaterial lenses,” J. Nanophoton. 5, 053509 (2011).
[CrossRef]

Ankonina, G.

Arriaga, J.

A.A. Krokhin, P. Halevi, J. Arriaga, “Long-wavelength limit (homogenization) for two-dimensional photonic crystals,” Phys. Rev. B 65, 115208 (2002).
[CrossRef]

Atkinson, R.

A.V. Kabashin, P. Evans, S. Pastkovsky, W. Hendren, G.A. Wurtz, R. Atkinson, R. Pollard, V.A. Podolskiy, A.V. Zayats, “Plasmonic nanorod metamaterials for biosensing,” Nature Mater. 8, 867–871 (2009).
[CrossRef]

Atrashchenko, A.

Atrashchenko, A.A.

C.R. Simovski, P.A. Belov, A.A. Atrashchenko, Y.S. Kivshar, “Wire metamaterials: Physics and applications,” Adv. Mater. 24, 4229–4248 (2012).
[CrossRef] [PubMed]

Barnakov, Yu. A.

L.V. Alekseyev, E.E. Narimarov, T. Tumkur, H. Li, Yu. A. Barnakov, M.A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. 97, 131107 (2010).
[CrossRef]

Barnakov, Yu.A.

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

Bartal, G.

J. Yao, Z Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A.M. Stacy, X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

Belov, P.

P. Belov, Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B 73, 113110 (2006).
[CrossRef]

Belov, P.A.

P. Ginzburg, F. J. Rodriguez Fortuno, G.A. Wurtz, W. Dickson, A. Murphy, F. Morgan, R.J. Pollard, I. Iorsh, A. Atrashchenko, P.A. Belov, Y.S. Kivshar, A. Nevet, G. Ankonina, M. Orenstein, A.V. Zayats, “Manipulating polarization of light with ultrathin epsilon-near-zero metamaterials,” Opt. Express 21, 14907–14917 (2013).
[CrossRef] [PubMed]

C.R. Simovski, P.A. Belov, A.A. Atrashchenko, Y.S. Kivshar, “Wire metamaterials: Physics and applications,” Adv. Mater. 24, 4229–4248 (2012).
[CrossRef] [PubMed]

A.A. Orlov, P.M. Voroshilov, P.A. Belov, Y.S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84, 045424 (2011).
[CrossRef]

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

P.A. Belov, R. Marques, S.I. Maslovski, I.S. Nefedov, M. Silveirinha, C.R. Simovski, S.A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B 67, 113103 (2003).
[CrossRef]

Bloemer, M.J.

M. Scalora, M.J. Bloemer, A.S. Pethel, J.P. Dowling, C.M. Bowden, A.S. Manka, “Transparent, metallodielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377–2383 (1998).
[CrossRef]

Bowden, C.M.

M. Scalora, M.J. Bloemer, A.S. Pethel, J.P. Dowling, C.M. Bowden, A.S. Manka, “Transparent, metallodielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377–2383 (1998).
[CrossRef]

Burokur, N.

N. Dubrovina, L.O. Le Cunff, N. Burokur, R. Ghasemi, A. Degiron, A. De Lustrac, A. Vial, G. Leronded, A. Lupu, “Single metafilm effective medium behavior in optical domain: Maxwell-Garnett approximation and beyond,” Appl. Phys. A 109, 901–906 (2012).
[CrossRef]

Chebykin, A.V.

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

Chen, K.R.

A.V. Goncharenko, V.U. Nazarov, K.R. Chen, “Nanostructured metamaterials with broadband optical properties,” Opt. Mater. Express 3, 143–156 (2013).
[CrossRef]

A.V. Goncharenko, V.U. Nazarov, K.R. Chen, “Development of metamaterials with desired broadband optical properties,” Appl. Phys. Lett. 101, 071907 (2012).
[CrossRef]

A.V. Goncharenko, K.R. Chen, “Strategy for designing epsilon-near-zero nanostructured metamaterials over a frequency range,” J. Nanophoton. 4, 041530 (2010).
[CrossRef]

Christensen, J.

C. David, N.A. Mortensen, J. Christensen, “Perfect imaging, epsilon-near zero phenomena and waveguiding in the scope of nonlocal effects,” Sci. Rept. 3, 02526 (2013).

David, C.

C. David, N.A. Mortensen, J. Christensen, “Perfect imaging, epsilon-near zero phenomena and waveguiding in the scope of nonlocal effects,” Sci. Rept. 3, 02526 (2013).

De Lustrac, A.

N. Dubrovina, L.O. Le Cunff, N. Burokur, R. Ghasemi, A. Degiron, A. De Lustrac, A. Vial, G. Leronded, A. Lupu, “Single metafilm effective medium behavior in optical domain: Maxwell-Garnett approximation and beyond,” Appl. Phys. A 109, 901–906 (2012).
[CrossRef]

Degiron, A.

N. Dubrovina, L.O. Le Cunff, N. Burokur, R. Ghasemi, A. Degiron, A. De Lustrac, A. Vial, G. Leronded, A. Lupu, “Single metafilm effective medium behavior in optical domain: Maxwell-Garnett approximation and beyond,” Appl. Phys. A 109, 901–906 (2012).
[CrossRef]

Dickson, W.

Dowling, J.P.

M. Scalora, M.J. Bloemer, A.S. Pethel, J.P. Dowling, C.M. Bowden, A.S. Manka, “Transparent, metallodielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377–2383 (1998).
[CrossRef]

Dubrovina, N.

N. Dubrovina, L.O. Le Cunff, N. Burokur, R. Ghasemi, A. Degiron, A. De Lustrac, A. Vial, G. Leronded, A. Lupu, “Single metafilm effective medium behavior in optical domain: Maxwell-Garnett approximation and beyond,” Appl. Phys. A 109, 901–906 (2012).
[CrossRef]

Elser, J.

J. Elser, R. Wangberg, V.A. Podolskiy, E.E. Narimanov, “Nanowire metamaterials with extreme optical anisotropy,” Appl. Phys. Lett. 89, 261102 (2006).
[CrossRef]

Evans, P.

A.V. Kabashin, P. Evans, S. Pastkovsky, W. Hendren, G.A. Wurtz, R. Atkinson, R. Pollard, V.A. Podolskiy, A.V. Zayats, “Plasmonic nanorod metamaterials for biosensing,” Nature Mater. 8, 867–871 (2009).
[CrossRef]

Gabitov, I.R.

Ghasemi, R.

N. Dubrovina, L.O. Le Cunff, N. Burokur, R. Ghasemi, A. Degiron, A. De Lustrac, A. Vial, G. Leronded, A. Lupu, “Single metafilm effective medium behavior in optical domain: Maxwell-Garnett approximation and beyond,” Appl. Phys. A 109, 901–906 (2012).
[CrossRef]

Gibson, E.A.

Ginzburg, P.

Goncharenko, A.V.

A.V. Goncharenko, V.U. Nazarov, K.R. Chen, “Nanostructured metamaterials with broadband optical properties,” Opt. Mater. Express 3, 143–156 (2013).
[CrossRef]

A.V. Goncharenko, V.U. Nazarov, K.R. Chen, “Development of metamaterials with desired broadband optical properties,” Appl. Phys. Lett. 101, 071907 (2012).
[CrossRef]

A.V. Goncharenko, K.R. Chen, “Strategy for designing epsilon-near-zero nanostructured metamaterials over a frequency range,” J. Nanophoton. 4, 041530 (2010).
[CrossRef]

A.V. Goncharenko, “Limiting geometries and dielectric tensor of superlattices,” Tech. Phys. Lett. 26(7), 594–596 (2000).
[CrossRef]

Halevi, P.

A.A. Krokhin, P. Halevi, J. Arriaga, “Long-wavelength limit (homogenization) for two-dimensional photonic crystals,” Phys. Rev. B 65, 115208 (2002).
[CrossRef]

Hao, Y.

P. Belov, Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B 73, 113110 (2006).
[CrossRef]

Hendren, W.

A.V. Kabashin, P. Evans, S. Pastkovsky, W. Hendren, G.A. Wurtz, R. Atkinson, R. Pollard, V.A. Podolskiy, A.V. Zayats, “Plasmonic nanorod metamaterials for biosensing,” Nature Mater. 8, 867–871 (2009).
[CrossRef]

Iorsh, I.

Jacob, Z.

Kabashin, A.V.

A.V. Kabashin, P. Evans, S. Pastkovsky, W. Hendren, G.A. Wurtz, R. Atkinson, R. Pollard, V.A. Podolskiy, A.V. Zayats, “Plasmonic nanorod metamaterials for biosensing,” Nature Mater. 8, 867–871 (2009).
[CrossRef]

Keller, J.B.

J.B. Keller, “A theorem on the conductivity of a composite medium,” J. Math. Phys. 5, 548–549 (1964).
[CrossRef]

Kisel, V.N.

Kivshar, Y.S.

P. Ginzburg, F. J. Rodriguez Fortuno, G.A. Wurtz, W. Dickson, A. Murphy, F. Morgan, R.J. Pollard, I. Iorsh, A. Atrashchenko, P.A. Belov, Y.S. Kivshar, A. Nevet, G. Ankonina, M. Orenstein, A.V. Zayats, “Manipulating polarization of light with ultrathin epsilon-near-zero metamaterials,” Opt. Express 21, 14907–14917 (2013).
[CrossRef] [PubMed]

C.R. Simovski, P.A. Belov, A.A. Atrashchenko, Y.S. Kivshar, “Wire metamaterials: Physics and applications,” Adv. Mater. 24, 4229–4248 (2012).
[CrossRef] [PubMed]

A.A. Orlov, P.M. Voroshilov, P.A. Belov, Y.S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84, 045424 (2011).
[CrossRef]

Kivshar, Yu.S.

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

Kolinko, P.

Krokhin, A.A.

A.A. Krokhin, P. Halevi, J. Arriaga, “Long-wavelength limit (homogenization) for two-dimensional photonic crystals,” Phys. Rev. B 65, 115208 (2002).
[CrossRef]

Lagarkov, A.N.

Laroche, T.

A. Vial, T. Laroche, “Comparison of gold and silver dispersion laws suitable for FDTD simulations,” Appl. Phys. B 93, 139–143 (2008).
[CrossRef]

Le Cunff, L.O.

N. Dubrovina, L.O. Le Cunff, N. Burokur, R. Ghasemi, A. Degiron, A. De Lustrac, A. Vial, G. Leronded, A. Lupu, “Single metafilm effective medium behavior in optical domain: Maxwell-Garnett approximation and beyond,” Appl. Phys. A 109, 901–906 (2012).
[CrossRef]

Leronded, G.

N. Dubrovina, L.O. Le Cunff, N. Burokur, R. Ghasemi, A. Degiron, A. De Lustrac, A. Vial, G. Leronded, A. Lupu, “Single metafilm effective medium behavior in optical domain: Maxwell-Garnett approximation and beyond,” Appl. Phys. A 109, 901–906 (2012).
[CrossRef]

Li, H.

L.V. Alekseyev, E.E. Narimarov, T. Tumkur, H. Li, Yu. A. Barnakov, M.A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. 97, 131107 (2010).
[CrossRef]

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

Litchinitser, N.M.

Liu, X.X.

X.X. Liu, A. Alu, “Limitations and potential of metamaterial lenses,” J. Nanophoton. 5, 053509 (2011).
[CrossRef]

Liu, Y.

J. Yao, Z Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A.M. Stacy, X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

Liu, Z

J. Yao, Z Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A.M. Stacy, X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

Lupu, A.

N. Dubrovina, L.O. Le Cunff, N. Burokur, R. Ghasemi, A. Degiron, A. De Lustrac, A. Vial, G. Leronded, A. Lupu, “Single metafilm effective medium behavior in optical domain: Maxwell-Garnett approximation and beyond,” Appl. Phys. A 109, 901–906 (2012).
[CrossRef]

Maimistov, A.I.

Manka, A.S.

M. Scalora, M.J. Bloemer, A.S. Pethel, J.P. Dowling, C.M. Bowden, A.S. Manka, “Transparent, metallodielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377–2383 (1998).
[CrossRef]

Marques, R.

P.A. Belov, R. Marques, S.I. Maslovski, I.S. Nefedov, M. Silveirinha, C.R. Simovski, S.A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B 67, 113103 (2003).
[CrossRef]

Maslovski, S.I.

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

S.I. Maslovski, M.G. Silveirinha, “Nonlocal permittivity from a quasistatic model for a class of wire media,” Phys. Rev. B 80, 245101 (2009).
[CrossRef]

P.A. Belov, R. Marques, S.I. Maslovski, I.S. Nefedov, M. Silveirinha, C.R. Simovski, S.A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B 67, 113103 (2003).
[CrossRef]

McPedran, R.C.

W.T. Perrins, R.C. McPedran, “Metamaterials and the homogenization of composite materials,” Metamaterials 4, 24–31 (2010).
[CrossRef]

Milton, G. W.

G. W. Milton, The Theory of Composites (Cambridge University, 2002).
[CrossRef]

Morgan, F.

Mortensen, N.A.

C. David, N.A. Mortensen, J. Christensen, “Perfect imaging, epsilon-near zero phenomena and waveguiding in the scope of nonlocal effects,” Sci. Rept. 3, 02526 (2013).

W. Yan, N.A. Mortensen, M. Wubs, “Hypebolic metamaterial lens with hydrodynamic nonlocal response,” Opt. Express 21, 15026–15036 (2013).
[CrossRef] [PubMed]

Murphy, A.

Myslivets, S.A.

A.K. Popov, S.A. Myslivets, “Transformable broad-band transparency and amplification in negative-index films,” Appl. Phys. Lett. 93, 191117 (2008).
[CrossRef]

Narimanov, E.

Narimanov, E.E.

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

J. Elser, R. Wangberg, V.A. Podolskiy, E.E. Narimanov, “Nanowire metamaterials with extreme optical anisotropy,” Appl. Phys. Lett. 89, 261102 (2006).
[CrossRef]

Narimarov, E.E.

L.V. Alekseyev, E.E. Narimarov, T. Tumkur, H. Li, Yu. A. Barnakov, M.A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. 97, 131107 (2010).
[CrossRef]

Nazarov, V.U.

A.V. Goncharenko, V.U. Nazarov, K.R. Chen, “Nanostructured metamaterials with broadband optical properties,” Opt. Mater. Express 3, 143–156 (2013).
[CrossRef]

A.V. Goncharenko, V.U. Nazarov, K.R. Chen, “Development of metamaterials with desired broadband optical properties,” Appl. Phys. Lett. 101, 071907 (2012).
[CrossRef]

Nefedov, I.S.

P.A. Belov, R. Marques, S.I. Maslovski, I.S. Nefedov, M. Silveirinha, C.R. Simovski, S.A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B 67, 113103 (2003).
[CrossRef]

Nevet, A.

Noginov, M.A.

L.V. Alekseyev, E.E. Narimarov, T. Tumkur, H. Li, Yu. A. Barnakov, M.A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. 97, 131107 (2010).
[CrossRef]

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

Orenstein, M.

Orlov, A.A.

A.A. Orlov, P.M. Voroshilov, P.A. Belov, Y.S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84, 045424 (2011).
[CrossRef]

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

Pastkovsky, S.

A.V. Kabashin, P. Evans, S. Pastkovsky, W. Hendren, G.A. Wurtz, R. Atkinson, R. Pollard, V.A. Podolskiy, A.V. Zayats, “Plasmonic nanorod metamaterials for biosensing,” Nature Mater. 8, 867–871 (2009).
[CrossRef]

Perrins, W.T.

W.T. Perrins, R.C. McPedran, “Metamaterials and the homogenization of composite materials,” Metamaterials 4, 24–31 (2010).
[CrossRef]

Pethel, A.S.

M. Scalora, M.J. Bloemer, A.S. Pethel, J.P. Dowling, C.M. Bowden, A.S. Manka, “Transparent, metallodielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377–2383 (1998).
[CrossRef]

Podolskiy, V.A.

A.V. Kabashin, P. Evans, S. Pastkovsky, W. Hendren, G.A. Wurtz, R. Atkinson, R. Pollard, V.A. Podolskiy, A.V. Zayats, “Plasmonic nanorod metamaterials for biosensing,” Nature Mater. 8, 867–871 (2009).
[CrossRef]

J. Elser, R. Wangberg, V.A. Podolskiy, E.E. Narimanov, “Nanowire metamaterials with extreme optical anisotropy,” Appl. Phys. Lett. 89, 261102 (2006).
[CrossRef]

Pollard, R.

A.V. Kabashin, P. Evans, S. Pastkovsky, W. Hendren, G.A. Wurtz, R. Atkinson, R. Pollard, V.A. Podolskiy, A.V. Zayats, “Plasmonic nanorod metamaterials for biosensing,” Nature Mater. 8, 867–871 (2009).
[CrossRef]

Pollard, R.J.

Popov, A.K.

A.K. Popov, S.A. Myslivets, “Transformable broad-band transparency and amplification in negative-index films,” Appl. Phys. Lett. 93, 191117 (2008).
[CrossRef]

Rauh, H.

H. Rauh, G.I. Yampolskaya, S.V. Yampolskii, “Optical transmittance of photonic structures with linearly graded dielectric constituents,” New J. Phys. 12, 073033 (2010).
[CrossRef]

Reuss, A.

A. Reuss, “Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle,” Z. Angew. Math. Mech. 9, 49–58 (1929).
[CrossRef]

Rodriguez Fortuno, F. J.

Sahimi, M.

M. Sahimi, Heterogeneous Materials. I. Linear Transport and Optical Properties (Springer, 2003).

Sarychev, A.K.

Scalora, M.

M. Scalora, M.J. Bloemer, A.S. Pethel, J.P. Dowling, C.M. Bowden, A.S. Manka, “Transparent, metallodielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377–2383 (1998).
[CrossRef]

Schurig, D.

Silveirinha, M.

P.A. Belov, R. Marques, S.I. Maslovski, I.S. Nefedov, M. Silveirinha, C.R. Simovski, S.A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B 67, 113103 (2003).
[CrossRef]

Silveirinha, M.G.

S.I. Maslovski, M.G. Silveirinha, “Nonlocal permittivity from a quasistatic model for a class of wire media,” Phys. Rev. B 80, 245101 (2009).
[CrossRef]

Simovski, C.R.

C.R. Simovski, P.A. Belov, A.A. Atrashchenko, Y.S. Kivshar, “Wire metamaterials: Physics and applications,” Adv. Mater. 24, 4229–4248 (2012).
[CrossRef] [PubMed]

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

P.A. Belov, R. Marques, S.I. Maslovski, I.S. Nefedov, M. Silveirinha, C.R. Simovski, S.A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B 67, 113103 (2003).
[CrossRef]

Smith, D.R.

Stacy, A.M.

J. Yao, Z Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A.M. Stacy, X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

Sun, C.

J. Yao, Z Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A.M. Stacy, X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

Sun, L.

Tretyakov, S.A.

P.A. Belov, R. Marques, S.I. Maslovski, I.S. Nefedov, M. Silveirinha, C.R. Simovski, S.A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B 67, 113103 (2003).
[CrossRef]

Tumkur, T.

L.V. Alekseyev, E.E. Narimarov, T. Tumkur, H. Li, Yu. A. Barnakov, M.A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. 97, 131107 (2010).
[CrossRef]

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

Vial, A.

N. Dubrovina, L.O. Le Cunff, N. Burokur, R. Ghasemi, A. Degiron, A. De Lustrac, A. Vial, G. Leronded, A. Lupu, “Single metafilm effective medium behavior in optical domain: Maxwell-Garnett approximation and beyond,” Appl. Phys. A 109, 901–906 (2012).
[CrossRef]

A. Vial, T. Laroche, “Comparison of gold and silver dispersion laws suitable for FDTD simulations,” Appl. Phys. B 93, 139–143 (2008).
[CrossRef]

Voight, W.

W. Voight, Lehrbuch der Kristallphysik (Teubner-Verlag, 1928).

Voroshilov, P.M.

A.A. Orlov, P.M. Voroshilov, P.A. Belov, Y.S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84, 045424 (2011).
[CrossRef]

Vozianova, A.V.

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

Wang, Y.

J. Yao, Z Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A.M. Stacy, X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

Wangberg, R.

J. Elser, R. Wangberg, V.A. Podolskiy, E.E. Narimanov, “Nanowire metamaterials with extreme optical anisotropy,” Appl. Phys. Lett. 89, 261102 (2006).
[CrossRef]

Wubs, M.

Wurtz, G.A.

Yampolskaya, G.I.

H. Rauh, G.I. Yampolskaya, S.V. Yampolskii, “Optical transmittance of photonic structures with linearly graded dielectric constituents,” New J. Phys. 12, 073033 (2010).
[CrossRef]

Yampolskii, S.V.

H. Rauh, G.I. Yampolskaya, S.V. Yampolskii, “Optical transmittance of photonic structures with linearly graded dielectric constituents,” New J. Phys. 12, 073033 (2010).
[CrossRef]

Yan, W.

Yang, X.

Yao, J.

J. Yao, Z Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A.M. Stacy, X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

Yu, K.W.

Zayats, A.V.

Zhang, X.

J. Yao, Z Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A.M. Stacy, X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

Zhu, G.

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

Adv. Mater. (1)

C.R. Simovski, P.A. Belov, A.A. Atrashchenko, Y.S. Kivshar, “Wire metamaterials: Physics and applications,” Adv. Mater. 24, 4229–4248 (2012).
[CrossRef] [PubMed]

Appl. Phys. A (1)

N. Dubrovina, L.O. Le Cunff, N. Burokur, R. Ghasemi, A. Degiron, A. De Lustrac, A. Vial, G. Leronded, A. Lupu, “Single metafilm effective medium behavior in optical domain: Maxwell-Garnett approximation and beyond,” Appl. Phys. A 109, 901–906 (2012).
[CrossRef]

Appl. Phys. B (1)

A. Vial, T. Laroche, “Comparison of gold and silver dispersion laws suitable for FDTD simulations,” Appl. Phys. B 93, 139–143 (2008).
[CrossRef]

Appl. Phys. Lett. (5)

A.K. Popov, S.A. Myslivets, “Transformable broad-band transparency and amplification in negative-index films,” Appl. Phys. Lett. 93, 191117 (2008).
[CrossRef]

J. Elser, R. Wangberg, V.A. Podolskiy, E.E. Narimanov, “Nanowire metamaterials with extreme optical anisotropy,” Appl. Phys. Lett. 89, 261102 (2006).
[CrossRef]

L.V. Alekseyev, E.E. Narimarov, T. Tumkur, H. Li, Yu. A. Barnakov, M.A. Noginov, “Uniaxial epsilon-near-zero metamaterial for angular filtering and polarization control,” Appl. Phys. Lett. 97, 131107 (2010).
[CrossRef]

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

A.V. Goncharenko, V.U. Nazarov, K.R. Chen, “Development of metamaterials with desired broadband optical properties,” Appl. Phys. Lett. 101, 071907 (2012).
[CrossRef]

J. Appl. Phys. (1)

M. Scalora, M.J. Bloemer, A.S. Pethel, J.P. Dowling, C.M. Bowden, A.S. Manka, “Transparent, metallodielectric, one-dimensional, photonic band-gap structures,” J. Appl. Phys. 83, 2377–2383 (1998).
[CrossRef]

J. Math. Phys. (1)

J.B. Keller, “A theorem on the conductivity of a composite medium,” J. Math. Phys. 5, 548–549 (1964).
[CrossRef]

J. Nanophoton. (2)

X.X. Liu, A. Alu, “Limitations and potential of metamaterial lenses,” J. Nanophoton. 5, 053509 (2011).
[CrossRef]

A.V. Goncharenko, K.R. Chen, “Strategy for designing epsilon-near-zero nanostructured metamaterials over a frequency range,” J. Nanophoton. 4, 041530 (2010).
[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. B (3)

Metamaterials (1)

W.T. Perrins, R.C. McPedran, “Metamaterials and the homogenization of composite materials,” Metamaterials 4, 24–31 (2010).
[CrossRef]

Nature Mater. (1)

A.V. Kabashin, P. Evans, S. Pastkovsky, W. Hendren, G.A. Wurtz, R. Atkinson, R. Pollard, V.A. Podolskiy, A.V. Zayats, “Plasmonic nanorod metamaterials for biosensing,” Nature Mater. 8, 867–871 (2009).
[CrossRef]

New J. Phys. (1)

H. Rauh, G.I. Yampolskaya, S.V. Yampolskii, “Optical transmittance of photonic structures with linearly graded dielectric constituents,” New J. Phys. 12, 073033 (2010).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Opt. Mater. Express (1)

Phys. Rev. B (6)

P. Belov, Y. Hao, “Subwavelength imaging at optical frequencies using a transmission device formed by a periodic layered metal-dielectric structure operating in the canalization regime,” Phys. Rev. B 73, 113110 (2006).
[CrossRef]

A.A. Krokhin, P. Halevi, J. Arriaga, “Long-wavelength limit (homogenization) for two-dimensional photonic crystals,” Phys. Rev. B 65, 115208 (2002).
[CrossRef]

A.A. Orlov, P.M. Voroshilov, P.A. Belov, Y.S. Kivshar, “Engineered optical nonlocality in nanostructured metamaterials,” Phys. Rev. B 84, 045424 (2011).
[CrossRef]

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

P.A. Belov, R. Marques, S.I. Maslovski, I.S. Nefedov, M. Silveirinha, C.R. Simovski, S.A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B 67, 113103 (2003).
[CrossRef]

S.I. Maslovski, M.G. Silveirinha, “Nonlocal permittivity from a quasistatic model for a class of wire media,” Phys. Rev. B 80, 245101 (2009).
[CrossRef]

Sci. Rept. (1)

C. David, N.A. Mortensen, J. Christensen, “Perfect imaging, epsilon-near zero phenomena and waveguiding in the scope of nonlocal effects,” Sci. Rept. 3, 02526 (2013).

Science (1)

J. Yao, Z Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A.M. Stacy, X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321, 930 (2008).
[CrossRef] [PubMed]

Tech. Phys. Lett. (1)

A.V. Goncharenko, “Limiting geometries and dielectric tensor of superlattices,” Tech. Phys. Lett. 26(7), 594–596 (2000).
[CrossRef]

Z. Angew. Math. Mech. (1)

A. Reuss, “Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle,” Z. Angew. Math. Mech. 9, 49–58 (1929).
[CrossRef]

Other (3)

M. Sahimi, Heterogeneous Materials. I. Linear Transport and Optical Properties (Springer, 2003).

G. W. Milton, The Theory of Composites (Cambridge University, 2002).
[CrossRef]

W. Voight, Lehrbuch der Kristallphysik (Teubner-Verlag, 1928).

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Figures (11)

Fig. 1
Fig. 1

Schematic of the unit cell representing geometry under consideration. The arrows on the right side of the figure show possible orientations of the electric field.

Fig. 2
Fig. 2

The linear and parabolic profiles of the metal phase distribution along the y-axis.

Fig. 3
Fig. 3

The real part of the effective permittivity of designed MMs fitted to zero over the bands of 570 – 670 nm (1), 570 – 720 nm (2), and 570 – 770 nm (3). The fitted parameters are: y′0 = 0.248, fmin = 0.043, fmax = 0.627(1); y′0 = 0.25, fmin = 0.035, fmax = 0.668(2);y′0 = 0.252, fmin = 0.03, fmax = 0.699(3).

Fig. 4
Fig. 4

The effective permittivity of two designed MMs with Reεy fitted to zero over the band of 700 – 800 nm. The fitted parameters are: y′0 = 0.259, fmin = 0.021, fmax = 0.975 (blue curve) and y′0 = 0.177, fmin = 0.063, fmax = 0.103 (red curve).

Fig. 5
Fig. 5

The effective refractive index of designed MMs fitted to n* = 0.25 over the band of 620 – 720 nm. The fitted parameters are: y′0 = 0.492, fmin = 0, fmax = 0.769 (1); y′0 = 0.498, fmin = 0.9425, fmax = 1 (2); y′0 = 0.396, fmin = 0, fmax = 0.1615 (3); y′0 = 0.436, fmin = 0.09, fmax = 0.15 (4).

Fig. 6
Fig. 6

The four branches for the root mean square curves vs n*.

Fig. 7
Fig. 7

The effective refractive index of designed MMs fitted to n* = 4.25 over the band of 620 – 720 nm. The fitted parameters are: y′0 = 0.063, fmin = 0.817, fmax = 0.936 (solid line); y′0 = 0.065, fmin = 0.805, fmax = 0.918 (dashed line). The root mean square is about 0.017 for both curves.

Fig. 8
Fig. 8

The three branches for the root mean square curves vs n* in the HRI regime.

Fig. 9
Fig. 9

The effective refractive index of designed MMs fitted to n* = 5 over the band of 660 – 860 nm (rms = 0.006) and to n* = 5.5 over the band of 700 – 900 nm (rms = 0.0091). The fitted parameters are: y′0 = 0.042, fmin = 0.856, fmax = 0.956 (solid line); y′0 = 0.041, fmin = 0.87, fmax = 0.953 (dashed line).

Fig. 10
Fig. 10

The frequency dependencies of n min = Ren u ( f min * ) calculated with the use of the exact Eqs. (15) and (16) (solid curve), with the use of the approximate Eq. (18) (dashed curve), and n max = Ren l ( f max * ) (dotted curve).

Fig. 11
Fig. 11

The upper (solid curves) and lower (dashed curves) Wiener’s bounds in the complex (Ren, Imn) plane for λ = 620, 670, and 720 nm.

Equations (23)

Equations on this page are rendered with MathJax. Learn more.

ε x = S y 1 0 S y ε ( y ) d y ,
ε y 1 = S y 1 0 S y d y ε ( y ) ,
f ( y max ) f ( y min ) y max y min S x 1 ,
ε | | ( y ) f ( y ) ε 1 + [ 1 f ( y ) ] ε 2 ,
ε ( y ) [ f ( y ) ε 1 + 1 f ( y ) ε 2 ] 1 .
ε x ε 1 ε 2 ε 2 ε 1 0 1 d y f ( y ) + ε 1 ε 2 ε 1 ,
ε y 1 1 ε 1 ε 2 0 1 d y f ( y ) + ε 2 ε 1 ε 2 ,
ε x ( ε 1 , ε 2 ) ε y ( ε 2 , ε 1 ) = ε 1 ε 2 .
f ( y ) = { f max for 0 y y 0 f min + ( f max f min ) ( y 1 / 2 ) ( y 0 1 / 2 ) for y 0 y 1 / 2 f min ( f max f min ) ( y 1 / 2 ) ( y 0 1 / 2 ) for 1 / 2 y 1 y 0 f max for 1 y 0 y 1
f ( y ) = { f max for 0 y y 0 f min + ( f max f min ) ( y 1 / 2 ) 2 ( y 0 1 / 2 ) 2 for y 0 y 1 y 0 f max for 1 y 0 y 1
ε x 2 ε 2 s [ y 0 f max + s + y 0 1 / 2 f max f min ln s + f min s + f max ]
ε y ε 2 2 s [ y 0 f max + s + y 0 1 / 2 f max f min ln s + f min s + f max ] 1 ,
ε x 2 ε 2 s [ y 0 f max + s + 1 / 2 y 0 ( f max f min ) ( f min + s ) arctan f max f min f min + s ]
ε y ε 2 2 s [ y 0 f max + s + 1 / 2 y 0 ( f max f min ) ( f min + s ) arctan f max f min f min + s ] 1 .
Ren u ( f ) = 1 2 ( f Δ + ε 2 ) 2 + f 2 ε 1 2 + f Δ + ε 2 ,
f min * = 2 Δ ε 2 Δ 2 + ε 1 2 .
( f m ε 1 f m Δ + ε 2 ) 2 < < 1 ,
n min Δ ε 1 ε 1 2 Δ ε 2 Δ 2 ε 1 2 Δ 2 + ε 1 2 .
f max * = | ε 1 | 2 tan ( ϕ ) ε 2 ε 1 + ( | ε 1 | 2 ε 2 ε 1 ) tan ( ϕ )
2 S x S y f max f min 1 2 y 0 1 .
ε y 1 = S y 1 0 S y d y ε | | ( y ) ,
ε x = S y 1 0 S y ε x ( y ) d y ,
ε z = S y 1 0 S y ε z ( y ) d y ,

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