Abstract

The analytic expressions for frequency locations of the zero-effective-phase photonic band edge of the photonic multilayers containing single-negative materials are derived based on the effective medium theory. By adopting the derived band-edge formula, the properties, especially the thickness- and angular-dependence of the zero- effective-phase photonic bandgap are investigated in detail. The obtained results are consistent with the predictions in the transfer-matrix method. Moreover, the potential application of the band-edge formula in extending zero-effective-phase photonic bandgaps is proposed, which is verified in the photonic heterostructure realized by using composite right-/left-handed transmission line.

© 2011 Optical Society of America

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2010 (1)

2009 (4)

2008 (1)

2007 (3)

Y. J. Xiang, X. Y. Dai, S. C. Wen, and D. Y. Fan, “Enlargement of zero averaged refractive index gaps in the photonic heterostructures containing negative-index materials,” Phys. Rev. E 76, 056604 (2007).
[CrossRef]

Y. H. Lu, M. D. Huang, S. Y. Park, P. J. Kim, T. U. Nahm, Y. P. Lee, and J. Y. Rhee, “Controllable switching behavior of defect modes in one-dimensional heterostructure photonic crystals,” J. Appl. Phys. 101, 036110 (2007).
[CrossRef]

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

2006 (4)

N. Krumbholz, K. Gerlach, F. Rutz, M. Koch, R. Piesiewicz, T. Kürner, and D. Mittleman, “Omnidirectional terahertz mirrors: A key element for future terahertz communication systems,” Appl. Phys. Lett. 88, 202905 (2006).
[CrossRef]

S. K. Awasthi, U. Malaviya, and S. P. Ojha, “Enhancement of omnidirectional total-reflection wavelength range by using one-dimensional ternary photonic bandgap material,” J. Opt. Soc. Am. B 23, 2566–2571 (2006).
[CrossRef]

S. M. Wang, C. J. Tang, T. Pan, and L. Gao, “Effectively negatively refractive material made of negative-permittivity and negative-permeability bilayer,” Phys. Lett. A 351, 391–397(2006).
[CrossRef]

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

2004 (2)

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

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

2003 (3)

J. S. 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] [PubMed]

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]

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef] [PubMed]

1998 (2)

N. Winn, Y. Fink, S. Fan, and J. D. Joannopoulos, “Omnidirectional reflection from a one-dimensional photonic crystal,” Opt. Lett. 23, 1573–1575 (1998).
[CrossRef]

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

1987 (2)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2063(1987).
[CrossRef] [PubMed]

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

1968 (1)

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

Andrés, P.

Awasthi, S. K.

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Caloz, C.

C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications(Wiley, 2006).

Chan, C. T.

J. S. 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] [PubMed]

Chen, C.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

Chen, H.

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

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

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

Chen, Y. H.

Dai, X. Y.

Y. J. Xiang, X. Y. Dai, S. C. Wen, and D. Y. Fan, “Independently tunable omnidirectional multichannel filters based on the fractal multilayers containing negative-index materials,” Opt. Lett. 33, 1255–1257 (2008).
[CrossRef] [PubMed]

Y. J. Xiang, X. Y. Dai, S. C. Wen, and D. Y. Fan, “Enlargement of zero averaged refractive index gaps in the photonic heterostructures containing negative-index materials,” Phys. Rev. E 76, 056604 (2007).
[CrossRef]

Deng, X. H.

Depine, R. A.

Fan, D. Y.

Y. J. Xiang, X. Y. Dai, S. C. Wen, and D. Y. Fan, “Independently tunable omnidirectional multichannel filters based on the fractal multilayers containing negative-index materials,” Opt. Lett. 33, 1255–1257 (2008).
[CrossRef] [PubMed]

Y. J. Xiang, X. Y. Dai, S. C. Wen, and D. Y. Fan, “Enlargement of zero averaged refractive index gaps in the photonic heterostructures containing negative-index materials,” Phys. Rev. E 76, 056604 (2007).
[CrossRef]

Fan, S.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

N. Winn, Y. Fink, S. Fan, and J. D. Joannopoulos, “Omnidirectional reflection from a one-dimensional photonic crystal,” Opt. Lett. 23, 1573–1575 (1998).
[CrossRef]

Fink, Y.

N. Winn, Y. Fink, S. Fan, and J. D. Joannopoulos, “Omnidirectional reflection from a one-dimensional photonic crystal,” Opt. Lett. 23, 1573–1575 (1998).
[CrossRef]

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

Gao, L.

S. M. Wang, C. J. Tang, T. Pan, and L. Gao, “Effectively negatively refractive material made of negative-permittivity and negative-permeability bilayer,” Phys. Lett. A 351, 391–397(2006).
[CrossRef]

Gerlach, K.

N. Krumbholz, K. Gerlach, F. Rutz, M. Koch, R. Piesiewicz, T. Kürner, and D. Mittleman, “Omnidirectional terahertz mirrors: A key element for future terahertz communication systems,” Appl. Phys. Lett. 88, 202905 (2006).
[CrossRef]

He, L.

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

Huang, M. D.

Y. H. Lu, M. D. Huang, S. Y. Park, P. J. Kim, T. U. Nahm, Y. P. Lee, and J. Y. Rhee, “Controllable switching behavior of defect modes in one-dimensional heterostructure photonic crystals,” J. Appl. Phys. 101, 036110 (2007).
[CrossRef]

Itoh, T.

C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications(Wiley, 2006).

Jiang, H. T.

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

Joannopoulos, J. D.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

N. Winn, Y. Fink, S. Fan, and J. D. Joannopoulos, “Omnidirectional reflection from a one-dimensional photonic crystal,” Opt. Lett. 23, 1573–1575 (1998).
[CrossRef]

John, S.

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

Kim, P. J.

Y. H. Lu, M. D. Huang, S. Y. Park, P. J. Kim, T. U. Nahm, Y. P. Lee, and J. Y. Rhee, “Controllable switching behavior of defect modes in one-dimensional heterostructure photonic crystals,” J. Appl. Phys. 101, 036110 (2007).
[CrossRef]

Koch, M.

N. Krumbholz, K. Gerlach, F. Rutz, M. Koch, R. Piesiewicz, T. Kürner, and D. Mittleman, “Omnidirectional terahertz mirrors: A key element for future terahertz communication systems,” Appl. Phys. Lett. 88, 202905 (2006).
[CrossRef]

Kochergin, V.

V. Kochergin, Omnidirectional Optical Filters (Kluwer, 2003).

Krumbholz, N.

N. Krumbholz, K. Gerlach, F. Rutz, M. Koch, R. Piesiewicz, T. Kürner, and D. Mittleman, “Omnidirectional terahertz mirrors: A key element for future terahertz communication systems,” Appl. Phys. Lett. 88, 202905 (2006).
[CrossRef]

Kürner, T.

N. Krumbholz, K. Gerlach, F. Rutz, M. Koch, R. Piesiewicz, T. Kürner, and D. Mittleman, “Omnidirectional terahertz mirrors: A key element for future terahertz communication systems,” Appl. Phys. Lett. 88, 202905 (2006).
[CrossRef]

Lee, Y. P.

Y. H. Lu, M. D. Huang, S. Y. Park, P. J. Kim, T. U. Nahm, Y. P. Lee, and J. Y. Rhee, “Controllable switching behavior of defect modes in one-dimensional heterostructure photonic crystals,” J. Appl. Phys. 101, 036110 (2007).
[CrossRef]

Li, H. Q.

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

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

Li, J. S.

J. S. 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] [PubMed]

Liao, Q. H.

Liu, J. T.

Liu, N. H.

Liu, W. H.

H. Y. Zhang, Y. P. Zhang, W. H. Liu, Y. Q. Wang, and J. G. Yang, “Zero-averaged refractive-index gaps extension by using photonic heterostructures containing negative-index materials,” Appl. Phys. B 96, 67–70 (2009).
[CrossRef]

Lu, Y. H.

Y. H. Lu, M. D. Huang, S. Y. Park, P. J. Kim, T. U. Nahm, Y. P. Lee, and J. Y. Rhee, “Controllable switching behavior of defect modes in one-dimensional heterostructure photonic crystals,” J. Appl. Phys. 101, 036110 (2007).
[CrossRef]

Malaviya, U.

Martínez-Ricci, M. L.

Michel, J.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

Mittleman, D.

N. Krumbholz, K. Gerlach, F. Rutz, M. Koch, R. Piesiewicz, T. Kürner, and D. Mittleman, “Omnidirectional terahertz mirrors: A key element for future terahertz communication systems,” Appl. Phys. Lett. 88, 202905 (2006).
[CrossRef]

Monsoriu, J. A.

Nahm, T. U.

Y. H. Lu, M. D. Huang, S. Y. Park, P. J. Kim, T. U. Nahm, Y. P. Lee, and J. Y. Rhee, “Controllable switching behavior of defect modes in one-dimensional heterostructure photonic crystals,” J. Appl. Phys. 101, 036110 (2007).
[CrossRef]

Ojha, S. P.

Pan, T.

S. M. Wang, C. J. Tang, T. Pan, and L. Gao, “Effectively negatively refractive material made of negative-permittivity and negative-permeability bilayer,” Phys. Lett. A 351, 391–397(2006).
[CrossRef]

Park, S. Y.

Y. H. Lu, M. D. Huang, S. Y. Park, P. J. Kim, T. U. Nahm, Y. P. Lee, and J. Y. Rhee, “Controllable switching behavior of defect modes in one-dimensional heterostructure photonic crystals,” J. Appl. Phys. 101, 036110 (2007).
[CrossRef]

Piesiewicz, R.

N. Krumbholz, K. Gerlach, F. Rutz, M. Koch, R. Piesiewicz, T. Kürner, and D. Mittleman, “Omnidirectional terahertz mirrors: A key element for future terahertz communication systems,” Appl. Phys. Lett. 88, 202905 (2006).
[CrossRef]

Rhee, J. Y.

Y. H. Lu, M. D. Huang, S. Y. Park, P. J. Kim, T. U. Nahm, Y. P. Lee, and J. Y. Rhee, “Controllable switching behavior of defect modes in one-dimensional heterostructure photonic crystals,” J. Appl. Phys. 101, 036110 (2007).
[CrossRef]

Rutz, F.

N. Krumbholz, K. Gerlach, F. Rutz, M. Koch, R. Piesiewicz, T. Kürner, and D. Mittleman, “Omnidirectional terahertz mirrors: A key element for future terahertz communication systems,” Appl. Phys. Lett. 88, 202905 (2006).
[CrossRef]

Schurig, D.

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef] [PubMed]

Shalaev, V. M.

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

Sheng, P.

J. S. 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] [PubMed]

Silvestre, E.

Smith, D. R.

D. R. Smith and D. Schurig, “Electromagnetic wave propagation in media with indefinite permittivity and permeability tensors,” Phys. Rev. Lett. 90, 077405 (2003).
[CrossRef] [PubMed]

Tang, C. J.

S. M. Wang, C. J. Tang, T. Pan, and L. Gao, “Effectively negatively refractive material made of negative-permittivity and negative-permeability bilayer,” Phys. Lett. A 351, 391–397(2006).
[CrossRef]

Thomas, E. L.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

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]

Wang, L. G.

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

Wang, S. M.

S. M. Wang, C. J. Tang, T. Pan, and L. Gao, “Effectively negatively refractive material made of negative-permittivity and negative-permeability bilayer,” Phys. Lett. A 351, 391–397(2006).
[CrossRef]

Wang, Y. Q.

H. Y. Zhang, Y. P. Zhang, W. H. Liu, Y. Q. Wang, and J. G. Yang, “Zero-averaged refractive-index gaps extension by using photonic heterostructures containing negative-index materials,” Appl. Phys. B 96, 67–70 (2009).
[CrossRef]

Wen, S. C.

Y. J. Xiang, X. Y. Dai, S. C. Wen, and D. Y. Fan, “Independently tunable omnidirectional multichannel filters based on the fractal multilayers containing negative-index materials,” Opt. Lett. 33, 1255–1257 (2008).
[CrossRef] [PubMed]

Y. J. Xiang, X. Y. Dai, S. C. Wen, and D. Y. Fan, “Enlargement of zero averaged refractive index gaps in the photonic heterostructures containing negative-index materials,” Phys. Rev. E 76, 056604 (2007).
[CrossRef]

Winn, J. N.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679–1682 (1998).
[CrossRef] [PubMed]

Winn, N.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University, 1999).

Xiang, Y. J.

Y. J. Xiang, X. Y. Dai, S. C. Wen, and D. Y. Fan, “Independently tunable omnidirectional multichannel filters based on the fractal multilayers containing negative-index materials,” Opt. Lett. 33, 1255–1257 (2008).
[CrossRef] [PubMed]

Y. J. Xiang, X. Y. Dai, S. C. Wen, and D. Y. Fan, “Enlargement of zero averaged refractive index gaps in the photonic heterostructures containing negative-index materials,” Phys. Rev. E 76, 056604 (2007).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2063(1987).
[CrossRef] [PubMed]

Yang, J. G.

H. Y. Zhang, Y. P. Zhang, W. H. Liu, Y. Q. Wang, and J. G. Yang, “Zero-averaged refractive-index gaps extension by using photonic heterostructures containing negative-index materials,” Appl. Phys. B 96, 67–70 (2009).
[CrossRef]

Yu, T. B.

Zhang, H. Y.

H. Y. Zhang, Y. P. Zhang, W. H. Liu, Y. Q. Wang, and J. G. Yang, “Zero-averaged refractive-index gaps extension by using photonic heterostructures containing negative-index materials,” Appl. Phys. B 96, 67–70 (2009).
[CrossRef]

Zhang, L. W.

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

Zhang, Y. P.

H. Y. Zhang, Y. P. Zhang, W. H. Liu, Y. Q. Wang, and J. G. Yang, “Zero-averaged refractive-index gaps extension by using photonic heterostructures containing negative-index materials,” Appl. Phys. B 96, 67–70 (2009).
[CrossRef]

Zhang, Y. W.

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[CrossRef]

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[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Schematic of PMs [ A B ] M composed of alternating layers of MNG (A) and ENG material (B) in the air background.

Fig. 2
Fig. 2

(a) Dependence of the upper and lower band-edge frequencies at normal incidence on the thickness ratio ρ. For comparison, the projected band structure obtained by TMM is also plotted, where the black region is the PBG. (b) The dispersion curves for different thickness ratio ρ, where ε 1 = μ 2 = 1 , ε 2 = μ 1 = 3 , α = 100 GHz 2 , β = 50 GHz 2 , and a = 12 mm .

Fig. 3
Fig. 3

Reflectance spectra of PM1 for the TE polarization when the losses of the ENG and MNG materials are considered: (a)  γ = γ α = γ β = 0.1 GHz and (b)  γ = γ α = γ β = 0.5 GHz . The upper and lower frequency limits on the incident angle are also shown.

Fig. 4
Fig. 4

(a) Dependence of the upper and lower band-edge frequencies on β / α at normal incidence. The projected band structure obtained by TMM is also plotted. (b), (c), and (d) show the transmittance for PM1, PM2, and PM 1 + PM 2 , respectively, where α = 100 GHz 2 , β = 50 GHz 2 in PM1 and α = 80 GHz 2 , β = 250 GHz 2 in PM2, b = 10 mm , and the other parameters have the same values as in Fig. 2.

Fig. 5
Fig. 5

Dependence of the upper and lower band-edges frequencies on the incident angle. The projected band structure is also displayed. (a) TE and (b) TM polarizations for PM1; (c) TE and (d) TM polarizations for PM2. The parameters have the same values as in the Fig. 2b. δ Ω 1 and δ Ω 2 are the frequency range of the omnidirectional PBG of the PM1 and PM2 respectively.

Fig. 6
Fig. 6

Dependences of the upper and lower band-edge frequencies on the incident angle of the heterostructure PM 1 + PM 2 for TE polarization (a) and TM polarization (b). For comparison, we also have plotted the projected band structure. δ Ω is the frequency range of the omnidirectional PBG.

Fig. 7
Fig. 7

The transmittance for (a) TL PM1 [ ( ENG 1 MNG 1 ) 10 ], (b) TL PM2 [ ( ENG 2 MNG 2 ) 10 ] and (c) TL PM 1 + PM 2 [ ( ENG 1 MNG 1 ) 10 ( ENG 2 MNG 2 ) 10 ], the parameters have been given in the Table 1.

Tables (1)

Tables Icon

Table 1 Physical Parameters of the TLs in the Heterostructure TL PM 1 + PM 2 [ ( ENG 1 MNG 1 ) 10 ( ENG 2 MNG 2 ) 10 ]

Equations (30)

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M i ( Δ z , ω ) = ( cosh ( | k i z | Δ z ) 1 q i z sinh ( | k i z | Δ z ) q i z sinh ( | k i z | Δ z ) cosh ( | k i z | Δ z ) ) ,
X ( ω ) = i = 1 2 M M i ( Δ z , ω ) = ( x 11 ( ω ) x 12 ( ω ) x 21 ( ω ) x 22 ( ω ) ) .
t ( ω ) = 2 q 0 [ q 0 x 22 ( ω ) + q s x 11 ( ω ) ] [ q 0 q s x 12 ( ω ) + x 21 ( ω ) ] .
cos ( K d ) = cosh ( | k a z | a ) cosh ( | k b z | b ) 1 2 ( | q a z | | q b z | + | q b z | | q a z | ) sinh ( | k a z | a ) sinh ( | k b z | b ) ,
M a ( ω ) = ( 1 M a ( 1 , 2 ) M a ( 2 , 1 ) 1 ) , M b ( ω ) = ( 1 M b ( 1 , 2 ) M b ( 2 , 1 ) 1 ) ,
M d ( ω ) = M a ( ω ) M b ( ω ) ( 1 M d ( 1 , 2 ) M d ( 2 , 1 ) 1 ) ,
ε eff = ( ε eff x 0 0 0 ε eff y 0 0 0 ε eff z ) , μ eff = ( μ eff x 0 0 0 μ eff y 0 0 0 μ eff z ) .
k eff z 2 = ε eff y μ eff x ( ω 2 / c 2 ) ( μ eff x / μ eff z ) k eff x 2 ,
M eff ( ω ) ( 1 M eff ( 1 , 2 ) M eff ( 2 , 1 ) 1 ) ,
μ eff x = μ a f a + μ b f b , ε eff y = ε a f a + ε b f b , μ eff z = ( f a / μ a + f b / μ b ) 1 .
ε eff x = ε a f a + ε b f b , μ eff y = μ a f a + μ b f b , ε eff z = ( f a / ε a + f b / ε b ) 1 .
k eff z 2 = 0 .
ε ¯ μ ¯ 1 sin 2 θ = 0 and μ ¯ = 0 ,
ε ¯ = 0 and μ ¯ ε ¯ 1 sin 2 θ = 0 ,
ε ¯ = 0 and μ ¯ = 0.
ω 1 ( ε ¯ = 0 ) = α ε 1 + ε 2 ( b / a ) ,
ω 2 ( μ ¯ = 0 ) = β μ 2 + μ 1 ( a / b ) .
ρ = [ ( α μ 2 ε 1 β ) + ( α μ 2 ε 1 β ) 2 + 4 α β ε 2 μ 1 ] / ( 2 β ε 2 ) ,
β / α = ( μ 2 + μ 1 / ρ ) / ( ε 1 + ε 2 ρ ) ,
ε 2 / ε 1 = ( α / β ) [ μ 2 / ( ε 1 ρ ) + μ 1 / ( ε 1 ρ 2 ) ] 1 / ρ ,
μ 2 / μ 1 = ( β / α ) ( ε 1 / μ 1 + ε 2 ρ / μ 1 ) 1 / ρ .
ω 2 ( μ ¯ = 0 ) = β μ 2 + μ 1 ( a / b ) ,
A ω 1 4 B ω 1 2 + C = 0 ,
ε i ( ω ) = ( C 0 1 ( ω 2 i γ ε ω ) L i d i ) / ( ε 0 p ) ,
μ i ( ω ) = p ( L 0 1 ( ω 2 i γ μ ω ) C i d i ) / μ 0 ,
ω a = 1 L 0 C d i and ω b = 1 L C 0 d i .
ε i ( ω ) = ε i 0 α i / ω 2 ,
μ i ( ω ) = μ i 0 β i / ω 2 ,
ω 1 ( ε ¯ = 0 ) = α E 1 + α M 1 υ ε E 10 + ε M 10 υ ,
ω 2 ( μ ¯ = 0 ) = β E 1 + β M 1 υ μ E 10 + μ M 10 υ ,

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