Abstract

We achieve one-dimensional cloaking effects in index-near-zero and epsilon-near-zero metamaterials embedded with just dielectric Bragg-fiber-structured defects. The dielectric Bragg-fiber structure (BFS) within the bandgap region can act as a perfect electric or magnetic conductor which prevents electromagnetic waves from penetrating into the core. As a result, one can hide objects inside the dielectric core without leading large distortion to the transmission. Moreover, both narrow and broad Fano profiles with variations between the total reflections and total transmissions are obtained, corresponding to the core resonances and the BFS resonances, respectively. The conditions of the total reflections and total transmissions are derived. Our work may have potential applications including applications in ultrasensitive sensors and switches.

© 2013 Optical Society of America

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2013

K. Zhang, J. Fu, L.-Y. Xiao, Q. Wu, and L.-W. Li, “Total transmission and total reflection of electromagnetic waves by anisotropic epsilon-near-zero metamaterials embedded with dielectric defects,” J. Appl. Phys. 113, 084908 (2013).
[CrossRef]

2012

J. Luo, P. Xu, L. Gao, Y. Lai, and H. Chen, “Manipulate the transmissions using index-near-zero or epsilon-near-zero metamaterials with coated defects,” Plasmonics 7, 353–358 (2012).
[CrossRef]

W. Zhu, I. D. Rukhlenko, and M. Premaratne, “Light amplification in zero-index metamaterial with gain inserts,” Appl. Phys. Lett. 101, 031907 (2012).
[CrossRef]

H. F. Ma, J. H. Shi, B. G. Cai, and T. J. Cui, “Total transmission and super reflection realized by anisotropic zero-index materials,” New J. Phys. 14, 123010 (2012).
[CrossRef]

Y. Sun, B. Edwards, A. Alù, and N. Engheta, “Experimental realization of optical lumped nanocircuits at infrared wavelengths,” Nat. Mater. 11, 208–212 (2012).
[CrossRef]

B. Edwards and N. Engheta, “Experimental verification of displacement-current conduits in metamaterials-inspired optical circuitry,” Phys. Rev. Lett. 108, 193902 (2012).
[CrossRef]

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100, 221903 (2012).
[CrossRef]

H. F. Ma, J. H. Shi, W. X. Jiang, and T. J. Cui, “Experimental realization of bending waveguide using anisotropic zero-index materials,” Appl. Phys. Lett. 101, 253513 (2012).
[CrossRef]

S. Feng, “Loss-induced omnidirectional bending to the normal in ε-near-zero metamaterials,” Phys. Rev. Lett. 108, 193904 (2012).
[CrossRef]

Q. Cheng, W. X. Jiang, and T. J. Cui, “Spatial power combination for omnidirectional radiation via anisotropic metamaterials,” Phys. Rev. Lett. 108, 213903 (2012).
[CrossRef]

J. Luo, P. Xu, and L. Gao, “Directive emission based on one-dimensional metal heterostructures,” J. Opt. Soc. Am. B 29, 35–39 (2012).
[CrossRef]

J. Luo, P. Xu, and L. Gao, “Electrically controllable unidirectional transmission in a heterostructure composed of a photonic crystal and a deformable liquid droplet,” Solid State Commun. 152, 577–580 (2012).
[CrossRef]

2011

D. C. Adams, S. Inampudi, T. Ribaudo, D. Slocum, S. Vangala, N. A. Kuhta, W. D. Goodhue, V. A. Podolskiy, and D. Wasserman, “Funneling light through a subwavelength aperture with epsilon-near-zero materials,” Phys. Rev. Lett. 107, 133901 (2011).
[CrossRef]

J. Luo, P. Xu, and L. Gao, “Controllable switching behavior of optical Tamm state based on nematic liquid crystal,” Solid State Commun. 151, 993–995 (2011).
[CrossRef]

Q. Cheng, W. X. Jiang, and T. J. Cui, “Multi-beam generations at pre-designed directions based on anisotropic zero-index metamaterials,” Appl. Phys. Lett. 99, 131913 (2011).
[CrossRef]

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

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]

2010

J. Hao, W. Yan, and M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96, 101109 (2010).
[CrossRef]

Y. Jin and S. He, “Enhancing and suppressing radiation with some permeability-near-zero structures,” Opt. Express 18, 16587–16593 (2010).
[CrossRef]

V. C. Nguyen, L. Chen, and K. Halterman, “Total transmission and total reflection by zero index metamaterials with defects,” Phys. Rev. Lett. 105, 233908 (2010).
[CrossRef]

Y. Jin, P. Zhang, and S. He, “Squeezing electromagnetic energy with a dielectric split ring inside a permeability-near-zero metamaterial,” Phys. Rev. B 81, 085117 (2010).
[CrossRef]

Q. Cheng, W. X. Jiang, and T. J. Cui, “Radiation of planar electromagnetic waves by a line source in anisotropic metamaterials,” J. Phys. D 43, 335406 (2010).
[CrossRef]

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

2009

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

L. Shen, T.-J. Yang, and Y.-F. Chau, “Effect of internal period on the optical dispersion of indefinite-medium materials,” Phys. Rev. B 77, 205124 (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]

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]

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical Tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett. 101, 113902 (2008).
[CrossRef]

2007

M. 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).

N. Engheta, “Circuits with light at nanoscales: optical nanocircuits inspired by metamaterials,” Science 317, 1698–1702 (2007).
[CrossRef]

2006

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]

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]

2005

N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett. 95, 095504 (2005).
[CrossRef]

2004

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[CrossRef]

2002

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

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

2000

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

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef]

1995

1978

1972

1966

1961

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961).
[CrossRef]

Adams, D. C.

D. C. Adams, S. Inampudi, T. Ribaudo, D. Slocum, S. Vangala, N. A. Kuhta, W. D. Goodhue, V. A. Podolskiy, and D. Wasserman, “Funneling light through a subwavelength aperture with epsilon-near-zero materials,” Phys. Rev. Lett. 107, 133901 (2011).
[CrossRef]

Alù, A.

Y. Sun, B. Edwards, A. Alù, and N. Engheta, “Experimental realization of optical lumped nanocircuits at infrared wavelengths,” Nat. Mater. 11, 208–212 (2012).
[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]

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]

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).

N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett. 95, 095504 (2005).
[CrossRef]

Baryshev, A. V.

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical Tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett. 101, 113902 (2008).
[CrossRef]

Berreman, D. W.

Cai, B. G.

H. F. Ma, J. H. Shi, B. G. Cai, and T. J. Cui, “Total transmission and super reflection realized by anisotropic zero-index materials,” New J. Phys. 14, 123010 (2012).
[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]

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

Chau, Y.-F.

L. Shen, T.-J. Yang, and Y.-F. Chau, “Effect of internal period on the optical dispersion of indefinite-medium materials,” Phys. Rev. B 77, 205124 (2008).
[CrossRef]

Chen, H.

J. Luo, P. Xu, L. Gao, Y. Lai, and H. Chen, “Manipulate the transmissions using index-near-zero or epsilon-near-zero metamaterials with coated defects,” Plasmonics 7, 353–358 (2012).
[CrossRef]

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100, 221903 (2012).
[CrossRef]

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

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

Chen, L.

V. C. Nguyen, L. Chen, and K. Halterman, “Total transmission and total reflection by zero index metamaterials with defects,” Phys. Rev. Lett. 105, 233908 (2010).
[CrossRef]

Cheng, Q.

Q. Cheng, W. X. Jiang, and T. J. Cui, “Spatial power combination for omnidirectional radiation via anisotropic metamaterials,” Phys. Rev. Lett. 108, 213903 (2012).
[CrossRef]

Q. Cheng, W. X. Jiang, and T. J. Cui, “Multi-beam generations at pre-designed directions based on anisotropic zero-index metamaterials,” Appl. Phys. Lett. 99, 131913 (2011).
[CrossRef]

Q. Cheng, W. X. Jiang, and T. J. Cui, “Radiation of planar electromagnetic waves by a line source in anisotropic metamaterials,” J. Phys. D 43, 335406 (2010).
[CrossRef]

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]

Cui, T. J.

Q. Cheng, W. X. Jiang, and T. J. Cui, “Spatial power combination for omnidirectional radiation via anisotropic metamaterials,” Phys. Rev. Lett. 108, 213903 (2012).
[CrossRef]

H. F. Ma, J. H. Shi, B. G. Cai, and T. J. Cui, “Total transmission and super reflection realized by anisotropic zero-index materials,” New J. Phys. 14, 123010 (2012).
[CrossRef]

H. F. Ma, J. H. Shi, W. X. Jiang, and T. J. Cui, “Experimental realization of bending waveguide using anisotropic zero-index materials,” Appl. Phys. Lett. 101, 253513 (2012).
[CrossRef]

Q. Cheng, W. X. Jiang, and T. J. Cui, “Multi-beam generations at pre-designed directions based on anisotropic zero-index metamaterials,” Appl. Phys. Lett. 99, 131913 (2011).
[CrossRef]

Q. Cheng, W. X. Jiang, and T. J. Cui, “Radiation of planar electromagnetic waves by a line source in anisotropic metamaterials,” J. Phys. D 43, 335406 (2010).
[CrossRef]

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]

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]

Dorofeenko, A. V.

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical Tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett. 101, 113902 (2008).
[CrossRef]

Edwards, B.

Y. Sun, B. Edwards, A. Alù, and N. Engheta, “Experimental realization of optical lumped nanocircuits at infrared wavelengths,” Nat. Mater. 11, 208–212 (2012).
[CrossRef]

B. Edwards and N. Engheta, “Experimental verification of displacement-current conduits in metamaterials-inspired optical circuitry,” Phys. Rev. Lett. 108, 193902 (2012).
[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]

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]

Engheta, N.

B. Edwards and N. Engheta, “Experimental verification of displacement-current conduits in metamaterials-inspired optical circuitry,” Phys. Rev. Lett. 108, 193902 (2012).
[CrossRef]

Y. Sun, B. Edwards, A. Alù, and N. Engheta, “Experimental realization of optical lumped nanocircuits at infrared wavelengths,” Nat. Mater. 11, 208–212 (2012).
[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]

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]

M. 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).

N. Engheta, “Circuits with light at nanoscales: optical nanocircuits inspired by metamaterials,” Science 317, 1698–1702 (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]

N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett. 95, 095504 (2005).
[CrossRef]

Enoch, S.

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

Fano, U.

U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev. 124, 1866–1878 (1961).
[CrossRef]

Feng, S.

S. Feng, “Loss-induced omnidirectional bending to the normal in ε-near-zero metamaterials,” Phys. Rev. Lett. 108, 193904 (2012).
[CrossRef]

Frankena, H. J.

Fu, J.

K. Zhang, J. Fu, L.-Y. Xiao, Q. Wu, and L.-W. Li, “Total transmission and total reflection of electromagnetic waves by anisotropic epsilon-near-zero metamaterials embedded with dielectric defects,” J. Appl. Phys. 113, 084908 (2013).
[CrossRef]

Gao, L.

J. Luo, P. Xu, L. Gao, Y. Lai, and H. Chen, “Manipulate the transmissions using index-near-zero or epsilon-near-zero metamaterials with coated defects,” Plasmonics 7, 353–358 (2012).
[CrossRef]

J. Luo, P. Xu, and L. Gao, “Electrically controllable unidirectional transmission in a heterostructure composed of a photonic crystal and a deformable liquid droplet,” Solid State Commun. 152, 577–580 (2012).
[CrossRef]

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100, 221903 (2012).
[CrossRef]

J. Luo, P. Xu, and L. Gao, “Directive emission based on one-dimensional metal heterostructures,” J. Opt. Soc. Am. B 29, 35–39 (2012).
[CrossRef]

J. Luo, P. Xu, and L. Gao, “Controllable switching behavior of optical Tamm state based on nematic liquid crystal,” Solid State Commun. 151, 993–995 (2011).
[CrossRef]

Goodhue, W. D.

D. C. Adams, S. Inampudi, T. Ribaudo, D. Slocum, S. Vangala, N. A. Kuhta, W. D. Goodhue, V. A. Podolskiy, and D. Wasserman, “Funneling light through a subwavelength aperture with epsilon-near-zero materials,” Phys. Rev. Lett. 107, 133901 (2011).
[CrossRef]

Goto, T.

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical Tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett. 101, 113902 (2008).
[CrossRef]

Granovsky, A. B.

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical Tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett. 101, 113902 (2008).
[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]

Halterman, K.

V. C. Nguyen, L. Chen, and K. Halterman, “Total transmission and total reflection by zero index metamaterials with defects,” Phys. Rev. Lett. 105, 233908 (2010).
[CrossRef]

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]

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, J.

J. Hao, W. Yan, and M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96, 101109 (2010).
[CrossRef]

He, S.

Y. Jin and S. He, “Enhancing and suppressing radiation with some permeability-near-zero structures,” Opt. Express 18, 16587–16593 (2010).
[CrossRef]

Y. Jin, P. Zhang, and S. He, “Squeezing electromagnetic energy with a dielectric split ring inside a permeability-near-zero metamaterial,” Phys. Rev. B 81, 085117 (2010).
[CrossRef]

Hou, B.

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100, 221903 (2012).
[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]

Inampudi, S.

D. C. Adams, S. Inampudi, T. Ribaudo, D. Slocum, S. Vangala, N. A. Kuhta, W. D. Goodhue, V. A. Podolskiy, and D. Wasserman, “Funneling light through a subwavelength aperture with epsilon-near-zero materials,” Phys. Rev. Lett. 107, 133901 (2011).
[CrossRef]

Inoue, M.

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical Tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett. 101, 113902 (2008).
[CrossRef]

Jiang, W. X.

H. F. Ma, J. H. Shi, W. X. Jiang, and T. J. Cui, “Experimental realization of bending waveguide using anisotropic zero-index materials,” Appl. Phys. Lett. 101, 253513 (2012).
[CrossRef]

Q. Cheng, W. X. Jiang, and T. J. Cui, “Spatial power combination for omnidirectional radiation via anisotropic metamaterials,” Phys. Rev. Lett. 108, 213903 (2012).
[CrossRef]

Q. Cheng, W. X. Jiang, and T. J. Cui, “Multi-beam generations at pre-designed directions based on anisotropic zero-index metamaterials,” Appl. Phys. Lett. 99, 131913 (2011).
[CrossRef]

Q. Cheng, W. X. Jiang, and T. J. Cui, “Radiation of planar electromagnetic waves by a line source in anisotropic metamaterials,” J. Phys. D 43, 335406 (2010).
[CrossRef]

Jin, Y.

Y. Jin and S. He, “Enhancing and suppressing radiation with some permeability-near-zero structures,” Opt. Express 18, 16587–16593 (2010).
[CrossRef]

Y. Jin, P. Zhang, and S. He, “Squeezing electromagnetic energy with a dielectric split ring inside a permeability-near-zero metamaterial,” Phys. Rev. B 81, 085117 (2010).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Johnson, S. G.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Kuhta, N. A.

D. C. Adams, S. Inampudi, T. Ribaudo, D. Slocum, S. Vangala, N. A. Kuhta, W. D. Goodhue, V. A. Podolskiy, and D. Wasserman, “Funneling light through a subwavelength aperture with epsilon-near-zero materials,” Phys. Rev. Lett. 107, 133901 (2011).
[CrossRef]

Laan, C. J. v. d.

Lai, Y.

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100, 221903 (2012).
[CrossRef]

J. Luo, P. Xu, L. Gao, Y. Lai, and H. Chen, “Manipulate the transmissions using index-near-zero or epsilon-near-zero metamaterials with coated defects,” Plasmonics 7, 353–358 (2012).
[CrossRef]

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]

Leonhardt, U.

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

Li, L.-W.

K. Zhang, J. Fu, L.-Y. Xiao, Q. Wu, and L.-W. Li, “Total transmission and total reflection of electromagnetic waves by anisotropic epsilon-near-zero metamaterials embedded with dielectric defects,” J. Appl. Phys. 113, 084908 (2013).
[CrossRef]

Lisyansky, A. A.

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical Tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett. 101, 113902 (2008).
[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]

Luo, J.

J. Luo, P. Xu, and L. Gao, “Directive emission based on one-dimensional metal heterostructures,” J. Opt. Soc. Am. B 29, 35–39 (2012).
[CrossRef]

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100, 221903 (2012).
[CrossRef]

J. Luo, P. Xu, L. Gao, Y. Lai, and H. Chen, “Manipulate the transmissions using index-near-zero or epsilon-near-zero metamaterials with coated defects,” Plasmonics 7, 353–358 (2012).
[CrossRef]

J. Luo, P. Xu, and L. Gao, “Electrically controllable unidirectional transmission in a heterostructure composed of a photonic crystal and a deformable liquid droplet,” Solid State Commun. 152, 577–580 (2012).
[CrossRef]

J. Luo, P. Xu, and L. Gao, “Controllable switching behavior of optical Tamm state based on nematic liquid crystal,” Solid State Commun. 151, 993–995 (2011).
[CrossRef]

Ma, H. F.

H. F. Ma, J. H. Shi, W. X. Jiang, and T. J. Cui, “Experimental realization of bending waveguide using anisotropic zero-index materials,” Appl. Phys. Lett. 101, 253513 (2012).
[CrossRef]

H. F. Ma, J. H. Shi, B. G. Cai, and T. J. Cui, “Total transmission and super reflection realized by anisotropic zero-index materials,” New J. Phys. 14, 123010 (2012).
[CrossRef]

Markos, P.

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Merzlikin, A. M.

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical Tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett. 101, 113902 (2008).
[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]

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef]

Nguyen, V. C.

V. C. Nguyen, L. Chen, and K. Halterman, “Total transmission and total reflection by zero index metamaterials with defects,” Phys. Rev. Lett. 105, 233908 (2010).
[CrossRef]

Ohmer, M. C.

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef]

Pendry, J. B.

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

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[CrossRef]

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

Podolskiy, V. A.

D. C. Adams, S. Inampudi, T. Ribaudo, D. Slocum, S. Vangala, N. A. Kuhta, W. D. Goodhue, V. A. Podolskiy, and D. Wasserman, “Funneling light through a subwavelength aperture with epsilon-near-zero materials,” Phys. Rev. Lett. 107, 133901 (2011).
[CrossRef]

Premaratne, M.

W. Zhu, I. D. Rukhlenko, and M. Premaratne, “Light amplification in zero-index metamaterial with gain inserts,” Appl. Phys. Lett. 101, 031907 (2012).
[CrossRef]

Qiu, M.

J. Hao, W. Yan, and M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96, 101109 (2010).
[CrossRef]

Ribaudo, T.

D. C. Adams, S. Inampudi, T. Ribaudo, D. Slocum, S. Vangala, N. A. Kuhta, W. D. Goodhue, V. A. Podolskiy, and D. Wasserman, “Funneling light through a subwavelength aperture with epsilon-near-zero materials,” Phys. Rev. Lett. 107, 133901 (2011).
[CrossRef]

Rukhlenko, I. D.

W. Zhu, I. D. Rukhlenko, and M. Premaratne, “Light amplification in zero-index metamaterial with gain inserts,” Appl. Phys. Lett. 101, 031907 (2012).
[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]

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).

N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett. 95, 095504 (2005).
[CrossRef]

Schultz, S.

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef]

Schurig, D.

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

Shen, L.

L. Shen, T.-J. Yang, and Y.-F. Chau, “Effect of internal period on the optical dispersion of indefinite-medium materials,” Phys. Rev. B 77, 205124 (2008).
[CrossRef]

Sheng, P.

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

Shi, J. H.

H. F. Ma, J. H. Shi, B. G. Cai, and T. J. Cui, “Total transmission and super reflection realized by anisotropic zero-index materials,” New J. Phys. 14, 123010 (2012).
[CrossRef]

H. F. Ma, J. H. Shi, W. X. Jiang, and T. J. Cui, “Experimental realization of bending waveguide using anisotropic zero-index materials,” Appl. Phys. Lett. 101, 253513 (2012).
[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]

M. 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]

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]

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).

Slocum, D.

D. C. Adams, S. Inampudi, T. Ribaudo, D. Slocum, S. Vangala, N. A. Kuhta, W. D. Goodhue, V. A. Podolskiy, and D. Wasserman, “Funneling light through a subwavelength aperture with epsilon-near-zero materials,” Phys. Rev. Lett. 107, 133901 (2011).
[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]

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

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[CrossRef]

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[CrossRef]

Soukoulis, C. M.

D. R. Smith, S. Schultz, P. Markos, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Sun, Y.

Y. Sun, B. Edwards, A. Alù, and N. Engheta, “Experimental realization of optical lumped nanocircuits at infrared wavelengths,” Nat. Mater. 11, 208–212 (2012).
[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]

Thelen, A.

Vangala, S.

D. C. Adams, S. Inampudi, T. Ribaudo, D. Slocum, S. Vangala, N. A. Kuhta, W. D. Goodhue, V. A. Podolskiy, and D. Wasserman, “Funneling light through a subwavelength aperture with epsilon-near-zero materials,” Phys. Rev. Lett. 107, 133901 (2011).
[CrossRef]

Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000).
[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]

Vinogradov, A. P.

T. Goto, A. V. Dorofeenko, A. M. Merzlikin, A. V. Baryshev, A. P. Vinogradov, M. Inoue, A. A. Lisyansky, and A. B. Granovsky, “Optical Tamm states in one-dimensional magnetophotonic structures,” Phys. Rev. Lett. 101, 113902 (2008).
[CrossRef]

Wasserman, D.

D. C. Adams, S. Inampudi, T. Ribaudo, D. Slocum, S. Vangala, N. A. Kuhta, W. D. Goodhue, V. A. Podolskiy, and D. Wasserman, “Funneling light through a subwavelength aperture with epsilon-near-zero materials,” Phys. Rev. Lett. 107, 133901 (2011).
[CrossRef]

Wiltshire, M. C. K.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305, 788–792 (2004).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Wu, Q.

K. Zhang, J. Fu, L.-Y. Xiao, Q. Wu, and L.-W. Li, “Total transmission and total reflection of electromagnetic waves by anisotropic epsilon-near-zero metamaterials embedded with dielectric defects,” J. Appl. Phys. 113, 084908 (2013).
[CrossRef]

Xiao, L.-Y.

K. Zhang, J. Fu, L.-Y. Xiao, Q. Wu, and L.-W. Li, “Total transmission and total reflection of electromagnetic waves by anisotropic epsilon-near-zero metamaterials embedded with dielectric defects,” J. Appl. Phys. 113, 084908 (2013).
[CrossRef]

Xu, P.

J. Luo, P. Xu, H. Chen, B. Hou, L. Gao, and Y. Lai, “Realizing almost perfect bending waveguides with anisotropic epsilon-near-zero metamaterials,” Appl. Phys. Lett. 100, 221903 (2012).
[CrossRef]

J. Luo, P. Xu, L. Gao, Y. Lai, and H. Chen, “Manipulate the transmissions using index-near-zero or epsilon-near-zero metamaterials with coated defects,” Plasmonics 7, 353–358 (2012).
[CrossRef]

J. Luo, P. Xu, and L. Gao, “Electrically controllable unidirectional transmission in a heterostructure composed of a photonic crystal and a deformable liquid droplet,” Solid State Commun. 152, 577–580 (2012).
[CrossRef]

J. Luo, P. Xu, and L. Gao, “Directive emission based on one-dimensional metal heterostructures,” J. Opt. Soc. Am. B 29, 35–39 (2012).
[CrossRef]

J. Luo, P. Xu, and L. Gao, “Controllable switching behavior of optical Tamm state based on nematic liquid crystal,” Solid State Commun. 151, 993–995 (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, W.

J. Hao, W. Yan, and M. Qiu, “Super-reflection and cloaking based on zero index metamaterial,” Appl. Phys. Lett. 96, 101109 (2010).
[CrossRef]

Yang, T.-J.

L. Shen, T.-J. Yang, and Y.-F. Chau, “Effect of internal period on the optical dispersion of indefinite-medium materials,” Phys. Rev. B 77, 205124 (2008).
[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]

Zhang, K.

K. Zhang, J. Fu, L.-Y. Xiao, Q. Wu, and L.-W. Li, “Total transmission and total reflection of electromagnetic waves by anisotropic epsilon-near-zero metamaterials embedded with dielectric defects,” J. Appl. Phys. 113, 084908 (2013).
[CrossRef]

Zhang, P.

Y. Jin, P. Zhang, and S. He, “Squeezing electromagnetic energy with a dielectric split ring inside a permeability-near-zero metamaterial,” Phys. Rev. B 81, 085117 (2010).
[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]

Zhu, W.

W. Zhu, I. D. Rukhlenko, and M. Premaratne, “Light amplification in zero-index metamaterial with gain inserts,” Appl. Phys. Lett. 101, 031907 (2012).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

Q. Cheng, W. X. Jiang, and T. J. Cui, “Multi-beam generations at pre-designed directions based on anisotropic zero-index metamaterials,” Appl. Phys. Lett. 99, 131913 (2011).
[CrossRef]

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Nat. Mater.

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Figures (5)

Fig. 1.
Fig. 1.

Schematic drawing of a 2D waveguide structure consisting of three parts: region 0 (air), region 1 (metamaterial with a BFS defect), and region 2 (air). The parallel lines in the x direction are PEC walls of the waveguide.

Fig. 2.
Fig. 2.

Dependence of the transmission bands on nA in an ideal A-B multilayer with dA=λ0/6, nB=2.5, and dB=λ0/10 under normal incidence. The red (green) regions denote the stop (pass) bands.

Fig. 3.
Fig. 3.

Variance of transmittance with different nA. (a) INZ metamaterials with BFSA (red and green lines) and BFSB (blue lines) defects. (b) ENZ metamaterials (μm=1) having BFSA (red and green trace) and BFSB (blue trace) defects. C1 and C2 represent the core with refractive indexes of 2.5 and 2.6, respectively.

Fig. 4.
Fig. 4.

(a) Snapshots of a magnetic field in an INZ metamaterial with a BFSB defect having nC=2.5 and nA=1.5272. (b) An irregular PMC is hidden inside the core. The amplitude of the magnetic field when a BFSA defect is embedded with (c) nC=2.5, nA=1.4342 (total reflection), (d) nC=2.5, nA=1.4406 (total transmission), (e) nC=2.6, nA=1.2883 (total reflection), and (f) nC=2.6, nA=1.2897 (total transmission).

Fig. 5.
Fig. 5.

Amplitude of the magnetic field inside an ENZ metamaterial with a BFSA defect. (a) nA=1.4342. (b) nA=1.4375. (c) nA=1.5312. (d) nA=1.6048. The refractive index of the core is selected as 2.5.

Equations (19)

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H1=H1z^.
Hc=AcJ0(kcr)z^inside the core,
Hp=[BpJ0(kpr)+CpN0(kpr)]z^in thepth layer,
Ec=iμcεcAcJ1(kcr)θ^inside the core,
Ep=iμpεp[BpJ1(kpr)+CpN1(kpr)]θ^in the pth layer.
(BpCp)=Mp,p1Mp+1,p(Bp+1Cp+1),
Mp,p=(J0(kprp)N0(kprp)μpεp+1J1(kprp)μpεp+1N1(kprp))
Mp+1,p=(J0(kp+1rp)N0(kp+1rp)μp+1εpJ1(kp+1rp)μp+1εpN1(kp+1rp)).
(B1C1)=M(BnCn),
(μcε1J0(k1rc)J1(kcrc)μ1εcJ0(kcrc)J1(k1rc)μcε1N0(k1rc)J1(kcrc)μ1εcJ0(kcrc)N1(k1rc))T(B1C1)=X(B1C1)=0,
(X11M11+X12M21X11M12+X12M22J0(knrn)N0(knrn))(BnCn)=Y(BnCn)=(0H1).
Bn=H1Y12|Y|andCn=H1Y11|Y|.
H0=H0(eik0x+Reik0x)z^,E0=k0ωε0H0(eik0xReik0x)y^,
H2=H0Teik0xz^,E2=k0ωε0H0Teik0(xd)y^,
1+R=T.
T=11[ik0μm(SSd)2h+iπhL],
L=μnεnrn[BnH1J1(knrn)+CnH1N1(knrn)]
T=11[ik0μm(Sj=1NSdj)2h+iπhj=1NLj].
T=11iπhj=1NLj.

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