Abstract

We propose a new scheme for an efficient photonic crystal 90° bend and power splitter. First we design a 90° bend by placing a super defect at the junction of two orthogonal photonic crystal waveguides. We changed the parameters of the super defect to optimize the transmission coefficient of the bent structure. Our two-dimensional (2D) simulations show more than 85% efficiency over a wide range of wavelengths. Then using this new scheme and adding another waveguide we design a new power splitter and optimize it with the same procedure. We used the coupled mode theory to analytically investigate the structures and finite difference time domain to simulate and optimize the structures performances.

© 2008 Optical Society of America

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References

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  1. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143-149 (1997).
    [CrossRef]
  2. M. Thorhauge, L. H. Frandsen, and P. I. Borel, “Efficient photonic crystal directional couplers,” Opt. Lett. 28, 1525-1527 (2003).
    [CrossRef] [PubMed]
  3. Y. Sugimoto, Y. Tanaka, N. Ikeda, K. Kanamoto, Y. Nakamura, S. Ohkouchi, H. Nakamura, K. Inoue, H. Sasaki, Y. Watanabe, K. Ishida, H. Ishikawa, and K. Asakawa, “Two dimensional semiconductor-based photonic crystal slab waveguides for ultra-fast optical signal processing devices,” IEICE Trans. Electron. E87-C, 316-327 (2004).
  4. A. Miroshnichenko and Y. Kivshar, “Sharp bends in photonic crystal waveguides as nonlinear Fano resonators,” Opt. Express 13, 3969-3976 (2005).
    [CrossRef] [PubMed]
  5. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987).
    [CrossRef] [PubMed]
  6. S. Y. Lin, E. Chow, S. G. Johnson, and J. D. Joannopoulos, “Demonstration of highly efficient waveguiding in a photonic crystal slab at the 1.5-μm wavelength,” Opt. Lett. 25, 1297-1299 (2000).
    [CrossRef]
  7. E. Chow, S. Y. Lin, S. G. Johnson, and J. D. Joannopoulos, “Quantitative analysis of bending efficiency in photonic-crystal waveguide bends at=1.55 mm wavelengths,” Opt. Lett. 26, 286-288 (2001).
    [CrossRef]
  8. P. R. Villeneuve, S. Fan, S. G. Johnson, and J. D. Joannopoulos, “Three-dimensional photon confinement in photonic crystals of low-dimensional periodicity,” Proc. IEEE 145, 384-390 (1998).
  9. E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983-986 (2000).
    [CrossRef] [PubMed]
  10. S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751-5758 (1999).
    [CrossRef]
  11. S. Y. Lin, E. Chow, J. Bur, S. G. Johnson, and J. D. Joannopoulos, “Low-loss, wide-angle Y splitter at 1.6-mm wavelengths built with a two-dimensional photonic crystal,” Opt. Lett. 27, 1400-1402 (2002).
    [CrossRef]
  12. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).
  13. S. Olivier, H. Benisty, C. Weisbuch, C. J. M. Smith, T. F. Krauss, R. Houdre, and U. Oesterle, “Improved 60 degrees bend transmission of submicron-width waveguides defined in two-dimensional photonic crystals,” J. Lightwave Technol. 20, 1198-1203 (2002).
    [CrossRef]
  14. D. Leunberger, “Experimental and numerical investigation of two dimensional photonic crystals for application in integrated optics,” Ph.D. dissertation [Ecole Polytechnique Federale de Lausanne (EPFL), 2004].
  15. S. Fan, S. G. Johnson, J. D. Joannopoulos, C. Manolatou, and H. A. Haus, “Waveguide branches in photonic crystals,” J. Opt. Soc. Am. B 18, 162-165 (2001).
    [CrossRef]

2005 (1)

2004 (1)

Y. Sugimoto, Y. Tanaka, N. Ikeda, K. Kanamoto, Y. Nakamura, S. Ohkouchi, H. Nakamura, K. Inoue, H. Sasaki, Y. Watanabe, K. Ishida, H. Ishikawa, and K. Asakawa, “Two dimensional semiconductor-based photonic crystal slab waveguides for ultra-fast optical signal processing devices,” IEICE Trans. Electron. E87-C, 316-327 (2004).

2003 (1)

2002 (2)

2001 (2)

2000 (2)

S. Y. Lin, E. Chow, S. G. Johnson, and J. D. Joannopoulos, “Demonstration of highly efficient waveguiding in a photonic crystal slab at the 1.5-μm wavelength,” Opt. Lett. 25, 1297-1299 (2000).
[CrossRef]

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983-986 (2000).
[CrossRef] [PubMed]

1999 (1)

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751-5758 (1999).
[CrossRef]

1998 (1)

P. R. Villeneuve, S. Fan, S. G. Johnson, and J. D. Joannopoulos, “Three-dimensional photon confinement in photonic crystals of low-dimensional periodicity,” Proc. IEEE 145, 384-390 (1998).

1997 (1)

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

1987 (1)

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

Alleman, A.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983-986 (2000).
[CrossRef] [PubMed]

Asakawa, K.

Y. Sugimoto, Y. Tanaka, N. Ikeda, K. Kanamoto, Y. Nakamura, S. Ohkouchi, H. Nakamura, K. Inoue, H. Sasaki, Y. Watanabe, K. Ishida, H. Ishikawa, and K. Asakawa, “Two dimensional semiconductor-based photonic crystal slab waveguides for ultra-fast optical signal processing devices,” IEICE Trans. Electron. E87-C, 316-327 (2004).

Benisty, H.

Borel, P. I.

Bur, J.

Chow, E.

Fan, S.

S. Fan, S. G. Johnson, J. D. Joannopoulos, C. Manolatou, and H. A. Haus, “Waveguide branches in photonic crystals,” J. Opt. Soc. Am. B 18, 162-165 (2001).
[CrossRef]

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751-5758 (1999).
[CrossRef]

P. R. Villeneuve, S. Fan, S. G. Johnson, and J. D. Joannopoulos, “Three-dimensional photon confinement in photonic crystals of low-dimensional periodicity,” Proc. IEEE 145, 384-390 (1998).

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

Frandsen, L. H.

Haus, H. A.

Hou, H.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983-986 (2000).
[CrossRef] [PubMed]

Houdre, R.

Ikeda, N.

Y. Sugimoto, Y. Tanaka, N. Ikeda, K. Kanamoto, Y. Nakamura, S. Ohkouchi, H. Nakamura, K. Inoue, H. Sasaki, Y. Watanabe, K. Ishida, H. Ishikawa, and K. Asakawa, “Two dimensional semiconductor-based photonic crystal slab waveguides for ultra-fast optical signal processing devices,” IEICE Trans. Electron. E87-C, 316-327 (2004).

Inoue, K.

Y. Sugimoto, Y. Tanaka, N. Ikeda, K. Kanamoto, Y. Nakamura, S. Ohkouchi, H. Nakamura, K. Inoue, H. Sasaki, Y. Watanabe, K. Ishida, H. Ishikawa, and K. Asakawa, “Two dimensional semiconductor-based photonic crystal slab waveguides for ultra-fast optical signal processing devices,” IEICE Trans. Electron. E87-C, 316-327 (2004).

Ishida, K.

Y. Sugimoto, Y. Tanaka, N. Ikeda, K. Kanamoto, Y. Nakamura, S. Ohkouchi, H. Nakamura, K. Inoue, H. Sasaki, Y. Watanabe, K. Ishida, H. Ishikawa, and K. Asakawa, “Two dimensional semiconductor-based photonic crystal slab waveguides for ultra-fast optical signal processing devices,” IEICE Trans. Electron. E87-C, 316-327 (2004).

Ishikawa, H.

Y. Sugimoto, Y. Tanaka, N. Ikeda, K. Kanamoto, Y. Nakamura, S. Ohkouchi, H. Nakamura, K. Inoue, H. Sasaki, Y. Watanabe, K. Ishida, H. Ishikawa, and K. Asakawa, “Two dimensional semiconductor-based photonic crystal slab waveguides for ultra-fast optical signal processing devices,” IEICE Trans. Electron. E87-C, 316-327 (2004).

Joannopoulos, J. D.

S. Y. Lin, E. Chow, J. Bur, S. G. Johnson, and J. D. Joannopoulos, “Low-loss, wide-angle Y splitter at 1.6-mm wavelengths built with a two-dimensional photonic crystal,” Opt. Lett. 27, 1400-1402 (2002).
[CrossRef]

S. Fan, S. G. Johnson, J. D. Joannopoulos, C. Manolatou, and H. A. Haus, “Waveguide branches in photonic crystals,” J. Opt. Soc. Am. B 18, 162-165 (2001).
[CrossRef]

E. Chow, S. Y. Lin, S. G. Johnson, and J. D. Joannopoulos, “Quantitative analysis of bending efficiency in photonic-crystal waveguide bends at=1.55 mm wavelengths,” Opt. Lett. 26, 286-288 (2001).
[CrossRef]

S. Y. Lin, E. Chow, S. G. Johnson, and J. D. Joannopoulos, “Demonstration of highly efficient waveguiding in a photonic crystal slab at the 1.5-μm wavelength,” Opt. Lett. 25, 1297-1299 (2000).
[CrossRef]

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983-986 (2000).
[CrossRef] [PubMed]

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751-5758 (1999).
[CrossRef]

P. R. Villeneuve, S. Fan, S. G. Johnson, and J. D. Joannopoulos, “Three-dimensional photon confinement in photonic crystals of low-dimensional periodicity,” Proc. IEEE 145, 384-390 (1998).

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

Johnson, S. G.

S. Y. Lin, E. Chow, J. Bur, S. G. Johnson, and J. D. Joannopoulos, “Low-loss, wide-angle Y splitter at 1.6-mm wavelengths built with a two-dimensional photonic crystal,” Opt. Lett. 27, 1400-1402 (2002).
[CrossRef]

S. Fan, S. G. Johnson, J. D. Joannopoulos, C. Manolatou, and H. A. Haus, “Waveguide branches in photonic crystals,” J. Opt. Soc. Am. B 18, 162-165 (2001).
[CrossRef]

E. Chow, S. Y. Lin, S. G. Johnson, and J. D. Joannopoulos, “Quantitative analysis of bending efficiency in photonic-crystal waveguide bends at=1.55 mm wavelengths,” Opt. Lett. 26, 286-288 (2001).
[CrossRef]

S. Y. Lin, E. Chow, S. G. Johnson, and J. D. Joannopoulos, “Demonstration of highly efficient waveguiding in a photonic crystal slab at the 1.5-μm wavelength,” Opt. Lett. 25, 1297-1299 (2000).
[CrossRef]

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983-986 (2000).
[CrossRef] [PubMed]

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751-5758 (1999).
[CrossRef]

P. R. Villeneuve, S. Fan, S. G. Johnson, and J. D. Joannopoulos, “Three-dimensional photon confinement in photonic crystals of low-dimensional periodicity,” Proc. IEEE 145, 384-390 (1998).

Kanamoto, K.

Y. Sugimoto, Y. Tanaka, N. Ikeda, K. Kanamoto, Y. Nakamura, S. Ohkouchi, H. Nakamura, K. Inoue, H. Sasaki, Y. Watanabe, K. Ishida, H. Ishikawa, and K. Asakawa, “Two dimensional semiconductor-based photonic crystal slab waveguides for ultra-fast optical signal processing devices,” IEICE Trans. Electron. E87-C, 316-327 (2004).

Kivshar, Y.

Kolodziejski, L. A.

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751-5758 (1999).
[CrossRef]

Krauss, T. F.

Leunberger, D.

D. Leunberger, “Experimental and numerical investigation of two dimensional photonic crystals for application in integrated optics,” Ph.D. dissertation [Ecole Polytechnique Federale de Lausanne (EPFL), 2004].

Lin, S. Y.

Manolatou, C.

Miroshnichenko, A.

Nakamura, H.

Y. Sugimoto, Y. Tanaka, N. Ikeda, K. Kanamoto, Y. Nakamura, S. Ohkouchi, H. Nakamura, K. Inoue, H. Sasaki, Y. Watanabe, K. Ishida, H. Ishikawa, and K. Asakawa, “Two dimensional semiconductor-based photonic crystal slab waveguides for ultra-fast optical signal processing devices,” IEICE Trans. Electron. E87-C, 316-327 (2004).

Nakamura, Y.

Y. Sugimoto, Y. Tanaka, N. Ikeda, K. Kanamoto, Y. Nakamura, S. Ohkouchi, H. Nakamura, K. Inoue, H. Sasaki, Y. Watanabe, K. Ishida, H. Ishikawa, and K. Asakawa, “Two dimensional semiconductor-based photonic crystal slab waveguides for ultra-fast optical signal processing devices,” IEICE Trans. Electron. E87-C, 316-327 (2004).

Oesterle, U.

Ohkouchi, S.

Y. Sugimoto, Y. Tanaka, N. Ikeda, K. Kanamoto, Y. Nakamura, S. Ohkouchi, H. Nakamura, K. Inoue, H. Sasaki, Y. Watanabe, K. Ishida, H. Ishikawa, and K. Asakawa, “Two dimensional semiconductor-based photonic crystal slab waveguides for ultra-fast optical signal processing devices,” IEICE Trans. Electron. E87-C, 316-327 (2004).

Olivier, S.

Sasaki, H.

Y. Sugimoto, Y. Tanaka, N. Ikeda, K. Kanamoto, Y. Nakamura, S. Ohkouchi, H. Nakamura, K. Inoue, H. Sasaki, Y. Watanabe, K. Ishida, H. Ishikawa, and K. Asakawa, “Two dimensional semiconductor-based photonic crystal slab waveguides for ultra-fast optical signal processing devices,” IEICE Trans. Electron. E87-C, 316-327 (2004).

Smith, C. J. M.

Sugimoto, Y.

Y. Sugimoto, Y. Tanaka, N. Ikeda, K. Kanamoto, Y. Nakamura, S. Ohkouchi, H. Nakamura, K. Inoue, H. Sasaki, Y. Watanabe, K. Ishida, H. Ishikawa, and K. Asakawa, “Two dimensional semiconductor-based photonic crystal slab waveguides for ultra-fast optical signal processing devices,” IEICE Trans. Electron. E87-C, 316-327 (2004).

Tanaka, Y.

Y. Sugimoto, Y. Tanaka, N. Ikeda, K. Kanamoto, Y. Nakamura, S. Ohkouchi, H. Nakamura, K. Inoue, H. Sasaki, Y. Watanabe, K. Ishida, H. Ishikawa, and K. Asakawa, “Two dimensional semiconductor-based photonic crystal slab waveguides for ultra-fast optical signal processing devices,” IEICE Trans. Electron. E87-C, 316-327 (2004).

Thorhauge, M.

Vawter, G. A.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983-986 (2000).
[CrossRef] [PubMed]

Villeneuve, P. R.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983-986 (2000).
[CrossRef] [PubMed]

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751-5758 (1999).
[CrossRef]

P. R. Villeneuve, S. Fan, S. G. Johnson, and J. D. Joannopoulos, “Three-dimensional photon confinement in photonic crystals of low-dimensional periodicity,” Proc. IEEE 145, 384-390 (1998).

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

Watanabe, Y.

Y. Sugimoto, Y. Tanaka, N. Ikeda, K. Kanamoto, Y. Nakamura, S. Ohkouchi, H. Nakamura, K. Inoue, H. Sasaki, Y. Watanabe, K. Ishida, H. Ishikawa, and K. Asakawa, “Two dimensional semiconductor-based photonic crystal slab waveguides for ultra-fast optical signal processing devices,” IEICE Trans. Electron. E87-C, 316-327 (2004).

Weisbuch, C.

Wendt, J. R.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983-986 (2000).
[CrossRef] [PubMed]

Yablonovitch, E.

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

Zubrzycki, W.

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983-986 (2000).
[CrossRef] [PubMed]

IEICE Trans. Electron. (1)

Y. Sugimoto, Y. Tanaka, N. Ikeda, K. Kanamoto, Y. Nakamura, S. Ohkouchi, H. Nakamura, K. Inoue, H. Sasaki, Y. Watanabe, K. Ishida, H. Ishikawa, and K. Asakawa, “Two dimensional semiconductor-based photonic crystal slab waveguides for ultra-fast optical signal processing devices,” IEICE Trans. Electron. E87-C, 316-327 (2004).

J. Lightwave Technol. (1)

J. Opt. Soc. Am. B (1)

Nature (2)

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

E. Chow, S. Y. Lin, S. G. Johnson, P. R. Villeneuve, J. D. Joannopoulos, J. R. Wendt, G. A. Vawter, W. Zubrzycki, H. Hou, and A. Alleman, “Three-dimensional control of light in a two-dimensional photonic crystal slab,” Nature 407, 983-986 (2000).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. B (1)

S. G. Johnson, S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and L. A. Kolodziejski, “Guided modes in photonic crystal slabs,” Phys. Rev. B 60, 5751-5758 (1999).
[CrossRef]

Phys. Rev. Lett. (1)

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

Proc. IEEE (1)

P. R. Villeneuve, S. Fan, S. G. Johnson, and J. D. Joannopoulos, “Three-dimensional photon confinement in photonic crystals of low-dimensional periodicity,” Proc. IEEE 145, 384-390 (1998).

Other (2)

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).

D. Leunberger, “Experimental and numerical investigation of two dimensional photonic crystals for application in integrated optics,” Ph.D. dissertation [Ecole Polytechnique Federale de Lausanne (EPFL), 2004].

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Figures (12)

Fig. 1
Fig. 1

Scheme of the coupled mode model for a resonant bend.

Fig. 2
Fig. 2

PC bent structure proposed in this paper.

Fig. 3
Fig. 3

Super defect introduced in this paper; scatterer rods and the middle rod are distinguished in the circles and the square, respectively.

Fig. 4
Fig. 4

Normalized transmission spectra for the bend structure in the absence of middle and scatterer rods.

Fig. 5
Fig. 5

Normalized transmission spectra for the bend structure with Rm = 0.8 and Rs = 1 .

Fig. 6
Fig. 6

Snapshot of the electric field pattern in our new bend at wavelength ( λ 1 = 1450 nm ) .

Fig. 7
Fig. 7

Schematic of the coupled mode model of a resonant cavity splitter.

Fig. 8
Fig. 8

Transmission coefficient into port 2 as a function of the ratio between the decay rates into the input and an output waveguide.

Fig. 9
Fig. 9

PC power splitter proposed in this paper.

Fig. 10
Fig. 10

Normalized transmission spectra for the power splitter in the absence of middle and scatterer rods.

Fig. 11
Fig. 11

(a) Normalized transmission spectra for the power splitter structure with Rm = 1.2 r and Rs = r ; the normalized transmission is over 91% in a wide range. (b) Normalized transmission spectra for the bend structure with Rm = 2 r and Rs = r .

Fig. 12
Fig. 12

Snapshots of the electric field pattern in our new power splitter at wavelength ( λ 1 = 1500 nm ) .

Equations (11)

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

a ( t ) e j ω 0 t e t τ 0 e t τ 1 e t τ 2 .
d a ( t ) d t = j ω 0 a ( t ) ( 1 τ 0 + 1 τ 1 + 1 τ 2 ) a ( t ) + 2 τ 1 S + 1 + 2 τ 2 S + 2 .
S 2 2 = 2 τ 2 a 2 .
a ( ω ) = 2 τ 1 s + 1 j ( ω ω 0 ) + 1 τ 0 + 1 τ 1 + 1 τ 2 .
T = S 2 S + 1 2 = 2 τ 2 2 τ 1 j ( ω ω 0 ) + 1 τ 1 + 1 τ 2 + 1 τ 0 2 .
R = S 1 S + 1 2 = j ( ω ω 0 ) + 1 τ 1 1 τ 2 1 τ 0 j ( ω ω 0 ) + 1 τ 1 + 1 τ 2 + 1 τ 0 2 .
1 τ 1 1 τ 2 1 τ 0 = 0 1 τ 1 = 1 τ 2 + 1 τ 0 .
d a ( t ) d t = j ω 0 a ( t ) ( 1 τ 0 + 1 τ 1 + 1 τ 2 + 1 τ 3 ) a ( t ) + 2 τ 1 S + 1 + 2 τ 2 S + 2 + 2 τ 3 S + 3 .
R = S 1 S + 1 2 = j ( ω ω 0 ) + 1 τ 1 1 τ 2 1 τ 3 1 τ 0 j ( ω ω 0 ) + 1 τ 1 + 1 τ 2 + 1 τ 3 + 1 τ 0 2 ,
T = S 2 S + 1 2 = 2 τ 1 τ 2 j ( ω ω 0 ) + 1 τ 1 + 1 τ 2 + 1 τ 3 + 1 τ 0 2 .
1 τ 1 = 1 τ 2 + 1 τ 3 .

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