Abstract

It is shown that the defect modes inside the zero-n¯ gap can be sensitive to the incident angle, the same as those inside the Bragg gap. An expression of the dispersion of the defect modes (i.e., the incident angle dependence of defect modes) inside the Bragg gap or the zero-n¯ gap of one-dimensional photonic crystals containing negative-index materials is derived. It is found from the expression that the dispersion of the defect mode approaches zero when the phase change on reflection from periodic stacks of the photonic crystals can cancel out the change of the optical phase thickness of the defect layer. However, if they cannot cancel each other out, the dispersion will be of either the positive or the negative type. Practical designs for omnidirectional defect modes are given according to the conditions of the near-zero dispersion. In addition, a narrow frequency and sharp angular defect mode are designed by combining two photonic crystals with defect modes of the positive and the negative types of dispersion, respectively.

© 2006 Optical Society of America

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References

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  1. E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (1987).
    [CrossRef] [PubMed]
  2. S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486-2489 (1987).
    [CrossRef] [PubMed]
  3. J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic band gap guidance in optical fibers," Science 282, 1476-1478 (1998).
    [CrossRef] [PubMed]
  4. C. M. Bowden, J. P. Dowling, and H. O. Everitt, "Development and applications of materials exhibiting photonic band gaps: introduction," J. Opt. Soc. Am. B 10, 280-413 (1993).
  5. Q. Qin, H. Lu, S. N. Zhu, C. S. Yuan, Y. Y. Zhu, and N. B. Ming, "Resonance transmission modes in dual-periodical dielectric multilayer films," Appl. Phys. Lett. 82, 4654-4656 (2003).
    [CrossRef]
  6. Z. S. Wang, L. Wang, Y. G. Wu, and L. Y. Chen, "Multiple channeled phenomena in heterostructures with defects mode," Appl. Phys. Lett. 84, 1629-1631 (2004).
    [CrossRef]
  7. V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of permittivity and permeability," Sov. Phys. Usp. 10, 509-514 (1968).
    [CrossRef]
  8. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
    [CrossRef] [PubMed]
  9. J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
    [CrossRef]
  10. R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental verification of a negative index of refraction," Science 292, 77-79 (2001).
    [CrossRef] [PubMed]
  11. 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] [PubMed]
  12. H. T. Jiang, H. Chen, H. Q. Li, and Y. W. Zhang, "Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials," Appl. Phys. Lett. 83, 5386-5388 (2003).
    [CrossRef]
  13. G. Q. Liang, P. Han, and H. Z. Wang, "Narrow frequency and sharp angular defect mode in one-dimensional photonic crystals from a photonic heterostructure," Opt. Lett. 29, 192-194 (2004).
    [CrossRef] [PubMed]
  14. C. R. Pidgeon and S. D. Smith, "Resolving power of multilayer filters in nonparallel light," J. Opt. Soc. Am. 54, 1459-1466 (1964).
    [CrossRef]
  15. H. A. Macleod, Thin-Film Optical Filters (Adam Hilger, 1989).
  16. J. S. Seeley, "Resolving power of multilayer filters," J. Opt. Soc. Am. 54, 342-346 (1962).
  17. H. Nemec and P. Kuzel, "Defect modes caused by twinning in one-dimensional photonic crystals," J. Opt. Soc. Am. B 21, 548-553 (2004).
    [CrossRef]
  18. H. Y. Sang, Z. Y. Li, and B. Y. Gu, "Stack-sequence dependent defect modes in one-dimensional photonic crystals," Phys. Lett. A 331, 414-422 (2004).
    [CrossRef]
  19. K. Y. Xu, X. G. Zheng, and W. L. She, "Properties of defect modes in one-dimensional photonic crystals containing a defect layer with a negative refractive index," Appl. Phys. Lett. 85, 6089-6091 (2004).
    [CrossRef]

2004 (5)

Z. S. Wang, L. Wang, Y. G. Wu, and L. Y. Chen, "Multiple channeled phenomena in heterostructures with defects mode," Appl. Phys. Lett. 84, 1629-1631 (2004).
[CrossRef]

G. Q. Liang, P. Han, and H. Z. Wang, "Narrow frequency and sharp angular defect mode in one-dimensional photonic crystals from a photonic heterostructure," Opt. Lett. 29, 192-194 (2004).
[CrossRef] [PubMed]

H. Nemec and P. Kuzel, "Defect modes caused by twinning in one-dimensional photonic crystals," J. Opt. Soc. Am. B 21, 548-553 (2004).
[CrossRef]

H. Y. Sang, Z. Y. Li, and B. Y. Gu, "Stack-sequence dependent defect modes in one-dimensional photonic crystals," Phys. Lett. A 331, 414-422 (2004).
[CrossRef]

K. Y. Xu, X. G. Zheng, and W. L. She, "Properties of defect modes in one-dimensional photonic crystals containing a defect layer with a negative refractive index," Appl. Phys. Lett. 85, 6089-6091 (2004).
[CrossRef]

2003 (3)

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] [PubMed]

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

Q. Qin, H. Lu, S. N. Zhu, C. S. Yuan, Y. Y. Zhu, and N. B. Ming, "Resonance transmission modes in dual-periodical dielectric multilayer films," Appl. Phys. Lett. 82, 4654-4656 (2003).
[CrossRef]

2001 (1)

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

1999 (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

1998 (1)

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic band gap guidance in optical fibers," Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

1996 (1)

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

1993 (1)

1987 (2)

E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059-2062 (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 permittivity and permeability," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

1964 (1)

1962 (1)

Birks, T. A.

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic band gap guidance in optical fibers," Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

Bowden, C. M.

Broeng, J.

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic band gap guidance in optical fibers," Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

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] [PubMed]

Chen, H.

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

Chen, L. Y.

Z. S. Wang, L. Wang, Y. G. Wu, and L. Y. Chen, "Multiple channeled phenomena in heterostructures with defects mode," Appl. Phys. Lett. 84, 1629-1631 (2004).
[CrossRef]

Dowling, J. P.

Everitt, H. O.

Gu, B. Y.

H. Y. Sang, Z. Y. Li, and B. Y. Gu, "Stack-sequence dependent defect modes in one-dimensional photonic crystals," Phys. Lett. A 331, 414-422 (2004).
[CrossRef]

Han, P.

Holden, A. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Jiang, H. T.

H. T. Jiang, H. Chen, H. Q. Li, and Y. W. Zhang, "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] [PubMed]

Knight, J. C.

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic band gap guidance in optical fibers," Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

Kuzel, P.

Li, H. Q.

H. T. Jiang, H. Chen, H. Q. Li, and Y. W. Zhang, "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] [PubMed]

Li, Z. Y.

H. Y. Sang, Z. Y. Li, and B. Y. Gu, "Stack-sequence dependent defect modes in one-dimensional photonic crystals," Phys. Lett. A 331, 414-422 (2004).
[CrossRef]

Liang, G. Q.

Lu, H.

Q. Qin, H. Lu, S. N. Zhu, C. S. Yuan, Y. Y. Zhu, and N. B. Ming, "Resonance transmission modes in dual-periodical dielectric multilayer films," Appl. Phys. Lett. 82, 4654-4656 (2003).
[CrossRef]

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (Adam Hilger, 1989).

Ming, N. B.

Q. Qin, H. Lu, S. N. Zhu, C. S. Yuan, Y. Y. Zhu, and N. B. Ming, "Resonance transmission modes in dual-periodical dielectric multilayer films," Appl. Phys. Lett. 82, 4654-4656 (2003).
[CrossRef]

Nemec, H.

Pendry, J. B.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Pidgeon, C. R.

Qin, Q.

Q. Qin, H. Lu, S. N. Zhu, C. S. Yuan, Y. Y. Zhu, and N. B. Ming, "Resonance transmission modes in dual-periodical dielectric multilayer films," Appl. Phys. Lett. 82, 4654-4656 (2003).
[CrossRef]

Robbins, D. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

Russell, P. St. J.

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic band gap guidance in optical fibers," Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

Sang, H. Y.

H. Y. Sang, Z. Y. Li, and B. Y. Gu, "Stack-sequence dependent defect modes in one-dimensional photonic crystals," Phys. Lett. A 331, 414-422 (2004).
[CrossRef]

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] [PubMed]

Seeley, J. S.

She, W. L.

K. Y. Xu, X. G. Zheng, and W. L. She, "Properties of defect modes in one-dimensional photonic crystals containing a defect layer with a negative refractive index," Appl. Phys. Lett. 85, 6089-6091 (2004).
[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] [PubMed]

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] [PubMed]

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] [PubMed]

Smith, S. D.

Stewart, W. J.

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Veselago, V. G.

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of permittivity and permeability," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Wang, H. Z.

Wang, L.

Z. S. Wang, L. Wang, Y. G. Wu, and L. Y. Chen, "Multiple channeled phenomena in heterostructures with defects mode," Appl. Phys. Lett. 84, 1629-1631 (2004).
[CrossRef]

Wang, Z. S.

Z. S. Wang, L. Wang, Y. G. Wu, and L. Y. Chen, "Multiple channeled phenomena in heterostructures with defects mode," Appl. Phys. Lett. 84, 1629-1631 (2004).
[CrossRef]

Wu, Y. G.

Z. S. Wang, L. Wang, Y. G. Wu, and L. Y. Chen, "Multiple channeled phenomena in heterostructures with defects mode," Appl. Phys. Lett. 84, 1629-1631 (2004).
[CrossRef]

Xu, K. Y.

K. Y. Xu, X. G. Zheng, and W. L. She, "Properties of defect modes in one-dimensional photonic crystals containing a defect layer with a negative refractive index," Appl. Phys. Lett. 85, 6089-6091 (2004).
[CrossRef]

Yablonovitch, E.

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

Youngs, I.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

Yuan, C. S.

Q. Qin, H. Lu, S. N. Zhu, C. S. Yuan, Y. Y. Zhu, and N. B. Ming, "Resonance transmission modes in dual-periodical dielectric multilayer films," Appl. Phys. Lett. 82, 4654-4656 (2003).
[CrossRef]

Zhang, Y. W.

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

Zheng, X. G.

K. Y. Xu, X. G. Zheng, and W. L. She, "Properties of defect modes in one-dimensional photonic crystals containing a defect layer with a negative refractive index," Appl. Phys. Lett. 85, 6089-6091 (2004).
[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] [PubMed]

Zhu, S. N.

Q. Qin, H. Lu, S. N. Zhu, C. S. Yuan, Y. Y. Zhu, and N. B. Ming, "Resonance transmission modes in dual-periodical dielectric multilayer films," Appl. Phys. Lett. 82, 4654-4656 (2003).
[CrossRef]

Zhu, Y. Y.

Q. Qin, H. Lu, S. N. Zhu, C. S. Yuan, Y. Y. Zhu, and N. B. Ming, "Resonance transmission modes in dual-periodical dielectric multilayer films," Appl. Phys. Lett. 82, 4654-4656 (2003).
[CrossRef]

Appl. Phys. Lett. (4)

Q. Qin, H. Lu, S. N. Zhu, C. S. Yuan, Y. Y. Zhu, and N. B. Ming, "Resonance transmission modes in dual-periodical dielectric multilayer films," Appl. Phys. Lett. 82, 4654-4656 (2003).
[CrossRef]

Z. S. Wang, L. Wang, Y. G. Wu, and L. Y. Chen, "Multiple channeled phenomena in heterostructures with defects mode," Appl. Phys. Lett. 84, 1629-1631 (2004).
[CrossRef]

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

K. Y. Xu, X. G. Zheng, and W. L. She, "Properties of defect modes in one-dimensional photonic crystals containing a defect layer with a negative refractive index," Appl. Phys. Lett. 85, 6089-6091 (2004).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

J. B. Pendry, A. J. Holden, D. J. Robbins, and W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech. 47, 2075-2084 (1999).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. B (2)

Opt. Lett. (1)

Phys. Lett. A (1)

H. Y. Sang, Z. Y. Li, and B. Y. Gu, "Stack-sequence dependent defect modes in one-dimensional photonic crystals," Phys. Lett. A 331, 414-422 (2004).
[CrossRef]

Phys. Rev. Lett. (4)

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] [PubMed]

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, "Extremely low frequency plasmons in metallic mesostructures," Phys. Rev. Lett. 76, 4773-4776 (1996).
[CrossRef] [PubMed]

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

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

Science (2)

J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, "Photonic band gap guidance in optical fibers," Science 282, 1476-1478 (1998).
[CrossRef] [PubMed]

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

Sov. Phys. Usp. (1)

V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of permittivity and permeability," Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Other (1)

H. A. Macleod, Thin-Film Optical Filters (Adam Hilger, 1989).

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Figures (5)

Fig. 1
Fig. 1

Transmission spectra of a defective PC stacked with alternate PIM and NIM at different incident angles for (a) TE waves and (b) TM waves. Structure parameters: n P = n D = 2 , d P = 12 mm , ε N ( ν ) = 1.21 100 ( 2 π ν ) 2 , μ N ( ν ) = 1.0 100 ( 2 π ν ) 2 , d N = 6 mm , and d D = 90 mm .

Fig. 2
Fig. 2

Dependence of the defect modes on the incident angle for different polarizations. The parameters are the same as those in Fig. 1.

Fig. 3
Fig. 3

Dispersion relations of defect modes in structures: (a) ( N P ) s 2 D ( P N ) s , with μ P = 1 , n P = 3 , μ D = 1 , n D = 4.65 , μ N = 1.0 100 ( 2 π ν ) 2 , ε N = 1.21 100 ( 2 π ν ) 2 , ν 0 = 0.985 GHz , d P = 25.5 mm , d N = 51 mm , and d D = 33 mm ; (b) ( P 1 P 2 ) s 2 D ( P 2 P 1 ) s , with μ P 1 = μ P 2 = 1 , n P 1 = 3 , n P 2 = 1.5 , μ D = 1.0 100 ( 2 π ν ) 2 , ε D = 1.21 100 ( 2 π ν ) 2 , ν 0 = 0.821 GHz , d P 1 = 61 mm , d P 2 = 30.5 mm , and d D = 69 mm . Here the number s is assumed to be infinite. Parameters in both (a) and (b) satisfy the near-zero dispersion condition, hence the frequency of the transmission peak of the defect mode is almost independent of the incident angle.

Fig. 4
Fig. 4

Dependence of the defect modes in structure of ( N P ) 4 2 D ( P N ) 4 on the incident angle for different polarizations. The triangles, circles, and squares represent the defect modes corresponding to the defect layers with n D = 2 , 4.65, and 5, respectively. The other parameters are the same as those in Fig. 3a. Here ν 0 = 0.985 GHz .

Fig. 5
Fig. 5

Frequency and incident angle-dependent transmittance of the heterostructure P ( N P ) 4 2 D 1 ( P N ) 4 P N P ( N P ) 4 2 D 1 ( P N ) 4 P , with μ P = 1 , n P = 3 , μ N = 1.0 100 ( 2 π ν ) 2 , ε N = 1.21 100 ( 2 π ν ) 2 , d P = 25.5 mm , d N = 51 mm , ν 0 = 0.985 GHz , μ D 1 = μ D 2 = 1 , n D 1 = 2 , n D 2 = 5 , d D 1 = 77 mm , and d D 21 = 31 mm .

Tables (1)

Tables Icon

Table 1 Product M and the Expressions of J for Different Types of PC Stacks

Equations (16)

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

T ( ν , θ e ) = T 1 ( ν , θ e ) T 2 ( ν , θ e ) { 1 [ R 1 ( ν , θ e ) R 2 ( ν , θ e ) ] 1 2 } 2 + 4 [ R 1 ( ν , θ e ) R 2 ( ν , θ e ) ] 1 2 sin 2 ( 1 2 δ ) ,
δ = ± ϕ 1 ( ν , θ e ) ± ϕ 2 ( ν , θ e ) + 4 π ν n D d D cos θ D = 2 m π
M i [ G i ± j Z i ± j Z i G i ] ,
M = i M i = [ m 11 j m 12 j m 21 m 22 ] .
Δ ϕ ( ν , θ e ) = ϕ ( ν , θ e ) ϕ ( ν 0 , 0 ) = k π ( ν ν 0 ν 0 ) + J sin 2 θ e = k π ( Δ ν ν 0 ) + J sin 2 θ e ,
J I = 2 Z D m 21 m I I ,
k I = g Z D Z i L ( Z r + 1 ) .
J I I = 2 m 12 m 22 Z D ,
k I I = Z i H g Z D ( Z r + 1 ) ,
Δ δ ( ν , θ e ) = δ ( ν , θ e ) δ ( ν 0 , 0 ) = ± 2 [ ϕ ( ν , θ e ) ϕ ( ν 0 , 0 ) ] + 2 g π ( cos ϕ 1 ) = ± ( k π Δ ν ν 0 + 2 J j sin 2 θ e ) g π n D 2 sin 2 θ e ,
± 2 k π Δ ν ν 0 = ( ± 2 J i g π n D 2 ) sin 2 θ e .
± 2 J i g π n D 2 = 0 ,
J I = π 2 1 Z D ( Z r 2 1 ) ( Z N n N 2 + Z N 2 n P 2 Z P ) ,
g n D 2 = 2 J I π .
J I I = π 2 Z D ( Z r 2 1 ) ( 1 n 1 2 Z 1 + Z 2 n 2 2 Z 1 2 ) ,
g n D 2 = 2 J I I π .

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