Abstract

We show that, under appropriate oblique-incidence and polarization conditions, the inherent opaqueness of a homogeneous, isotropic single-negative slab may be perfectly compensated (in the ideal lossless case) by a homogeneous, anisotropic (uniaxial) double-positive slab, so that complete tunneling (with total transmission and zero phase delay) occurs. We present an analytical and numerical study aimed at deriving the basic design rules, elucidating the underlying physical mechanisms, and exploring the role of the various involved parameters.

© 2011 Optical Society of America

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  1. D. R. Fredkin and A. Ron, “Effective left-handed (negative index) composite material,” Appl. Phys. Lett. 81, 1753–1755(2002).
    [CrossRef]
  2. A. Lakhtakia and C. M. Krowne, “Restricted equivalence of paired epsilon-negative and mu-negative layers to a negative phase-velocity material (alias left-handed material),” Optik 114, 305–307 (2003).
    [CrossRef]
  3. A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558–2571 (2003).
    [CrossRef]
  4. J. B. Pendry and S. A. Ramakrishna, “Focusing light using negative refraction,” J. Phys. 15, 6345–6364 (2003).
    [CrossRef]
  5. H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, and S. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
    [CrossRef]
  6. L.-G. Wang, H. Chen, and S.-Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials,” Phys. Rev. B 70, 245102 (2004).
    [CrossRef]
  7. L. Zhou, W. Wen, C. T. Chan, and P. Sheng, “Electromagnetic-wave tunneling through negative-permittivity media with high magnetic fields,” Phys. Rev. Lett. 94, 243905 (2005).
    [CrossRef]
  8. B. Hou, H. Wen, Y. Leng, and W. Wen, “Electromagnetic wave transmission through subwavelength metallic meshes sandwiched between split rings,” Appl. Phys. Lett. 87, 201114 (2005).
    [CrossRef]
  9. G. Guan, H. Jiang, H. Li, Y. Zhang, H. Chen, and S. Zhu, “Tunneling modes of photonic heterostructures consisting of single-negative materials,” Appl. Phys. Lett. 88, 211112 (2006).
    [CrossRef]
  10. L. Zhang, Y. Zhang, L. He, H. Li, and H. Chen, “Experimental study of photonic crystals consisting of ε-negative and μ-negative materials,” Phys. Rev. E 74, 056615(2006).
    [CrossRef]
  11. X. Zhou and G. Hu, “Total transmission condition for photon tunnelling in a layered structure with metamaterials,” J. Opt. A 9, 60–65 (2007).
    [CrossRef]
  12. Y. Chen, “Defect modes merging in one-dimensional photonic crystals with multiple single-negative material defects,” Appl. Phys. Lett. 92, 011925 (2008).
    [CrossRef]
  13. K.-Y. Kim and B. Lee, “Complete tunneling of light through impedance-mismatched barrier layers,” Phys. Rev. A 77, 023822(2008).
    [CrossRef]
  14. Y. Fang and S. He, “Transparent structure consisting of metamaterial layers and matching layers,” Phys. Rev. A 78, 023813(2008).
    [CrossRef]
  15. T. Feng, Y. Li, H. Jiang, Y. Sun, L. He, H. Li, Y. Zhang, Y. Shi, and H. Chen, “Electromagnetic tunneling in a sandwich structure containing single negative media,” Phys. Rev. E 79, 026601(2009).
    [CrossRef]
  16. H. Oraizi and A. Abdolali, “Mathematical formulation for zero reflection from multilayer metamaterial structures and their notable applications,” IET Microw. Antennas Propag. 3, 987–996 (2009).
    [CrossRef]
  17. Y. Ding, Y. Li, H. Jiang, and H. Chen, “Electromagnetic tunneling in nonconjugated epsilon-negative and mu-negative metamaterial pair,” PIERS Online 6, 109–112 (2010).
    [CrossRef]
  18. C. A. M. Butler, I. R. Hooper, A. P. Hibbins, J. R. Sambles, and P. A. Hobson, “Metamaterial tunnel barrier gives broadband microwave transmission,” J. Appl. Phys. 109, 013104 (2011).
    [CrossRef]
  19. 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]
  20. G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Electromagnetic tunneling through a single-negative slab paired with a double-positive bilayer,” Phys. Rev. B 83, 081105 (2011).
    [CrossRef]
  21. E. Cojocaru, “Electromagnetic tunneling in lossless trilayer stacks containing single-negative metamaterials,” Prog. Electromagn. Res. 113, 227–249 (2011).
    [CrossRef]
  22. V. Lomakin and E. Michielssen, “Enhanced transmission through metallic plates perforated by arrays of subwavelength holes and sandwiched between dielectric slabs,” Phys. Rev. B 71, 235117 (2005).
    [CrossRef]
  23. It can be verified that the reflection-coefficient denominator is always nonzero.
  24. R. Marqués, J. Martel, F. Mesa, and F. Medina, “Left-handed-media simulation and transmission of EM waves in subwavelength split-ring-resonator-loaded metallic waveguides,” Phys. Rev. Lett. 89, 183901 (2002).
    [CrossRef] [PubMed]
  25. J. Esteban, C. Camacho-Peñalosa, J. E. Page, T. M. Martín-Guerrero, and E. Márquez-Segura, “Simulation of negative permittivity and negative permeability by means of evanescent waveguide modes—theory and experiment,” IEEE Trans. Microw. Theory Tech. 53, 1506–1514 (2005).
    [CrossRef]
  26. P. A. Belov, R. Marqués, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and S. A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B 67, 113103 (2003).
    [CrossRef]
  27. A. Sihvola, Electromagnetic Mixing Formulas and Applications (IEE Publishing, 1999).
    [CrossRef]
  28. A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410(2007).
    [CrossRef]
  29. Note that the transmittance peak in the angular response of the stand-alone ENG slab for near-grazing incidence (see inset in Fig. ) is attributable to a pseudo-Brewster condition.
  30. I. R. Hooper, T. W. Preist, and J. R. Sambles, “Making tunnel barriers (including metals) transparent,” Phys. Rev. Lett. 97, 053902 (2006).
    [CrossRef] [PubMed]
  31. L. Jelinek, J. D. Baena, J. Voves, and R. Marques, “Metamaterial-inspired perfect tunneling in semiconductor heterostructures,” New J. Phys 13, 083011 (2010).
    [CrossRef]

2011 (4)

C. A. M. Butler, I. R. Hooper, A. P. Hibbins, J. R. Sambles, and P. A. Hobson, “Metamaterial tunnel barrier gives broadband microwave transmission,” J. Appl. Phys. 109, 013104 (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]

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Electromagnetic tunneling through a single-negative slab paired with a double-positive bilayer,” Phys. Rev. B 83, 081105 (2011).
[CrossRef]

E. Cojocaru, “Electromagnetic tunneling in lossless trilayer stacks containing single-negative metamaterials,” Prog. Electromagn. Res. 113, 227–249 (2011).
[CrossRef]

2010 (2)

Y. Ding, Y. Li, H. Jiang, and H. Chen, “Electromagnetic tunneling in nonconjugated epsilon-negative and mu-negative metamaterial pair,” PIERS Online 6, 109–112 (2010).
[CrossRef]

L. Jelinek, J. D. Baena, J. Voves, and R. Marques, “Metamaterial-inspired perfect tunneling in semiconductor heterostructures,” New J. Phys 13, 083011 (2010).
[CrossRef]

2009 (2)

T. Feng, Y. Li, H. Jiang, Y. Sun, L. He, H. Li, Y. Zhang, Y. Shi, and H. Chen, “Electromagnetic tunneling in a sandwich structure containing single negative media,” Phys. Rev. E 79, 026601(2009).
[CrossRef]

H. Oraizi and A. Abdolali, “Mathematical formulation for zero reflection from multilayer metamaterial structures and their notable applications,” IET Microw. Antennas Propag. 3, 987–996 (2009).
[CrossRef]

2008 (3)

Y. Chen, “Defect modes merging in one-dimensional photonic crystals with multiple single-negative material defects,” Appl. Phys. Lett. 92, 011925 (2008).
[CrossRef]

K.-Y. Kim and B. Lee, “Complete tunneling of light through impedance-mismatched barrier layers,” Phys. Rev. A 77, 023822(2008).
[CrossRef]

Y. Fang and S. He, “Transparent structure consisting of metamaterial layers and matching layers,” Phys. Rev. A 78, 023813(2008).
[CrossRef]

2007 (2)

X. Zhou and G. Hu, “Total transmission condition for photon tunnelling in a layered structure with metamaterials,” J. Opt. A 9, 60–65 (2007).
[CrossRef]

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410(2007).
[CrossRef]

2006 (3)

I. R. Hooper, T. W. Preist, and J. R. Sambles, “Making tunnel barriers (including metals) transparent,” Phys. Rev. Lett. 97, 053902 (2006).
[CrossRef] [PubMed]

G. Guan, H. Jiang, H. Li, Y. Zhang, H. Chen, and S. Zhu, “Tunneling modes of photonic heterostructures consisting of single-negative materials,” Appl. Phys. Lett. 88, 211112 (2006).
[CrossRef]

L. Zhang, Y. Zhang, L. He, H. Li, and H. Chen, “Experimental study of photonic crystals consisting of ε-negative and μ-negative materials,” Phys. Rev. E 74, 056615(2006).
[CrossRef]

2005 (4)

L. Zhou, W. Wen, C. T. Chan, and P. Sheng, “Electromagnetic-wave tunneling through negative-permittivity media with high magnetic fields,” Phys. Rev. Lett. 94, 243905 (2005).
[CrossRef]

B. Hou, H. Wen, Y. Leng, and W. Wen, “Electromagnetic wave transmission through subwavelength metallic meshes sandwiched between split rings,” Appl. Phys. Lett. 87, 201114 (2005).
[CrossRef]

V. Lomakin and E. Michielssen, “Enhanced transmission through metallic plates perforated by arrays of subwavelength holes and sandwiched between dielectric slabs,” Phys. Rev. B 71, 235117 (2005).
[CrossRef]

J. Esteban, C. Camacho-Peñalosa, J. E. Page, T. M. Martín-Guerrero, and E. Márquez-Segura, “Simulation of negative permittivity and negative permeability by means of evanescent waveguide modes—theory and experiment,” IEEE Trans. Microw. Theory Tech. 53, 1506–1514 (2005).
[CrossRef]

2004 (2)

H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, and S. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

L.-G. Wang, H. Chen, and S.-Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials,” Phys. Rev. B 70, 245102 (2004).
[CrossRef]

2003 (4)

A. Lakhtakia and C. M. Krowne, “Restricted equivalence of paired epsilon-negative and mu-negative layers to a negative phase-velocity material (alias left-handed material),” Optik 114, 305–307 (2003).
[CrossRef]

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558–2571 (2003).
[CrossRef]

J. B. Pendry and S. A. Ramakrishna, “Focusing light using negative refraction,” J. Phys. 15, 6345–6364 (2003).
[CrossRef]

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

2002 (2)

R. Marqués, J. Martel, F. Mesa, and F. Medina, “Left-handed-media simulation and transmission of EM waves in subwavelength split-ring-resonator-loaded metallic waveguides,” Phys. Rev. Lett. 89, 183901 (2002).
[CrossRef] [PubMed]

D. R. Fredkin and A. Ron, “Effective left-handed (negative index) composite material,” Appl. Phys. Lett. 81, 1753–1755(2002).
[CrossRef]

Abdolali, A.

H. Oraizi and A. Abdolali, “Mathematical formulation for zero reflection from multilayer metamaterial structures and their notable applications,” IET Microw. Antennas Propag. 3, 987–996 (2009).
[CrossRef]

Alù, A.

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]

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Electromagnetic tunneling through a single-negative slab paired with a double-positive bilayer,” Phys. Rev. B 83, 081105 (2011).
[CrossRef]

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410(2007).
[CrossRef]

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558–2571 (2003).
[CrossRef]

Baena, J. D.

L. Jelinek, J. D. Baena, J. Voves, and R. Marques, “Metamaterial-inspired perfect tunneling in semiconductor heterostructures,” New J. Phys 13, 083011 (2010).
[CrossRef]

Belov, P. A.

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

Butler, C. A. M.

C. A. M. Butler, I. R. Hooper, A. P. Hibbins, J. R. Sambles, and P. A. Hobson, “Metamaterial tunnel barrier gives broadband microwave transmission,” J. Appl. Phys. 109, 013104 (2011).
[CrossRef]

Camacho-Peñalosa, C.

J. Esteban, C. Camacho-Peñalosa, J. E. Page, T. M. Martín-Guerrero, and E. Márquez-Segura, “Simulation of negative permittivity and negative permeability by means of evanescent waveguide modes—theory and experiment,” IEEE Trans. Microw. Theory Tech. 53, 1506–1514 (2005).
[CrossRef]

Castaldi, G.

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Electromagnetic tunneling through a single-negative slab paired with a double-positive bilayer,” Phys. Rev. B 83, 081105 (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]

Chan, C. T.

L. Zhou, W. Wen, C. T. Chan, and P. Sheng, “Electromagnetic-wave tunneling through negative-permittivity media with high magnetic fields,” Phys. Rev. Lett. 94, 243905 (2005).
[CrossRef]

Chen, H.

Y. Ding, Y. Li, H. Jiang, and H. Chen, “Electromagnetic tunneling in nonconjugated epsilon-negative and mu-negative metamaterial pair,” PIERS Online 6, 109–112 (2010).
[CrossRef]

T. Feng, Y. Li, H. Jiang, Y. Sun, L. He, H. Li, Y. Zhang, Y. Shi, and H. Chen, “Electromagnetic tunneling in a sandwich structure containing single negative media,” Phys. Rev. E 79, 026601(2009).
[CrossRef]

L. Zhang, Y. Zhang, L. He, H. Li, and H. Chen, “Experimental study of photonic crystals consisting of ε-negative and μ-negative materials,” Phys. Rev. E 74, 056615(2006).
[CrossRef]

G. Guan, H. Jiang, H. Li, Y. Zhang, H. Chen, and S. Zhu, “Tunneling modes of photonic heterostructures consisting of single-negative materials,” Appl. Phys. Lett. 88, 211112 (2006).
[CrossRef]

H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, and S. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

L.-G. Wang, H. Chen, and S.-Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials,” Phys. Rev. B 70, 245102 (2004).
[CrossRef]

Chen, Y.

Y. Chen, “Defect modes merging in one-dimensional photonic crystals with multiple single-negative material defects,” Appl. Phys. Lett. 92, 011925 (2008).
[CrossRef]

Cojocaru, E.

E. Cojocaru, “Electromagnetic tunneling in lossless trilayer stacks containing single-negative metamaterials,” Prog. Electromagn. Res. 113, 227–249 (2011).
[CrossRef]

Ding, Y.

Y. Ding, Y. Li, H. Jiang, and H. Chen, “Electromagnetic tunneling in nonconjugated epsilon-negative and mu-negative metamaterial pair,” PIERS Online 6, 109–112 (2010).
[CrossRef]

Engheta, N.

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]

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Electromagnetic tunneling through a single-negative slab paired with a double-positive bilayer,” Phys. Rev. B 83, 081105 (2011).
[CrossRef]

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410(2007).
[CrossRef]

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558–2571 (2003).
[CrossRef]

Esteban, J.

J. Esteban, C. Camacho-Peñalosa, J. E. Page, T. M. Martín-Guerrero, and E. Márquez-Segura, “Simulation of negative permittivity and negative permeability by means of evanescent waveguide modes—theory and experiment,” IEEE Trans. Microw. Theory Tech. 53, 1506–1514 (2005).
[CrossRef]

Fang, Y.

Y. Fang and S. He, “Transparent structure consisting of metamaterial layers and matching layers,” Phys. Rev. A 78, 023813(2008).
[CrossRef]

Feng, T.

T. Feng, Y. Li, H. Jiang, Y. Sun, L. He, H. Li, Y. Zhang, Y. Shi, and H. Chen, “Electromagnetic tunneling in a sandwich structure containing single negative media,” Phys. Rev. E 79, 026601(2009).
[CrossRef]

Fredkin, D. R.

D. R. Fredkin and A. Ron, “Effective left-handed (negative index) composite material,” Appl. Phys. Lett. 81, 1753–1755(2002).
[CrossRef]

Galdi, V.

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]

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Electromagnetic tunneling through a single-negative slab paired with a double-positive bilayer,” Phys. Rev. B 83, 081105 (2011).
[CrossRef]

Gallina, I.

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Electromagnetic tunneling through a single-negative slab paired with a double-positive bilayer,” Phys. Rev. B 83, 081105 (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]

Guan, G.

G. Guan, H. Jiang, H. Li, Y. Zhang, H. Chen, and S. Zhu, “Tunneling modes of photonic heterostructures consisting of single-negative materials,” Appl. Phys. Lett. 88, 211112 (2006).
[CrossRef]

He, L.

T. Feng, Y. Li, H. Jiang, Y. Sun, L. He, H. Li, Y. Zhang, Y. Shi, and H. Chen, “Electromagnetic tunneling in a sandwich structure containing single negative media,” Phys. Rev. E 79, 026601(2009).
[CrossRef]

L. Zhang, Y. Zhang, L. He, H. Li, and H. Chen, “Experimental study of photonic crystals consisting of ε-negative and μ-negative materials,” Phys. Rev. E 74, 056615(2006).
[CrossRef]

He, S.

Y. Fang and S. He, “Transparent structure consisting of metamaterial layers and matching layers,” Phys. Rev. A 78, 023813(2008).
[CrossRef]

Hibbins, A. P.

C. A. M. Butler, I. R. Hooper, A. P. Hibbins, J. R. Sambles, and P. A. Hobson, “Metamaterial tunnel barrier gives broadband microwave transmission,” J. Appl. Phys. 109, 013104 (2011).
[CrossRef]

Hobson, P. A.

C. A. M. Butler, I. R. Hooper, A. P. Hibbins, J. R. Sambles, and P. A. Hobson, “Metamaterial tunnel barrier gives broadband microwave transmission,” J. Appl. Phys. 109, 013104 (2011).
[CrossRef]

Hooper, I. R.

C. A. M. Butler, I. R. Hooper, A. P. Hibbins, J. R. Sambles, and P. A. Hobson, “Metamaterial tunnel barrier gives broadband microwave transmission,” J. Appl. Phys. 109, 013104 (2011).
[CrossRef]

I. R. Hooper, T. W. Preist, and J. R. Sambles, “Making tunnel barriers (including metals) transparent,” Phys. Rev. Lett. 97, 053902 (2006).
[CrossRef] [PubMed]

Hou, B.

B. Hou, H. Wen, Y. Leng, and W. Wen, “Electromagnetic wave transmission through subwavelength metallic meshes sandwiched between split rings,” Appl. Phys. Lett. 87, 201114 (2005).
[CrossRef]

Hu, G.

X. Zhou and G. Hu, “Total transmission condition for photon tunnelling in a layered structure with metamaterials,” J. Opt. A 9, 60–65 (2007).
[CrossRef]

Jelinek, L.

L. Jelinek, J. D. Baena, J. Voves, and R. Marques, “Metamaterial-inspired perfect tunneling in semiconductor heterostructures,” New J. Phys 13, 083011 (2010).
[CrossRef]

Jiang, H.

Y. Ding, Y. Li, H. Jiang, and H. Chen, “Electromagnetic tunneling in nonconjugated epsilon-negative and mu-negative metamaterial pair,” PIERS Online 6, 109–112 (2010).
[CrossRef]

T. Feng, Y. Li, H. Jiang, Y. Sun, L. He, H. Li, Y. Zhang, Y. Shi, and H. Chen, “Electromagnetic tunneling in a sandwich structure containing single negative media,” Phys. Rev. E 79, 026601(2009).
[CrossRef]

G. Guan, H. Jiang, H. Li, Y. Zhang, H. Chen, and S. Zhu, “Tunneling modes of photonic heterostructures consisting of single-negative materials,” Appl. Phys. Lett. 88, 211112 (2006).
[CrossRef]

H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, and S. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

Kim, K.-Y.

K.-Y. Kim and B. Lee, “Complete tunneling of light through impedance-mismatched barrier layers,” Phys. Rev. A 77, 023822(2008).
[CrossRef]

Krowne, C. M.

A. Lakhtakia and C. M. Krowne, “Restricted equivalence of paired epsilon-negative and mu-negative layers to a negative phase-velocity material (alias left-handed material),” Optik 114, 305–307 (2003).
[CrossRef]

Lakhtakia, A.

A. Lakhtakia and C. M. Krowne, “Restricted equivalence of paired epsilon-negative and mu-negative layers to a negative phase-velocity material (alias left-handed material),” Optik 114, 305–307 (2003).
[CrossRef]

Lee, B.

K.-Y. Kim and B. Lee, “Complete tunneling of light through impedance-mismatched barrier layers,” Phys. Rev. A 77, 023822(2008).
[CrossRef]

Leng, Y.

B. Hou, H. Wen, Y. Leng, and W. Wen, “Electromagnetic wave transmission through subwavelength metallic meshes sandwiched between split rings,” Appl. Phys. Lett. 87, 201114 (2005).
[CrossRef]

Li, H.

T. Feng, Y. Li, H. Jiang, Y. Sun, L. He, H. Li, Y. Zhang, Y. Shi, and H. Chen, “Electromagnetic tunneling in a sandwich structure containing single negative media,” Phys. Rev. E 79, 026601(2009).
[CrossRef]

L. Zhang, Y. Zhang, L. He, H. Li, and H. Chen, “Experimental study of photonic crystals consisting of ε-negative and μ-negative materials,” Phys. Rev. E 74, 056615(2006).
[CrossRef]

G. Guan, H. Jiang, H. Li, Y. Zhang, H. Chen, and S. Zhu, “Tunneling modes of photonic heterostructures consisting of single-negative materials,” Appl. Phys. Lett. 88, 211112 (2006).
[CrossRef]

H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, and S. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

Li, Y.

Y. Ding, Y. Li, H. Jiang, and H. Chen, “Electromagnetic tunneling in nonconjugated epsilon-negative and mu-negative metamaterial pair,” PIERS Online 6, 109–112 (2010).
[CrossRef]

T. Feng, Y. Li, H. Jiang, Y. Sun, L. He, H. Li, Y. Zhang, Y. Shi, and H. Chen, “Electromagnetic tunneling in a sandwich structure containing single negative media,” Phys. Rev. E 79, 026601(2009).
[CrossRef]

Lomakin, V.

V. Lomakin and E. Michielssen, “Enhanced transmission through metallic plates perforated by arrays of subwavelength holes and sandwiched between dielectric slabs,” Phys. Rev. B 71, 235117 (2005).
[CrossRef]

Marques, R.

L. Jelinek, J. D. Baena, J. Voves, and R. Marques, “Metamaterial-inspired perfect tunneling in semiconductor heterostructures,” New J. Phys 13, 083011 (2010).
[CrossRef]

Marqués, R.

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

R. Marqués, J. Martel, F. Mesa, and F. Medina, “Left-handed-media simulation and transmission of EM waves in subwavelength split-ring-resonator-loaded metallic waveguides,” Phys. Rev. Lett. 89, 183901 (2002).
[CrossRef] [PubMed]

Márquez-Segura, E.

J. Esteban, C. Camacho-Peñalosa, J. E. Page, T. M. Martín-Guerrero, and E. Márquez-Segura, “Simulation of negative permittivity and negative permeability by means of evanescent waveguide modes—theory and experiment,” IEEE Trans. Microw. Theory Tech. 53, 1506–1514 (2005).
[CrossRef]

Martel, J.

R. Marqués, J. Martel, F. Mesa, and F. Medina, “Left-handed-media simulation and transmission of EM waves in subwavelength split-ring-resonator-loaded metallic waveguides,” Phys. Rev. Lett. 89, 183901 (2002).
[CrossRef] [PubMed]

Martín-Guerrero, T. M.

J. Esteban, C. Camacho-Peñalosa, J. E. Page, T. M. Martín-Guerrero, and E. Márquez-Segura, “Simulation of negative permittivity and negative permeability by means of evanescent waveguide modes—theory and experiment,” IEEE Trans. Microw. Theory Tech. 53, 1506–1514 (2005).
[CrossRef]

Maslovski, S. I.

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

Medina, F.

R. Marqués, J. Martel, F. Mesa, and F. Medina, “Left-handed-media simulation and transmission of EM waves in subwavelength split-ring-resonator-loaded metallic waveguides,” Phys. Rev. Lett. 89, 183901 (2002).
[CrossRef] [PubMed]

Mesa, F.

R. Marqués, J. Martel, F. Mesa, and F. Medina, “Left-handed-media simulation and transmission of EM waves in subwavelength split-ring-resonator-loaded metallic waveguides,” Phys. Rev. Lett. 89, 183901 (2002).
[CrossRef] [PubMed]

Michielssen, E.

V. Lomakin and E. Michielssen, “Enhanced transmission through metallic plates perforated by arrays of subwavelength holes and sandwiched between dielectric slabs,” Phys. Rev. B 71, 235117 (2005).
[CrossRef]

Nefedov, I. S.

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

Oraizi, H.

H. Oraizi and A. Abdolali, “Mathematical formulation for zero reflection from multilayer metamaterial structures and their notable applications,” IET Microw. Antennas Propag. 3, 987–996 (2009).
[CrossRef]

Page, J. E.

J. Esteban, C. Camacho-Peñalosa, J. E. Page, T. M. Martín-Guerrero, and E. Márquez-Segura, “Simulation of negative permittivity and negative permeability by means of evanescent waveguide modes—theory and experiment,” IEEE Trans. Microw. Theory Tech. 53, 1506–1514 (2005).
[CrossRef]

Pendry, J. B.

J. B. Pendry and S. A. Ramakrishna, “Focusing light using negative refraction,” J. Phys. 15, 6345–6364 (2003).
[CrossRef]

Preist, T. W.

I. R. Hooper, T. W. Preist, and J. R. Sambles, “Making tunnel barriers (including metals) transparent,” Phys. Rev. Lett. 97, 053902 (2006).
[CrossRef] [PubMed]

Ramakrishna, S. A.

J. B. Pendry and S. A. Ramakrishna, “Focusing light using negative refraction,” J. Phys. 15, 6345–6364 (2003).
[CrossRef]

Ron, A.

D. R. Fredkin and A. Ron, “Effective left-handed (negative index) composite material,” Appl. Phys. Lett. 81, 1753–1755(2002).
[CrossRef]

Salandrino, A.

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410(2007).
[CrossRef]

Sambles, J. R.

C. A. M. Butler, I. R. Hooper, A. P. Hibbins, J. R. Sambles, and P. A. Hobson, “Metamaterial tunnel barrier gives broadband microwave transmission,” J. Appl. Phys. 109, 013104 (2011).
[CrossRef]

I. R. Hooper, T. W. Preist, and J. R. Sambles, “Making tunnel barriers (including metals) transparent,” Phys. Rev. Lett. 97, 053902 (2006).
[CrossRef] [PubMed]

Sheng, P.

L. Zhou, W. Wen, C. T. Chan, and P. Sheng, “Electromagnetic-wave tunneling through negative-permittivity media with high magnetic fields,” Phys. Rev. Lett. 94, 243905 (2005).
[CrossRef]

Shi, Y.

T. Feng, Y. Li, H. Jiang, Y. Sun, L. He, H. Li, Y. Zhang, Y. Shi, and H. Chen, “Electromagnetic tunneling in a sandwich structure containing single negative media,” Phys. Rev. E 79, 026601(2009).
[CrossRef]

Sihvola, A.

A. Sihvola, Electromagnetic Mixing Formulas and Applications (IEE Publishing, 1999).
[CrossRef]

Silveirinha, M.

P. A. Belov, R. Marqués, S. I. Maslovski, I. S. Nefedov, M. Silveirinha, C. R. Simovski, and 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.

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410(2007).
[CrossRef]

Simovski, C. R.

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

Sun, Y.

T. Feng, Y. Li, H. Jiang, Y. Sun, L. He, H. Li, Y. Zhang, Y. Shi, and H. Chen, “Electromagnetic tunneling in a sandwich structure containing single negative media,” Phys. Rev. E 79, 026601(2009).
[CrossRef]

Tretyakov, S. A.

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

Voves, J.

L. Jelinek, J. D. Baena, J. Voves, and R. Marques, “Metamaterial-inspired perfect tunneling in semiconductor heterostructures,” New J. Phys 13, 083011 (2010).
[CrossRef]

Wang, L.-G.

L.-G. Wang, H. Chen, and S.-Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials,” Phys. Rev. B 70, 245102 (2004).
[CrossRef]

Wen, H.

B. Hou, H. Wen, Y. Leng, and W. Wen, “Electromagnetic wave transmission through subwavelength metallic meshes sandwiched between split rings,” Appl. Phys. Lett. 87, 201114 (2005).
[CrossRef]

Wen, W.

B. Hou, H. Wen, Y. Leng, and W. Wen, “Electromagnetic wave transmission through subwavelength metallic meshes sandwiched between split rings,” Appl. Phys. Lett. 87, 201114 (2005).
[CrossRef]

L. Zhou, W. Wen, C. T. Chan, and P. Sheng, “Electromagnetic-wave tunneling through negative-permittivity media with high magnetic fields,” Phys. Rev. Lett. 94, 243905 (2005).
[CrossRef]

Zhang, L.

L. Zhang, Y. Zhang, L. He, H. Li, and H. Chen, “Experimental study of photonic crystals consisting of ε-negative and μ-negative materials,” Phys. Rev. E 74, 056615(2006).
[CrossRef]

Zhang, Y.

T. Feng, Y. Li, H. Jiang, Y. Sun, L. He, H. Li, Y. Zhang, Y. Shi, and H. Chen, “Electromagnetic tunneling in a sandwich structure containing single negative media,” Phys. Rev. E 79, 026601(2009).
[CrossRef]

L. Zhang, Y. Zhang, L. He, H. Li, and H. Chen, “Experimental study of photonic crystals consisting of ε-negative and μ-negative materials,” Phys. Rev. E 74, 056615(2006).
[CrossRef]

G. Guan, H. Jiang, H. Li, Y. Zhang, H. Chen, and S. Zhu, “Tunneling modes of photonic heterostructures consisting of single-negative materials,” Appl. Phys. Lett. 88, 211112 (2006).
[CrossRef]

H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, and S. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

Zhou, L.

L. Zhou, W. Wen, C. T. Chan, and P. Sheng, “Electromagnetic-wave tunneling through negative-permittivity media with high magnetic fields,” Phys. Rev. Lett. 94, 243905 (2005).
[CrossRef]

Zhou, X.

X. Zhou and G. Hu, “Total transmission condition for photon tunnelling in a layered structure with metamaterials,” J. Opt. A 9, 60–65 (2007).
[CrossRef]

Zhu, S.

G. Guan, H. Jiang, H. Li, Y. Zhang, H. Chen, and S. Zhu, “Tunneling modes of photonic heterostructures consisting of single-negative materials,” Appl. Phys. Lett. 88, 211112 (2006).
[CrossRef]

H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, and S. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

Zhu, S.-Y.

L.-G. Wang, H. Chen, and S.-Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials,” Phys. Rev. B 70, 245102 (2004).
[CrossRef]

Zi, J.

H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, and S. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

Appl. Phys. Lett. (4)

D. R. Fredkin and A. Ron, “Effective left-handed (negative index) composite material,” Appl. Phys. Lett. 81, 1753–1755(2002).
[CrossRef]

B. Hou, H. Wen, Y. Leng, and W. Wen, “Electromagnetic wave transmission through subwavelength metallic meshes sandwiched between split rings,” Appl. Phys. Lett. 87, 201114 (2005).
[CrossRef]

G. Guan, H. Jiang, H. Li, Y. Zhang, H. Chen, and S. Zhu, “Tunneling modes of photonic heterostructures consisting of single-negative materials,” Appl. Phys. Lett. 88, 211112 (2006).
[CrossRef]

Y. Chen, “Defect modes merging in one-dimensional photonic crystals with multiple single-negative material defects,” Appl. Phys. Lett. 92, 011925 (2008).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

A. Alù and N. Engheta, “Pairing an epsilon-negative slab with a mu-negative slab: resonance, tunneling and transparency,” IEEE Trans. Antennas Propag. 51, 2558–2571 (2003).
[CrossRef]

IEEE Trans. Microw. Theory Tech. (1)

J. Esteban, C. Camacho-Peñalosa, J. E. Page, T. M. Martín-Guerrero, and E. Márquez-Segura, “Simulation of negative permittivity and negative permeability by means of evanescent waveguide modes—theory and experiment,” IEEE Trans. Microw. Theory Tech. 53, 1506–1514 (2005).
[CrossRef]

IET Microw. Antennas Propag. (1)

H. Oraizi and A. Abdolali, “Mathematical formulation for zero reflection from multilayer metamaterial structures and their notable applications,” IET Microw. Antennas Propag. 3, 987–996 (2009).
[CrossRef]

J. Appl. Phys. (1)

C. A. M. Butler, I. R. Hooper, A. P. Hibbins, J. R. Sambles, and P. A. Hobson, “Metamaterial tunnel barrier gives broadband microwave transmission,” J. Appl. Phys. 109, 013104 (2011).
[CrossRef]

J. Opt. (1)

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]

J. Opt. A (1)

X. Zhou and G. Hu, “Total transmission condition for photon tunnelling in a layered structure with metamaterials,” J. Opt. A 9, 60–65 (2007).
[CrossRef]

J. Phys. (1)

J. B. Pendry and S. A. Ramakrishna, “Focusing light using negative refraction,” J. Phys. 15, 6345–6364 (2003).
[CrossRef]

New J. Phys (1)

L. Jelinek, J. D. Baena, J. Voves, and R. Marques, “Metamaterial-inspired perfect tunneling in semiconductor heterostructures,” New J. Phys 13, 083011 (2010).
[CrossRef]

Optik (1)

A. Lakhtakia and C. M. Krowne, “Restricted equivalence of paired epsilon-negative and mu-negative layers to a negative phase-velocity material (alias left-handed material),” Optik 114, 305–307 (2003).
[CrossRef]

Phys. Rev. A (2)

K.-Y. Kim and B. Lee, “Complete tunneling of light through impedance-mismatched barrier layers,” Phys. Rev. A 77, 023822(2008).
[CrossRef]

Y. Fang and S. He, “Transparent structure consisting of metamaterial layers and matching layers,” Phys. Rev. A 78, 023813(2008).
[CrossRef]

Phys. Rev. B (5)

G. Castaldi, I. Gallina, V. Galdi, A. Alù, and N. Engheta, “Electromagnetic tunneling through a single-negative slab paired with a double-positive bilayer,” Phys. Rev. B 83, 081105 (2011).
[CrossRef]

L.-G. Wang, H. Chen, and S.-Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals with single-negative materials,” Phys. Rev. B 70, 245102 (2004).
[CrossRef]

A. Alù, M. G. Silveirinha, A. Salandrino, and N. Engheta, “Epsilon-near-zero metamaterials and electromagnetic sources: tailoring the radiation phase pattern,” Phys. Rev. B 75, 155410(2007).
[CrossRef]

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

V. Lomakin and E. Michielssen, “Enhanced transmission through metallic plates perforated by arrays of subwavelength holes and sandwiched between dielectric slabs,” Phys. Rev. B 71, 235117 (2005).
[CrossRef]

Phys. Rev. E (3)

L. Zhang, Y. Zhang, L. He, H. Li, and H. Chen, “Experimental study of photonic crystals consisting of ε-negative and μ-negative materials,” Phys. Rev. E 74, 056615(2006).
[CrossRef]

H. Jiang, H. Chen, H. Li, Y. Zhang, J. Zi, and S. Zhu, “Properties of one-dimensional photonic crystals containing single-negative materials,” Phys. Rev. E 69, 066607 (2004).
[CrossRef]

T. Feng, Y. Li, H. Jiang, Y. Sun, L. He, H. Li, Y. Zhang, Y. Shi, and H. Chen, “Electromagnetic tunneling in a sandwich structure containing single negative media,” Phys. Rev. E 79, 026601(2009).
[CrossRef]

Phys. Rev. Lett. (3)

L. Zhou, W. Wen, C. T. Chan, and P. Sheng, “Electromagnetic-wave tunneling through negative-permittivity media with high magnetic fields,” Phys. Rev. Lett. 94, 243905 (2005).
[CrossRef]

R. Marqués, J. Martel, F. Mesa, and F. Medina, “Left-handed-media simulation and transmission of EM waves in subwavelength split-ring-resonator-loaded metallic waveguides,” Phys. Rev. Lett. 89, 183901 (2002).
[CrossRef] [PubMed]

I. R. Hooper, T. W. Preist, and J. R. Sambles, “Making tunnel barriers (including metals) transparent,” Phys. Rev. Lett. 97, 053902 (2006).
[CrossRef] [PubMed]

PIERS Online (1)

Y. Ding, Y. Li, H. Jiang, and H. Chen, “Electromagnetic tunneling in nonconjugated epsilon-negative and mu-negative metamaterial pair,” PIERS Online 6, 109–112 (2010).
[CrossRef]

Prog. Electromagn. Res. (1)

E. Cojocaru, “Electromagnetic tunneling in lossless trilayer stacks containing single-negative metamaterials,” Prog. Electromagn. Res. 113, 227–249 (2011).
[CrossRef]

Other (3)

It can be verified that the reflection-coefficient denominator is always nonzero.

A. Sihvola, Electromagnetic Mixing Formulas and Applications (IEE Publishing, 1999).
[CrossRef]

Note that the transmittance peak in the angular response of the stand-alone ENG slab for near-grazing incidence (see inset in Fig. ) is attributable to a pseudo-Brewster condition.

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Figures (10)

Fig. 1
Fig. 1

Problem schematic in the associated Cartesian reference system: we consider a homogeneous, isotropic slab of ENG material of thickness d 1 and relative permittivity ε 1 [with Re ( ε 1 ) < 0 ] paired with a homogeneous, anisotropic (uniaxial) DPS slab of thickness d 2 and relative permittivity tensor ε 2 ̲ ̲ given in Eq. (1). The bilayer is immersed in vacuum, and it is illuminated by an obliquely incident, TM-polarized plane wave.

Fig. 2
Fig. 2

(a) Intensity and (b) phase distributions of TM (solid) and TE (dashed) fields (normalized by the incident values) at resonance, for an ideal, lossless bilayer as in Fig. 1, with ε 1 = 3 , d 1 = d 2 = 0.1 λ 0 , ε 2 = 0.12 , and ε 2 = 3 , under TM-polarized, obliquely incident illumination with θ i = θ i 0 = 30 ° .

Fig. 3
Fig. 3

Transmittance (for θ i = θ i 0 = 30 ° ) frequency response, for TM (solid blue curve) and TE (dashed red curve), pertaining to the parameter configuration as in Fig. 2, but considering for the ENG medium the Drude-type model [Eq. (18)] in with ω p 1 = 2 ω 0 , γ 1 = 3.75 · 10 3 ω p 1 (so that Re [ ε 1 ( ω 0 ) ] 3 ), and for the uniaxial-DPS medium the mixing rules in Eqs. (19, 20), with τ = 0.252 , ω p a = 0.984 ω 0 , γ a = 3.24 · 10 4 ω p a (so that Re [ ε 2 ( ω 0 ) ] 0.12 and Re [ ε 2 ( ω 0 ) ] 3 ). Also shown as a reference (in the inset) is the response of the stand-alone ENG slab.

Fig. 4
Fig. 4

As in Fig. 3, but with an angular response at resonance.

Fig. 5
Fig. 5

(a) Magnitude and (b) phase of the reflection-coefficient frequency response (for θ i = θ i 0 = 30 ° ) pertaining to the stand-alone uniaxial-DPS slab (solid blue curve) with parameters as in Figs. 2, 3 (for TM polarization, and assuming zero losses), compared with that of an effective (homogeneous, isotropic) matched [see Eq. (21)] MNG slab (dashed red curve), with the relative permeability described by a lossless Drude-type model μ 2 e ( ω ) = 1 2 ω 0 2 / ω 2 .

Fig. 6
Fig. 6

As in Fig. 5, but with an angular response at resonance.

Fig. 7
Fig. 7

As in Fig. 3, but for θ i = θ i 0 = 15 ° , and an ENG slab [see Eq. (18)] with ω p 1 = 10.05 ω 0 , γ 1 = 9.85 · 10 4 ω p 1 (i.e., Re [ ε 1 ( ω 0 ) ] 100 ), and d 1 = λ 0 / 100 , and a matched uniaxial-DPS slab [see Eqs. (19) and (20)] with τ = 0.251 , ω p a = 0.992 ω 0 , γ a = 1.69 · 10 4 ω p a (i.e., Re [ ε 2 ( ω 0 ) ] 0.065 and Re [ ε 2 ( ω 0 ) ] 3 ), and d 2 = λ 0 / 3 .

Fig. 8
Fig. 8

As in Fig. 7, but with an angular response at resonance.

Fig. 9
Fig. 9

As in Fig. 3, but for an ENG slab [see Eq. (18)] with ω p 1 = 1.58 ω 0 , γ 1 = 3.8 · 10 5 ω p 1 (i.e., Re [ ε 1 ( ω 0 ) ] 1.5 ), and d 1 = λ 0 / 2 , and a matched uniaxial-DPS slab [see Eqs. (19, 20)] with τ = 0.637 , ω p a = 0.962 ω 0 , γ a = 8.34 · 10 6 ω p a (i.e., Re [ ε 2 ( ω 0 ) ] 0.115 and Re [ ε 2 ( ω 0 ) ] 1.5 ), ε b = 4 ( 1 + 10 4 i ) , and d 2 = λ 0 / 2 . Note the semilog scale and the narrower frequency range considered.

Fig. 10
Fig. 10

As in Fig. 9, but with an angular response at resonance. Note the semilog scale and the narrower angular range considered.

Equations (21)

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ε 2 ̲ ̲ = [ ε 2 0 0 0 ε 2 0 0 0 ε 2 ] , Re ( ε 2 ) > 0 , Re ( ε 2 ) > 0 .
H z i ( x , y ) = exp [ i k ( x cos θ i + y sin θ i ) ] ,
H z ( x , y ) = { H z i + B 0 exp [ i k ( x cos θ i + y sin θ i ) ] , x < d 1 , exp ( i k y sin θ i ) [ A 1 cosh ( α 1 x ) + B 1 sinh ( α 1 x ) ] , d 1 < x < 0 , exp ( i k y sin θ i ) [ A 2 cosh ( α 2 x ) + B 2 sinh ( α 2 x ) ] , 0 < x < d 2 , A 3 exp [ i k ( x cos θ i + y sin θ i ) ] , x > d 2 ,
α 1 = k | ε 1 | + sin 2 θ i ,
α 2 = k ε 2 ( sin 2 θ i ε 2 1 ) , sin 2 θ i > ε 2 ,
ε 2 α 2 ( α 1 2 + ε 1 2 k 2 cos 2 θ i ) tanh ( α 1 d 1 ) + ε 1 α 1 ( α 2 2 + ε 2 2 k 2 cos 2 θ i ) tanh ( α 2 d 2 ) + i k cos θ i ( ε 1 2 α 2 2 ε 2 2 α 1 2 ) tanh ( α 1 d 1 ) tanh ( α 2 d 2 ) = 0 .
ε 2 = ε 1 2 sin 2 θ i ε 2 ( | ε 1 | + sin 2 θ i ) + ε 1 2 ,
ε 2 2 α 1 ( α 1 2 + ε 1 2 k 2 cos 2 θ i ) | ε 1 | cosh ( α 1 d 1 ε 1 ) cosh ( ε 2 α 1 d 1 d 2 ε 1 ) sinh [ α 1 ( d 1 d 2 ε 2 | ε 1 | ) ] = 0 ,
| ε 1 | d 1 = ε 2 d 2 .
A 3 = exp [ i k cos θ i ( d 1 + d 2 ) ] ,
| H z ( x ) | 2 = { { cosh 2 [ α 1 ( x + d 1 ) ] + ε 1 2 k 2 cos 2 θ i α 1 2 sinh 2 [ α 1 ( x + d 1 ) ] } , d 1 < x < 0 , { cosh 2 [ α 1 ε 2 ( x d 2 ) ε 1 ] + ε 1 2 k 2 cos 2 θ i α 1 2 sinh 2 [ α 1 ε 2 ( x d 2 ) ε 1 ] } , 0 < x < d 2 ,
{ ε 2 e = d 1 d 2 | ε 1 | , μ 2 e = ( 1 ε 2 e ε 2 e ε 1 2 ) sin 2 θ i ε 2 e | ε 1 | ,
| sin θ i | ε 2 e | ε 1 | ε 1 2 ε 2 e 2 .
α 2 e = k sin 2 θ i ε 2 e μ 2 e ,
η 2 e = i η α 2 e k ε 2 e ,
η 2 = i η α 2 k ε 2 .
{ ε 2 = ε 2 e , α 2 = α 2 e , η 2 = η 2 e .
ε 1 ( ω ) = 1 ω p 1 2 ω ( ω + i γ 1 ) ,
{ ε 2 ( ω ) = [ τ ε a ( ω ) + 1 τ ε b ] 1 , ε 2 ( ω ) = τ ε a ( ω ) + ( 1 τ ) ε b ,
ε a ( ω ) = 1 ω p a 2 ω ( ω + i γ a ) , ε b = 4 ( 1 + 10 3 i ) ,
ε 2 e = 3 , μ 2 e ( ω 0 ) = 1 ,

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