Abstract

The Bloch impedance is studied and used to understand the properties of the absorption loss in one-dimensional photonic crystals (PCs) composed of air and metal-based double-negative metamaterials. We find that as the frequency increases across the zero-n¯ gap of the considered structures, the modulus of the Bloch impedance always decreases from a maximum to a minimum value. On the other hand, the frequency dependence of the phase angle of the Bloch impedance is greatly influenced by the ratio of the electric to the magnetic damping coefficient γe/γm of the metamaterials. When the phase angle of the Bloch impedance reaches maximum inside the zero-n¯ gap, the impedance mismatch between the incident medium and the considered PC becomes greatest, the reflection will be strongest and a minimum absorption will be observed. As γe/γm increases, the frequency corresponding to the minimum absorption shifts from the lower to the upper gap edge. We also show that the main characteristics of both the Bloch impedance and the absorption loss are insensitive to the geometrical parameters. Our study offers a valuable reference in the designs of zero-n¯ gap with optimized properties.

© 2013 Optical Society of America

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C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics 5, 523–530 (2011).
[CrossRef]

2010

B. D. F. Casse, W. T. Lu, Y. J. Huang, E. Gultepe, L. Menon, and S. Sridhar, “Super-resolution imaging using a three-dimensional metamaterials nanolens,” Appl. Phys. Lett. 96, 023114 (2010).
[CrossRef]

2009

F. J. Lawrence, L. C. Botten, K. B. Dossou, C. M. de Sterke, and R. C. McPhedran, “Impedance of square and triangular lattice photonic crystals,” Phys. Rev. A 80, 023826 (2009).
[CrossRef]

Q. V. Tran, S. Combrié, P. Colman, and A. De Rossi, “Photonic crystal membrane waveguides with low insertion losses,” Appl. Phys. Lett. 95, 061105 (2009).
[CrossRef]

2007

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41–48 (2007).
[CrossRef]

C. M. Soukoulis, S. Linden, and M. Wegener, “Negative refractive index at optical wavelengths,” Science 315, 47–49 (2007).
[CrossRef]

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1, 449–458 (2007).
[CrossRef]

2006

H. Daninthe, S. Foteinopoulou, and C. M. Soukoulis, “Omni-reflectance and enhanced resonant tunneling from multilayers containing left-handed materials,” Photon. Nanostr. Fundam. Appl. 4, 123–131 (2006).
[CrossRef]

Y. H. Chen, J. W. Dong, and H. Z. Wang, “Conditions of near-zero dispersion of defect modes in one-dimensional photonic crystals containing negative-index materials,” J. Opt. Soc. Am. B 23, 776–781 (2006).
[CrossRef]

2004

R. Biswas, Z. Y. Li, and K. M. Ho, “Impedance of photonic crystals and photonic crystal waveguides,” Appl. Phys. Lett. 84, 1254–1256 (2004).
[CrossRef]

2003

Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E 67, 046607 (2003).
[CrossRef]

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[CrossRef]

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef]

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials,” Appl. Phys. Lett. 83, 5386–5388 (2003).
[CrossRef]

2002

A. Grbic and G. V. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial,” J. Appl. Phys. 92, 5930–5935 (2002).
[CrossRef]

S. Boscolo, M. Midrio, and T. F. Krauss, “Y junctions in photonic crystal channel waveguides: high transmission and impedance matching,” Opt. Lett. 27, 1001–1003 (2002).
[CrossRef]

2001

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef]

2000

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]

B. Gralak, S. Enoch, and G. Tayeb, “Anomalous refractive properties of photonic crystals,” J. Opt. Soc. Am. A 17, 1012–1020 (2000).
[CrossRef]

1996

1989

1987

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef]

1968

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[CrossRef]

Akahane, Y.

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[CrossRef]

Asano, T.

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1, 449–458 (2007).
[CrossRef]

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[CrossRef]

Bethune, D. S.

Biswas, R.

R. Biswas, Z. Y. Li, and K. M. Ho, “Impedance of photonic crystals and photonic crystal waveguides,” Appl. Phys. Lett. 84, 1254–1256 (2004).
[CrossRef]

Boscolo, S.

Botten, L. C.

F. J. Lawrence, L. C. Botten, K. B. Dossou, C. M. de Sterke, and R. C. McPhedran, “Impedance of square and triangular lattice photonic crystals,” Phys. Rev. A 80, 023826 (2009).
[CrossRef]

Casse, B. D. F.

B. D. F. Casse, W. T. Lu, Y. J. Huang, E. Gultepe, L. Menon, and S. Sridhar, “Super-resolution imaging using a three-dimensional metamaterials nanolens,” Appl. Phys. Lett. 96, 023114 (2010).
[CrossRef]

Chan, C. T.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef]

Chen, H.

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials,” Appl. Phys. Lett. 83, 5386–5388 (2003).
[CrossRef]

Chen, Y. H.

Colman, P.

Q. V. Tran, S. Combrié, P. Colman, and A. De Rossi, “Photonic crystal membrane waveguides with low insertion losses,” Appl. Phys. Lett. 95, 061105 (2009).
[CrossRef]

Combrié, S.

Q. V. Tran, S. Combrié, P. Colman, and A. De Rossi, “Photonic crystal membrane waveguides with low insertion losses,” Appl. Phys. Lett. 95, 061105 (2009).
[CrossRef]

Daninthe, H.

H. Daninthe, S. Foteinopoulou, and C. M. Soukoulis, “Omni-reflectance and enhanced resonant tunneling from multilayers containing left-handed materials,” Photon. Nanostr. Fundam. Appl. 4, 123–131 (2006).
[CrossRef]

De Rossi, A.

Q. V. Tran, S. Combrié, P. Colman, and A. De Rossi, “Photonic crystal membrane waveguides with low insertion losses,” Appl. Phys. Lett. 95, 061105 (2009).
[CrossRef]

de Sterke, C. M.

F. J. Lawrence, L. C. Botten, K. B. Dossou, C. M. de Sterke, and R. C. McPhedran, “Impedance of square and triangular lattice photonic crystals,” Phys. Rev. A 80, 023826 (2009).
[CrossRef]

Dong, J. W.

Dossou, K. B.

F. J. Lawrence, L. C. Botten, K. B. Dossou, C. M. de Sterke, and R. C. McPhedran, “Impedance of square and triangular lattice photonic crystals,” Phys. Rev. A 80, 023826 (2009).
[CrossRef]

Eleftheriades, G. V.

A. Grbic and G. V. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial,” J. Appl. Phys. 92, 5930–5935 (2002).
[CrossRef]

Enoch, S.

Foteinopoulou, S.

H. Daninthe, S. Foteinopoulou, and C. M. Soukoulis, “Omni-reflectance and enhanced resonant tunneling from multilayers containing left-handed materials,” Photon. Nanostr. Fundam. Appl. 4, 123–131 (2006).
[CrossRef]

Fujita, M.

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1, 449–458 (2007).
[CrossRef]

Gralak, B.

Grbic, A.

A. Grbic and G. V. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial,” J. Appl. Phys. 92, 5930–5935 (2002).
[CrossRef]

Gultepe, E.

B. D. F. Casse, W. T. Lu, Y. J. Huang, E. Gultepe, L. Menon, and S. Sridhar, “Super-resolution imaging using a three-dimensional metamaterials nanolens,” Appl. Phys. Lett. 96, 023114 (2010).
[CrossRef]

Ho, K. M.

R. Biswas, Z. Y. Li, and K. M. Ho, “Impedance of photonic crystals and photonic crystal waveguides,” Appl. Phys. Lett. 84, 1254–1256 (2004).
[CrossRef]

Huang, Y. J.

B. D. F. Casse, W. T. Lu, Y. J. Huang, E. Gultepe, L. Menon, and S. Sridhar, “Super-resolution imaging using a three-dimensional metamaterials nanolens,” Appl. Phys. Lett. 96, 023114 (2010).
[CrossRef]

Jiang, H. T.

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials,” Appl. Phys. Lett. 83, 5386–5388 (2003).
[CrossRef]

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef]

Krauss, T. F.

Lawrence, F. J.

F. J. Lawrence, L. C. Botten, K. B. Dossou, C. M. de Sterke, and R. C. McPhedran, “Impedance of square and triangular lattice photonic crystals,” Phys. Rev. A 80, 023826 (2009).
[CrossRef]

Li, H. Q.

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials,” Appl. Phys. Lett. 83, 5386–5388 (2003).
[CrossRef]

Li, J.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef]

Li, Z. Y.

R. Biswas, Z. Y. Li, and K. M. Ho, “Impedance of photonic crystals and photonic crystal waveguides,” Appl. Phys. Lett. 84, 1254–1256 (2004).
[CrossRef]

Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E 67, 046607 (2003).
[CrossRef]

Lin, L. L.

Z. Y. Li and L. L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E 67, 046607 (2003).
[CrossRef]

Linden, S.

C. M. Soukoulis, S. Linden, and M. Wegener, “Negative refractive index at optical wavelengths,” Science 315, 47–49 (2007).
[CrossRef]

Lu, W. T.

B. D. F. Casse, W. T. Lu, Y. J. Huang, E. Gultepe, L. Menon, and S. Sridhar, “Super-resolution imaging using a three-dimensional metamaterials nanolens,” Appl. Phys. Lett. 96, 023114 (2010).
[CrossRef]

Markoš, P.

P. Markoš and C. M. Soukoulis, Wave Propagation: From Electrons to Photonic crystals and Left-handed Materials (Princeton, 2008).

McPhedran, R. C.

F. J. Lawrence, L. C. Botten, K. B. Dossou, C. M. de Sterke, and R. C. McPhedran, “Impedance of square and triangular lattice photonic crystals,” Phys. Rev. A 80, 023826 (2009).
[CrossRef]

Menon, L.

B. D. F. Casse, W. T. Lu, Y. J. Huang, E. Gultepe, L. Menon, and S. Sridhar, “Super-resolution imaging using a three-dimensional metamaterials nanolens,” Appl. Phys. Lett. 96, 023114 (2010).
[CrossRef]

Midrio, M.

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]

Noda, S.

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1, 449–458 (2007).
[CrossRef]

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[CrossRef]

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]

Pozar, D. M.

D. M. Pozar, Microwave Engineering (Wiley, 1998).

Sakoda, K.

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[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]

Shalaev, V. M.

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41–48 (2007).
[CrossRef]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[CrossRef]

Sheng, P.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef]

Smith, D. R.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77–79 (2001).
[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]

Song, B.-S.

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[CrossRef]

Soukoulis, C. M.

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics 5, 523–530 (2011).
[CrossRef]

C. M. Soukoulis, S. Linden, and M. Wegener, “Negative refractive index at optical wavelengths,” Science 315, 47–49 (2007).
[CrossRef]

H. Daninthe, S. Foteinopoulou, and C. M. Soukoulis, “Omni-reflectance and enhanced resonant tunneling from multilayers containing left-handed materials,” Photon. Nanostr. Fundam. Appl. 4, 123–131 (2006).
[CrossRef]

P. Markoš and C. M. Soukoulis, Wave Propagation: From Electrons to Photonic crystals and Left-handed Materials (Princeton, 2008).

Sridhar, S.

B. D. F. Casse, W. T. Lu, Y. J. Huang, E. Gultepe, L. Menon, and S. Sridhar, “Super-resolution imaging using a three-dimensional metamaterials nanolens,” Appl. Phys. Lett. 96, 023114 (2010).
[CrossRef]

Tayeb, G.

Tran, P.

Tran, Q. V.

Q. V. Tran, S. Combrié, P. Colman, and A. De Rossi, “Photonic crystal membrane waveguides with low insertion losses,” Appl. Phys. Lett. 95, 061105 (2009).
[CrossRef]

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10, 509–514 (1968).
[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]

Wang, H. Z.

Wegener, M.

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics 5, 523–530 (2011).
[CrossRef]

C. M. Soukoulis, S. Linden, and M. Wegener, “Negative refractive index at optical wavelengths,” Science 315, 47–49 (2007).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef]

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystal, Propagation and Control of Laser Radiation (Wiley, 1984).

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystal, Propagation and Control of Laser Radiation (Wiley, 1984).

Zhang, Y. W.

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials,” Appl. Phys. Lett. 83, 5386–5388 (2003).
[CrossRef]

Zhou, L.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic band gap from a stack of positive and negative index materials,” Phys. Rev. Lett. 90, 083901 (2003).
[CrossRef]

Zhu, S. Y.

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials,” Appl. Phys. Lett. 83, 5386–5388 (2003).
[CrossRef]

Appl. Phys. Lett.

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials,” Appl. Phys. Lett. 83, 5386–5388 (2003).
[CrossRef]

B. D. F. Casse, W. T. Lu, Y. J. Huang, E. Gultepe, L. Menon, and S. Sridhar, “Super-resolution imaging using a three-dimensional metamaterials nanolens,” Appl. Phys. Lett. 96, 023114 (2010).
[CrossRef]

R. Biswas, Z. Y. Li, and K. M. Ho, “Impedance of photonic crystals and photonic crystal waveguides,” Appl. Phys. Lett. 84, 1254–1256 (2004).
[CrossRef]

Q. V. Tran, S. Combrié, P. Colman, and A. De Rossi, “Photonic crystal membrane waveguides with low insertion losses,” Appl. Phys. Lett. 95, 061105 (2009).
[CrossRef]

J. Appl. Phys.

A. Grbic and G. V. Eleftheriades, “Experimental verification of backward-wave radiation from a negative refractive index metamaterial,” J. Appl. Phys. 92, 5930–5935 (2002).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Nat. Photonics

C. M. Soukoulis and M. Wegener, “Past achievements and future challenges in the development of three-dimensional photonic metamaterials,” Nat. Photonics 5, 523–530 (2011).
[CrossRef]

V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1, 41–48 (2007).
[CrossRef]

S. Noda, M. Fujita, and T. Asano, “Spontaneous-emission control by photonic crystals and nanocavities,” Nat. Photonics 1, 449–458 (2007).
[CrossRef]

Nature

Y. Akahane, T. Asano, B.-S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425, 944–947 (2003).
[CrossRef]

Opt. Lett.

Photon. Nanostr. Fundam. Appl.

H. Daninthe, S. Foteinopoulou, and C. M. Soukoulis, “Omni-reflectance and enhanced resonant tunneling from multilayers containing left-handed materials,” Photon. Nanostr. Fundam. Appl. 4, 123–131 (2006).
[CrossRef]

Phys. Rev. A

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Figures (9)

Fig. 1.
Fig. 1.

Schematic representation of the DPS/DNG multilayered structure.

Fig. 2.
Fig. 2.

(a) n¯, (b) KB, and (c) |ZB| and θZB as a function of the angular frequency ω in infinite DPS/DNG PC with d1=6mm, d2=12mm, and γe=γm=0. The gray areas correspond to the zero-n¯ gap.

Fig. 3.
Fig. 3.

EM field distributions in lossless DPS/DNG PC with d1=6mm, and d2=12mm at (a) ω=7.68GHz and (b) ω=8.14GHz near the lower and upper gap edge, respectively.

Fig. 4.
Fig. 4.

Dependence of absorptance on |ZB| under different values of θZB.

Fig. 5.
Fig. 5.

|ZB|, θZB and absorptance versus frequency in DPS/DNG PC under different values of γe/γm: (a) and (b) γe/γm=10 with γe=0.01GHz, γm=0.001GHz; (c) and (d) γe/γm=1 with γe=0.01GHz, γm=0.01GHz; and (e) and (f) γe/γm=0.1 with γe=0.001GHz, γm=0.01GHz. The thicknesses of the layers are d1=6mm and d2=12mm. The gray areas represent the zero-n¯ gap.

Fig. 6.
Fig. 6.

ωmin as a function of γe/γm when γm is fixed to be 0.001 GHz. The gray area corresponds to the zero-n¯ gap. The thicknesses of the layers are d1=6mm and d2=12mm.

Fig. 7.
Fig. 7.

(a) |ZB|, (b) θZB, and (c) absorptance versus ω under different γm in DPS/DNG PC with γe/γm=1. (d) ωmin as a function of γm under different values of γe/γm. The thicknesses of the layers are d1=6mm and d2=12mm. The gray area corresponds to the zero-n¯ gap.

Fig. 8.
Fig. 8.

(a) |ZB|, (b) θZB, and (c) absorptance versus ω under different d1 for DPS/DNG PC structures with d1/d2=1/2 and γe=γm=0.001GHz. (d) ωmin as a function of d1 in PC structures with d1/d2=1/2.

Fig. 9.
Fig. 9.

(a) |ZB|, (b) θZB, and (c) absorptance versus ω under different d1/d2 for DPS/DNG PC structures with d1=6mm and γe=0.01GHz, γm=0.001GHz. The squares, circles, and triangles correspond to d2=12, 10, and 8 mm, respectively. (d) ωmin as a function of d1/d2 when d1=6mm.

Equations (13)

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ε2(ω)=ε0ωep2ω(ω+iγe),
μ2(ω)=μ0ωmp2ω(ω+iγm),
Mj=[cos(kjdj)i/qjsin(kjdj)iqjsin(kjdj)cos(kjdj)],
M=M1M2=[u11u12u21u22],
[EnHn]=[u11u12u21u22][En+1Hn+1].
[u11eiKBdu12u21u22eiKBd][En+1Hn+1]=0,
cos(KBd)=(u11+u22)/2.
eiKBd=(u11+u22)±(u11+u22)242.
ZB=En+1Hn+1Z0.
ZB=u12u11eiKBdZ0.
ZB=±2u12u11u22+(u11+u22)24Z0.
R=|1ZB/Z01+ZB/Z0|2,
A=1R=1|1ZB/Z01+ZB/Z0|2.

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