Abstract

High order plasmonic Bragg reflection in the metal-insulator-metal (MIM) waveguide Bragg grating (WBG) and its applications are proposed and demonstrated numerically. With the effective index method and the standard transfer matrix method, we reveal that there exist high order plasmonic Bragg reflections in MIM WBG and corresponding Bragg wavelengths can be obtained. Contrary to the high order Bragg wavelengths in the case of the conventional dielectric slab waveguide, the results of the MIM WBG exhibit red shifts of tens of nanometers. We also propose a method to design a MIM WBG to have high order plasmonic Bragg reflection at a desired wavelength. The MIM WBG operating in visible spectral regime, which requires quite accurate fabrication process with grating period of 100 to 200 nm for the fundamental Bragg reflection, can be implemented by using the higher order plasmonic Bragg reflection with grating period of 400 to 600 nm. It is shown that the higher order plasmonic Bragg reflection can be employed to implement a narrow reflection bandwidth as well. We also address the dependence of the filling factor upon the bandgap and discuss the quarter-wave stack condition and the second bandgap closing.

©2008 Optical Society of America

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References

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  1. H. Rather, Surface Plasmons (Springer-Verlag, Berlin, 1988).
  2. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
    [Crossref] [PubMed]
  3. T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyi, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. 85, 5833–5835 (2004).
    [Crossref]
  4. A. Boltasseva, T. Nikolajsen, K. Leosson, K. Kjaer, M. S. Larsen, and S. I. Bozhevolnyi, “Integrated optical components utilizing long-range surface plasmon polaritons,” J. Lightwave Technol. 23, 413–422 (2005).
    [Crossref]
  5. S. I. Bozhevolnyi, V. S. Volkov, E. Devaus, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
    [Crossref] [PubMed]
  6. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B 61, 10484–10503 (2000).
    [Crossref]
  7. E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182, 539–554 (1969).
    [Crossref]
  8. J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B 33, 5186–5201 (1986).
    [Crossref]
  9. R. Zia, M. D. Selker, P. B. Catrysse, and M. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A 21, 2442–2446 (2004).
    [Crossref]
  10. J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Planar metal plasmon waveguides: Frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B 72, 075405 (2005).
    [Crossref]
  11. 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]
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    [Crossref]
  14. B. Wang and G. P. Wang, “Plasmon Bragg reflectors and nanocavities on flat metallic surfaces,” Appl. Phys. Lett. 87, 013107 (2005).
    [Crossref]
  15. A. Hosseini and Y. Massoud, “A low-loss metal-insulator-metal plasmonic Bragg reflector,” Opt. Express 14, 11318–11323 (2006).
    [Crossref]
  16. 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]
  17. A. Hosseini and Y. Massoud, “Subwavelength plasmonic Bragg reflector structures for on-chip optoelectronic applications,” International Symposium on Circuits and Systems, New Orleans, LA, 2283–2286 (2007).
  18. J. A. Dionne, H. J. Lezec, and H. A. Atwater, “Highly confined photon transport in subwavelength metallic slot waveguides,” Nano Letters 6, 1928–1932 (2006).
    [Crossref] [PubMed]
  19. D. Z. Lin, C. K. Chang, Y. C. Chen, D. L. Yang, M. W. Lin, J. T. Yeh, J. M. Liu, C. H. Kuan, C. S. Yeh, and C. K. D. Lee, “Beaming light from a subwavelength metal slit surrounded by dielectric surface gratings,” Opt. Express 14, 3503–3511 (2006).
    [Crossref] [PubMed]
  20. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. A 71, 811–818 (1981).
    [Crossref]
  21. M. G. Moharam, E. B. Grann, and D. A. Pommet, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1067–1076 (1995).
    [Crossref]
  22. P. Lalanne, “Improved formulation of the coupled-wave method for two-dimensional gratings,” J. Opt. Soc. Am. A 14, 1592–1598 (1997).
    [Crossref]
  23. H. Kim, I.-M. Lee, and B. Lee, “Extended scattering-matrix method for efficient full parallel implementation of rigorous coupled-wave analysis,” J. Opt. Soc. Am. A 24, 2313–2327 (2007).
    [Crossref]
  24. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley Intersceince, Hoboken, NJ, 2007).

2007 (3)

2006 (5)

2005 (4)

B. Wang and G. P. Wang, “Plasmon Bragg reflectors and nanocavities on flat metallic surfaces,” Appl. Phys. Lett. 87, 013107 (2005).
[Crossref]

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Planar metal plasmon waveguides: Frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B 72, 075405 (2005).
[Crossref]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaus, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[Crossref] [PubMed]

A. Boltasseva, T. Nikolajsen, K. Leosson, K. Kjaer, M. S. Larsen, and S. I. Bozhevolnyi, “Integrated optical components utilizing long-range surface plasmon polaritons,” J. Lightwave Technol. 23, 413–422 (2005).
[Crossref]

2004 (2)

R. Zia, M. D. Selker, P. B. Catrysse, and M. Brongersma, “Geometries and materials for subwavelength surface plasmon modes,” J. Opt. Soc. Am. A 21, 2442–2446 (2004).
[Crossref]

T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyi, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. 85, 5833–5835 (2004).
[Crossref]

2003 (1)

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

2000 (1)

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B 61, 10484–10503 (2000).
[Crossref]

1997 (1)

1995 (1)

M. G. Moharam, E. B. Grann, and D. A. Pommet, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1067–1076 (1995).
[Crossref]

1986 (1)

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B 33, 5186–5201 (1986).
[Crossref]

1981 (1)

M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. A 71, 811–818 (1981).
[Crossref]

1969 (1)

E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182, 539–554 (1969).
[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]

J. A. Dionne, H. J. Lezec, and H. A. Atwater, “Highly confined photon transport in subwavelength metallic slot waveguides,” Nano Letters 6, 1928–1932 (2006).
[Crossref] [PubMed]

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Planar metal plasmon waveguides: Frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B 72, 075405 (2005).
[Crossref]

Barnes, W. L.

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

Berini, P.

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B 61, 10484–10503 (2000).
[Crossref]

Boltasseva, A.

Bozhevolnyi, S. I.

A. Boltasseva, S. I. Bozhevolnyi, T. Nikolajsen, and K. Leosson, “Compact Bragg gratings for long-range surface plasmon polaritons,” J. Lightwave Technol. 24, 912–918 (2006).
[Crossref]

A. Boltasseva, T. Nikolajsen, K. Leosson, K. Kjaer, M. S. Larsen, and S. I. Bozhevolnyi, “Integrated optical components utilizing long-range surface plasmon polaritons,” J. Lightwave Technol. 23, 413–422 (2005).
[Crossref]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaus, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[Crossref] [PubMed]

T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyi, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. 85, 5833–5835 (2004).
[Crossref]

Brongersma, M.

Burke, J. J.

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B 33, 5186–5201 (1986).
[Crossref]

Catrysse, P. B.

Chang, C. K.

Chen, Y. C.

Dereux, A.

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

Devaus, E.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaus, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[Crossref] [PubMed]

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]

J. A. Dionne, H. J. Lezec, and H. A. Atwater, “Highly confined photon transport in subwavelength metallic slot waveguides,” Nano Letters 6, 1928–1932 (2006).
[Crossref] [PubMed]

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Planar metal plasmon waveguides: Frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B 72, 075405 (2005).
[Crossref]

Ebbesen, T. W.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaus, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[Crossref] [PubMed]

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

Economou, E. N.

E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182, 539–554 (1969).
[Crossref]

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]

Gaylord, T. K.

M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. A 71, 811–818 (1981).
[Crossref]

Grann, E. B.

M. G. Moharam, E. B. Grann, and D. A. Pommet, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1067–1076 (1995).
[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, 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]

Hosseini, A.

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

A. Hosseini and Y. Massoud, “Subwavelength plasmonic Bragg reflector structures for on-chip optoelectronic applications,” International Symposium on Circuits and Systems, New Orleans, LA, 2283–2286 (2007).

Jung, J.

Kim, H.

Kjaer, K.

Kuan, C. H.

Lalanne, P.

Larsen, M. S.

Lee, B.

Lee, C. K. D.

Lee, I.-M.

Leosson, K.

Lezec, H. J.

J. A. Dionne, H. J. Lezec, and H. A. Atwater, “Highly confined photon transport in subwavelength metallic slot waveguides,” Nano Letters 6, 1928–1932 (2006).
[Crossref] [PubMed]

Lin, D. Z.

Lin, M. W.

Liu, J. M.

Massoud, Y.

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

A. Hosseini and Y. Massoud, “Subwavelength plasmonic Bragg reflector structures for on-chip optoelectronic applications,” International Symposium on Circuits and Systems, New Orleans, LA, 2283–2286 (2007).

Moharam, M. G.

M. G. Moharam, E. B. Grann, and D. A. Pommet, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1067–1076 (1995).
[Crossref]

M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. A 71, 811–818 (1981).
[Crossref]

Nikolajsen, T.

Park, J.

Pommet, D. A.

M. G. Moharam, E. B. Grann, and D. A. Pommet, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1067–1076 (1995).
[Crossref]

Rather, H.

H. Rather, Surface Plasmons (Springer-Verlag, Berlin, 1988).

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley Intersceince, Hoboken, NJ, 2007).

Selker, M. D.

Stegeman, G. I.

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B 33, 5186–5201 (1986).
[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]

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Planar metal plasmon waveguides: Frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B 72, 075405 (2005).
[Crossref]

Tamir, T.

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B 33, 5186–5201 (1986).
[Crossref]

Teich, M. C.

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley Intersceince, Hoboken, NJ, 2007).

Volkov, V. S.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaus, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[Crossref] [PubMed]

Wang, B.

B. Wang and G. P. Wang, “Plasmon Bragg reflectors and nanocavities on flat metallic surfaces,” Appl. Phys. Lett. 87, 013107 (2005).
[Crossref]

Wang, G. P.

B. Wang and G. P. Wang, “Plasmon Bragg reflectors and nanocavities on flat metallic surfaces,” Appl. Phys. Lett. 87, 013107 (2005).
[Crossref]

Yang, D. L.

Yeh, C. S.

Yeh, J. T.

Zia, R.

Appl. Phys. Lett. (2)

T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyi, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. 85, 5833–5835 (2004).
[Crossref]

B. Wang and G. P. Wang, “Plasmon Bragg reflectors and nanocavities on flat metallic surfaces,” Appl. Phys. Lett. 87, 013107 (2005).
[Crossref]

IEEE Photon. Technol. Lett. (1)

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]

J. Lightwave Technol. (2)

J. Opt. Soc. Am. A (5)

Nano Letters (1)

J. A. Dionne, H. J. Lezec, and H. A. Atwater, “Highly confined photon transport in subwavelength metallic slot waveguides,” Nano Letters 6, 1928–1932 (2006).
[Crossref] [PubMed]

Nature (1)

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

Opt. Express (3)

Phys. Rev. (1)

E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182, 539–554 (1969).
[Crossref]

Phys. Rev. B (4)

J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B 33, 5186–5201 (1986).
[Crossref]

J. A. Dionne, L. A. Sweatlock, and H. A. Atwater, “Planar metal plasmon waveguides: Frequency-dependent dispersion, propagation, localization, and loss beyond the free electron model,” Phys. Rev. B 72, 075405 (2005).
[Crossref]

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]

P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: bound modes of symmetric structures,” Phys. Rev. B 61, 10484–10503 (2000).
[Crossref]

Phys. Rev. Lett. (1)

S. I. Bozhevolnyi, V. S. Volkov, E. Devaus, and T. W. Ebbesen, “Channel plasmon-polariton guiding by subwavelength metal grooves,” Phys. Rev. Lett. 95, 046802 (2005).
[Crossref] [PubMed]

Other (3)

H. Rather, Surface Plasmons (Springer-Verlag, Berlin, 1988).

A. Hosseini and Y. Massoud, “Subwavelength plasmonic Bragg reflector structures for on-chip optoelectronic applications,” International Symposium on Circuits and Systems, New Orleans, LA, 2283–2286 (2007).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 2nd ed. (Wiley Intersceince, Hoboken, NJ, 2007).

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Figures (10)

Fig. 1.
Fig. 1. Schematic diagram of a metal-insulator-metal waveguide

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Fig. 2.
Fig. 2. Basic properties of the anti-symmetric mode in the MIM waveguide. (a) Dispersion relation (t=50 nm, εd =1, and ωp =9eV). Dependence of the effective refractive index at λ=633 nm upon (b) the refractive index of the dielectric core (t=50 nm, ωp =9eV), (c) the bulk plasma frequency of the metal cladding (t=50 nm, εd =1), and (d) the thickness of the dielectric core (εd =1, ωp =9eV)

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Fig. 3.
Fig. 3. Schematic diagram of a metal-insulator-metal waveguide Bragg grating with periodic modulation of the core index

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Fig. 4.
Fig. 4. (a) Graphical method to find the high order plasmonic Bragg wavelengths. (b) Transmission spectrum (solid line) and reflection spectrum (dashed line). t=50nm, ε d1=1.00, ε d2= 1.44, Λ=510nm, and f=0.5.

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Fig. 5.
Fig. 5. Intensity of the tangential component of the electric field (|Ex |2) of the MIM WBG with (a) the fundamental Bragg wavelength, (b) the second order Bragg wavelength, (c) the third order Bragg wavelength, and (d) wavelength of 1000 nm in the passband. The geometrical parameters are the same as those in the Fig. 4.

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Fig. 6.
Fig. 6. Dispersion diagram of the alternatively stacked MIM waveguide structure

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Fig. 7.
Fig. 7. (a) Reflection and (b) transmission spectra of the MIM WBG with the fundamental Bragg reflection (dashed red line) and the third order Bragg reflection (solid blue line)

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Fig. 8.
Fig. 8. (a) Reflection and (b) transmission spectra of the MIM WBG with the low index contrast fundamental Bragg reflection (solid blue line), the high index contrast third order Bragg reflection (dashed red line), and the high index contrast fundamental Bragg reflection with reduced cell number (dash-dotted green line)

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Fig. 9.
Fig. 9. (a) Bandgap diagram of the infinite periodic structure as a function of filling factor. (b) Ratio between n eff,1 and n eff,2 as a function of the operating wavelength

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Fig. 10.
Fig. 10. (a) Reflection (R) and transmission (T) spectra and (b) Bloch diagram for the MIM WBG with the filling factor of 0.548

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Tables (2)

Tables Icon

Table 1. Bandgap properties of the MIM WBG having bandgap at 532 nm

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Table 2. Bandgap properties of the MIM WBG having bandgap at 1550 nm

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Equations (9)

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

κ m ε m = − κ d ε d tanh ( t 2 κ d ) ,
ε silver ( ω ) = ε ∞ − ω p 2 ω ( ω + i γ ) ,
− κ d 2 + β 2 = ε d k 0 2 = ε d ( ω c 0 ) 2 ,
− κ m 2 + β 2 = ε m k 0 2 = ε m ( ω c 0 ) 2 ,
q λ B , q = 2 Λ n ˜ eff ∣ λ B , 1 ,
n ˜ eff ∣ λ B , 1 = d 1 n eff , 1 ∣ λ B , 1 + d 2 n eff , 2 ∣ λ B , 1 d 1 + d 2 .
n eff ( λ ) = q λ 2 Λ
cos ( K Λ ) = Re { 1 t ( ω ) } = ( n eff , 1 + n eff , 2 ) 2 4 n eff , 1 n eff , 2 cos ( φ 1 + φ 2 ) − ( n eff , 1 − n eff , 2 ) 2 4 n eff , 1 n eff , 2 cos ( φ 1 − φ 2 ) ,
Δ n norm = ∣ n eff , 2 − n eff , 1 n eff , 2 + n eff , 1 ∣

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