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]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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2005 (5)

2004 (4)

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]

Sangin Kim, Ikmo Park, Hanjo Lim, and Chul-Sik Kee, “Highly efficient photonic crystal-based multichannel 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]

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]

Steven G. 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)

Yong Xu, Yi Li, Reginald K. Lee, and Amnon 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]

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]

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]

A, Forchel.

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]

Akahane, Y.

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]

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]

Akahane, Yoshihiro

Asano, T.

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]

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]

Asano, Takashi

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]

Griol, Amadeu

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

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, Kyu H.

J, 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]

Joannopoulos, J. D.

Steven G. Johnson and J. D. Joannopoulos, “Designing synthetic optical media: photonic crystals,” Acta Materialia. 51, 5823 (2003).
[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]

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]

Johnson, Steven G.

Steven G. Johnson and J. D. Joannopoulos, “Designing synthetic optical media: photonic crystals,” Acta Materialia. 51, 5823 (2003).
[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, Chul-Sik

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, Sangin

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, Eiichi

Lee, Reginald K.

Yong Xu, Yi Li, Reginald K. Lee, and Amnon Yariv, “Sattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E. 62, 7389 (2000).
[CrossRef]

Li, Yi

Yong Xu, Yi Li, Reginald K. Lee, and Amnon Yariv, “Sattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E. 62, 7389 (2000).
[CrossRef]

Lim, Hanjo

M, Kamp.

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]

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]

Marti, Javier

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

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]

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, Satoshi

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.

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]

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]

Noda, Susumu

Notomi, M.

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]

Notomi, Masaya

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, Ikmo

Prather, D.W.

Qiu, Min

Ziyang Zhang and Min 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.or g/abstract.cfm?URI=OPEX-13-7-2596.
[CrossRef] [PubMed]

R, März

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]

Sakoda, Kazuaki

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

Sanchis, Pablo

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

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, Bong-Shik

Song, G. Hugh

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]

Takana, Yoshinori

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]

Takano, Hitomichi

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]

Xu, Yong

Yong Xu, Yi Li, Reginald K. Lee, and Amnon Yariv, “Sattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E. 62, 7389 (2000).
[CrossRef]

Yablonovitch, E.

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

Yariv, Amnon

Yong Xu, Yi Li, Reginald K. Lee, and Amnon Yariv, “Sattering-theory analysis of waveguide-resonator coupling,” Phys. Rev. E. 62, 7389 (2000).
[CrossRef]

Zhang, Ziyang

Ziyang Zhang and Min 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.or g/abstract.cfm?URI=OPEX-13-7-2596.
[CrossRef] [PubMed]

Acta Materialia. (1)

Steven G. 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)

Ziyang Zhang and Min 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.or g/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)

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[CrossRef]

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[CrossRef]

Science (1)

B. S. Song, S. Noda, and T. Asano, “Photonic devices based on in-plane hetero photonic crystals,” Science,  300, 1537 (2003).
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Other (1)

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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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