Abstract

In the paper, a novel three-port channel drop filter in two dimensional photonic crystals (2D PCs) with a wavelength-selective reflection micro-cavity is proposed. In the structure, two micro-cavities are used. One is used for a resonant tunneling-based channel drop filter. The other is used to realize wavelength-selective reflection feedback in the bus wave-guide, which consists of a point defect micro-cavity side-coupled to a line defect waveguide based on photonic crystals. Using coupled mode theory in time, the conditions to achieve 100% drop efficiency are derived thoroughly. The simulation results by using the finite-difference time-domain (FDTD) method imply that the design is feasible.

© 2006 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  3. K. Sakoda, Optical properties of photonic crystals (NY: Springer-Verlag Berlin Heidelberg, New York, 2004.)
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    [CrossRef]
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  6. H. Takahashi, S. Suzuki and I. Nishi, "Wavelength multiplexer based on {SiO2}-{Ta2O5} arrayed-waveguide grating," IEEE J.Lightwave Technol. 12,989 (1994).
    [CrossRef]
  7. Zimmermann. J , Kamp. M , Forchel. A , and März, R , "Photonic crystal waveguide directional couplers as wavelength selective optical filters," Opt. Commun. 230, 387 (2004).
    [CrossRef]
  8. A. Martinez, Amadeu Griol, Pablo Sanchis, and Javier Marti, "Mach Zehnder interferometer employing coupled-resonator optical waveguides," Opt.Lett. 28,405 (2003).
    [CrossRef] [PubMed]
  9. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58,R10096 (1998).
    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  13. B. K. Min, J. E. Kim and H. Y. Park, "Channel drop filters using resonant tunneling processes in two-dimensional triangular lattice photonic crystal slabs," Opt. Commun. 237,59 (2004).
    [CrossRef]
  14. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, "Theoretical investigation of channel drop tunneling processes," Phys. Rev. B. 59,15882 (1999).
    [CrossRef]
  15. B. S. Song, S. Noda and T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science,  300,1537 (2003).
    [CrossRef] [PubMed]
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    [CrossRef]
  23. O. Painter, J. Vuckovic, and A. Scherer, "Defect modes of a two-dimensional photonic crystal in an optically thin dielectric slab," J. Opt. Soc. Am B 16,275 (1999).
    [CrossRef]

2005 (5)

2004 (4)

S. Kim, I. Park, H. Lim, and C.-S. Kee, "Highly efficient photonic crystal-based multi-channel drop filters of three-port system with reflection feedback," Opt. Express 12,5518 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-22-5518
[CrossRef] [PubMed]

H. Takano, Y. Akahane, T. Asano, and S. Noda, "In-plane-type channel drop filter in a two-dimensional photonic crystal slab," Appl. Phys. Lett. 84,2226-2228 (2004).
[CrossRef]

B. K. Min, J. E. Kim and H. Y. Park, "Channel drop filters using resonant tunneling processes in two-dimensional triangular lattice photonic crystal slabs," Opt. Commun. 237,59 (2004).
[CrossRef]

Zimmermann. J , Kamp. M , Forchel. A , and März, R , "Photonic crystal waveguide directional couplers as wavelength selective optical filters," Opt. Commun. 230, 387 (2004).
[CrossRef]

2003 (3)

A. Martinez, Amadeu Griol, Pablo Sanchis, and Javier Marti, "Mach Zehnder interferometer employing coupled-resonator optical waveguides," Opt.Lett. 28,405 (2003).
[CrossRef] [PubMed]

StevenG. Johnson and J. D. Joannopoulos, "Designing synthetic optical media: photonic crystals," Acta Materialia. 51,5823 (2003).
[CrossRef]

B. S. Song, S. Noda and T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science,  300,1537 (2003).
[CrossRef] [PubMed]

2001 (1)

2000 (1)

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, "Sattering-theory analysis of waveguide-resonator coupling," Phys. Rev. E. 62, 7389 (2000).
[CrossRef]

1999 (3)

O. Painter, J. Vuckovic, and A. Scherer, "Defect modes of a two-dimensional photonic crystal in an optically thin dielectric slab," J. Opt. Soc. Am B 16,275 (1999).
[CrossRef]

C. Manolatou, M.J. Khan, S. Fan, P.R. Villeneuve, H.A. Haus, and J.D. Joannopoulos, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35,1322 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, "Theoretical investigation of channel drop tunneling processes," Phys. Rev. B. 59,15882 (1999).
[CrossRef]

1998 (2)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58,R10096 (1998).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, "Channel drop tunneling through localized states," Phys. Rev. Lett. 80,960 (1998).
[CrossRef]

1994 (1)

H. Takahashi, S. Suzuki and I. Nishi, "Wavelength multiplexer based on {SiO2}-{Ta2O5} arrayed-waveguide grating," IEEE J.Lightwave Technol. 12,989 (1994).
[CrossRef]

1987 (2)

E. Yablonovitch, "Inhibited Spontaneous Emission in Solid-State Physics and Electronics," Phys. Rev. Lett. 58,2059 (1987).
[CrossRef] [PubMed]

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

Akahane, Y.

Asano, T.

H. Takano, Y. Akahane, T. Asano, and S. Noda, "In-plane-type channel drop filter in a two-dimensional photonic crystal slab," Appl. Phys. Lett. 84,2226-2228 (2004).
[CrossRef]

B. S. Song, S. Noda and T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science,  300,1537 (2003).
[CrossRef] [PubMed]

Bong-Shik Song,

Bong-Shik Song, T. Asano, Y. Akahane, and S.Noda, "Role of interfaces in hetero photonic crystals for manipulation of photons," Phys. Rev. B. 71,195101 (2005).
[CrossRef]

Fan, S.

C. Manolatou, M.J. Khan, S. Fan, P.R. Villeneuve, H.A. Haus, and J.D. Joannopoulos, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35,1322 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, "Theoretical investigation of channel drop tunneling processes," Phys. Rev. B. 59,15882 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, "Channel drop tunneling through localized states," Phys. Rev. Lett. 80,960 (1998).
[CrossRef]

Forchel, M

Zimmermann. J , Kamp. M , Forchel. A , and März, R , "Photonic crystal waveguide directional couplers as wavelength selective optical filters," Opt. Commun. 230, 387 (2004).
[CrossRef]

Haus, H. A.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, "Theoretical investigation of channel drop tunneling processes," Phys. Rev. B. 59,15882 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, "Channel drop tunneling through localized states," Phys. Rev. Lett. 80,960 (1998).
[CrossRef]

Haus, H.A.

C. Manolatou, M.J. Khan, S. Fan, P.R. Villeneuve, H.A. Haus, and J.D. Joannopoulos, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35,1322 (1999).
[CrossRef]

Hwang, K. H.

Joannopoulos, J. D.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, "Theoretical investigation of channel drop tunneling processes," Phys. Rev. B. 59,15882 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, "Channel drop tunneling through localized states," Phys. Rev. Lett. 80,960 (1998).
[CrossRef]

Joannopoulos, J.D.

C. Manolatou, M.J. Khan, S. Fan, P.R. Villeneuve, H.A. Haus, and J.D. Joannopoulos, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35,1322 (1999).
[CrossRef]

John, S.

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

Kamp, J

Zimmermann. J , Kamp. M , Forchel. A , and März, R , "Photonic crystal waveguide directional couplers as wavelength selective optical filters," Opt. Commun. 230, 387 (2004).
[CrossRef]

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58,R10096 (1998).
[CrossRef]

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58,R10096 (1998).
[CrossRef]

Kee, C.-S.

Khan, M. J.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, "Theoretical investigation of channel drop tunneling processes," Phys. Rev. B. 59,15882 (1999).
[CrossRef]

Khan, M.J.

C. Manolatou, M.J. Khan, S. Fan, P.R. Villeneuve, H.A. Haus, and J.D. Joannopoulos, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35,1322 (1999).
[CrossRef]

Kim, J. E.

B. K. Min, J. E. Kim and H. Y. Park, "Channel drop filters using resonant tunneling processes in two-dimensional triangular lattice photonic crystal slabs," Opt. Commun. 237,59 (2004).
[CrossRef]

Kim, S.

Kosaka, H.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58,R10096 (1998).
[CrossRef]

Kuramochi, E.

Lim, H.

Manolatou, C.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, "Theoretical investigation of channel drop tunneling processes," Phys. Rev. B. 59,15882 (1999).
[CrossRef]

C. Manolatou, M.J. Khan, S. Fan, P.R. Villeneuve, H.A. Haus, and J.D. Joannopoulos, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35,1322 (1999).
[CrossRef]

Martinez, A.

A. Martinez, Amadeu Griol, Pablo Sanchis, and Javier Marti, "Mach Zehnder interferometer employing coupled-resonator optical waveguides," Opt.Lett. 28,405 (2003).
[CrossRef] [PubMed]

März, A

Zimmermann. J , Kamp. M , Forchel. A , and März, R , "Photonic crystal waveguide directional couplers as wavelength selective optical filters," Opt. Commun. 230, 387 (2004).
[CrossRef]

Min, B. K.

B. K. Min, J. E. Kim and H. Y. Park, "Channel drop filters using resonant tunneling processes in two-dimensional triangular lattice photonic crystal slabs," Opt. Commun. 237,59 (2004).
[CrossRef]

Mitsugi, S.

Nishi, I.

H. Takahashi, S. Suzuki and I. Nishi, "Wavelength multiplexer based on {SiO2}-{Ta2O5} arrayed-waveguide grating," IEEE J.Lightwave Technol. 12,989 (1994).
[CrossRef]

Noda, S.

H. Takano, Y. Akahane, T. Asano, and S. Noda, "In-plane-type channel drop filter in a two-dimensional photonic crystal slab," Appl. Phys. Lett. 84,2226-2228 (2004).
[CrossRef]

B. S. Song, S. Noda and T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science,  300,1537 (2003).
[CrossRef] [PubMed]

Notomi, M.

Painter, O.

O. Painter, J. Vuckovic, and A. Scherer, "Defect modes of a two-dimensional photonic crystal in an optically thin dielectric slab," J. Opt. Soc. Am B 16,275 (1999).
[CrossRef]

Park, H. Y.

B. K. Min, J. E. Kim and H. Y. Park, "Channel drop filters using resonant tunneling processes in two-dimensional triangular lattice photonic crystal slabs," Opt. Commun. 237,59 (2004).
[CrossRef]

Park, I.

Prather, D.W.

Qiu, M.

Z. Zhang and M. Qiu, "Compact in-plane channel drop filter design using a single cavity with two degenerate modes in 2D photonic crystal slabs," Opt.Express 13,2596 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-7-2596
[CrossRef] [PubMed]

Sato, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58,R10096 (1998).
[CrossRef]

Scherer, A.

O. Painter, J. Vuckovic, and A. Scherer, "Defect modes of a two-dimensional photonic crystal in an optically thin dielectric slab," J. Opt. Soc. Am B 16,275 (1999).
[CrossRef]

Sharkawy, A.

Shi, S.

Shinya, A.

Song, B. S.

B. S. Song, S. Noda and T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science,  300,1537 (2003).
[CrossRef] [PubMed]

Song, G.

Steven,

StevenG. Johnson and J. D. Joannopoulos, "Designing synthetic optical media: photonic crystals," Acta Materialia. 51,5823 (2003).
[CrossRef]

Suzuki, S.

H. Takahashi, S. Suzuki and I. Nishi, "Wavelength multiplexer based on {SiO2}-{Ta2O5} arrayed-waveguide grating," IEEE J.Lightwave Technol. 12,989 (1994).
[CrossRef]

Takahashi, H.

H. Takahashi, S. Suzuki and I. Nishi, "Wavelength multiplexer based on {SiO2}-{Ta2O5} arrayed-waveguide grating," IEEE J.Lightwave Technol. 12,989 (1994).
[CrossRef]

Takano, H.

H. Takano, Y. Akahane, T. Asano, and S. Noda, "In-plane-type channel drop filter in a two-dimensional photonic crystal slab," Appl. Phys. Lett. 84,2226-2228 (2004).
[CrossRef]

Tamamura, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58,R10096 (1998).
[CrossRef]

Tomita, A.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58,R10096 (1998).
[CrossRef]

Villeneuve, P. R.

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, "Theoretical investigation of channel drop tunneling processes," Phys. Rev. B. 59,15882 (1999).
[CrossRef]

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, "Channel drop tunneling through localized states," Phys. Rev. Lett. 80,960 (1998).
[CrossRef]

Villeneuve, P.R.

C. Manolatou, M.J. Khan, S. Fan, P.R. Villeneuve, H.A. Haus, and J.D. Joannopoulos, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35,1322 (1999).
[CrossRef]

Vuckovic, J.

O. Painter, J. Vuckovic, and A. Scherer, "Defect modes of a two-dimensional photonic crystal in an optically thin dielectric slab," J. Opt. Soc. Am B 16,275 (1999).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, "Inhibited Spontaneous Emission in Solid-State Physics and Electronics," Phys. Rev. Lett. 58,2059 (1987).
[CrossRef] [PubMed]

Zhang, Z.

Z. Zhang and M. Qiu, "Compact in-plane channel drop filter design using a single cavity with two degenerate modes in 2D photonic crystal slabs," Opt.Express 13,2596 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-7-2596
[CrossRef] [PubMed]

Zimmermann,

Zimmermann. J , Kamp. M , Forchel. A , and März, R , "Photonic crystal waveguide directional couplers as wavelength selective optical filters," Opt. Commun. 230, 387 (2004).
[CrossRef]

Acta Materialia. (1)

StevenG. Johnson and J. D. Joannopoulos, "Designing synthetic optical media: photonic crystals," Acta Materialia. 51,5823 (2003).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

H. Takano, Y. Akahane, T. Asano, and S. Noda, "In-plane-type channel drop filter in a two-dimensional photonic crystal slab," Appl. Phys. Lett. 84,2226-2228 (2004).
[CrossRef]

IEEE J. Quantum Electron. (1)

C. Manolatou, M.J. Khan, S. Fan, P.R. Villeneuve, H.A. Haus, and J.D. Joannopoulos, "Coupling of modes analysis of resonant channel add-drop filters," IEEE J. Quantum Electron. 35,1322 (1999).
[CrossRef]

IEEE J.Lightwave Technol. (1)

H. Takahashi, S. Suzuki and I. Nishi, "Wavelength multiplexer based on {SiO2}-{Ta2O5} arrayed-waveguide grating," IEEE J.Lightwave Technol. 12,989 (1994).
[CrossRef]

J. Opt. Soc. Am B (1)

O. Painter, J. Vuckovic, and A. Scherer, "Defect modes of a two-dimensional photonic crystal in an optically thin dielectric slab," J. Opt. Soc. Am B 16,275 (1999).
[CrossRef]

Opt. Commun. (2)

Zimmermann. J , Kamp. M , Forchel. A , and März, R , "Photonic crystal waveguide directional couplers as wavelength selective optical filters," Opt. Commun. 230, 387 (2004).
[CrossRef]

B. K. Min, J. E. Kim and H. Y. Park, "Channel drop filters using resonant tunneling processes in two-dimensional triangular lattice photonic crystal slabs," Opt. Commun. 237,59 (2004).
[CrossRef]

Opt. Express (4)

Opt.Express (1)

Z. Zhang and M. Qiu, "Compact in-plane channel drop filter design using a single cavity with two degenerate modes in 2D photonic crystal slabs," Opt.Express 13,2596 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-7-2596
[CrossRef] [PubMed]

Opt.Lett. (1)

A. Martinez, Amadeu Griol, Pablo Sanchis, and Javier Marti, "Mach Zehnder interferometer employing coupled-resonator optical waveguides," Opt.Lett. 28,405 (2003).
[CrossRef] [PubMed]

Phys. Rev. B (1)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, "Superprism phenomena in photonic crystals," Phys. Rev. B 58,R10096 (1998).
[CrossRef]

Phys. Rev. B. (2)

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, M. J. Khan, C. Manolatou, and H. A. Haus, "Theoretical investigation of channel drop tunneling processes," Phys. Rev. B. 59,15882 (1999).
[CrossRef]

Bong-Shik Song, T. Asano, Y. Akahane, and S.Noda, "Role of interfaces in hetero photonic crystals for manipulation of photons," Phys. Rev. B. 71,195101 (2005).
[CrossRef]

Phys. Rev. E. (1)

Y. Xu, Y. Li, R. K. Lee, and A. Yariv, "Sattering-theory analysis of waveguide-resonator coupling," Phys. Rev. E. 62, 7389 (2000).
[CrossRef]

Phys. Rev. Lett. (3)

S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and H. A. Haus, "Channel drop tunneling through localized states," Phys. Rev. Lett. 80,960 (1998).
[CrossRef]

E. Yablonovitch, "Inhibited Spontaneous Emission in Solid-State Physics and Electronics," Phys. Rev. Lett. 58,2059 (1987).
[CrossRef] [PubMed]

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

Science (1)

B. S. Song, S. Noda and T. Asano, "Photonic devices based on in-plane hetero photonic crystals," Science,  300,1537 (2003).
[CrossRef] [PubMed]

Other (1)

K. Sakoda, Optical properties of photonic crystals (NY: Springer-Verlag Berlin Heidelberg, New York, 2004.)

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Figures (5)

Fig. 1.
Fig. 1.

Basic structure of a wave-guide side-coupled to a cavity

Fig. 2.
Fig. 2.

The three-port channel drop filter with a wavelength-selective reflection cavity. The cavity a is used to realize the wavelength-selective reflection function, and the other cavity b is used for a resonant tunneling-based channel drop operation.

Fig. 3.
Fig. 3.

(a) The curve of drop efficiency as a function of phase ϕ for k= 1, 2 and 3 (donated by the thick solid, thin solid, and dotted curves, respectively), supposing 100% reflection feedback. (b) The values of drop efficiency versus the quality factor ratio k at ϕ = (2n + 1)π, assuming 100% reflection feedback. (c) At k = 2, the dependence of drop efficiency on the various value of l (l=1, 0.98, 0.9, represented by the thin solid, dotted, and thick solid curves, respectively). (d)The curve of the drop efficiency as a function of p at ϕ = (2n + 1)π and k = 2 under 100% reflection feedback. The drop efficiencies in the cases of (a), (b), (c) and (d) are all calculated under the assumption ω0a = ω0b · (e) At ω 0a =(1 - sin ϕ>/2Q 1)ω 0b , the curve of the drop efficiency as a function of phase ϕ for the case of k =1, 2 and 3(donated by the thick solid, thin solid, and dotted curves, respectively). (f) The drop efficiency as a function of the frequency detuning coefficient m for Q 1 =103,104 and 105 (donated by the dotted, thick solid, and thin solid curves, respectively). In the cases of (e) and (f), the phase term, quality factor ratio and 100% reflection feedback are well satisfied.

Fig. 4.
Fig. 4.

(a) The three-port channel drop filter with the channel drop wave-guide perpendicular to the bus waveguide. The smaller rod that defines the defect of the channel drop wave-guide has a radius of 0.042a, and the two rods at the interface between the cavity and channel drop wave-guide or the cavity and bus wave-guide have a radius of 0.211a. They all have the same dielectric constant as the background rods. (b) The transmission characteristics calculated by the FDTD method.

Fig. 5.
Fig. 5.

(a) The three-port channel drop filter with the wavelength-selective reflection cavity. The parameters of the channel drop cavity are same as that in Fig. 4(a). The smaller rod of the wavelength-selective reflection cavity has a radius of 0.042a, and the rod between the cavity and the bus waveguide has a radius of 0.23a. (b) The intensity spectra calculated by the FDTD method. (c) The wave propagation at the resonant frequency shown in Fig. 5(a).

Equations (44)

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da dt = ( j ω 0 ω 0 Q 0 ω 0 2 Q e ) a + ω 0 2 Q e e S + 1 + ω 0 2 Q e e S + 2 ,
S 1 = S + 2 e ω 0 2 Q e a ,
S 2 = S + 1 e ω 0 2 Q e a .
R = S 1 S + 1 = 1 2 Q e j ( ω ω 0 1 ) + 1 Q o + 1 2 Q e
T = S 2 S + 1 = j ( ω ω 0 1 ) + 1 Q o j ( ω ω 0 1 ) + 1 Q o + 1 2 Q e .
da dt = ( j ω 0 a ω 0 a Q oa ω 0 a 2 Q 3 ) + ω 0 a 2 Q 3 e j θ 3 S + 3 + ω 0 a 2 Q 3 e j θ 3 S + 3 ,
db dt = ( j ω 0 b ω 0 b Q ob ω 0 b 2 Q 1 ω 0 b 2 Q 2 ) b + e j θ 1 ω 0 b 2 Q 1 S + 1 + e j θ 1 ω 0 b 2 Q 1 S + 1 + e j θ 2 ω 0 b Q 2 S + 2 ,
S 3 = S + 3 e j θ 3 ω 0 a 2 Q 3 a ,
S′ 3 = S + 3 e j θ 3 ω 0 a 2 Q 3 a ,
S + 3 = S 1 e jβd ,
S + 1 = S 3 e jβd ,
S 1 = S + 1 e j θ 1 ω 0 b 2 Q 1 b ,
S 1 = S + 1 e j θ 1 ω 0 b 2 Q 1 b ,
S 2 = S + 2 + e j θ 2 ω 0 b Q 2 b ,
S + 1 = 1 2 Q 3 j ( ω ω 0 a 1 ) + 1 2 Q oa + 1 2 Q 3 e j 2 βd ( S + 1 e j θ 1 ω 0 b 2 Q 1 b ) .
r = 1 2 Q 3 j ( ω ω 0 a 1 ) + 1 Q oa + 1 2 Q 3 ,
ϕ = 2 βd ,
b = e j θ 1 ω 0 b 2 Q 1 [ 1 r ( cos ϕ j sin ϕ ) ] S + 1 j ω 0 b ( ω ω 0 b 1 + r 2 Q 1 sin ϕ ) + ω 0 b Q ob + ω 0 b 2 Q 2 + ω 0 b 2 Q 1 ( 1 r cos ϕ ) .
T 1 = S 2 S + 1 = e ( j θ 1 j θ 2 ) 1 2 Q 1 Q 2 [ 1 r ( cos ϕ j sin ϕ ) ] j ( ω ω 0 b 1 + r 2 Q 1 sin ϕ ) + 1 Q ob + 1 2 Q 2 + 1 2 Q 1 ( 1 r cos ϕ ) ,
T 2 = S 3 S + 1 = ( 1 r ) ( 1 1 2 Q 1 [ 1 r ( cos ϕ j sin ϕ ) ] j ( ω ω 0 b 1 + r 2 Q 1 sin ϕ ) + 1 Q ob + 1 2 Q 2 + 1 2 Q 1 ( 1 r cos ϕ ) ) e / 2 ,
R′ = S 1 S + 1 = r ( cos ϕ j sin ϕ ) [ 1 r ( cos ϕ j sin ϕ ) ] 2 1 2 Q 1 j ( ω ω 0 b 1 + r 2 Q 1 sin ϕ ) + 1 Q ob + 1 2 Q 2 + 1 2 Q 1 ( 1 r cos ϕ ) ,
η = T 1 2 = m 4 ( ω ω 0 ω 0 ) 4 + m 3 ( ω ω 0 ω 0 ) 3 + m 2 ( ω ω 0 ω 0 ) 2 + m 1 ( ω ω 0 ω 0 ) + m 0 ( ω ω 0 ω 0 ) 6 + n 4 ( ω ω 0 ω 0 ) 4 + n 3 ( ω ω 0 ω 0 ) 3 + n 2 ( ω ω 0 ω 0 ) 2 + n 1 ( ω ω 0 ω 0 ) + n 0 ,
m 0 = 1 cos ϕ 16 Q 1 Q 2 Q 3 4 ,
m 1 = sin ϕ 8 Q 1 Q 2 Q 3 3 ,
m 2 = 3 2 cos ϕ 8 Q 1 Q 2 Q 3 2 ,
m 3 = sin ϕ 2 Q 1 Q 2 Q 3 ,
m 4 = 1 2 Q 1 Q 2 ,
n 0 = 1 16 Q 3 4 [ ( 1 2 Q 1 + 1 2 Q 2 ) 2 + 1 4 Q 1 2 cos ϕ 2 Q 1 Q 2 cos ϕ 2 Q 1 2 ] ,
n 1 = ( 1 16 Q 1 Q 3 4 + 1 16 Q 1 Q 2 Q 3 3 + 1 16 Q 1 2 Q 3 3 ) sin ϕ ,
n 2 = ( 1 8 Q 1 Q 3 3 1 8 Q 1 Q 2 Q 3 2 1 8 Q 1 2 Q 3 2 ) cos ϕ + ( 1 2 Q 1 + 1 2 Q 2 ) 2 1 2 Q 3 2 + 1 16 Q 3 4 ,
n 3 = ( 1 4 Q 1 Q 3 2 + 1 4 Q 1 Q 2 Q 3 + 1 4 Q 1 2 Q 3 ) sin ϕ ,
n 4 = cos ϕ 2 Q 1 Q 3 + 1 4 Q 3 2 + ( 1 2 Q 1 + 1 2 Q 2 ) 2 .
η 0 = T 1 2 = m 0 n 0 = 1 cos ϕ Q 1 Q 2 ( 1 2 Q 1 + 1 2 Q 2 ) 2 + 1 4 Q 1 2 cos ϕ 2 Q 1 Q 2 cos ϕ 2 Q 1 2 .
η 0 = 4 k ( 1 cos ϕ ) k 2 + 2 k ( 1 cos ϕ ) + 2 ( 1 cos ϕ ) .
l = 1 2 Q 3 1 Q oa + 1 2 Q 3 ,
η 0 = T 1 2 = 1 2 Q 1 Q 2 ( 1 l cos ϕ + jl sin ϕ ) j l sin ϕ 2 Q 1 + 1 2 Q 2 + 1 2 Q 1 ( 1 cos ϕ ) 2 .
η 0 = 4 ( 1 + l 2 2 l cos ϕ ) 9 + l 2 6 l cos ϕ .
Q 1 Q ob = p ,
η 0 = 2 Q 1 Q 2 ( 1 Q ob + 1 Q 1 + 1 2 Q 2 ) 2 = 4 ( p + 2 ) 2 .
ω 0 a = ( 1 sin ϕ 2 Q 1 ) ω 0 b ,
η 0 = T 1 2 = 1 2 Q 1 Q 2 [ ( 1 cos ϕ ) 2 + sin 2 ϕ ] [ 1 Q ob + 1 2 Q 2 + 1 2 Q 1 ( 1 cos ϕ ) ] 2
η 0 = 4 k ( 1 cos ϕ ) k 2 + ( 1 cos ϕ ) 2 + 2 k ( 1 cos ϕ ) ,
ω 0 a ω 0 b = m ,
η 0 = 2 Q 1 Q 2 ( m 1 ) 2 + ( 1 Q 1 + 1 2 Q 2 ) 2 = 4 Q 1 2 ( m 1 ) 2 + 4 .

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