Abstract

With the method of adding two point defects on modulated surface, novel photonic crystal (PC) waveguide-based beam splitters were presented. The modulated surface layer supports surface states, and introduced discrete point defects can serve as discrete light emitters. The finite-difference time-domain (FDTD) simulations show that the number of beams is sensitive to the distance of two point defects. By adjusting the positions of the point defects, 1-to-N beam splitters can be realized. These simple, easy-to-fabricate and controllable structures have important potential applications in integrated optical circuits.

© 2010 OSA

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  1. L. Y. Jiang, W. Jia, H. P. Li, X. Y. Li, C. X. Cong, and Z. X. Shen, “Inverse design for directional emitter and power splitter based on photonic crystal waveguide with surface corrugations,” J. Opt. Soc. Am. B 26(11), 2157–2160 (2009).
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    [CrossRef]
  4. M. Q. Xin, L. Zhang, C. Eng Png, J. H. Teng, and J. Aaron Danner, “Asymmetric open cavities for beam steering and switching from line-defect photonic crystals,” J. Opt. Soc. Am. B 27(6), 1153–1157 (2010).
    [CrossRef]
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    [CrossRef] [PubMed]
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2010 (2)

W. Jia, L. Y. Jiang, K. Chen, and X. Y. Li, “Design of photonic crystal power beam splitters via corrugated and gratinglike surfaces,” Opt. Commun. 283(20), 4078–4084 (2010).
[CrossRef]

M. Q. Xin, L. Zhang, C. Eng Png, J. H. Teng, and J. Aaron Danner, “Asymmetric open cavities for beam steering and switching from line-defect photonic crystals,” J. Opt. Soc. Am. B 27(6), 1153–1157 (2010).
[CrossRef]

2009 (2)

2008 (2)

Z. F. Li, K. Aydin, and E. Ozbay, “Wide band width directional beaming via waveguide ports in photonic crystals,” J. Phys. D Appl. Phys. 41(15), 155115 (2008).
[CrossRef]

H. Caglayan, I. Bulu, and E. Ozbay, “Off-axis directional beaming via photonic crystal surface modes,” Appl. Phys. Lett. 92(9), 092114 (2008).
[CrossRef]

2007 (3)

2006 (3)

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. Lee, “Beaming light from a subwavelength metal slit surrounded by dielectric surface gratings,” Opt. Express 14(8), 3503–3511 (2006), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-8-3503 .
[CrossRef] [PubMed]

R. Mousse, Th. Koschny, and C. M. Soukoulis, “Excitation of surface waves in a photonic crystal with negative refraction: The role of surface termination,” Phys. Rev. B 74(11), 115111 (2006).
[CrossRef]

B. Wang, W. Dai, A. Fang, L. Zhang, G. Tuttle, Th. Koschny, and C. M. Soukoulis, “Surface waves in photonic crystal slabs,” Phys. Rev. B 74(19), 195104 (2006).
[CrossRef]

2005 (1)

S. K. Morrison and Y. S. Kivshar, “Engineering of directional emission from photonic-crystal waveguide,” Appl. Phys. Lett. 86(8), 081110 (2005).
[CrossRef]

2004 (1)

E. Moreno, F. J. García-Vidal, and L. Martín-Moreno, “Enhanced transmission and beaming of light via photonic crystal surface modes,” Phys. Rev. B 69(12), 121402 (2004).
[CrossRef]

2003 (1)

F. J. García-Vidal, L. Martín-Moreno, H. J. Lezec, and T. W. Ebbesen, “Focusing light with a single subwavelength aperture flanked by surface corrugations,” Appl. Phys. Lett. 83(22), 4500–4502 (2003).
[CrossRef]

2002 (1)

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002).
[CrossRef] [PubMed]

1994 (1)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
[CrossRef]

1991 (1)

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B Condens. Matter 44(19), 10961–10964 (1991).
[CrossRef] [PubMed]

Aaron Danner, J.

Aydin, K.

Z. F. Li, K. Aydin, and E. Ozbay, “Wide band width directional beaming via waveguide ports in photonic crystals,” J. Phys. D Appl. Phys. 41(15), 155115 (2008).
[CrossRef]

Berenger, J. P.

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
[CrossRef]

Brommer, K. D.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B Condens. Matter 44(19), 10961–10964 (1991).
[CrossRef] [PubMed]

Bulu, I.

H. Caglayan, I. Bulu, and E. Ozbay, “Off-axis directional beaming via photonic crystal surface modes,” Appl. Phys. Lett. 92(9), 092114 (2008).
[CrossRef]

Caglayan, H.

H. Caglayan, I. Bulu, and E. Ozbay, “Off-axis directional beaming via photonic crystal surface modes,” Appl. Phys. Lett. 92(9), 092114 (2008).
[CrossRef]

Chang, C. K.

Chen, K.

W. Jia, L. Y. Jiang, K. Chen, and X. Y. Li, “Design of photonic crystal power beam splitters via corrugated and gratinglike surfaces,” Opt. Commun. 283(20), 4078–4084 (2010).
[CrossRef]

Chen, Y. C.

Cong, C. X.

Dai, W.

B. Wang, W. Dai, A. Fang, L. Zhang, G. Tuttle, Th. Koschny, and C. M. Soukoulis, “Surface waves in photonic crystal slabs,” Phys. Rev. B 74(19), 195104 (2006).
[CrossRef]

Degiron, A.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002).
[CrossRef] [PubMed]

Devaux, E.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002).
[CrossRef] [PubMed]

Dong, J. W.

Ebbesen, T. W.

F. J. García-Vidal, L. Martín-Moreno, H. J. Lezec, and T. W. Ebbesen, “Focusing light with a single subwavelength aperture flanked by surface corrugations,” Appl. Phys. Lett. 83(22), 4500–4502 (2003).
[CrossRef]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002).
[CrossRef] [PubMed]

Eng Png, C.

Fang, A.

B. Wang, W. Dai, A. Fang, L. Zhang, G. Tuttle, Th. Koschny, and C. M. Soukoulis, “Surface waves in photonic crystal slabs,” Phys. Rev. B 74(19), 195104 (2006).
[CrossRef]

Garcia-Vidal, F. J.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002).
[CrossRef] [PubMed]

García-Vidal, F. J.

E. Moreno, F. J. García-Vidal, and L. Martín-Moreno, “Enhanced transmission and beaming of light via photonic crystal surface modes,” Phys. Rev. B 69(12), 121402 (2004).
[CrossRef]

F. J. García-Vidal, L. Martín-Moreno, H. J. Lezec, and T. W. Ebbesen, “Focusing light with a single subwavelength aperture flanked by surface corrugations,” Appl. Phys. Lett. 83(22), 4500–4502 (2003).
[CrossRef]

Jia, W.

W. Jia, L. Y. Jiang, K. Chen, and X. Y. Li, “Design of photonic crystal power beam splitters via corrugated and gratinglike surfaces,” Opt. Commun. 283(20), 4078–4084 (2010).
[CrossRef]

L. Y. Jiang, W. Jia, H. P. Li, X. Y. Li, C. X. Cong, and Z. X. Shen, “Inverse design for directional emitter and power splitter based on photonic crystal waveguide with surface corrugations,” J. Opt. Soc. Am. B 26(11), 2157–2160 (2009).
[CrossRef]

Jiang, L. Y.

W. Jia, L. Y. Jiang, K. Chen, and X. Y. Li, “Design of photonic crystal power beam splitters via corrugated and gratinglike surfaces,” Opt. Commun. 283(20), 4078–4084 (2010).
[CrossRef]

L. Y. Jiang, W. Jia, H. P. Li, X. Y. Li, C. X. Cong, and Z. X. Shen, “Inverse design for directional emitter and power splitter based on photonic crystal waveguide with surface corrugations,” J. Opt. Soc. Am. B 26(11), 2157–2160 (2009).
[CrossRef]

Jiang, X. Y.

Y. L. Zhang, D. Y. Zhao, C. H. Zhou, and X. Y. Jiang, “Directional light emission through a metallic nanostructure,” J. Appl. Phys. 105(11), 113124 (2009).
[CrossRef]

Joannopoulos, J. D.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B Condens. Matter 44(19), 10961–10964 (1991).
[CrossRef] [PubMed]

Kivshar, Y. S.

S. K. Morrison and Y. S. Kivshar, “Engineering of directional emission from photonic-crystal waveguide,” Appl. Phys. Lett. 86(8), 081110 (2005).
[CrossRef]

Koschny, Th.

R. Moussa, B. Wang, G. Tuttle, Th. Koschny, and C. M. Soukoulis, “Effect of beaming and enhanced transmission in photonic crystals,” Phys. Rev. B 76(23), 235417 (2007).
[CrossRef]

R. Mousse, Th. Koschny, and C. M. Soukoulis, “Excitation of surface waves in a photonic crystal with negative refraction: The role of surface termination,” Phys. Rev. B 74(11), 115111 (2006).
[CrossRef]

B. Wang, W. Dai, A. Fang, L. Zhang, G. Tuttle, Th. Koschny, and C. M. Soukoulis, “Surface waves in photonic crystal slabs,” Phys. Rev. B 74(19), 195104 (2006).
[CrossRef]

Kuan, C. H.

Lee, C. K.

Lezec, H. J.

F. J. García-Vidal, L. Martín-Moreno, H. J. Lezec, and T. W. Ebbesen, “Focusing light with a single subwavelength aperture flanked by surface corrugations,” Appl. Phys. Lett. 83(22), 4500–4502 (2003).
[CrossRef]

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002).
[CrossRef] [PubMed]

Li, H. P.

Li, X. Y.

W. Jia, L. Y. Jiang, K. Chen, and X. Y. Li, “Design of photonic crystal power beam splitters via corrugated and gratinglike surfaces,” Opt. Commun. 283(20), 4078–4084 (2010).
[CrossRef]

L. Y. Jiang, W. Jia, H. P. Li, X. Y. Li, C. X. Cong, and Z. X. Shen, “Inverse design for directional emitter and power splitter based on photonic crystal waveguide with surface corrugations,” J. Opt. Soc. Am. B 26(11), 2157–2160 (2009).
[CrossRef]

Li, Z. F.

Z. F. Li, K. Aydin, and E. Ozbay, “Wide band width directional beaming via waveguide ports in photonic crystals,” J. Phys. D Appl. Phys. 41(15), 155115 (2008).
[CrossRef]

Liang, W. Y.

Lin, D. Z.

Lin, M. W.

Linke, R. A.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002).
[CrossRef] [PubMed]

Liu, J. M.

Martin-Moreno, L.

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002).
[CrossRef] [PubMed]

Martín-Moreno, L.

E. Moreno, F. J. García-Vidal, and L. Martín-Moreno, “Enhanced transmission and beaming of light via photonic crystal surface modes,” Phys. Rev. B 69(12), 121402 (2004).
[CrossRef]

F. J. García-Vidal, L. Martín-Moreno, H. J. Lezec, and T. W. Ebbesen, “Focusing light with a single subwavelength aperture flanked by surface corrugations,” Appl. Phys. Lett. 83(22), 4500–4502 (2003).
[CrossRef]

Meade, R. D.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B Condens. Matter 44(19), 10961–10964 (1991).
[CrossRef] [PubMed]

Moreno, E.

E. Moreno, F. J. García-Vidal, and L. Martín-Moreno, “Enhanced transmission and beaming of light via photonic crystal surface modes,” Phys. Rev. B 69(12), 121402 (2004).
[CrossRef]

Morrison, S. K.

S. K. Morrison and Y. S. Kivshar, “Engineering of directional emission from photonic-crystal waveguide,” Appl. Phys. Lett. 86(8), 081110 (2005).
[CrossRef]

Moussa, R.

R. Moussa, B. Wang, G. Tuttle, Th. Koschny, and C. M. Soukoulis, “Effect of beaming and enhanced transmission in photonic crystals,” Phys. Rev. B 76(23), 235417 (2007).
[CrossRef]

Mousse, R.

R. Mousse, Th. Koschny, and C. M. Soukoulis, “Excitation of surface waves in a photonic crystal with negative refraction: The role of surface termination,” Phys. Rev. B 74(11), 115111 (2006).
[CrossRef]

Ozbay, E.

Z. F. Li, K. Aydin, and E. Ozbay, “Wide band width directional beaming via waveguide ports in photonic crystals,” J. Phys. D Appl. Phys. 41(15), 155115 (2008).
[CrossRef]

H. Caglayan, I. Bulu, and E. Ozbay, “Off-axis directional beaming via photonic crystal surface modes,” Appl. Phys. Lett. 92(9), 092114 (2008).
[CrossRef]

Qiu, M.

Rappe, A. M.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B Condens. Matter 44(19), 10961–10964 (1991).
[CrossRef] [PubMed]

Shen, Z. X.

Soukoulis, C. M.

R. Moussa, B. Wang, G. Tuttle, Th. Koschny, and C. M. Soukoulis, “Effect of beaming and enhanced transmission in photonic crystals,” Phys. Rev. B 76(23), 235417 (2007).
[CrossRef]

B. Wang, W. Dai, A. Fang, L. Zhang, G. Tuttle, Th. Koschny, and C. M. Soukoulis, “Surface waves in photonic crystal slabs,” Phys. Rev. B 74(19), 195104 (2006).
[CrossRef]

R. Mousse, Th. Koschny, and C. M. Soukoulis, “Excitation of surface waves in a photonic crystal with negative refraction: The role of surface termination,” Phys. Rev. B 74(11), 115111 (2006).
[CrossRef]

Teng, J. H.

Tuttle, G.

R. Moussa, B. Wang, G. Tuttle, Th. Koschny, and C. M. Soukoulis, “Effect of beaming and enhanced transmission in photonic crystals,” Phys. Rev. B 76(23), 235417 (2007).
[CrossRef]

B. Wang, W. Dai, A. Fang, L. Zhang, G. Tuttle, Th. Koschny, and C. M. Soukoulis, “Surface waves in photonic crystal slabs,” Phys. Rev. B 74(19), 195104 (2006).
[CrossRef]

Wang, B.

R. Moussa, B. Wang, G. Tuttle, Th. Koschny, and C. M. Soukoulis, “Effect of beaming and enhanced transmission in photonic crystals,” Phys. Rev. B 76(23), 235417 (2007).
[CrossRef]

B. Wang, W. Dai, A. Fang, L. Zhang, G. Tuttle, Th. Koschny, and C. M. Soukoulis, “Surface waves in photonic crystal slabs,” Phys. Rev. B 74(19), 195104 (2006).
[CrossRef]

Wang, H. Z.

Xiao, S. S.

Xin, M. Q.

Yang, D. L.

Yeh, C. S.

Yeh, J. T.

Zhang, L.

M. Q. Xin, L. Zhang, C. Eng Png, J. H. Teng, and J. Aaron Danner, “Asymmetric open cavities for beam steering and switching from line-defect photonic crystals,” J. Opt. Soc. Am. B 27(6), 1153–1157 (2010).
[CrossRef]

B. Wang, W. Dai, A. Fang, L. Zhang, G. Tuttle, Th. Koschny, and C. M. Soukoulis, “Surface waves in photonic crystal slabs,” Phys. Rev. B 74(19), 195104 (2006).
[CrossRef]

Zhang, Y. L.

Y. L. Zhang, D. Y. Zhao, C. H. Zhou, and X. Y. Jiang, “Directional light emission through a metallic nanostructure,” J. Appl. Phys. 105(11), 113124 (2009).
[CrossRef]

Zhao, D. Y.

Y. L. Zhang, D. Y. Zhao, C. H. Zhou, and X. Y. Jiang, “Directional light emission through a metallic nanostructure,” J. Appl. Phys. 105(11), 113124 (2009).
[CrossRef]

Zhou, C. H.

Y. L. Zhang, D. Y. Zhao, C. H. Zhou, and X. Y. Jiang, “Directional light emission through a metallic nanostructure,” J. Appl. Phys. 105(11), 113124 (2009).
[CrossRef]

Appl. Phys. Lett. (3)

S. K. Morrison and Y. S. Kivshar, “Engineering of directional emission from photonic-crystal waveguide,” Appl. Phys. Lett. 86(8), 081110 (2005).
[CrossRef]

H. Caglayan, I. Bulu, and E. Ozbay, “Off-axis directional beaming via photonic crystal surface modes,” Appl. Phys. Lett. 92(9), 092114 (2008).
[CrossRef]

F. J. García-Vidal, L. Martín-Moreno, H. J. Lezec, and T. W. Ebbesen, “Focusing light with a single subwavelength aperture flanked by surface corrugations,” Appl. Phys. Lett. 83(22), 4500–4502 (2003).
[CrossRef]

J. Appl. Phys. (1)

Y. L. Zhang, D. Y. Zhao, C. H. Zhou, and X. Y. Jiang, “Directional light emission through a metallic nanostructure,” J. Appl. Phys. 105(11), 113124 (2009).
[CrossRef]

J. Comput. Phys. (1)

J. P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114(2), 185–200 (1994).
[CrossRef]

J. Opt. Soc. Am. B (3)

J. Phys. D Appl. Phys. (1)

Z. F. Li, K. Aydin, and E. Ozbay, “Wide band width directional beaming via waveguide ports in photonic crystals,” J. Phys. D Appl. Phys. 41(15), 155115 (2008).
[CrossRef]

Opt. Commun. (1)

W. Jia, L. Y. Jiang, K. Chen, and X. Y. Li, “Design of photonic crystal power beam splitters via corrugated and gratinglike surfaces,” Opt. Commun. 283(20), 4078–4084 (2010).
[CrossRef]

Opt. Express (2)

Phys. Rev. B (4)

R. Mousse, Th. Koschny, and C. M. Soukoulis, “Excitation of surface waves in a photonic crystal with negative refraction: The role of surface termination,” Phys. Rev. B 74(11), 115111 (2006).
[CrossRef]

B. Wang, W. Dai, A. Fang, L. Zhang, G. Tuttle, Th. Koschny, and C. M. Soukoulis, “Surface waves in photonic crystal slabs,” Phys. Rev. B 74(19), 195104 (2006).
[CrossRef]

E. Moreno, F. J. García-Vidal, and L. Martín-Moreno, “Enhanced transmission and beaming of light via photonic crystal surface modes,” Phys. Rev. B 69(12), 121402 (2004).
[CrossRef]

R. Moussa, B. Wang, G. Tuttle, Th. Koschny, and C. M. Soukoulis, “Effect of beaming and enhanced transmission in photonic crystals,” Phys. Rev. B 76(23), 235417 (2007).
[CrossRef]

Phys. Rev. B Condens. Matter (1)

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, “Electromagnetic Bloch waves at the surface of a photonic crystal,” Phys. Rev. B Condens. Matter 44(19), 10961–10964 (1991).
[CrossRef] [PubMed]

Science (1)

H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002).
[CrossRef] [PubMed]

Other (4)

A. Degiron and T. W. Ebbesen, “Analysis of the transmission process through single apertures surrounded by periodic corrugations,” Opt. Express 12, 3694–3700 (2004). http://www.opticsinfobase.org/oe/abstract. cfm?URI=oe-12-16-3694 .
[CrossRef] [PubMed]

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

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic crystal: Molding the flow of light (Princeton University Press, Princeton, 1995).

A. Taflove, and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Boston, 2000).

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Figures (14)

Fig. 1
Fig. 1

Schematic diagram of the photonic crystal waveguide structures, output surface cylinders with rs = 0.09a (denoted by red dots).

Fig. 2
Fig. 2

Simulated spatial distributions of |Ey|2 for the structure illustrated in Fig. 1. The frequency and width of the incident beam are ω = 0.408 (2πc/a) and a, respectively. (a) Unchanged surface (no surface modes allowed), (b) surface cylinders with rs = 0.09a.

Fig. 3
Fig. 3

Schematic drawing of the photonic crystal waveguide structures under study. (a) Structure with Ns modified surface cylinders rs = 0.09a (denoted by red dots) but without point defect. A part of surface cylinders (denoted by blue dots) are not modified, (b) structure with Ns modified surface cylinders rs = 0.09a (denoted by red dots) and two point defect at position (xs , 24.5), (-xs , 24.5), L = 2xs .

Fig. 4
Fig. 4

Simulated spatial distributions of |Ey|2 for the structure illustrated in Fig. 3(a). (a) Ns = 2, (b) Ns = 3, (c) Ns = 4, (d) Ns = 5.

Fig. 5
Fig. 5

Simulated spatial distributions of |Ey|2 for the structure illustrated in Fig. 3(b), (a)L = 4a (b)L = 6a (c)L = 8a (d) L = 10a.

Fig. 6
Fig. 6

Calculated angular distribution of |Ey|2 for radiation pattern plotted in Fig. 5 (along a half circle of r = 20λ). All the above results are obtained at a frequency of ω = 0.408 (2πc/a).

Fig. 7
Fig. 7

Poynting field Sz(x, z = 24.5a) along the PC-air interface for the case of L = 4a, 6a, 8a, 10a.

Fig. 8
Fig. 8

Schematic drawing of the photonic crystal waveguide structure with Ns + Nb modified surface cylinders rs = 0.09a (denoted by red dots) and two point defects at position (xs , 24.5a), (-xs , 24.5a), L = 2xs .

Fig. 9
Fig. 9

Spatial distributions of |Ey|2 for the structure illustrated in Fig. 8, (a) L = 4a, Nb = 0. (b) L = 4a, Nb = 1. (c) L = 4a, Nb = 2. (d) L = 6a, Nb = 0. (e) L = 6a, Nb = 1. (f) L = 6a, Nb = 2. (g) L = 8a, Nb = 0 (h) L = 8a, Nb = 1. (i) L = 8a, Nb = 2. (j) L = 10a, Nb = 0. (k) L = 10a, Nb = 1. (l) L = 10a, Nb = 2.

Fig. 10
Fig. 10

Calculated angular distribution of |Ey|2 for radiation pattern plotted in Fig. 9 (along a half circle of r = 20λ), (a) L = 4a. (b) L = 6a. (c) L = 8a. (d) L = 10a.

Fig. 11
Fig. 11

Poynting field Sz(x, z = 24.5a) along the PC-air interface for Nb = 0, 1, 2, (a) L = 4a. (b) L = 6a. (c) L = 8a. (d) L = 10a.

Fig. 12
Fig. 12

Schematic drawing of the photonic crystal waveguide structure with two extra point defects are introduced to the adjacent layer of output surface. These two extra point defects at position (xs -1, zs -1), here zs -1 = 23.5a, xs -1 = xs + a.

Fig. 13
Fig. 13

Spatial distributions of |Ey|2 for the structure illustrated in Fig. 12, (a) L = 4a, without A. (b) L = 4a, with A. (c) L = 6a, without A. (d) L = 6a, with A. (e) L = 8a, without A. (f) L = 8a, with A. (g) L = 10a, without A. (h) L = 10a, with A.

Fig. 14
Fig. 14

Calculated angular distribution of |Ey|2 for radiation pattern plotted in Fig. 12 (along a half circle of r = 20λ). (a) L = 4a (b) L = 6a. (c) L = 8a. (d) L = 10a.

Equations (3)

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E y ( x , y ) = i = 1 3 E y , i ( x , y )
δ i j = k a i r ( r i r j ) + Δ ϕ i j i , j = 1 , 2 , 3
Δ ϕ 12 = Δ ϕ 32 k λ f f L / 2 , and Δ ϕ 13 = 0

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