Abstract

In this paper, we applied the band structure theory to investigate the plasmonic band (PB) structures and optical properties of subwavelength metal/dielectric/metal Bragg waveguides in the near infrared range with either dielectric or geometric modulation. The Bloch wave vector, density of states, slowdown factor, propagation length and transmittance are calculated and analyzed. Both the modulations are in favor of manipulating surface-plasmon-polariton (SPP) waves. For the dielectric modulation, the PB structure is mainly formed by SPP modes and possesses a “regular pattern” in which the bands and gaps have a relatively even distribution. For the geometric modulation, due to the strong transverse scattering, the contributions of higher modes have to be considered and the gap widths have a significant increase compared to the dielectric modulation. A larger slowdown factor may emerge at the band edge; especially for the geometric modulation, the group velocity can be reduced to 1/100 of light, and negative group velocity is observed as well. While inside the bands, the slowdown factor is smaller and the bands are flat. The contribution of each eigenmode to the PB structure is analyzed.

© 2012 OSA

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  33. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
    [Crossref]
  34. L. Li, “Note on the S-matrix propagation algorithm,” J. Opt. Soc. Am. A 20, 655–660 (2003).
    [Crossref]
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  38. D. Y. Fedyanin, A. V. Arsenin, V. G. Leiman, and A. D. Gladun, “Backward waves in planar insulator-metal-insulator waveguide structures,” J. Opt. 12, 015002 (2010).
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    [Crossref]

2012 (1)

C. Li, Y. S. Zhou, and H. Y. Wang, “Scattering mechanism in a step-modulated subwavelength metal slit: a multi-mode multi-reflection analysis,” Eur. Phys. J. D 66, 8 (2012).
[Crossref]

2011 (4)

C. Li, Y. S. Zhou, H. Y. Wang, and F. H. Wang, “Investigation of the wave behaviors inside a step-modulated subwavelength metal slit,” Opt. Express 19, 10073–10087 (2011).
[Crossref] [PubMed]

X. L. Zhong, Z. Y. Li, C. Wang, and Y. S. Zhou, “Analytical single-mode model for subwavelength metallic Bragg waveguides,” J. Appl. Phys. 109, 093115 (2011).
[Crossref]

Y. Xu, A. E. Miroshnichenko, S. Lan, Q. Guo, and L. J. Wu, “Impedance matching induce high transmissionand flat response band-pass plasmonic waveguides,” Plasmonics 6, 337–343 (2011).
[Crossref]

E. P. Fitrakis, T. Kamalakis, and T. Sphicopoulos, “Slow light in insulator-metal-insulator plasmonic waveguides,” J. Opt. Soc. Am. B 28, 2159–2164 (2011).
[Crossref]

2010 (6)

2009 (3)

2008 (4)

2007 (3)

Z. Han, E. Forsberg, and S. He, “Surface plasmon Bragg gratings formed in metal-insulator-metal waveguides,” IEEE Photon. Technol. Lett. 19, 91–93 (2007).
[Crossref]

Y. Kurokawa and H. T. Miyazaki, “Metal-insulator-metal plasmon nanocavities: analysis of optical properties,” Phys. Rev. B 75, 035411 (2007).
[Crossref]

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007).
[Crossref]

2006 (5)

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature (London) 440, 508–511 (2006).
[Crossref]

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[Crossref] [PubMed]

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[Crossref]

R. Gordon, “Light in a subwavelength slit in a metal: propagation and reflection,” Phys. Rev. B 73, 153405 (2006).
[Crossref]

A. Hossieni and Y. Massoud, “A low-loss metal-insulator-metal plasmonic bragg reflector,” Opt. Express 14, 11318–11323 (2006).
[Crossref] [PubMed]

2003 (4)

L. Li, “Note on the S-matrix propagation algorithm,” J. Opt. Soc. Am. A 20, 655–660 (2003).
[Crossref]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature (London) 424, 824–830 (2003).
[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]

Z. Y. Li and K. M. Ho, “Analytic modal solution to light propagation through layer-by-layer metallic photonic crystals,” Phys. Rev. B 67, 165104 (2003).
[Crossref]

2001 (1)

F. Villa, T. Lopez-Rios, and L. E. Regalado, “Electromagnetic modes in metal-insulator-metal structures,” Phys. Rev. B 63, 165103 (2001).
[Crossref]

1997 (1)

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) 386, 143–149 (1997).
[Crossref]

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]

1972 (1)

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Arsenin, A. V.

D. Y. Fedyanin, A. V. Arsenin, V. G. Leiman, and A. D. Gladun, “Backward waves in planar insulator-metal-insulator waveguide structures,” J. Opt. 12, 015002 (2010).
[Crossref]

Atwater, H. A.

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[Crossref]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature (London) 424, 824–830 (2003).
[Crossref]

Bozhevolnyi, S. I.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature (London) 440, 508–511 (2006).
[Crossref]

Brillouin, L.

L. Brillouin, Wave Propagation and Group Velocity (Academic, 1960).

Cai, L.

Christy, R. W.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature (London) 424, 824–830 (2003).
[Crossref]

Devaux, E.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature (London) 440, 508–511 (2006).
[Crossref]

Dionne, J. A.

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[Crossref]

Ebbesen, T. W.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature (London) 440, 508–511 (2006).
[Crossref]

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature (London) 424, 824–830 (2003).
[Crossref]

Fan, S.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) 386, 143–149 (1997).
[Crossref]

Fang, G.

Fedyanin, D. Y.

D. Y. Fedyanin, A. V. Arsenin, V. G. Leiman, and A. D. Gladun, “Backward waves in planar insulator-metal-insulator waveguide structures,” J. Opt. 12, 015002 (2010).
[Crossref]

Fitrakis, E. P.

Forsberg, E.

Z. Han, E. Forsberg, and S. He, “Surface plasmon Bragg gratings formed in metal-insulator-metal waveguides,” IEEE Photon. Technol. Lett. 19, 91–93 (2007).
[Crossref]

Gladun, A. D.

D. Y. Fedyanin, A. V. Arsenin, V. G. Leiman, and A. D. Gladun, “Backward waves in planar insulator-metal-insulator waveguide structures,” J. Opt. 12, 015002 (2010).
[Crossref]

Gordon, R.

R. Gordon, “Light in a subwavelength slit in a metal: propagation and reflection,” Phys. Rev. B 73, 153405 (2006).
[Crossref]

Gorkunov, M.

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007).
[Crossref]

Gu, B. Y.

Y. S. Zhou, B. Y. Gu, and H. Y. Wang, “Band-gap structures of surface-plasmon polaritons in a subwavelength metal slit filled with periodic dielectrics,” Phys. Rev. A 81, 015801 (2010).
[Crossref]

Y. S. Zhou, B. Y. Gu, S. Lan, and L. M. Zhao, “Time-domain analysis of mechanism of plasmon-assisted extraordinary optical transmission,” Phys. Rev. B 78, 081404 (2008).
[Crossref]

Guo, Q.

Y. Xu, A. E. Miroshnichenko, S. Lan, Q. Guo, and L. J. Wu, “Impedance matching induce high transmissionand flat response band-pass plasmonic waveguides,” Plasmonics 6, 337–343 (2011).
[Crossref]

Han, B.

B. Han and C. Jiang, “Plasmonic slow light waveguide and cavity,” Appl. Phys. B: Lasers Opt. 95, 97–103 (2009).
[Crossref]

Han, Z.

Z. Han, E. Forsberg, and S. He, “Surface plasmon Bragg gratings formed in metal-insulator-metal waveguides,” IEEE Photon. Technol. Lett. 19, 91–93 (2007).
[Crossref]

He, M. D.

He, S.

Z. Han, E. Forsberg, and S. He, “Surface plasmon Bragg gratings formed in metal-insulator-metal waveguides,” IEEE Photon. Technol. Lett. 19, 91–93 (2007).
[Crossref]

Ho, K. M.

Z. Y. Li and K. M. Ho, “Analytic modal solution to light propagation through layer-by-layer metallic photonic crystals,” Phys. Rev. B 67, 165104 (2003).
[Crossref]

Hosseini, A.

Hossieni, A.

Huang, W. Q.

Jiang, C.

B. Han and C. Jiang, “Plasmonic slow light waveguide and cavity,” Appl. Phys. B: Lasers Opt. 95, 97–103 (2009).
[Crossref]

Joannopoulos, J. D.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) 386, 143–149 (1997).
[Crossref]

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press1995).

John, S.

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

Johnson, P. B.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Kamalakis, T.

Kang, Z. W.

Kim, J.

Krauss, T. F.

T. F. Krauss, “Why do we need slow light?” Nat. Photonics 2, 448–450 (2008).
[Crossref]

Kurokawa, Y.

Y. Kurokawa and H. T. Miyazaki, “Metal-insulator-metal plasmon nanocavities: analysis of optical properties,” Phys. Rev. B 75, 035411 (2007).
[Crossref]

Laluet, J. Y.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature (London) 440, 508–511 (2006).
[Crossref]

Lan, S.

Y. Xu, A. E. Miroshnichenko, S. Lan, Q. Guo, and L. J. Wu, “Impedance matching induce high transmissionand flat response band-pass plasmonic waveguides,” Plasmonics 6, 337–343 (2011).
[Crossref]

Y. S. Zhou, B. Y. Gu, S. Lan, and L. M. Zhao, “Time-domain analysis of mechanism of plasmon-assisted extraordinary optical transmission,” Phys. Rev. B 78, 081404 (2008).
[Crossref]

Leiman, V. G.

D. Y. Fedyanin, A. V. Arsenin, V. G. Leiman, and A. D. Gladun, “Backward waves in planar insulator-metal-insulator waveguide structures,” J. Opt. 12, 015002 (2010).
[Crossref]

Li, C.

Li, G. Y.

Li, L.

Li, Z. Y.

X. L. Zhong, Z. Y. Li, C. Wang, and Y. S. Zhou, “Analytical single-mode model for subwavelength metallic Bragg waveguides,” J. Appl. Phys. 109, 093115 (2011).
[Crossref]

Z. Y. Li and K. M. Ho, “Analytic modal solution to light propagation through layer-by-layer metallic photonic crystals,” Phys. Rev. B 67, 165104 (2003).
[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]

Lin, W. H.

Liu, J.

Liu, J. Q.

Liu, S.

Liu, Y.

Lopez-Rios, T.

F. Villa, T. Lopez-Rios, and L. E. Regalado, “Electromagnetic modes in metal-insulator-metal structures,” Phys. Rev. B 63, 165103 (2001).
[Crossref]

Massoud, Y.

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press1995).

Min, C.

Miroshnichenko, A. E.

Y. Xu, A. E. Miroshnichenko, S. Lan, Q. Guo, and L. J. Wu, “Impedance matching induce high transmissionand flat response band-pass plasmonic waveguides,” Plasmonics 6, 337–343 (2011).
[Crossref]

Miyazaki, H. T.

Y. Kurokawa and H. T. Miyazaki, “Metal-insulator-metal plasmon nanocavities: analysis of optical properties,” Phys. Rev. B 75, 035411 (2007).
[Crossref]

Nejati, H.

Ozbay, E.

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[Crossref] [PubMed]

Pei, Y. J.

Podivilov, E.

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007).
[Crossref]

Raether, H.

H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, 1988).

Regalado, L. E.

F. Villa, T. Lopez-Rios, and L. E. Regalado, “Electromagnetic modes in metal-insulator-metal structures,” Phys. Rev. B 63, 165103 (2001).
[Crossref]

Sakoda, K.

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

Sphicopoulos, T.

Sturman, B.

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007).
[Crossref]

Sweatlock, L. A.

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Plasmon slot waveguides: towards chip-scale propagation with subwavelength-scale localization,” Phys. Rev. B 73, 035407 (2006).
[Crossref]

Veronis, G.

Villa, F.

F. Villa, T. Lopez-Rios, and L. E. Regalado, “Electromagnetic modes in metal-insulator-metal structures,” Phys. Rev. B 63, 165103 (2001).
[Crossref]

Villeneuve, P. R.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature (London) 386, 143–149 (1997).
[Crossref]

Volkov, V. S.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J. Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature (London) 440, 508–511 (2006).
[Crossref]

Wang, C.

X. L. Zhong, Z. Y. Li, C. Wang, and Y. S. Zhou, “Analytical single-mode model for subwavelength metallic Bragg waveguides,” J. Appl. Phys. 109, 093115 (2011).
[Crossref]

Wang, D. Y.

Wang, F. H.

Wang, G. P.

Wang, H. Y.

C. Li, Y. S. Zhou, and H. Y. Wang, “Scattering mechanism in a step-modulated subwavelength metal slit: a multi-mode multi-reflection analysis,” Eur. Phys. J. D 66, 8 (2012).
[Crossref]

C. Li, Y. S. Zhou, H. Y. Wang, and F. H. Wang, “Investigation of the wave behaviors inside a step-modulated subwavelength metal slit,” Opt. Express 19, 10073–10087 (2011).
[Crossref] [PubMed]

C. Li, Y. S. Zhou, H. Y. Wang, and F. H. Wang, “Wavelength squeeze of surface plasmon polariton in a subwavelength metal slit,” J. Opt. Soc. Am. B 27, 59–64 (2010).
[Crossref]

Y. S. Zhou, B. Y. Gu, and H. Y. Wang, “Band-gap structures of surface-plasmon polaritons in a subwavelength metal slit filled with periodic dielectrics,” Phys. Rev. A 81, 015801 (2010).
[Crossref]

Wang, L. L.

Wen, S. C.

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press1995).

Wu, L. J.

Y. Xu, A. E. Miroshnichenko, S. Lan, Q. Guo, and L. J. Wu, “Impedance matching induce high transmissionand flat response band-pass plasmonic waveguides,” Plasmonics 6, 337–343 (2011).
[Crossref]

Xiao, F.

Xu, A. S.

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C. Li, Y. S. Zhou, and H. Y. Wang, “Scattering mechanism in a step-modulated subwavelength metal slit: a multi-mode multi-reflection analysis,” Eur. Phys. J. D 66, 8 (2012).
[Crossref]

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C. Li, Y. S. Zhou, and H. Y. Wang, “Scattering mechanism in a step-modulated subwavelength metal slit: a multi-mode multi-reflection analysis,” Eur. Phys. J. D 66, 8 (2012).
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Y. S. Zhou, B. Y. Gu, and H. Y. Wang, “Band-gap structures of surface-plasmon polaritons in a subwavelength metal slit filled with periodic dielectrics,” Phys. Rev. A 81, 015801 (2010).
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Figures (7)

Fig. 1
Fig. 1

Sketches of two kinds of BWGs. Both structures possess a central symmetry and are confined in the x direction by two perfectly conducting walls with a confined width L = 2μm. Gray, white and slashed areas represent silver, air, and dielectric, respectively. In white and slashed areas, the dielectric constants ε are 1.0 and 9.0, respectively. The BWGs are along the y direction. The basic unit, with a fixed length h = 1μm, is composed of two adjacent layers. (a) Dielectric modulation: slit width, w(p−1) = w(p) = 0.1μm. (b) Geometric modulation: slit width, w(p−1) = 0.1μm, w(p) = 0.6μm. The band structure calculation is carried out for infinitely long BWGs. The calculation of transmission is done for finitely long BWGs.

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Fig. 2
Fig. 2

Numerical results of the dielectric modulation with (a) f = 0.5 and (b) f = 0.1. In each figure, the 1st (top) panel is the DOS; the 2nd panel is the transmittances of BWGs containing 5, 10 and 15 periods; the 3rd panel is the real part (left, green) and imaginary part (right, yellow) of Bloch wave vector kB; and the 4th (bottom) panel is the slowdown factor c/vg (left, green) and propagation length Lp (right, yellow). In the bottom panels, the green dotted horizontal lines are c/vg = 1, and the green circles represent the divergences caused by the abrupt changes of the real parts of kB.

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Fig. 3
Fig. 3

Variation of (a) slowdown factor c/vg and (b) propagation length Lp with f and λ in the dielectric modulation. The structural parameters were given in Fig. 1.

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Fig. 4
Fig. 4

Numerical results of the geometric modulation with (a) f = 0.5 and (b) f = 0.1. The gray areas in (b) represent the wavelength ranges for “local modes”. Other captions are the same as those in Fig. 2.

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Fig. 5
Fig. 5

Variation of (a) slowdown factor c/vg and (b) propagation length Lp with f and λ in the geometric modulation. The structural parameters were given in Fig. 1.

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Fig. 6
Fig. 6

(a) Imaginary part and (b) real part of the Bloch wave vector kB in the dielectric modulation for f = 0.1. Crosses: EIM; Circles: N = 1 MEM; Solid lines: N = 80 MEM.

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Fig. 7
Fig. 7

Bloch wave vectors kB in the cases of (a) f = 0.5 and (b) f = 0.1 in the geometric modulation.

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Equations (6)

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H z ( l ) ( x , y ) = ∑ n = 1 ∞ φ n ( l ) ( x ) [ e i k y n ( l ) ( y − Q ( l − 1 ) ) u n ( l ) + e − i k y n ( l ) ( y − Q ( l ) ) d n ( l ) ] , l = ( p − 1 )   and   p ,
{ [ e i k y m ( p − 1 ) q ( p − 1 ) u m ( p − 1 ) + d m ( p − 1 ) ] = ∑ n = 1 ∞ K m n ( p − 1 , p ) [ u n ( p ) + e i k y n ( p ) q ( p ) d n ( p ) ] ∑ n = 1 ∞ J m n ( p , p − 1 ) [ e i k y n ( p − 1 ) q ( p − 1 ) u n ( p − 1 ) − d n ( p − 1 ) ] = k y m ( p ) [ u m ( p ) − e i k y m ( p ) q ( p ) d m ( p ) ] e i k B h [ u m ( p − 1 ) + e i k y m ( p − 1 ) q ( p − 1 ) d m ( p − 1 ) ] = ∑ n = 1 ∞ K m n ( p − 1 , p ) [ e i k y n ( p ) q ( p ) u n ( p ) + d n ( p ) ] e i k B h ∑ n = 1 ∞ J m n ( p , p − 1 ) [ u n ( p − 1 ) − e i k y n ( p − 1 ) q ( p − 1 ) d n ( p − 1 ) ] = k y m ( p ) [ e i k y m ( p ) q ( p ) u m ( p ) − d m ( p ) ] ,
K mn ( p − 1 , p ) = ∫ L 1 ε ( p − 1 ) φ m ( p − 1 ) + ¯ φ n ( p ) d x , J m n ( p , p − 1 ) = ∫ L 1 ε ( p − 1 ) φ m ( p ) + ¯ φ n ( p − 1 ) d x .
[ S 11 ( p ) 0 S 21 ( p ) − I ] ( u ( p − 1 ) d ( p − 1 ) ) = e i k B h [ I − S 12 ( p ) 0 − S 22 ( p ) ] ( u ( p − 1 ) d ( p − 1 ) ) ,
cos ( k B h ) = cos ( k SPP ( p − 1 ) q ( p − 1 ) ) cos ( k S P P ( p ) q ( p ) ) − 1 2 ( k S P P ( p − 1 ) ε ˜ ( p ) k S P P ( p ) ε ˜ ( p − 1 ) + k S P P ( p ) ε ˜ ( p − 1 ) k S P P ( p − 1 ) ε ˜ ( p ) ) × sin ( k S P P ( p − 1 ) q ( p − 1 ) ) sin ( k S P P ( p ) q ( p ) ) ,
cos ( k B h ) = cos ( k 0 n eff ( p − 1 ) q ( p − 1 ) ) cos ( k 0 n eff ( p ) q ( p ) ) − 1 2 ( n eff ( p − 1 ) n eff ( p ) + n eff ( p ) n eff ( p − 1 ) ) × sin ( k 0 n eff ( p − 1 ) q ( p − 1 ) ) sin ( k 0 n eff ( p ) q ( p ) ) .

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