Abstract

We propose a method to realize directive emission using a layered metal heterostructure with around-zero average permittivity in the optical region. In the long-wavelength limit the heterostructure can be viewed as an epsilon-near-zero metamaterial. Our results show that when a transverse electric polarized source is placed in front of the heterostructure only the lights around the lines normal to the surfaces of the heterostructure can propagate through it. By exploiting constructive interference we can achieve good directivity and high transmission simultaneously. In addition, if the source is embedded in the heterostructure, the energy radiated by the source will be concentrated in a narrow cone.

© 2011 Optical Society of America

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  1. T. Akalin, J. Danglot, O. Vanbésien, and D. Lippens, “A highly directive dipole antenna embedded in a Fabry–Pérot type cavity,” IEEE Microw. Wireless Compon. Lett. 12, 48–50(2002).
    [CrossRef]
  2. B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K. M. Ho, “Photonic crystal-based resonant antenna with a very high directivity,” J. Appl. Phys. 87, 603–605(2000).
    [CrossRef]
  3. S. N. Burokur, J. P. Daniel, P. Ratajczak, and A. de Lustrac, “Tunable bilayered metasurface for frequency reconfigurable directive emissions,” Appl. Phys. Lett. 97, 064101 (2010).
    [CrossRef]
  4. N. V. Q. Tran, S. Combrié, and A. De Rossi, “Directive emission from high-Q photonic crystal cavities through band folding,” Phys. Rev. B 79, 041101 (2009).
    [CrossRef]
  5. N. V. Q. Tran, S. Combrié, P. Colman, A. De Rossi, and T. Mei, “Vertical high emission in photonic crystal nanocavities by band-folding design,” Phys. Rev. B 82, 075120 (2010).
    [CrossRef]
  6. L. Zhou, H. Q. Li, Y. Q. Qin, Z. Y. Wei, and C. T. Chan, “Directive emissions from subwavelength metamaterial-based cavities,” Appl. Phys. Lett. 86, 101101 (2005).
    [CrossRef]
  7. I. Bulu, H. Caglayan, and E. Ozbay, “Highly directive radiation from sources embedded inside photonic crystals,” Appl. Phys. Lett. 83, 3263–3265 (2003).
    [CrossRef]
  8. H. Caglayan, I. Bulu, and E. Ozbay, “Highly directional enhanced radiation from sources embedded inside three-dimensional photonic crystals,” Opt. Express 13, 7645–7652 (2005).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  19. A. Alù and N. Engheta, “Dielectric sensing in ε-near-zero narrow waveguide channels,” Phys. Rev. B 78, 045102 (2008).
    [CrossRef]
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    [CrossRef]
  21. B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100, 033903 (2008).
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  22. R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100, 023903 (2008).
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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2011 (2)

Y. Xu and H. Chen, “Total reflection and transmission by epsilon-near-zero metamaterials with defects,” Appl. Phys. Lett. 98, 113501 (2011).
[CrossRef]

J. Luo, P. Xu, T. Sun, and L. Gao, “Tunable beam splitting and optical negative refraction in heterostructure with metamaterial,” Appl. Phys. A 104, 1137–1142 (2011).
[CrossRef]

2010 (5)

C. C. Yan, D. H. Zhang, Y. A. Zhang, D. D. Li, and M. A. Fiddy, “Metal-dielectric composites for beam splitting and far-field deep sub-wavelength resolution for visible wavelengths,” Opt. Express 18, 14794–14801 (2010).
[CrossRef] [PubMed]

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

B. Wang and K. M. Huang, “Shaping the radiation pattern with mu and epsilon-near-zero metamaterials,” PIER 106, 107–119(2010).
[CrossRef]

S. N. Burokur, J. P. Daniel, P. Ratajczak, and A. de Lustrac, “Tunable bilayered metasurface for frequency reconfigurable directive emissions,” Appl. Phys. Lett. 97, 064101 (2010).
[CrossRef]

N. V. Q. Tran, S. Combrié, P. Colman, A. De Rossi, and T. Mei, “Vertical high emission in photonic crystal nanocavities by band-folding design,” Phys. Rev. B 82, 075120 (2010).
[CrossRef]

2009 (3)

N. V. Q. Tran, S. Combrié, and A. De Rossi, “Directive emission from high-Q photonic crystal cavities through band folding,” Phys. Rev. B 79, 041101 (2009).
[CrossRef]

F. M. Zhu, Z. Y. Wang, T. Jiang, L. F. Shen, and L. X. Ran, “Directive emission based on a new type of metamaterial,” Microwave Opt. Technol. Lett. 51, 2178–2180 (2009).
[CrossRef]

B. Edwards, A. Alù, M. G. Silveirinha, and N. Engheta, “Reflectionless sharp bends and corners in waveguides using epsilon-near-zero effects,” J. Appl. Phys. 105, 044905 (2009).
[CrossRef]

2008 (5)

Y. Yuan, L. F. Shen, L. X. Ran, T. Jiang, J. T. Huangfu, and J. A. Kong, “Directive emission based on anisotropic metamaterials,” Phys. Rev. A 77, 053821 (2008).
[CrossRef]

A. Alù and N. Engheta, “Dielectric sensing in ε-near-zero narrow waveguide channels,” Phys. Rev. B 78, 045102 (2008).
[CrossRef]

A. Alù, M. G. Silveirinha, and N. Engheta, “Transmission-line analysis of ε-near-zero–filled narrow channels,” Phys. Rev. E 78, 016604 (2008).
[CrossRef]

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100, 033903 (2008).
[CrossRef] [PubMed]

R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100, 023903 (2008).
[CrossRef] [PubMed]

2007 (6)

Z. B. Weng, Y. C. Jiao, G. Zhao, and F. S. Zhang, “Design and experiment of one dimension and two dimension metamaterial structure for directive emission,” PIER 70, 199–209 (2007).
[CrossRef]

M. G. Silveirinha and N. Engheta, “Theory of supercoupling, squeezing wave energy, and field confinement in narrow channels and tight bends using ε near-zero metamaterials,” Phys. Rev. B 76, 245109 (2007).
[CrossRef]

M. Silveirinha and N. Engheta, “Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media,” Phys. Rev. B 75, 075119 (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]

L. F. Shen, T. J. Yang, and Y. F. Chau, “50/50 beam splitter using a one-dimensional metal photonic crystal with parabolalike dispersion,” Appl. Phys. Lett. 90, 251909 (2007).
[CrossRef]

M. Scalora, G. D’Aguanno, N. Mattiucci, M. J. Bloemer, D. de Ceglia, M. Centini, Antonio Mandatori, C. Sibilia, N. Akozbek, M. G. Cappeddu, M. Fowler, and J. W. Haus, “Negative refraction and sub-wavelength focusing in the visible range using transparent metallodielectric stacks,” Opt. Express 15, 508–523 (2007).
[CrossRef] [PubMed]

2006 (3)

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett. 97, 073901 (2006).
[CrossRef] [PubMed]

H. C. Shin and S. H. Fan, “All-angle negative refraction and evanescent wave amplification using one-dimensional metallodielectric photonic crystals,” Appl. Phys. Lett. 89, 151102(2006).
[CrossRef]

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using “ε-near-zero materials,” Phys. Rev. Lett. 97, 157403 (2006).
[CrossRef] [PubMed]

2005 (3)

I. Bulu, H. Caglayan, K. Aydin, and E. Ozbay, “Compact size highly directive antennas based on the SRR metamaterial medium,” New J. Phys. 7, 223 (2005).
[CrossRef]

L. Zhou, H. Q. Li, Y. Q. Qin, Z. Y. Wei, and C. T. Chan, “Directive emissions from subwavelength metamaterial-based cavities,” Appl. Phys. Lett. 86, 101101 (2005).
[CrossRef]

H. Caglayan, I. Bulu, and E. Ozbay, “Highly directional enhanced radiation from sources embedded inside three-dimensional photonic crystals,” Opt. Express 13, 7645–7652 (2005).
[CrossRef] [PubMed]

2003 (1)

I. Bulu, H. Caglayan, and E. Ozbay, “Highly directive radiation from sources embedded inside photonic crystals,” Appl. Phys. Lett. 83, 3263–3265 (2003).
[CrossRef]

2002 (3)

S. Enoch, B. Gralak, and G. R. Tayeb, “Enhanced emission with angular confinement from photonic crystals,” Appl. Phys. Lett. 81, 1588–1590 (2002).
[CrossRef]

S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[CrossRef] [PubMed]

T. Akalin, J. Danglot, O. Vanbésien, and D. Lippens, “A highly directive dipole antenna embedded in a Fabry–Pérot type cavity,” IEEE Microw. Wireless Compon. Lett. 12, 48–50(2002).
[CrossRef]

2000 (1)

B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K. M. Ho, “Photonic crystal-based resonant antenna with a very high directivity,” J. Appl. Phys. 87, 603–605(2000).
[CrossRef]

1972 (1)

Akalin, T.

T. Akalin, J. Danglot, O. Vanbésien, and D. Lippens, “A highly directive dipole antenna embedded in a Fabry–Pérot type cavity,” IEEE Microw. Wireless Compon. Lett. 12, 48–50(2002).
[CrossRef]

Akozbek, N.

Alekseyev, L. V.

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

Alù, A.

B. Edwards, A. Alù, M. G. Silveirinha, and N. Engheta, “Reflectionless sharp bends and corners in waveguides using epsilon-near-zero effects,” J. Appl. Phys. 105, 044905 (2009).
[CrossRef]

A. Alù, M. G. Silveirinha, and N. Engheta, “Transmission-line analysis of ε-near-zero–filled narrow channels,” Phys. Rev. E 78, 016604 (2008).
[CrossRef]

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100, 033903 (2008).
[CrossRef] [PubMed]

A. Alù and N. Engheta, “Dielectric sensing in ε-near-zero narrow waveguide channels,” Phys. Rev. B 78, 045102 (2008).
[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]

Aydin, K.

I. Bulu, H. Caglayan, K. Aydin, and E. Ozbay, “Compact size highly directive antennas based on the SRR metamaterial medium,” New J. Phys. 7, 223 (2005).
[CrossRef]

Barnakov, Y. A.

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

Bayindir, M.

B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K. M. Ho, “Photonic crystal-based resonant antenna with a very high directivity,” J. Appl. Phys. 87, 603–605(2000).
[CrossRef]

Berreman, D. W.

Biswas, R.

B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K. M. Ho, “Photonic crystal-based resonant antenna with a very high directivity,” J. Appl. Phys. 87, 603–605(2000).
[CrossRef]

Bloemer, M. J.

Bulu, I.

H. Caglayan, I. Bulu, and E. Ozbay, “Highly directional enhanced radiation from sources embedded inside three-dimensional photonic crystals,” Opt. Express 13, 7645–7652 (2005).
[CrossRef] [PubMed]

I. Bulu, H. Caglayan, K. Aydin, and E. Ozbay, “Compact size highly directive antennas based on the SRR metamaterial medium,” New J. Phys. 7, 223 (2005).
[CrossRef]

I. Bulu, H. Caglayan, and E. Ozbay, “Highly directive radiation from sources embedded inside photonic crystals,” Appl. Phys. Lett. 83, 3263–3265 (2003).
[CrossRef]

Burokur, S. N.

S. N. Burokur, J. P. Daniel, P. Ratajczak, and A. de Lustrac, “Tunable bilayered metasurface for frequency reconfigurable directive emissions,” Appl. Phys. Lett. 97, 064101 (2010).
[CrossRef]

Caglayan, H.

H. Caglayan, I. Bulu, and E. Ozbay, “Highly directional enhanced radiation from sources embedded inside three-dimensional photonic crystals,” Opt. Express 13, 7645–7652 (2005).
[CrossRef] [PubMed]

I. Bulu, H. Caglayan, K. Aydin, and E. Ozbay, “Compact size highly directive antennas based on the SRR metamaterial medium,” New J. Phys. 7, 223 (2005).
[CrossRef]

I. Bulu, H. Caglayan, and E. Ozbay, “Highly directive radiation from sources embedded inside photonic crystals,” Appl. Phys. Lett. 83, 3263–3265 (2003).
[CrossRef]

Cappeddu, M. G.

Centini, M.

Chan, C. T.

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett. 97, 073901 (2006).
[CrossRef] [PubMed]

L. Zhou, H. Q. Li, Y. Q. Qin, Z. Y. Wei, and C. T. Chan, “Directive emissions from subwavelength metamaterial-based cavities,” Appl. Phys. Lett. 86, 101101 (2005).
[CrossRef]

Chau, Y. F.

L. F. Shen, T. J. Yang, and Y. F. Chau, “50/50 beam splitter using a one-dimensional metal photonic crystal with parabolalike dispersion,” Appl. Phys. Lett. 90, 251909 (2007).
[CrossRef]

Chen, H.

Y. Xu and H. Chen, “Total reflection and transmission by epsilon-near-zero metamaterials with defects,” Appl. Phys. Lett. 98, 113501 (2011).
[CrossRef]

Cheng, Q.

R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100, 023903 (2008).
[CrossRef] [PubMed]

Colman, P.

N. V. Q. Tran, S. Combrié, P. Colman, A. De Rossi, and T. Mei, “Vertical high emission in photonic crystal nanocavities by band-folding design,” Phys. Rev. B 82, 075120 (2010).
[CrossRef]

Combrié, S.

N. V. Q. Tran, S. Combrié, P. Colman, A. De Rossi, and T. Mei, “Vertical high emission in photonic crystal nanocavities by band-folding design,” Phys. Rev. B 82, 075120 (2010).
[CrossRef]

N. V. Q. Tran, S. Combrié, and A. De Rossi, “Directive emission from high-Q photonic crystal cavities through band folding,” Phys. Rev. B 79, 041101 (2009).
[CrossRef]

Cui, T. J.

R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100, 023903 (2008).
[CrossRef] [PubMed]

Cummer, S. A.

R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100, 023903 (2008).
[CrossRef] [PubMed]

D’Aguanno, G.

Danglot, J.

T. Akalin, J. Danglot, O. Vanbésien, and D. Lippens, “A highly directive dipole antenna embedded in a Fabry–Pérot type cavity,” IEEE Microw. Wireless Compon. Lett. 12, 48–50(2002).
[CrossRef]

Daniel, J. P.

S. N. Burokur, J. P. Daniel, P. Ratajczak, and A. de Lustrac, “Tunable bilayered metasurface for frequency reconfigurable directive emissions,” Appl. Phys. Lett. 97, 064101 (2010).
[CrossRef]

de Ceglia, D.

de Lustrac, A.

S. N. Burokur, J. P. Daniel, P. Ratajczak, and A. de Lustrac, “Tunable bilayered metasurface for frequency reconfigurable directive emissions,” Appl. Phys. Lett. 97, 064101 (2010).
[CrossRef]

De Rossi, A.

N. V. Q. Tran, S. Combrié, P. Colman, A. De Rossi, and T. Mei, “Vertical high emission in photonic crystal nanocavities by band-folding design,” Phys. Rev. B 82, 075120 (2010).
[CrossRef]

N. V. Q. Tran, S. Combrié, and A. De Rossi, “Directive emission from high-Q photonic crystal cavities through band folding,” Phys. Rev. B 79, 041101 (2009).
[CrossRef]

Edwards, B.

B. Edwards, A. Alù, M. G. Silveirinha, and N. Engheta, “Reflectionless sharp bends and corners in waveguides using epsilon-near-zero effects,” J. Appl. Phys. 105, 044905 (2009).
[CrossRef]

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100, 033903 (2008).
[CrossRef] [PubMed]

Engheta, N.

B. Edwards, A. Alù, M. G. Silveirinha, and N. Engheta, “Reflectionless sharp bends and corners in waveguides using epsilon-near-zero effects,” J. Appl. Phys. 105, 044905 (2009).
[CrossRef]

A. Alù, M. G. Silveirinha, and N. Engheta, “Transmission-line analysis of ε-near-zero–filled narrow channels,” Phys. Rev. E 78, 016604 (2008).
[CrossRef]

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100, 033903 (2008).
[CrossRef] [PubMed]

A. Alù and N. Engheta, “Dielectric sensing in ε-near-zero narrow waveguide channels,” Phys. Rev. B 78, 045102 (2008).
[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]

M. Silveirinha and N. Engheta, “Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media,” Phys. Rev. B 75, 075119 (2007).
[CrossRef]

M. G. Silveirinha and N. Engheta, “Theory of supercoupling, squeezing wave energy, and field confinement in narrow channels and tight bends using ε near-zero metamaterials,” Phys. Rev. B 76, 245109 (2007).
[CrossRef]

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using “ε-near-zero materials,” Phys. Rev. Lett. 97, 157403 (2006).
[CrossRef] [PubMed]

Enoch, S.

S. Enoch, B. Gralak, and G. R. Tayeb, “Enhanced emission with angular confinement from photonic crystals,” Appl. Phys. Lett. 81, 1588–1590 (2002).
[CrossRef]

S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[CrossRef] [PubMed]

Fan, S. H.

H. C. Shin and S. H. Fan, “All-angle negative refraction and evanescent wave amplification using one-dimensional metallodielectric photonic crystals,” Appl. Phys. Lett. 89, 151102(2006).
[CrossRef]

Fan, X.

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett. 97, 073901 (2006).
[CrossRef] [PubMed]

Fiddy, M. A.

Fowler, M.

Gao, L.

J. Luo, P. Xu, T. Sun, and L. Gao, “Tunable beam splitting and optical negative refraction in heterostructure with metamaterial,” Appl. Phys. A 104, 1137–1142 (2011).
[CrossRef]

Gralak, B.

S. Enoch, B. Gralak, and G. R. Tayeb, “Enhanced emission with angular confinement from photonic crystals,” Appl. Phys. Lett. 81, 1588–1590 (2002).
[CrossRef]

Guerin, N.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[CrossRef] [PubMed]

Hand, T.

R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100, 023903 (2008).
[CrossRef] [PubMed]

Haus, J. W.

Ho, K. M.

B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K. M. Ho, “Photonic crystal-based resonant antenna with a very high directivity,” J. Appl. Phys. 87, 603–605(2000).
[CrossRef]

Huang, K. M.

B. Wang and K. M. Huang, “Shaping the radiation pattern with mu and epsilon-near-zero metamaterials,” PIER 106, 107–119(2010).
[CrossRef]

Huangfu, J. T.

Y. Yuan, L. F. Shen, L. X. Ran, T. Jiang, J. T. Huangfu, and J. A. Kong, “Directive emission based on anisotropic metamaterials,” Phys. Rev. A 77, 053821 (2008).
[CrossRef]

Jiang, T.

F. M. Zhu, Z. Y. Wang, T. Jiang, L. F. Shen, and L. X. Ran, “Directive emission based on a new type of metamaterial,” Microwave Opt. Technol. Lett. 51, 2178–2180 (2009).
[CrossRef]

Y. Yuan, L. F. Shen, L. X. Ran, T. Jiang, J. T. Huangfu, and J. A. Kong, “Directive emission based on anisotropic metamaterials,” Phys. Rev. A 77, 053821 (2008).
[CrossRef]

Jiao, Y. C.

Z. B. Weng, Y. C. Jiao, G. Zhao, and F. S. Zhang, “Design and experiment of one dimension and two dimension metamaterial structure for directive emission,” PIER 70, 199–209 (2007).
[CrossRef]

Kong, J. A.

Y. Yuan, L. F. Shen, L. X. Ran, T. Jiang, J. T. Huangfu, and J. A. Kong, “Directive emission based on anisotropic metamaterials,” Phys. Rev. A 77, 053821 (2008).
[CrossRef]

Lee, J. C. W.

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett. 97, 073901 (2006).
[CrossRef] [PubMed]

Li, D. D.

Li, H.

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

Li, H. Q.

L. Zhou, H. Q. Li, Y. Q. Qin, Z. Y. Wei, and C. T. Chan, “Directive emissions from subwavelength metamaterial-based cavities,” Appl. Phys. Lett. 86, 101101 (2005).
[CrossRef]

Lippens, D.

T. Akalin, J. Danglot, O. Vanbésien, and D. Lippens, “A highly directive dipole antenna embedded in a Fabry–Pérot type cavity,” IEEE Microw. Wireless Compon. Lett. 12, 48–50(2002).
[CrossRef]

Liu, R.

R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100, 023903 (2008).
[CrossRef] [PubMed]

Luo, J.

J. Luo, P. Xu, T. Sun, and L. Gao, “Tunable beam splitting and optical negative refraction in heterostructure with metamaterial,” Appl. Phys. A 104, 1137–1142 (2011).
[CrossRef]

Mandatori, Antonio

Mattiucci, N.

Mei, T.

N. V. Q. Tran, S. Combrié, P. Colman, A. De Rossi, and T. Mei, “Vertical high emission in photonic crystal nanocavities by band-folding design,” Phys. Rev. B 82, 075120 (2010).
[CrossRef]

Mock, J. J.

R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100, 023903 (2008).
[CrossRef] [PubMed]

Narimanov, E. E.

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

Noginov, M. A.

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

Ozbay, E.

H. Caglayan, I. Bulu, and E. Ozbay, “Highly directional enhanced radiation from sources embedded inside three-dimensional photonic crystals,” Opt. Express 13, 7645–7652 (2005).
[CrossRef] [PubMed]

I. Bulu, H. Caglayan, K. Aydin, and E. Ozbay, “Compact size highly directive antennas based on the SRR metamaterial medium,” New J. Phys. 7, 223 (2005).
[CrossRef]

I. Bulu, H. Caglayan, and E. Ozbay, “Highly directive radiation from sources embedded inside photonic crystals,” Appl. Phys. Lett. 83, 3263–3265 (2003).
[CrossRef]

B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K. M. Ho, “Photonic crystal-based resonant antenna with a very high directivity,” J. Appl. Phys. 87, 603–605(2000).
[CrossRef]

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids(Academic, 1985).

Qin, Y. Q.

L. Zhou, H. Q. Li, Y. Q. Qin, Z. Y. Wei, and C. T. Chan, “Directive emissions from subwavelength metamaterial-based cavities,” Appl. Phys. Lett. 86, 101101 (2005).
[CrossRef]

Ran, L. X.

F. M. Zhu, Z. Y. Wang, T. Jiang, L. F. Shen, and L. X. Ran, “Directive emission based on a new type of metamaterial,” Microwave Opt. Technol. Lett. 51, 2178–2180 (2009).
[CrossRef]

Y. Yuan, L. F. Shen, L. X. Ran, T. Jiang, J. T. Huangfu, and J. A. Kong, “Directive emission based on anisotropic metamaterials,” Phys. Rev. A 77, 053821 (2008).
[CrossRef]

Ratajczak, P.

S. N. Burokur, J. P. Daniel, P. Ratajczak, and A. de Lustrac, “Tunable bilayered metasurface for frequency reconfigurable directive emissions,” Appl. Phys. Lett. 97, 064101 (2010).
[CrossRef]

Sabouroux, P.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[CrossRef] [PubMed]

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]

Scalora, M.

Shen, L. F.

F. M. Zhu, Z. Y. Wang, T. Jiang, L. F. Shen, and L. X. Ran, “Directive emission based on a new type of metamaterial,” Microwave Opt. Technol. Lett. 51, 2178–2180 (2009).
[CrossRef]

Y. Yuan, L. F. Shen, L. X. Ran, T. Jiang, J. T. Huangfu, and J. A. Kong, “Directive emission based on anisotropic metamaterials,” Phys. Rev. A 77, 053821 (2008).
[CrossRef]

L. F. Shen, T. J. Yang, and Y. F. Chau, “50/50 beam splitter using a one-dimensional metal photonic crystal with parabolalike dispersion,” Appl. Phys. Lett. 90, 251909 (2007).
[CrossRef]

Shin, H. C.

H. C. Shin and S. H. Fan, “All-angle negative refraction and evanescent wave amplification using one-dimensional metallodielectric photonic crystals,” Appl. Phys. Lett. 89, 151102(2006).
[CrossRef]

Sibilia, C.

Sigalas, M. M.

B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K. M. Ho, “Photonic crystal-based resonant antenna with a very high directivity,” J. Appl. Phys. 87, 603–605(2000).
[CrossRef]

Silveirinha, M.

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100, 033903 (2008).
[CrossRef] [PubMed]

M. Silveirinha and N. Engheta, “Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media,” Phys. Rev. B 75, 075119 (2007).
[CrossRef]

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using “ε-near-zero materials,” Phys. Rev. Lett. 97, 157403 (2006).
[CrossRef] [PubMed]

Silveirinha, M. G.

B. Edwards, A. Alù, M. G. Silveirinha, and N. Engheta, “Reflectionless sharp bends and corners in waveguides using epsilon-near-zero effects,” J. Appl. Phys. 105, 044905 (2009).
[CrossRef]

A. Alù, M. G. Silveirinha, and N. Engheta, “Transmission-line analysis of ε-near-zero–filled narrow channels,” Phys. Rev. E 78, 016604 (2008).
[CrossRef]

M. G. Silveirinha and N. Engheta, “Theory of supercoupling, squeezing wave energy, and field confinement in narrow channels and tight bends using ε near-zero metamaterials,” Phys. Rev. B 76, 245109 (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]

Smith, D. R.

R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100, 023903 (2008).
[CrossRef] [PubMed]

Sun, T.

J. Luo, P. Xu, T. Sun, and L. Gao, “Tunable beam splitting and optical negative refraction in heterostructure with metamaterial,” Appl. Phys. A 104, 1137–1142 (2011).
[CrossRef]

Tayeb, G.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[CrossRef] [PubMed]

Tayeb, G. R.

S. Enoch, B. Gralak, and G. R. Tayeb, “Enhanced emission with angular confinement from photonic crystals,” Appl. Phys. Lett. 81, 1588–1590 (2002).
[CrossRef]

Temelkuran, B.

B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K. M. Ho, “Photonic crystal-based resonant antenna with a very high directivity,” J. Appl. Phys. 87, 603–605(2000).
[CrossRef]

Tran, N. V. Q.

N. V. Q. Tran, S. Combrié, P. Colman, A. De Rossi, and T. Mei, “Vertical high emission in photonic crystal nanocavities by band-folding design,” Phys. Rev. B 82, 075120 (2010).
[CrossRef]

N. V. Q. Tran, S. Combrié, and A. De Rossi, “Directive emission from high-Q photonic crystal cavities through band folding,” Phys. Rev. B 79, 041101 (2009).
[CrossRef]

Tumkur, T.

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

Tuttle, G.

B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K. M. Ho, “Photonic crystal-based resonant antenna with a very high directivity,” J. Appl. Phys. 87, 603–605(2000).
[CrossRef]

Vanbésien, O.

T. Akalin, J. Danglot, O. Vanbésien, and D. Lippens, “A highly directive dipole antenna embedded in a Fabry–Pérot type cavity,” IEEE Microw. Wireless Compon. Lett. 12, 48–50(2002).
[CrossRef]

Vincent, P.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[CrossRef] [PubMed]

Wang, B.

B. Wang and K. M. Huang, “Shaping the radiation pattern with mu and epsilon-near-zero metamaterials,” PIER 106, 107–119(2010).
[CrossRef]

Wang, G. P.

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett. 97, 073901 (2006).
[CrossRef] [PubMed]

Wang, Z. Y.

F. M. Zhu, Z. Y. Wang, T. Jiang, L. F. Shen, and L. X. Ran, “Directive emission based on a new type of metamaterial,” Microwave Opt. Technol. Lett. 51, 2178–2180 (2009).
[CrossRef]

Wei, Z. Y.

L. Zhou, H. Q. Li, Y. Q. Qin, Z. Y. Wei, and C. T. Chan, “Directive emissions from subwavelength metamaterial-based cavities,” Appl. Phys. Lett. 86, 101101 (2005).
[CrossRef]

Weng, Z. B.

Z. B. Weng, Y. C. Jiao, G. Zhao, and F. S. Zhang, “Design and experiment of one dimension and two dimension metamaterial structure for directive emission,” PIER 70, 199–209 (2007).
[CrossRef]

Xu, P.

J. Luo, P. Xu, T. Sun, and L. Gao, “Tunable beam splitting and optical negative refraction in heterostructure with metamaterial,” Appl. Phys. A 104, 1137–1142 (2011).
[CrossRef]

Xu, Y.

Y. Xu and H. Chen, “Total reflection and transmission by epsilon-near-zero metamaterials with defects,” Appl. Phys. Lett. 98, 113501 (2011).
[CrossRef]

Yan, C. C.

Yang, T. J.

L. F. Shen, T. J. Yang, and Y. F. Chau, “50/50 beam splitter using a one-dimensional metal photonic crystal with parabolalike dispersion,” Appl. Phys. Lett. 90, 251909 (2007).
[CrossRef]

Young, M. E.

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100, 033903 (2008).
[CrossRef] [PubMed]

Yuan, Y.

Y. Yuan, L. F. Shen, L. X. Ran, T. Jiang, J. T. Huangfu, and J. A. Kong, “Directive emission based on anisotropic metamaterials,” Phys. Rev. A 77, 053821 (2008).
[CrossRef]

Zhang, D. H.

Zhang, F. S.

Z. B. Weng, Y. C. Jiao, G. Zhao, and F. S. Zhang, “Design and experiment of one dimension and two dimension metamaterial structure for directive emission,” PIER 70, 199–209 (2007).
[CrossRef]

Zhang, Y. A.

Zhao, G.

Z. B. Weng, Y. C. Jiao, G. Zhao, and F. S. Zhang, “Design and experiment of one dimension and two dimension metamaterial structure for directive emission,” PIER 70, 199–209 (2007).
[CrossRef]

Zhou, L.

L. Zhou, H. Q. Li, Y. Q. Qin, Z. Y. Wei, and C. T. Chan, “Directive emissions from subwavelength metamaterial-based cavities,” Appl. Phys. Lett. 86, 101101 (2005).
[CrossRef]

Zhu, F. M.

F. M. Zhu, Z. Y. Wang, T. Jiang, L. F. Shen, and L. X. Ran, “Directive emission based on a new type of metamaterial,” Microwave Opt. Technol. Lett. 51, 2178–2180 (2009).
[CrossRef]

Appl. Phys. A (1)

J. Luo, P. Xu, T. Sun, and L. Gao, “Tunable beam splitting and optical negative refraction in heterostructure with metamaterial,” Appl. Phys. A 104, 1137–1142 (2011).
[CrossRef]

Appl. Phys. Lett. (8)

H. C. Shin and S. H. Fan, “All-angle negative refraction and evanescent wave amplification using one-dimensional metallodielectric photonic crystals,” Appl. Phys. Lett. 89, 151102(2006).
[CrossRef]

L. F. Shen, T. J. Yang, and Y. F. Chau, “50/50 beam splitter using a one-dimensional metal photonic crystal with parabolalike dispersion,” Appl. Phys. Lett. 90, 251909 (2007).
[CrossRef]

Y. Xu and H. Chen, “Total reflection and transmission by epsilon-near-zero metamaterials with defects,” Appl. Phys. Lett. 98, 113501 (2011).
[CrossRef]

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

S. N. Burokur, J. P. Daniel, P. Ratajczak, and A. de Lustrac, “Tunable bilayered metasurface for frequency reconfigurable directive emissions,” Appl. Phys. Lett. 97, 064101 (2010).
[CrossRef]

L. Zhou, H. Q. Li, Y. Q. Qin, Z. Y. Wei, and C. T. Chan, “Directive emissions from subwavelength metamaterial-based cavities,” Appl. Phys. Lett. 86, 101101 (2005).
[CrossRef]

I. Bulu, H. Caglayan, and E. Ozbay, “Highly directive radiation from sources embedded inside photonic crystals,” Appl. Phys. Lett. 83, 3263–3265 (2003).
[CrossRef]

S. Enoch, B. Gralak, and G. R. Tayeb, “Enhanced emission with angular confinement from photonic crystals,” Appl. Phys. Lett. 81, 1588–1590 (2002).
[CrossRef]

IEEE Microw. Wireless Compon. Lett. (1)

T. Akalin, J. Danglot, O. Vanbésien, and D. Lippens, “A highly directive dipole antenna embedded in a Fabry–Pérot type cavity,” IEEE Microw. Wireless Compon. Lett. 12, 48–50(2002).
[CrossRef]

J. Appl. Phys. (2)

B. Temelkuran, M. Bayindir, E. Ozbay, R. Biswas, M. M. Sigalas, G. Tuttle, and K. M. Ho, “Photonic crystal-based resonant antenna with a very high directivity,” J. Appl. Phys. 87, 603–605(2000).
[CrossRef]

B. Edwards, A. Alù, M. G. Silveirinha, and N. Engheta, “Reflectionless sharp bends and corners in waveguides using epsilon-near-zero effects,” J. Appl. Phys. 105, 044905 (2009).
[CrossRef]

J. Opt. Soc. Am. (1)

Microwave Opt. Technol. Lett. (1)

F. M. Zhu, Z. Y. Wang, T. Jiang, L. F. Shen, and L. X. Ran, “Directive emission based on a new type of metamaterial,” Microwave Opt. Technol. Lett. 51, 2178–2180 (2009).
[CrossRef]

New J. Phys. (1)

I. Bulu, H. Caglayan, K. Aydin, and E. Ozbay, “Compact size highly directive antennas based on the SRR metamaterial medium,” New J. Phys. 7, 223 (2005).
[CrossRef]

Opt. Express (3)

Phys. Rev. A (1)

Y. Yuan, L. F. Shen, L. X. Ran, T. Jiang, J. T. Huangfu, and J. A. Kong, “Directive emission based on anisotropic metamaterials,” Phys. Rev. A 77, 053821 (2008).
[CrossRef]

Phys. Rev. B (6)

N. V. Q. Tran, S. Combrié, and A. De Rossi, “Directive emission from high-Q photonic crystal cavities through band folding,” Phys. Rev. B 79, 041101 (2009).
[CrossRef]

N. V. Q. Tran, S. Combrié, P. Colman, A. De Rossi, and T. Mei, “Vertical high emission in photonic crystal nanocavities by band-folding design,” Phys. Rev. B 82, 075120 (2010).
[CrossRef]

M. G. Silveirinha and N. Engheta, “Theory of supercoupling, squeezing wave energy, and field confinement in narrow channels and tight bends using ε near-zero metamaterials,” Phys. Rev. B 76, 245109 (2007).
[CrossRef]

M. Silveirinha and N. Engheta, “Design of matched zero-index metamaterials using nonmagnetic inclusions in epsilon-near-zero media,” Phys. Rev. B 75, 075119 (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]

A. Alù and N. Engheta, “Dielectric sensing in ε-near-zero narrow waveguide channels,” Phys. Rev. B 78, 045102 (2008).
[CrossRef]

Phys. Rev. E (1)

A. Alù, M. G. Silveirinha, and N. Engheta, “Transmission-line analysis of ε-near-zero–filled narrow channels,” Phys. Rev. E 78, 016604 (2008).
[CrossRef]

Phys. Rev. Lett. (5)

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100, 033903 (2008).
[CrossRef] [PubMed]

R. Liu, Q. Cheng, T. Hand, J. J. Mock, T. J. Cui, S. A. Cummer, and D. R. Smith, “Experimental demonstration of electromagnetic tunneling through an epsilon-near-zero metamaterial at microwave frequencies,” Phys. Rev. Lett. 100, 023903 (2008).
[CrossRef] [PubMed]

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using “ε-near-zero materials,” Phys. Rev. Lett. 97, 157403 (2006).
[CrossRef] [PubMed]

S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[CrossRef] [PubMed]

X. Fan, G. P. Wang, J. C. W. Lee, and C. T. Chan, “All-angle broadband negative refraction of metal waveguide arrays in the visible range: theoretical analysis and numerical demonstration,” Phys. Rev. Lett. 97, 073901 (2006).
[CrossRef] [PubMed]

PIER (2)

B. Wang and K. M. Huang, “Shaping the radiation pattern with mu and epsilon-near-zero metamaterials,” PIER 106, 107–119(2010).
[CrossRef]

Z. B. Weng, Y. C. Jiao, G. Zhao, and F. S. Zhang, “Design and experiment of one dimension and two dimension metamaterial structure for directive emission,” PIER 70, 199–209 (2007).
[CrossRef]

Other (1)

E. D. Palik, Handbook of Optical Constants of Solids(Academic, 1985).

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Figures (4)

Fig. 1
Fig. 1

Geometry of the one-dimensional metal heterostructure. The heterostructure is composed of silver and air layers.

Fig. 2
Fig. 2

Dispersion curves of the metal heterostructure with ε r = 1 , f m = 0.5 , and a = λ 0 / 4 for TE-polarized waves for the lossless case ( ε m = 1 ). Thick and thin solid lines are the real and imaginary part of k z calculated by Eq. (6) and circles and triangles represent the real and imaginary part of k z calculated by Eq. (8).

Fig. 3
Fig. 3

Electric field distribution in the metal heterostructure. The parameters ε r = 1 , f m = 0.5 , and a = λ 0 / 4 . (a) A circular line source placed to the left of the heterostructure for the lossless case ( ε m = 1 ). (b) Lossy case ( ε m = 1 + 0.58 i ). (c) A point source is embedded in the center of the metal heterostructure without loss. The white portions represent the electric field larger than the maximum magnitude or smaller than the minimum magnitude marked in the right color bar.

Fig. 4
Fig. 4

Transmission spectra for the ENZ metamaterial with parameter (a)  ε y = 10 5 and (b)  ε y = 0.01 . The thickness of the ENZ metamaterial decreases from 10 λ 0 to 3 λ 0 in (a). And in (b) the thickness is 5 λ 0 .

Equations (11)

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

ψ ( x + d l ) = [ cos φ l ( i u l / v l ) sin φ l ( i v l / u l ) sin φ l cos φ l ] ψ ( x ) = P ( d l ) ψ ( x ) ,
ψ ( x + a ) = e i k x a ψ ( x ) .
ψ ( x + a ) = P ( d m ) P ( d r ) ψ ( x ) = T ψ ( x ) ,
e i k x a = 1 2 [ T 11 + T 22 ± ( T 11 + T 22 ) 2 4 det ( T ) ] .
k x = 1 a arccos [ 1 2 ( T 11 + T 22 ) ] .
cos ( k x a ) = cos ( f m p x a ) cosh ( ( 1 f m ) q x a ) 1 2 ( μ m p x μ r q x μ r q x μ m p x ) sin ( f m p x a ) sinh ( ( 1 f m ) q x a ) ,
sin x = x 1 3 ! x 3 , and cos x = 1 1 2 ! x 2 + 1 4 ! x 4 .
k x 2 + k z 2 = a 2 12 { ( 1 f m ) 2 ε r 2 k 0 4 [ ( 1 f m ) 4 + f m 4 + 4 f m ( 1 f m ) ( ( 1 f m ) 2 + f m 2 ) ] k z 4 } .
sin x = x , and cos x = 1 1 2 ! x 2
k x 2 + k z 2 = ε y k 0 2 .
2 E y = 0 .

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