Abstract

Transformation optics (TO) has established itself as a powerful and versatile approach to the synthesis of metamaterials with prescribed field-manipulation capabilities, via suitable spatial modulation of their constitutive properties inspired by local distortions of the spatial coordinate reference frame. From the mathematical viewpoint, this approach can be reformulated in the frequency-wavenumber reciprocal phase space so as to engineer nonlocal interactions and spatial dispersion effects, which are becoming increasingly relevant in electrodynamics and optics. Here, we present a general nonlocal-TO framework, based on complex-valued, frequency-dependent wavenumber coordinate transformations, and explore its possible applications to scenarios of interest for dispersion engineering. A key attribute of our approach, similar to conventional TO, is the separation of the conceptual design (based on intuitive geometrical considerations) from the actual metamaterial synthesis (based on a suitable approximation of analytically derived constitutive “blueprints”). To illustrate the capabilities and potential of the proposed approach, we address the engineering (from the conceptual design to the actual synthesis) of multilayered metamaterials exhibiting various exotic dispersion effects, including “one-way” (nonreciprocal) propagation, “frozen-mode” regime, and Dirac-point conical singularities. Our approach may open up new perspectives in the systematic design of metamaterials with broad field-manipulation capabilities as well as complex spatiotemporal dispersion effects, with potential applications to nonreciprocal optics, topological photonics, and “computational metamaterials.”

© 2016 Optical Society of America

Full Article  |  PDF Article
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References

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  1. U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
    [Crossref]
  2. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
    [Crossref]
  3. D. H. Werner and D. H. Kwon, Transformation Electromagnetics and Metamaterials: Fundamental Principles and Applications (Springer, 2013).
  4. M. Kadic, T. Bückmann, R. Schittny, and M. Wegener, “Metamaterials beyond electromagnetism,” Rep. Progress Phys. 76, 126501 (2013).
    [Crossref]
  5. M. Moccia, G. Castaldi, S. Savo, Y. Sato, and V. Galdi, “Independent manipulation of heat and electrical current via bifunctional metamaterials,” Phys. Rev. X 4, 021025 (2014).
  6. S. A. Tretyakov, I. S. Nefedov, and P. Alitalo, “Generalized field-transforming metamaterials,” New J. Phys. 10, 115028 (2008).
    [Crossref]
  7. L. Bergamin, “Generalized transformation optics from triple spacetime metamaterials,” Phys. Rev. A 78, 043825 (2008).
    [Crossref]
  8. L. Bergamin, P. Alitalo, and S. A. Tretyakov, “Nonlinear transformation optics and engineering of the Kerr effect,” Phys. Rev. B 84, 205103 (2011).
    [Crossref]
  9. S. A. Cummer and R. T. Thompson, “Frequency conversion by exploiting time in transformation optics,” J. Opt. 13, 024007 (2011).
    [Crossref]
  10. G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Transformation-optics generalization of tunnelling effects in bi-layers made of paired pseudo-epsilon-negative/mu-negative media,” J. Opt. 13, 024011 (2011).
    [Crossref]
  11. R. T. Thompson, S. A. Cummer, and J. Frauendiener, “A completely covariant approach to transformation optics,” J. Opt. 13, 024008 (2011).
    [Crossref]
  12. B.-I. Popa and S. A. Cummer, “Complex coordinates in transformation optics,” Phys. Rev. A 84, 063837 (2011).
    [Crossref]
  13. O. Paul and M. Rahm, “Covariant description of transformation optics in nonlinear media,” Opt. Express 20, 8982–8997 (2012).
    [Crossref]
  14. G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Nonlocal transformation optics,” Phys. Rev. Lett. 108, 063902 (2012).
    [Crossref]
  15. G. Castaldi, S. Savoia, V. Galdi, A. Alù, and N. Engheta, “PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110, 173901 (2013).
    [Crossref]
  16. L. D. Landau, J. S. Bell, J. Kearsley, L. P. Pitaevskii, E. M. Lifshitz, and J. B. Sykes, Electrodynamics of Continuous Media, Vol. 8 of Course of Theoretical Physics (Elsevier, 1984).
  17. V. M. Agranovich and V. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons, Springer Series in Solid-State Sciences (Springer, 2013).
  18. S. M. Mikki and A. A. Kishk, “Nonlocal electromagnetic media: a paradigm for material engineering,” in Passive Microwave Components and Antennas, V. Zhurbenko, ed. (InTech, 2010), Chap. 4, pp. 73–94.
  19. G. A. Wurtz, R. Pollard, W. Hendren, G. Wiederrecht, D. Gosztola, V. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6, 107–111 (2011).
    [Crossref]
  20. J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90, 191109 (2007).
    [Crossref]
  21. M. G. Silveirinha, “Generalized Lorentz-Lorenz formulas for microstructured materials,” Phys. Rev. B 76, 245117 (2007).
    [Crossref]
  22. M. G. Silveirinha, “Time domain homogenization of metamaterials,” Phys. Rev. B 83, 165104 (2011).
    [Crossref]
  23. A. V. Chebykin, A. A. Orlov, A. V. Vozianova, S. I. Maslovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective medium model for multilayered metal-dielectric metamaterials,” Phys. Rev. B 84, 115438 (2011).
    [Crossref]
  24. A. V. Chebykin, A. A. Orlov, C. R. Simovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B 86, 115420 (2012).
    [Crossref]
  25. R.-L. Chern, “Spatial dispersion and nonlocal effective permittivity for periodic layered metamaterials,” Opt. Express 21, 16514–16527 (2013).
    [Crossref]
  26. B. M. Wells, A. V. Zayats, and V. A. Podolskiy, “Nonlocal optics of plasmonic nanowire metamaterials,” Phys. Rev. B 89, 035111 (2014).
    [Crossref]
  27. A. Ciattoni and C. Rizza, “Nonlocal homogenization theory in metamaterials: effective electromagnetic spatial dispersion and artificial chirality,” Phys. Rev. B 91, 184207 (2015).
    [Crossref]
  28. L. Sun, Z. Li, T. S. Luk, X. Yang, and J. Gao, “Nonlocal effective medium analysis in symmetric metal-dielectric multilayer metamaterials,” Phys. Rev. B 91, 195147 (2015).
    [Crossref]
  29. Z. Awan, “Nonlocal effective parameters of a coated sphere medium,” J. Mod. Opt. 62, 528–535 (2014).
    [Crossref]
  30. M. A. Gorlach and P. A. Belov, “Nonlocality in uniaxially polarizable media,” arXiv:1505.01064 (2015).
  31. T. Geng, S. Zhuang, J. Gao, and X. Yang, “Nonlocal effective medium approximation for metallic nanorod metamaterials,” arXiv:1506.00727 (2015).
  32. A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–163 (2014).
    [Crossref]
  33. E. H. Lock, “The properties of isofrequency dependences and the laws of geometrical optics,” Phys. Usp. 51, 375–394 (2008).
    [Crossref]
  34. A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7, 948–957 (2013).
    [Crossref]
  35. A. Figotin and I. Vitebskiy, “Electromagnetic unidirectionality in magnetic photonic crystals,” Phys. Rev. B 67, 165210 (2003).
    [Crossref]
  36. A. Figotin and I. Vitebskiy, “Slow wave phenomena in photonic crystals,” Laser Photon. Rev. 5, 201–213 (2011).
    [Crossref]
  37. K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
    [Crossref]
  38. Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100, 023902 (2008).
    [Crossref]
  39. A. B. Khanikaev and M. J. Steel, “Low-symmetry magnetic photonic crystals for nonreciprocal and unidirectional devices,” Opt. Express 17, 5265–5272 (2009).
    [Crossref]
  40. Z. Wang, Y. Chong, J. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
    [Crossref]
  41. X. Lin, Y. Xu, B. Zhang, R. Hao, H. Chen, and E. Li, “Unidirectional surface plasmons in nonreciprocal graphene,” New J. Phys. 15, 113003 (2013).
    [Crossref]
  42. U. K. Chettiar, A. R. Davoyan, and N. Engheta, “Hotspots from nonreciprocal surface waves,” Opt. Lett. 39, 1760–1763 (2014).
    [Crossref]
  43. A. Davoyan and N. Engheta, “Electrically controlled one-way photon flow in plasmonic nanostructures,” Nat. Commun. 5, 5250 (2014).
    [Crossref]
  44. A. Leviyev, B. Stein, T. Galfsky, H. Krishnamoorthy, I. L. Kuskovsky, V. Menon, and A. B. Khanikaev, “Nonreciprocity and one-way topological transitions in hyperbolic metamaterials,” arXiv:1505.05438 (2015).
  45. R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75, 063813 (2007).
    [Crossref]
  46. O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98, 103901 (2007).
    [Crossref]
  47. F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
    [Crossref]
  48. X. Zhang, “Observing Zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. 100, 113903 (2008).
    [Crossref]
  49. T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80, 155103 (2009).
    [Crossref]
  50. X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10, 582–586 (2011).
    [Crossref]
  51. L.-G. Wang, Z.-G. Wang, J.-X. Zhang, and S.-Y. Zhu, “Realization of Dirac point with double cones in optics,” Opt. Lett. 34, 1510–1512 (2009).
    [Crossref]
  52. Y. P. Bliokh, V. Freilikher, and F. Nori, “Ballistic charge transport in graphene and light propagation in periodic dielectric structures with metamaterials: a comparative study,” Phys. Rev. B 87, 245134 (2013).
    [Crossref]
  53. S. H. Nam, A. J. Taylor, and A. Efimov, “Diabolical point and conical-like diffraction in periodic plasmonic nanostructures,” Opt. Express 18, 10120–10126 (2010).
    [Crossref]
  54. S. H. Nam, J. Zhou, A. J. Taylor, and A. Efimov, “Dirac dynamics in one-dimensional graphene-like plasmonic crystals: pseudo-spin, chirality, and diffraction anomaly,” Opt. Express 18, 25329–25338 (2010).
    [Crossref]
  55. L. Sun, J. Gao, and X. Yang, “Giant optical nonlocality near the Dirac point in metal-dielectric multilayer metamaterials,” Opt. Express 21, 21542–21555 (2013).
    [Crossref]
  56. R. L. Haupt and D. H. Werner, Genetic Algorithms in Electromagnetics (Wiley, 2007).

2015 (2)

A. Ciattoni and C. Rizza, “Nonlocal homogenization theory in metamaterials: effective electromagnetic spatial dispersion and artificial chirality,” Phys. Rev. B 91, 184207 (2015).
[Crossref]

L. Sun, Z. Li, T. S. Luk, X. Yang, and J. Gao, “Nonlocal effective medium analysis in symmetric metal-dielectric multilayer metamaterials,” Phys. Rev. B 91, 195147 (2015).
[Crossref]

2014 (6)

Z. Awan, “Nonlocal effective parameters of a coated sphere medium,” J. Mod. Opt. 62, 528–535 (2014).
[Crossref]

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–163 (2014).
[Crossref]

B. M. Wells, A. V. Zayats, and V. A. Podolskiy, “Nonlocal optics of plasmonic nanowire metamaterials,” Phys. Rev. B 89, 035111 (2014).
[Crossref]

A. Davoyan and N. Engheta, “Electrically controlled one-way photon flow in plasmonic nanostructures,” Nat. Commun. 5, 5250 (2014).
[Crossref]

M. Moccia, G. Castaldi, S. Savo, Y. Sato, and V. Galdi, “Independent manipulation of heat and electrical current via bifunctional metamaterials,” Phys. Rev. X 4, 021025 (2014).

U. K. Chettiar, A. R. Davoyan, and N. Engheta, “Hotspots from nonreciprocal surface waves,” Opt. Lett. 39, 1760–1763 (2014).
[Crossref]

2013 (7)

R.-L. Chern, “Spatial dispersion and nonlocal effective permittivity for periodic layered metamaterials,” Opt. Express 21, 16514–16527 (2013).
[Crossref]

L. Sun, J. Gao, and X. Yang, “Giant optical nonlocality near the Dirac point in metal-dielectric multilayer metamaterials,” Opt. Express 21, 21542–21555 (2013).
[Crossref]

Y. P. Bliokh, V. Freilikher, and F. Nori, “Ballistic charge transport in graphene and light propagation in periodic dielectric structures with metamaterials: a comparative study,” Phys. Rev. B 87, 245134 (2013).
[Crossref]

X. Lin, Y. Xu, B. Zhang, R. Hao, H. Chen, and E. Li, “Unidirectional surface plasmons in nonreciprocal graphene,” New J. Phys. 15, 113003 (2013).
[Crossref]

G. Castaldi, S. Savoia, V. Galdi, A. Alù, and N. Engheta, “PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110, 173901 (2013).
[Crossref]

M. Kadic, T. Bückmann, R. Schittny, and M. Wegener, “Metamaterials beyond electromagnetism,” Rep. Progress Phys. 76, 126501 (2013).
[Crossref]

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7, 948–957 (2013).
[Crossref]

2012 (3)

A. V. Chebykin, A. A. Orlov, C. R. Simovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B 86, 115420 (2012).
[Crossref]

G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Nonlocal transformation optics,” Phys. Rev. Lett. 108, 063902 (2012).
[Crossref]

O. Paul and M. Rahm, “Covariant description of transformation optics in nonlinear media,” Opt. Express 20, 8982–8997 (2012).
[Crossref]

2011 (10)

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10, 582–586 (2011).
[Crossref]

M. G. Silveirinha, “Time domain homogenization of metamaterials,” Phys. Rev. B 83, 165104 (2011).
[Crossref]

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

G. A. Wurtz, R. Pollard, W. Hendren, G. Wiederrecht, D. Gosztola, V. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6, 107–111 (2011).
[Crossref]

L. Bergamin, P. Alitalo, and S. A. Tretyakov, “Nonlinear transformation optics and engineering of the Kerr effect,” Phys. Rev. B 84, 205103 (2011).
[Crossref]

S. A. Cummer and R. T. Thompson, “Frequency conversion by exploiting time in transformation optics,” J. Opt. 13, 024007 (2011).
[Crossref]

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Transformation-optics generalization of tunnelling effects in bi-layers made of paired pseudo-epsilon-negative/mu-negative media,” J. Opt. 13, 024011 (2011).
[Crossref]

R. T. Thompson, S. A. Cummer, and J. Frauendiener, “A completely covariant approach to transformation optics,” J. Opt. 13, 024008 (2011).
[Crossref]

B.-I. Popa and S. A. Cummer, “Complex coordinates in transformation optics,” Phys. Rev. A 84, 063837 (2011).
[Crossref]

A. Figotin and I. Vitebskiy, “Slow wave phenomena in photonic crystals,” Laser Photon. Rev. 5, 201–213 (2011).
[Crossref]

2010 (2)

2009 (4)

A. B. Khanikaev and M. J. Steel, “Low-symmetry magnetic photonic crystals for nonreciprocal and unidirectional devices,” Opt. Express 17, 5265–5272 (2009).
[Crossref]

L.-G. Wang, Z.-G. Wang, J.-X. Zhang, and S.-Y. Zhu, “Realization of Dirac point with double cones in optics,” Opt. Lett. 34, 1510–1512 (2009).
[Crossref]

Z. Wang, Y. Chong, J. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
[Crossref]

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80, 155103 (2009).
[Crossref]

2008 (6)

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100, 023902 (2008).
[Crossref]

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[Crossref]

X. Zhang, “Observing Zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. 100, 113903 (2008).
[Crossref]

E. H. Lock, “The properties of isofrequency dependences and the laws of geometrical optics,” Phys. Usp. 51, 375–394 (2008).
[Crossref]

S. A. Tretyakov, I. S. Nefedov, and P. Alitalo, “Generalized field-transforming metamaterials,” New J. Phys. 10, 115028 (2008).
[Crossref]

L. Bergamin, “Generalized transformation optics from triple spacetime metamaterials,” Phys. Rev. A 78, 043825 (2008).
[Crossref]

2007 (4)

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90, 191109 (2007).
[Crossref]

M. G. Silveirinha, “Generalized Lorentz-Lorenz formulas for microstructured materials,” Phys. Rev. B 76, 245117 (2007).
[Crossref]

R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75, 063813 (2007).
[Crossref]

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98, 103901 (2007).
[Crossref]

2006 (2)

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[Crossref]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref]

2005 (1)

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
[Crossref]

2003 (1)

A. Figotin and I. Vitebskiy, “Electromagnetic unidirectionality in magnetic photonic crystals,” Phys. Rev. B 67, 165210 (2003).
[Crossref]

Agranovich, V. M.

V. M. Agranovich and V. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons, Springer Series in Solid-State Sciences (Springer, 2013).

Alitalo, P.

L. Bergamin, P. Alitalo, and S. A. Tretyakov, “Nonlinear transformation optics and engineering of the Kerr effect,” Phys. Rev. B 84, 205103 (2011).
[Crossref]

S. A. Tretyakov, I. S. Nefedov, and P. Alitalo, “Generalized field-transforming metamaterials,” New J. Phys. 10, 115028 (2008).
[Crossref]

Alù, A.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–163 (2014).
[Crossref]

G. Castaldi, S. Savoia, V. Galdi, A. Alù, and N. Engheta, “PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110, 173901 (2013).
[Crossref]

G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Nonlocal transformation optics,” Phys. Rev. Lett. 108, 063902 (2012).
[Crossref]

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Transformation-optics generalization of tunnelling effects in bi-layers made of paired pseudo-epsilon-negative/mu-negative media,” J. Opt. 13, 024011 (2011).
[Crossref]

Avrutsky, I.

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90, 191109 (2007).
[Crossref]

Awan, Z.

Z. Awan, “Nonlocal effective parameters of a coated sphere medium,” J. Mod. Opt. 62, 528–535 (2014).
[Crossref]

Bartal, G.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98, 103901 (2007).
[Crossref]

Bazaliy, Y. B.

R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75, 063813 (2007).
[Crossref]

Beenakker, C. W. J.

R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75, 063813 (2007).
[Crossref]

Bell, J. S.

L. D. Landau, J. S. Bell, J. Kearsley, L. P. Pitaevskii, E. M. Lifshitz, and J. B. Sykes, Electrodynamics of Continuous Media, Vol. 8 of Course of Theoretical Physics (Elsevier, 1984).

Belov, P.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7, 948–957 (2013).
[Crossref]

Belov, P. A.

A. V. Chebykin, A. A. Orlov, C. R. Simovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B 86, 115420 (2012).
[Crossref]

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

M. A. Gorlach and P. A. Belov, “Nonlocality in uniaxially polarizable media,” arXiv:1505.01064 (2015).

Bergamin, L.

L. Bergamin, P. Alitalo, and S. A. Tretyakov, “Nonlinear transformation optics and engineering of the Kerr effect,” Phys. Rev. B 84, 205103 (2011).
[Crossref]

L. Bergamin, “Generalized transformation optics from triple spacetime metamaterials,” Phys. Rev. A 78, 043825 (2008).
[Crossref]

Bliokh, Y. P.

Y. P. Bliokh, V. Freilikher, and F. Nori, “Ballistic charge transport in graphene and light propagation in periodic dielectric structures with metamaterials: a comparative study,” Phys. Rev. B 87, 245134 (2013).
[Crossref]

Bückmann, T.

M. Kadic, T. Bückmann, R. Schittny, and M. Wegener, “Metamaterials beyond electromagnetism,” Rep. Progress Phys. 76, 126501 (2013).
[Crossref]

Castaldi, G.

M. Moccia, G. Castaldi, S. Savo, Y. Sato, and V. Galdi, “Independent manipulation of heat and electrical current via bifunctional metamaterials,” Phys. Rev. X 4, 021025 (2014).

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–163 (2014).
[Crossref]

G. Castaldi, S. Savoia, V. Galdi, A. Alù, and N. Engheta, “PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110, 173901 (2013).
[Crossref]

G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Nonlocal transformation optics,” Phys. Rev. Lett. 108, 063902 (2012).
[Crossref]

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Transformation-optics generalization of tunnelling effects in bi-layers made of paired pseudo-epsilon-negative/mu-negative media,” J. Opt. 13, 024011 (2011).
[Crossref]

Chan, C. T.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10, 582–586 (2011).
[Crossref]

Chebykin, A. V.

A. V. Chebykin, A. A. Orlov, C. R. Simovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B 86, 115420 (2012).
[Crossref]

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

Chen, H.

X. Lin, Y. Xu, B. Zhang, R. Hao, H. Chen, and E. Li, “Unidirectional surface plasmons in nonreciprocal graphene,” New J. Phys. 15, 113003 (2013).
[Crossref]

Chern, R.-L.

Chettiar, U. K.

Chong, Y.

Z. Wang, Y. Chong, J. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
[Crossref]

Christodoulides, D. N.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98, 103901 (2007).
[Crossref]

Ciattoni, A.

A. Ciattoni and C. Rizza, “Nonlocal homogenization theory in metamaterials: effective electromagnetic spatial dispersion and artificial chirality,” Phys. Rev. B 91, 184207 (2015).
[Crossref]

Cummer, S. A.

S. A. Cummer and R. T. Thompson, “Frequency conversion by exploiting time in transformation optics,” J. Opt. 13, 024007 (2011).
[Crossref]

B.-I. Popa and S. A. Cummer, “Complex coordinates in transformation optics,” Phys. Rev. A 84, 063837 (2011).
[Crossref]

R. T. Thompson, S. A. Cummer, and J. Frauendiener, “A completely covariant approach to transformation optics,” J. Opt. 13, 024008 (2011).
[Crossref]

Davoyan, A.

A. Davoyan and N. Engheta, “Electrically controlled one-way photon flow in plasmonic nanostructures,” Nat. Commun. 5, 5250 (2014).
[Crossref]

Davoyan, A. R.

Dubonos, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
[Crossref]

Efimov, A.

Elser, J.

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90, 191109 (2007).
[Crossref]

Engheta, N.

A. Davoyan and N. Engheta, “Electrically controlled one-way photon flow in plasmonic nanostructures,” Nat. Commun. 5, 5250 (2014).
[Crossref]

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–163 (2014).
[Crossref]

U. K. Chettiar, A. R. Davoyan, and N. Engheta, “Hotspots from nonreciprocal surface waves,” Opt. Lett. 39, 1760–1763 (2014).
[Crossref]

G. Castaldi, S. Savoia, V. Galdi, A. Alù, and N. Engheta, “PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110, 173901 (2013).
[Crossref]

G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Nonlocal transformation optics,” Phys. Rev. Lett. 108, 063902 (2012).
[Crossref]

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Transformation-optics generalization of tunnelling effects in bi-layers made of paired pseudo-epsilon-negative/mu-negative media,” J. Opt. 13, 024011 (2011).
[Crossref]

Fan, S.

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100, 023902 (2008).
[Crossref]

Figotin, A.

A. Figotin and I. Vitebskiy, “Slow wave phenomena in photonic crystals,” Laser Photon. Rev. 5, 201–213 (2011).
[Crossref]

A. Figotin and I. Vitebskiy, “Electromagnetic unidirectionality in magnetic photonic crystals,” Phys. Rev. B 67, 165210 (2003).
[Crossref]

Firsov, A. A.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
[Crossref]

Frauendiener, J.

R. T. Thompson, S. A. Cummer, and J. Frauendiener, “A completely covariant approach to transformation optics,” J. Opt. 13, 024008 (2011).
[Crossref]

Freedman, B.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98, 103901 (2007).
[Crossref]

Freilikher, V.

Y. P. Bliokh, V. Freilikher, and F. Nori, “Ballistic charge transport in graphene and light propagation in periodic dielectric structures with metamaterials: a comparative study,” Phys. Rev. B 87, 245134 (2013).
[Crossref]

Galdi, V.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–163 (2014).
[Crossref]

M. Moccia, G. Castaldi, S. Savo, Y. Sato, and V. Galdi, “Independent manipulation of heat and electrical current via bifunctional metamaterials,” Phys. Rev. X 4, 021025 (2014).

G. Castaldi, S. Savoia, V. Galdi, A. Alù, and N. Engheta, “PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110, 173901 (2013).
[Crossref]

G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Nonlocal transformation optics,” Phys. Rev. Lett. 108, 063902 (2012).
[Crossref]

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Transformation-optics generalization of tunnelling effects in bi-layers made of paired pseudo-epsilon-negative/mu-negative media,” J. Opt. 13, 024011 (2011).
[Crossref]

Galfsky, T.

A. Leviyev, B. Stein, T. Galfsky, H. Krishnamoorthy, I. L. Kuskovsky, V. Menon, and A. B. Khanikaev, “Nonreciprocity and one-way topological transitions in hyperbolic metamaterials,” arXiv:1505.05438 (2015).

Gallina, I.

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Transformation-optics generalization of tunnelling effects in bi-layers made of paired pseudo-epsilon-negative/mu-negative media,” J. Opt. 13, 024011 (2011).
[Crossref]

Gao, J.

L. Sun, Z. Li, T. S. Luk, X. Yang, and J. Gao, “Nonlocal effective medium analysis in symmetric metal-dielectric multilayer metamaterials,” Phys. Rev. B 91, 195147 (2015).
[Crossref]

L. Sun, J. Gao, and X. Yang, “Giant optical nonlocality near the Dirac point in metal-dielectric multilayer metamaterials,” Opt. Express 21, 21542–21555 (2013).
[Crossref]

T. Geng, S. Zhuang, J. Gao, and X. Yang, “Nonlocal effective medium approximation for metallic nanorod metamaterials,” arXiv:1506.00727 (2015).

Geim, A. K.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
[Crossref]

Geng, T.

T. Geng, S. Zhuang, J. Gao, and X. Yang, “Nonlocal effective medium approximation for metallic nanorod metamaterials,” arXiv:1506.00727 (2015).

Ginzburg, V.

V. M. Agranovich and V. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons, Springer Series in Solid-State Sciences (Springer, 2013).

Gorlach, M. A.

M. A. Gorlach and P. A. Belov, “Nonlocality in uniaxially polarizable media,” arXiv:1505.01064 (2015).

Gosztola, D.

G. A. Wurtz, R. Pollard, W. Hendren, G. Wiederrecht, D. Gosztola, V. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6, 107–111 (2011).
[Crossref]

Grigorieva, I. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
[Crossref]

Haldane, F. D. M.

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[Crossref]

Hang, Z. H.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10, 582–586 (2011).
[Crossref]

Hao, R.

X. Lin, Y. Xu, B. Zhang, R. Hao, H. Chen, and E. Li, “Unidirectional surface plasmons in nonreciprocal graphene,” New J. Phys. 15, 113003 (2013).
[Crossref]

Haupt, R. L.

R. L. Haupt and D. H. Werner, Genetic Algorithms in Electromagnetics (Wiley, 2007).

Hendren, W.

G. A. Wurtz, R. Pollard, W. Hendren, G. Wiederrecht, D. Gosztola, V. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6, 107–111 (2011).
[Crossref]

Huang, X.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10, 582–586 (2011).
[Crossref]

Iorsh, I.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7, 948–957 (2013).
[Crossref]

Jiang, D.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
[Crossref]

Joannopoulos, J.

Z. Wang, Y. Chong, J. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
[Crossref]

Kadic, M.

M. Kadic, T. Bückmann, R. Schittny, and M. Wegener, “Metamaterials beyond electromagnetism,” Rep. Progress Phys. 76, 126501 (2013).
[Crossref]

Katsnelson, M. I.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
[Crossref]

Kearsley, J.

L. D. Landau, J. S. Bell, J. Kearsley, L. P. Pitaevskii, E. M. Lifshitz, and J. B. Sykes, Electrodynamics of Continuous Media, Vol. 8 of Course of Theoretical Physics (Elsevier, 1984).

Khanikaev, A. B.

A. B. Khanikaev and M. J. Steel, “Low-symmetry magnetic photonic crystals for nonreciprocal and unidirectional devices,” Opt. Express 17, 5265–5272 (2009).
[Crossref]

A. Leviyev, B. Stein, T. Galfsky, H. Krishnamoorthy, I. L. Kuskovsky, V. Menon, and A. B. Khanikaev, “Nonreciprocity and one-way topological transitions in hyperbolic metamaterials,” arXiv:1505.05438 (2015).

Kishk, A. A.

S. M. Mikki and A. A. Kishk, “Nonlocal electromagnetic media: a paradigm for material engineering,” in Passive Microwave Components and Antennas, V. Zhurbenko, ed. (InTech, 2010), Chap. 4, pp. 73–94.

Kivshar, Y.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7, 948–957 (2013).
[Crossref]

Kivshar, Y. S.

A. V. Chebykin, A. A. Orlov, C. R. Simovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B 86, 115420 (2012).
[Crossref]

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

Krishnamoorthy, H.

A. Leviyev, B. Stein, T. Galfsky, H. Krishnamoorthy, I. L. Kuskovsky, V. Menon, and A. B. Khanikaev, “Nonreciprocity and one-way topological transitions in hyperbolic metamaterials,” arXiv:1505.05438 (2015).

Kuskovsky, I. L.

A. Leviyev, B. Stein, T. Galfsky, H. Krishnamoorthy, I. L. Kuskovsky, V. Menon, and A. B. Khanikaev, “Nonreciprocity and one-way topological transitions in hyperbolic metamaterials,” arXiv:1505.05438 (2015).

Kwon, D. H.

D. H. Werner and D. H. Kwon, Transformation Electromagnetics and Metamaterials: Fundamental Principles and Applications (Springer, 2013).

Lai, Y.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10, 582–586 (2011).
[Crossref]

Landau, L. D.

L. D. Landau, J. S. Bell, J. Kearsley, L. P. Pitaevskii, E. M. Lifshitz, and J. B. Sykes, Electrodynamics of Continuous Media, Vol. 8 of Course of Theoretical Physics (Elsevier, 1984).

Leonhardt, U.

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[Crossref]

Leviyev, A.

A. Leviyev, B. Stein, T. Galfsky, H. Krishnamoorthy, I. L. Kuskovsky, V. Menon, and A. B. Khanikaev, “Nonreciprocity and one-way topological transitions in hyperbolic metamaterials,” arXiv:1505.05438 (2015).

Li, E.

X. Lin, Y. Xu, B. Zhang, R. Hao, H. Chen, and E. Li, “Unidirectional surface plasmons in nonreciprocal graphene,” New J. Phys. 15, 113003 (2013).
[Crossref]

Li, Z.

L. Sun, Z. Li, T. S. Luk, X. Yang, and J. Gao, “Nonlocal effective medium analysis in symmetric metal-dielectric multilayer metamaterials,” Phys. Rev. B 91, 195147 (2015).
[Crossref]

Lifshitz, E. M.

L. D. Landau, J. S. Bell, J. Kearsley, L. P. Pitaevskii, E. M. Lifshitz, and J. B. Sykes, Electrodynamics of Continuous Media, Vol. 8 of Course of Theoretical Physics (Elsevier, 1984).

Lin, X.

X. Lin, Y. Xu, B. Zhang, R. Hao, H. Chen, and E. Li, “Unidirectional surface plasmons in nonreciprocal graphene,” New J. Phys. 15, 113003 (2013).
[Crossref]

Lock, E. H.

E. H. Lock, “The properties of isofrequency dependences and the laws of geometrical optics,” Phys. Usp. 51, 375–394 (2008).
[Crossref]

Luk, T. S.

L. Sun, Z. Li, T. S. Luk, X. Yang, and J. Gao, “Nonlocal effective medium analysis in symmetric metal-dielectric multilayer metamaterials,” Phys. Rev. B 91, 195147 (2015).
[Crossref]

Manela, O.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98, 103901 (2007).
[Crossref]

Maslovski, S. I.

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

Menon, V.

A. Leviyev, B. Stein, T. Galfsky, H. Krishnamoorthy, I. L. Kuskovsky, V. Menon, and A. B. Khanikaev, “Nonreciprocity and one-way topological transitions in hyperbolic metamaterials,” arXiv:1505.05438 (2015).

Mikki, S. M.

S. M. Mikki and A. A. Kishk, “Nonlocal electromagnetic media: a paradigm for material engineering,” in Passive Microwave Components and Antennas, V. Zhurbenko, ed. (InTech, 2010), Chap. 4, pp. 73–94.

Moccia, M.

M. Moccia, G. Castaldi, S. Savo, Y. Sato, and V. Galdi, “Independent manipulation of heat and electrical current via bifunctional metamaterials,” Phys. Rev. X 4, 021025 (2014).

Monticone, F.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–163 (2014).
[Crossref]

Morozov, S. V.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
[Crossref]

Nam, S. H.

Nefedov, I. S.

S. A. Tretyakov, I. S. Nefedov, and P. Alitalo, “Generalized field-transforming metamaterials,” New J. Phys. 10, 115028 (2008).
[Crossref]

Nori, F.

Y. P. Bliokh, V. Freilikher, and F. Nori, “Ballistic charge transport in graphene and light propagation in periodic dielectric structures with metamaterials: a comparative study,” Phys. Rev. B 87, 245134 (2013).
[Crossref]

Novoselov, K. S.

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
[Crossref]

Ochiai, T.

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80, 155103 (2009).
[Crossref]

Onoda, M.

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80, 155103 (2009).
[Crossref]

Orlov, A. A.

A. V. Chebykin, A. A. Orlov, C. R. Simovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B 86, 115420 (2012).
[Crossref]

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

Paul, O.

Peleg, O.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98, 103901 (2007).
[Crossref]

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref]

Pitaevskii, L. P.

L. D. Landau, J. S. Bell, J. Kearsley, L. P. Pitaevskii, E. M. Lifshitz, and J. B. Sykes, Electrodynamics of Continuous Media, Vol. 8 of Course of Theoretical Physics (Elsevier, 1984).

Poddubny, A.

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7, 948–957 (2013).
[Crossref]

Podolskiy, V.

G. A. Wurtz, R. Pollard, W. Hendren, G. Wiederrecht, D. Gosztola, V. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6, 107–111 (2011).
[Crossref]

Podolskiy, V. A.

B. M. Wells, A. V. Zayats, and V. A. Podolskiy, “Nonlocal optics of plasmonic nanowire metamaterials,” Phys. Rev. B 89, 035111 (2014).
[Crossref]

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90, 191109 (2007).
[Crossref]

Pollard, R.

G. A. Wurtz, R. Pollard, W. Hendren, G. Wiederrecht, D. Gosztola, V. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6, 107–111 (2011).
[Crossref]

Popa, B.-I.

B.-I. Popa and S. A. Cummer, “Complex coordinates in transformation optics,” Phys. Rev. A 84, 063837 (2011).
[Crossref]

Raghu, S.

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[Crossref]

Rahm, M.

Rizza, C.

A. Ciattoni and C. Rizza, “Nonlocal homogenization theory in metamaterials: effective electromagnetic spatial dispersion and artificial chirality,” Phys. Rev. B 91, 184207 (2015).
[Crossref]

Salakhutdinov, I.

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90, 191109 (2007).
[Crossref]

Sato, Y.

M. Moccia, G. Castaldi, S. Savo, Y. Sato, and V. Galdi, “Independent manipulation of heat and electrical current via bifunctional metamaterials,” Phys. Rev. X 4, 021025 (2014).

Savo, S.

M. Moccia, G. Castaldi, S. Savo, Y. Sato, and V. Galdi, “Independent manipulation of heat and electrical current via bifunctional metamaterials,” Phys. Rev. X 4, 021025 (2014).

Savoia, S.

G. Castaldi, S. Savoia, V. Galdi, A. Alù, and N. Engheta, “PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110, 173901 (2013).
[Crossref]

Schittny, R.

M. Kadic, T. Bückmann, R. Schittny, and M. Wegener, “Metamaterials beyond electromagnetism,” Rep. Progress Phys. 76, 126501 (2013).
[Crossref]

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref]

Segev, M.

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98, 103901 (2007).
[Crossref]

Sepkhanov, R. A.

R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75, 063813 (2007).
[Crossref]

Silva, A.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–163 (2014).
[Crossref]

Silveirinha, M. G.

M. G. Silveirinha, “Time domain homogenization of metamaterials,” Phys. Rev. B 83, 165104 (2011).
[Crossref]

M. G. Silveirinha, “Generalized Lorentz-Lorenz formulas for microstructured materials,” Phys. Rev. B 76, 245117 (2007).
[Crossref]

Simovski, C. R.

A. V. Chebykin, A. A. Orlov, C. R. Simovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B 86, 115420 (2012).
[Crossref]

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref]

Soljacic, M.

Z. Wang, Y. Chong, J. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
[Crossref]

Steel, M. J.

Stein, B.

A. Leviyev, B. Stein, T. Galfsky, H. Krishnamoorthy, I. L. Kuskovsky, V. Menon, and A. B. Khanikaev, “Nonreciprocity and one-way topological transitions in hyperbolic metamaterials,” arXiv:1505.05438 (2015).

Sun, L.

L. Sun, Z. Li, T. S. Luk, X. Yang, and J. Gao, “Nonlocal effective medium analysis in symmetric metal-dielectric multilayer metamaterials,” Phys. Rev. B 91, 195147 (2015).
[Crossref]

L. Sun, J. Gao, and X. Yang, “Giant optical nonlocality near the Dirac point in metal-dielectric multilayer metamaterials,” Opt. Express 21, 21542–21555 (2013).
[Crossref]

Sykes, J. B.

L. D. Landau, J. S. Bell, J. Kearsley, L. P. Pitaevskii, E. M. Lifshitz, and J. B. Sykes, Electrodynamics of Continuous Media, Vol. 8 of Course of Theoretical Physics (Elsevier, 1984).

Taylor, A. J.

Thompson, R. T.

R. T. Thompson, S. A. Cummer, and J. Frauendiener, “A completely covariant approach to transformation optics,” J. Opt. 13, 024008 (2011).
[Crossref]

S. A. Cummer and R. T. Thompson, “Frequency conversion by exploiting time in transformation optics,” J. Opt. 13, 024007 (2011).
[Crossref]

Tretyakov, S. A.

L. Bergamin, P. Alitalo, and S. A. Tretyakov, “Nonlinear transformation optics and engineering of the Kerr effect,” Phys. Rev. B 84, 205103 (2011).
[Crossref]

S. A. Tretyakov, I. S. Nefedov, and P. Alitalo, “Generalized field-transforming metamaterials,” New J. Phys. 10, 115028 (2008).
[Crossref]

Veronis, G.

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100, 023902 (2008).
[Crossref]

Vitebskiy, I.

A. Figotin and I. Vitebskiy, “Slow wave phenomena in photonic crystals,” Laser Photon. Rev. 5, 201–213 (2011).
[Crossref]

A. Figotin and I. Vitebskiy, “Electromagnetic unidirectionality in magnetic photonic crystals,” Phys. Rev. B 67, 165210 (2003).
[Crossref]

Vozianova, A. V.

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

Wang, L.-G.

Wang, Z.

Z. Wang, Y. Chong, J. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
[Crossref]

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100, 023902 (2008).
[Crossref]

Wang, Z.-G.

Wegener, M.

M. Kadic, T. Bückmann, R. Schittny, and M. Wegener, “Metamaterials beyond electromagnetism,” Rep. Progress Phys. 76, 126501 (2013).
[Crossref]

Wells, B. M.

B. M. Wells, A. V. Zayats, and V. A. Podolskiy, “Nonlocal optics of plasmonic nanowire metamaterials,” Phys. Rev. B 89, 035111 (2014).
[Crossref]

Werner, D. H.

D. H. Werner and D. H. Kwon, Transformation Electromagnetics and Metamaterials: Fundamental Principles and Applications (Springer, 2013).

R. L. Haupt and D. H. Werner, Genetic Algorithms in Electromagnetics (Wiley, 2007).

Wiederrecht, G.

G. A. Wurtz, R. Pollard, W. Hendren, G. Wiederrecht, D. Gosztola, V. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6, 107–111 (2011).
[Crossref]

Wurtz, G. A.

G. A. Wurtz, R. Pollard, W. Hendren, G. Wiederrecht, D. Gosztola, V. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6, 107–111 (2011).
[Crossref]

Xu, Y.

X. Lin, Y. Xu, B. Zhang, R. Hao, H. Chen, and E. Li, “Unidirectional surface plasmons in nonreciprocal graphene,” New J. Phys. 15, 113003 (2013).
[Crossref]

Yang, X.

L. Sun, Z. Li, T. S. Luk, X. Yang, and J. Gao, “Nonlocal effective medium analysis in symmetric metal-dielectric multilayer metamaterials,” Phys. Rev. B 91, 195147 (2015).
[Crossref]

L. Sun, J. Gao, and X. Yang, “Giant optical nonlocality near the Dirac point in metal-dielectric multilayer metamaterials,” Opt. Express 21, 21542–21555 (2013).
[Crossref]

T. Geng, S. Zhuang, J. Gao, and X. Yang, “Nonlocal effective medium approximation for metallic nanorod metamaterials,” arXiv:1506.00727 (2015).

Yu, Z.

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100, 023902 (2008).
[Crossref]

Zayats, A. V.

B. M. Wells, A. V. Zayats, and V. A. Podolskiy, “Nonlocal optics of plasmonic nanowire metamaterials,” Phys. Rev. B 89, 035111 (2014).
[Crossref]

G. A. Wurtz, R. Pollard, W. Hendren, G. Wiederrecht, D. Gosztola, V. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6, 107–111 (2011).
[Crossref]

Zhang, B.

X. Lin, Y. Xu, B. Zhang, R. Hao, H. Chen, and E. Li, “Unidirectional surface plasmons in nonreciprocal graphene,” New J. Phys. 15, 113003 (2013).
[Crossref]

Zhang, J.-X.

Zhang, X.

X. Zhang, “Observing Zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. 100, 113903 (2008).
[Crossref]

Zheng, H.

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10, 582–586 (2011).
[Crossref]

Zhou, J.

Zhu, S.-Y.

Zhuang, S.

T. Geng, S. Zhuang, J. Gao, and X. Yang, “Nonlocal effective medium approximation for metallic nanorod metamaterials,” arXiv:1506.00727 (2015).

Appl. Phys. Lett. (1)

J. Elser, V. A. Podolskiy, I. Salakhutdinov, and I. Avrutsky, “Nonlocal effects in effective-medium response of nanolayered metamaterials,” Appl. Phys. Lett. 90, 191109 (2007).
[Crossref]

J. Mod. Opt. (1)

Z. Awan, “Nonlocal effective parameters of a coated sphere medium,” J. Mod. Opt. 62, 528–535 (2014).
[Crossref]

J. Opt. (3)

S. A. Cummer and R. T. Thompson, “Frequency conversion by exploiting time in transformation optics,” J. Opt. 13, 024007 (2011).
[Crossref]

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Transformation-optics generalization of tunnelling effects in bi-layers made of paired pseudo-epsilon-negative/mu-negative media,” J. Opt. 13, 024011 (2011).
[Crossref]

R. T. Thompson, S. A. Cummer, and J. Frauendiener, “A completely covariant approach to transformation optics,” J. Opt. 13, 024008 (2011).
[Crossref]

Laser Photon. Rev. (1)

A. Figotin and I. Vitebskiy, “Slow wave phenomena in photonic crystals,” Laser Photon. Rev. 5, 201–213 (2011).
[Crossref]

Nat. Commun. (1)

A. Davoyan and N. Engheta, “Electrically controlled one-way photon flow in plasmonic nanostructures,” Nat. Commun. 5, 5250 (2014).
[Crossref]

Nat. Mater. (1)

X. Huang, Y. Lai, Z. H. Hang, H. Zheng, and C. T. Chan, “Dirac cones induced by accidental degeneracy in photonic crystals and zero-refractive-index materials,” Nat. Mater. 10, 582–586 (2011).
[Crossref]

Nat. Nanotechnol. (1)

G. A. Wurtz, R. Pollard, W. Hendren, G. Wiederrecht, D. Gosztola, V. Podolskiy, and A. V. Zayats, “Designed ultrafast optical nonlinearity in a plasmonic nanorod metamaterial enhanced by nonlocality,” Nat. Nanotechnol. 6, 107–111 (2011).
[Crossref]

Nat. Photonics (1)

A. Poddubny, I. Iorsh, P. Belov, and Y. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7, 948–957 (2013).
[Crossref]

Nature (2)

K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, “Two-dimensional gas of massless Dirac fermions in graphene,” Nature 438, 197–200 (2005).
[Crossref]

Z. Wang, Y. Chong, J. Joannopoulos, and M. Soljačić, “Observation of unidirectional backscattering-immune topological electromagnetic states,” Nature 461, 772–775 (2009).
[Crossref]

New J. Phys. (2)

X. Lin, Y. Xu, B. Zhang, R. Hao, H. Chen, and E. Li, “Unidirectional surface plasmons in nonreciprocal graphene,” New J. Phys. 15, 113003 (2013).
[Crossref]

S. A. Tretyakov, I. S. Nefedov, and P. Alitalo, “Generalized field-transforming metamaterials,” New J. Phys. 10, 115028 (2008).
[Crossref]

Opt. Express (6)

Opt. Lett. (2)

Phys. Rev. A (3)

R. A. Sepkhanov, Y. B. Bazaliy, and C. W. J. Beenakker, “Extremal transmission at the Dirac point of a photonic band structure,” Phys. Rev. A 75, 063813 (2007).
[Crossref]

B.-I. Popa and S. A. Cummer, “Complex coordinates in transformation optics,” Phys. Rev. A 84, 063837 (2011).
[Crossref]

L. Bergamin, “Generalized transformation optics from triple spacetime metamaterials,” Phys. Rev. A 78, 043825 (2008).
[Crossref]

Phys. Rev. B (11)

L. Bergamin, P. Alitalo, and S. A. Tretyakov, “Nonlinear transformation optics and engineering of the Kerr effect,” Phys. Rev. B 84, 205103 (2011).
[Crossref]

B. M. Wells, A. V. Zayats, and V. A. Podolskiy, “Nonlocal optics of plasmonic nanowire metamaterials,” Phys. Rev. B 89, 035111 (2014).
[Crossref]

A. Ciattoni and C. Rizza, “Nonlocal homogenization theory in metamaterials: effective electromagnetic spatial dispersion and artificial chirality,” Phys. Rev. B 91, 184207 (2015).
[Crossref]

L. Sun, Z. Li, T. S. Luk, X. Yang, and J. Gao, “Nonlocal effective medium analysis in symmetric metal-dielectric multilayer metamaterials,” Phys. Rev. B 91, 195147 (2015).
[Crossref]

A. Figotin and I. Vitebskiy, “Electromagnetic unidirectionality in magnetic photonic crystals,” Phys. Rev. B 67, 165210 (2003).
[Crossref]

M. G. Silveirinha, “Generalized Lorentz-Lorenz formulas for microstructured materials,” Phys. Rev. B 76, 245117 (2007).
[Crossref]

M. G. Silveirinha, “Time domain homogenization of metamaterials,” Phys. Rev. B 83, 165104 (2011).
[Crossref]

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

A. V. Chebykin, A. A. Orlov, C. R. Simovski, Y. S. Kivshar, and P. A. Belov, “Nonlocal effective parameters of multilayered metal-dielectric metamaterials,” Phys. Rev. B 86, 115420 (2012).
[Crossref]

Y. P. Bliokh, V. Freilikher, and F. Nori, “Ballistic charge transport in graphene and light propagation in periodic dielectric structures with metamaterials: a comparative study,” Phys. Rev. B 87, 245134 (2013).
[Crossref]

T. Ochiai and M. Onoda, “Photonic analog of graphene model and its extension: Dirac cone, symmetry, and edge states,” Phys. Rev. B 80, 155103 (2009).
[Crossref]

Phys. Rev. Lett. (6)

O. Peleg, G. Bartal, B. Freedman, O. Manela, M. Segev, and D. N. Christodoulides, “Conical diffraction and gap solitons in honeycomb photonic lattices,” Phys. Rev. Lett. 98, 103901 (2007).
[Crossref]

F. D. M. Haldane and S. Raghu, “Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry,” Phys. Rev. Lett. 100, 013904 (2008).
[Crossref]

X. Zhang, “Observing Zitterbewegung for photons near the Dirac point of a two-dimensional photonic crystal,” Phys. Rev. Lett. 100, 113903 (2008).
[Crossref]

G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Nonlocal transformation optics,” Phys. Rev. Lett. 108, 063902 (2012).
[Crossref]

G. Castaldi, S. Savoia, V. Galdi, A. Alù, and N. Engheta, “PT metamaterials via complex-coordinate transformation optics,” Phys. Rev. Lett. 110, 173901 (2013).
[Crossref]

Z. Yu, G. Veronis, Z. Wang, and S. Fan, “One-way electromagnetic waveguide formed at the interface between a plasmonic metal under a static magnetic field and a photonic crystal,” Phys. Rev. Lett. 100, 023902 (2008).
[Crossref]

Phys. Rev. X (1)

M. Moccia, G. Castaldi, S. Savo, Y. Sato, and V. Galdi, “Independent manipulation of heat and electrical current via bifunctional metamaterials,” Phys. Rev. X 4, 021025 (2014).

Phys. Usp. (1)

E. H. Lock, “The properties of isofrequency dependences and the laws of geometrical optics,” Phys. Usp. 51, 375–394 (2008).
[Crossref]

Rep. Progress Phys. (1)

M. Kadic, T. Bückmann, R. Schittny, and M. Wegener, “Metamaterials beyond electromagnetism,” Rep. Progress Phys. 76, 126501 (2013).
[Crossref]

Science (3)

U. Leonhardt, “Optical conformal mapping,” Science 312, 1777–1780 (2006).
[Crossref]

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312, 1780–1782 (2006).
[Crossref]

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343, 160–163 (2014).
[Crossref]

Other (8)

M. A. Gorlach and P. A. Belov, “Nonlocality in uniaxially polarizable media,” arXiv:1505.01064 (2015).

T. Geng, S. Zhuang, J. Gao, and X. Yang, “Nonlocal effective medium approximation for metallic nanorod metamaterials,” arXiv:1506.00727 (2015).

D. H. Werner and D. H. Kwon, Transformation Electromagnetics and Metamaterials: Fundamental Principles and Applications (Springer, 2013).

L. D. Landau, J. S. Bell, J. Kearsley, L. P. Pitaevskii, E. M. Lifshitz, and J. B. Sykes, Electrodynamics of Continuous Media, Vol. 8 of Course of Theoretical Physics (Elsevier, 1984).

V. M. Agranovich and V. Ginzburg, Crystal Optics with Spatial Dispersion, and Excitons, Springer Series in Solid-State Sciences (Springer, 2013).

S. M. Mikki and A. A. Kishk, “Nonlocal electromagnetic media: a paradigm for material engineering,” in Passive Microwave Components and Antennas, V. Zhurbenko, ed. (InTech, 2010), Chap. 4, pp. 73–94.

A. Leviyev, B. Stein, T. Galfsky, H. Krishnamoorthy, I. L. Kuskovsky, V. Menon, and A. B. Khanikaev, “Nonreciprocity and one-way topological transitions in hyperbolic metamaterials,” arXiv:1505.05438 (2015).

R. L. Haupt and D. H. Werner, Genetic Algorithms in Electromagnetics (Wiley, 2007).

Supplementary Material (1)

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Figures (5)

Fig. 1.
Fig. 1. Schematic of the extended NLTO framework. Starting from an auxiliary space (left panel) r(x,y,z) filled by a homogeneous medium (vacuum) with a given distribution of electric and magnetic sources (J and M, respectively) radiating an EM field (E, H), the associated reciprocal space (center panel) k(kx,ky,kz) is accessed via spatial Fourier transform. Via a frequency-dependent wavenumber transformation, this auxiliary reciprocal space is mapped onto a new reciprocal space k(kx,ky,kz) (right panel), with the transformed fields and sources related to the original ones via Eqs. (3). The implied nonlocal field-manipulation effects can be alternatively interpreted as pertaining to an undistorted reciprocal reference frame k associated with a physical space r filled with a homogeneous, anisotropic “transformation medium” with wavevector-dependent relative permittivity (ϵ̳˜) and permeability (μ̳˜) tensors given by Eq. (4). For notational compactness, the frequency dependence is not shown explicitly.
Fig. 2.
Fig. 2. Geometrical interpretation of the NLTO approach. (a)–(c) Dispersion surface, EFC (at a given ω=ω0), and dispersion diagram (at kx=0), respectively, pertaining to the auxiliary vacuum space [cf. Eq. (9)]. (d)–(f) Corresponding maps in the transformed space for a frequency-independent transformation with a purely imaginary component [cf. Eq. (11)], yielding a hyperbolic dispersion law. (g)–(i) Same as above, but for a frequency-independent, noncentrosymmetric transformation, yielding one-way propagation. (j)–(l) Same as above, but for a frequency-independent transformation yielding a cusp-like point in the EFC and an inflection point in the dispersion diagram (red dots), representative of a frozen mode. (m)–(o) Same as above, but for a two-valued, frequency-dependent transformation yielding Dirac-point conical singularities (red crosses).
Fig. 3.
Fig. 3. (a), (b) Constitutive parameters pertaining to a one-way propagation scenario at the design radian frequency ω=ω0, over the relevant wavenumber ranges. Red-dashed curves represent the NLTO blueprints, obtained from Eqs. (8), (7), (12), and (13) with p2=0.102, kz0=4.5k0, q1=9.452·102, and q2=2.014·102k01. Blue-solid curves pertain to the nonlocal effective model of the synthesized multilayered metamaterial [three-layer unit cell shown in the inset of panel (a)] with ϵ1=1.1, ϵ̳2 given in Eq. (15) [with ϵd=1.1 and ϵg=0.03], ϵ3=2.2, d1=0.08λ0, d2=0.04λ0, and d3=0.07λ0. (c) Prescribed NLTO EFCs compared with actual synthesis results (black-solid), with a magnified view around the nominal design point. (d) Finite-element-computed magnetic-field (magnitude) map in false-color scale (arbitrary units) in the unit cell excited by a magnetic line-source located at x=d1, z=0. The thin gray lines indicate the layer interfaces.
Fig. 4.
Fig. 4. (a), (b) Constitutive parameters pertaining to a frozen-mode scenario at the design radian frequency ω=ω0. Red-dashed curves represent the NLTO blueprints, obtained from Eqs. (8), (7), (12), and (17) with p2=20.349, ν=3, q3=6.068k01, and kz0=k0. Blue-solid curves pertain to the nonlocal effective model of the synthesized multilayered metamaterial [three-layer unit cell shown in the inset of panel (b)] with ϵ1=2.883, ϵ̳2 given in Eq. (15) [with ϵd=1.1 and ϵg=0.03], ϵ3=9.416, d1=0.188λ0, d2=0.049λ0, and d3=0.193λ0. The parameter matching is enforced at kx=0 and within the (green-shaded) region 0.9k0<kz<1.1k0. (c), (d) Prescribed NLTO EFCs and dispersion diagram (with a magnified view around the nominal design point) compared with actual synthesis results (black-solid). (e) Finite-element-computed magnetic-field (magnitude) map in false-color scale (normalized with respect to incident one) in the unit cell due to a plane-wave impinging from vacuum along the positive z direction. The black-dashed and thin gray lines indicate the vacuum–metamaterial and the layer interfaces, respectively. (f) Corresponding magnetic density energy (normalized with respect to incident one, and averaged along the x direction) at z=5.42λ0 as a function of the radian frequency (black-solid). Also shown (red-dashed), as a reference, is the fit in terms of the theoretically predicted behavior |ωω0|2/3.
Fig. 5.
Fig. 5. (a), (b) Constitutive parameters pertaining to a Dirac-point scenario at the design radian frequency ω=ω0. Red-dashed curves represent the NLTO blueprints, obtained from Eqs. (8), (7), (12), and (20) with p2=γx=0.25, γz=2.505·103, and kz0=0.778k0. Blue-solid curves pertain to the nonlocal effective model of the synthesized multilayered metamaterial [four-layer unit cell shown in the inset of panel (b)] with ϵ1=5.7, ωp1=7.025ω0, d1=0.058λ0, ϵ2=6.555, d2=0.277λ0, ϵ3=5.7, ωp3=6.935ω0, d3=0.060λ0, ϵ4=5.00, and d4=0.0965λ0. Due to symmetry, only positive values of kz are shown. (c), (d) Prescribed NLTO EFCs and dispersion diagrams at kz=kz0 and kx=0, respectively (with a magnified view around the nominal Dirac point), compared with actual synthesis results (black-solid). Green-shaded areas indicate the radian-frequency range 0.995ω0<ω<1.005ω0 over which the parameter matching is enforced. (e), (f) Finite-element-computed magnetic-field (magnitude) maps in false-color scale (arbitrary units) due to a collimated Gaussian beam (with waist of 42λ0) impinging from vacuum along the positive z direction, at ω=ω0 and ω=1.05ω0, respectively. The black-dashed line indicates the vacuum–metamaterial interface.

Equations (22)

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G˜(k)=G(r)exp(ik·r)dr.
k=F˜(k,ω)=Λ̳˜T(k,ω)·k,
{E˜,H˜}(k,ω)=Λ̳˜T(k,ω)·{E˜,H˜}[F˜(k,ω)],
{J˜,M˜}(k,ω)=det1[Λ̳˜(k,ω)]Λ̳˜(k,ω)·{J˜,M˜}[F˜(k,ω)],
ϵ̳˜(k,ω)=μ̳˜(k,ω)=det1[Λ̳˜(k,ω)]Λ̳˜(k,ω)·Λ̳˜T(k,ω).
α_(opt)=argminα_{ϵ̳˜(TO)(k,ω)ϵ̳˜(eff)(k,ω;α_)S2+μ̳˜(TO)(k,ω)μ̳˜(eff)(k,ω;α_)S2},
Λ̳˜(k,ω)=Λ̳˜(k,ω)
F˜x(kx,ω)=ωP˜(kx)W(ω),F˜z(kz,ω)=ωQ˜(kz)W(ω),
ϵ˜xx(kz,ω)=kz2W2(ω)ω2Q˜(kz),ϵ˜zz(kx,ω)=kx2W2(ω)ω2P˜(kx).
k·k=ω2c2,
F˜(k,ω)·F˜(k,ω)=ω2c2.
F˜x(kx,kz)=ikx,F˜z(kx,kz)=kz,
P˜(kx)=p2kx2,
Q˜(kz)=(kzkz0)(k02kz0+q1kz+q2kz2)+k02,W(ω)=ω,
q2>kz0q124k02.
ϵ̳2=[ϵd0iϵg010iϵg0ϵd].
nω(kz)n(kz0)=0,n=1,,ν1,νω(kz)ν(kz0)0,
Q˜(kz)=qν(kzkz0)ν+k02,W(ω)=ω,
γxkx2+γz(kzkz0)2=(ωω0)2c2,
Q˜z(kz)=γz(kzkz0)2,W(ω)=ωω0,
Q˜(kz)=γzkz24(kz2kz022),W(ω)=12(ω2ω02)2ω02γzc2kz02,
ϵ1(ω)=ϵ1ωp12ω2,ϵ3(ω)=ϵ3ωp32ω2,

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