Abstract

We study theoretically the focusing effects when a Gaussian beam passing through a graded photonic crystal consisting of air holes with gradually varying radii. The device is used to couple the light between the conventional and photonic crystal waveguides. The coupling efficient is about 4 times higher than that by the linear taper structure.

© 2006 Optical Society of America

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References

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  1. C. C. Chen, H. D. Chien, and P. G. Luan, "Photonic crystal beam splitters," Appl. Opt. 43, 6187-6190 (2004) http://www.opticsinfobase.org/abstract.cfm?URI=ao-43-33-6187.
    [CrossRef] [PubMed]
  2. H. T. Chien, C. C. Chen, and P. G. Luan, "Photonic crystal beam splitters," Opt. Commun. 259, 873-875 (2006).
    [CrossRef]
  3. C. C. Chen, Y. L. Tsai, C. L. Hsu, and J. Y. Chang, "Propagation loss reduction of photonic crystal slab waveguides by microspheres," Opt. Express 12, 3934-3939 (2004).
    [CrossRef] [PubMed]
  4. L. M. Chang, C. H. Hou, Y. C. Ting, C. C. Chen, C. L. Hsu, J. Y. Chang, C. C. Lee, G. T. Chen, and J. I. Chyi, "Laser emission from GaN photonic crystals," Appl. Phys. Lett. 89, 071116 (2006).
    [CrossRef]
  5. C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104 (2002).
    [CrossRef]
  6. A. Martínez and J. Martí, "Negative refraction in two-dimensional photonic crystals: Role of lattice orientation and interface termination," Phys. Rev. B 71, 235115 (2005).
    [CrossRef]
  7. H. T. Chien, H. T. Tang, C. H. Kuo, C.C. Chien, and Z. Ye, "Directed diffraction without negative refraction," Phys. Rev. B 70, 113101 (2004).
    [CrossRef]
  8. Z. Y. Li and L. L. Lin, "Evaluation of lensing in photonic crystal slabs exhibiting negative refraction," Phys. Rev. B 68, 245110 (2003).
    [CrossRef]
  9. P. Halevi, A. A. Krokhin, and J. Arriaga, "Photonic crystals as optical components," Appl. Phys. Lett. 75, 2725-2727 (1999).
    [CrossRef]
  10. L. Sanchis, A. Håkansson, D. López-Zanón, J. Bravo-Abad, and José Sánchez-Dehesa, " Integrated optical devices design by genetic algorithm," Appl. Phys. Lett. 84, 4460-4462 (2004).
    [CrossRef]
  11. E. Centeno and D. Cassagne, "Graded photonic crystals," Opt. Lett. 30, 2278-2280 (2005). http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-17-2278
    [CrossRef] [PubMed]
  12. E. Pshenay-Severin, C. C. Chen, T. Pertsch, M. Augustin, A. Chipouline, A. Tünnermann, "Photonic Crystal Lens for Photonic Crystal Waveguide Coupling," Proceeding of CLEO2006,  CTHK3, 2006.
  13. E. H. Khoo, A. Q. Liu, J. H. Wu, J. Li, and D. Pinjala, "Modified step-theory for investigating mode coupling mechanism in photonic crystal waveguide taper," Opt. Express 14, 6035-6054 (2006).
    [CrossRef] [PubMed]
  14. D. H. Chien, C. H. Tsai, S. S. Lo, C. C. Chen, J. Y. Chang, "Solid immersion lenses in planar waveguides," J. Lightwave Technol. 23, 2746-2748 (2005).
    [CrossRef]

2006 (4)

H. T. Chien, C. C. Chen, and P. G. Luan, "Photonic crystal beam splitters," Opt. Commun. 259, 873-875 (2006).
[CrossRef]

E. Pshenay-Severin, C. C. Chen, T. Pertsch, M. Augustin, A. Chipouline, A. Tünnermann, "Photonic Crystal Lens for Photonic Crystal Waveguide Coupling," Proceeding of CLEO2006,  CTHK3, 2006.

L. M. Chang, C. H. Hou, Y. C. Ting, C. C. Chen, C. L. Hsu, J. Y. Chang, C. C. Lee, G. T. Chen, and J. I. Chyi, "Laser emission from GaN photonic crystals," Appl. Phys. Lett. 89, 071116 (2006).
[CrossRef]

E. H. Khoo, A. Q. Liu, J. H. Wu, J. Li, and D. Pinjala, "Modified step-theory for investigating mode coupling mechanism in photonic crystal waveguide taper," Opt. Express 14, 6035-6054 (2006).
[CrossRef] [PubMed]

2005 (3)

2004 (3)

2003 (1)

Z. Y. Li and L. L. Lin, "Evaluation of lensing in photonic crystal slabs exhibiting negative refraction," Phys. Rev. B 68, 245110 (2003).
[CrossRef]

2002 (1)

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104 (2002).
[CrossRef]

1999 (1)

P. Halevi, A. A. Krokhin, and J. Arriaga, "Photonic crystals as optical components," Appl. Phys. Lett. 75, 2725-2727 (1999).
[CrossRef]

Arriaga, J.

P. Halevi, A. A. Krokhin, and J. Arriaga, "Photonic crystals as optical components," Appl. Phys. Lett. 75, 2725-2727 (1999).
[CrossRef]

Augustin, M.

E. Pshenay-Severin, C. C. Chen, T. Pertsch, M. Augustin, A. Chipouline, A. Tünnermann, "Photonic Crystal Lens for Photonic Crystal Waveguide Coupling," Proceeding of CLEO2006,  CTHK3, 2006.

Cassagne, D.

Centeno, E.

Chang, J. Y.

Chang, L. M.

L. M. Chang, C. H. Hou, Y. C. Ting, C. C. Chen, C. L. Hsu, J. Y. Chang, C. C. Lee, G. T. Chen, and J. I. Chyi, "Laser emission from GaN photonic crystals," Appl. Phys. Lett. 89, 071116 (2006).
[CrossRef]

Chen, C. C.

L. M. Chang, C. H. Hou, Y. C. Ting, C. C. Chen, C. L. Hsu, J. Y. Chang, C. C. Lee, G. T. Chen, and J. I. Chyi, "Laser emission from GaN photonic crystals," Appl. Phys. Lett. 89, 071116 (2006).
[CrossRef]

E. Pshenay-Severin, C. C. Chen, T. Pertsch, M. Augustin, A. Chipouline, A. Tünnermann, "Photonic Crystal Lens for Photonic Crystal Waveguide Coupling," Proceeding of CLEO2006,  CTHK3, 2006.

H. T. Chien, C. C. Chen, and P. G. Luan, "Photonic crystal beam splitters," Opt. Commun. 259, 873-875 (2006).
[CrossRef]

D. H. Chien, C. H. Tsai, S. S. Lo, C. C. Chen, J. Y. Chang, "Solid immersion lenses in planar waveguides," J. Lightwave Technol. 23, 2746-2748 (2005).
[CrossRef]

C. C. Chen, Y. L. Tsai, C. L. Hsu, and J. Y. Chang, "Propagation loss reduction of photonic crystal slab waveguides by microspheres," Opt. Express 12, 3934-3939 (2004).
[CrossRef] [PubMed]

C. C. Chen, H. D. Chien, and P. G. Luan, "Photonic crystal beam splitters," Appl. Opt. 43, 6187-6190 (2004) http://www.opticsinfobase.org/abstract.cfm?URI=ao-43-33-6187.
[CrossRef] [PubMed]

Chen, G. T.

L. M. Chang, C. H. Hou, Y. C. Ting, C. C. Chen, C. L. Hsu, J. Y. Chang, C. C. Lee, G. T. Chen, and J. I. Chyi, "Laser emission from GaN photonic crystals," Appl. Phys. Lett. 89, 071116 (2006).
[CrossRef]

Chien, C.C.

H. T. Chien, H. T. Tang, C. H. Kuo, C.C. Chien, and Z. Ye, "Directed diffraction without negative refraction," Phys. Rev. B 70, 113101 (2004).
[CrossRef]

Chien, D. H.

Chien, H. D.

Chien, H. T.

H. T. Chien, C. C. Chen, and P. G. Luan, "Photonic crystal beam splitters," Opt. Commun. 259, 873-875 (2006).
[CrossRef]

H. T. Chien, H. T. Tang, C. H. Kuo, C.C. Chien, and Z. Ye, "Directed diffraction without negative refraction," Phys. Rev. B 70, 113101 (2004).
[CrossRef]

Chipouline, A.

E. Pshenay-Severin, C. C. Chen, T. Pertsch, M. Augustin, A. Chipouline, A. Tünnermann, "Photonic Crystal Lens for Photonic Crystal Waveguide Coupling," Proceeding of CLEO2006,  CTHK3, 2006.

Chyi, J. I.

L. M. Chang, C. H. Hou, Y. C. Ting, C. C. Chen, C. L. Hsu, J. Y. Chang, C. C. Lee, G. T. Chen, and J. I. Chyi, "Laser emission from GaN photonic crystals," Appl. Phys. Lett. 89, 071116 (2006).
[CrossRef]

Halevi, P.

P. Halevi, A. A. Krokhin, and J. Arriaga, "Photonic crystals as optical components," Appl. Phys. Lett. 75, 2725-2727 (1999).
[CrossRef]

Hou, C. H.

L. M. Chang, C. H. Hou, Y. C. Ting, C. C. Chen, C. L. Hsu, J. Y. Chang, C. C. Lee, G. T. Chen, and J. I. Chyi, "Laser emission from GaN photonic crystals," Appl. Phys. Lett. 89, 071116 (2006).
[CrossRef]

Hsu, C. L.

L. M. Chang, C. H. Hou, Y. C. Ting, C. C. Chen, C. L. Hsu, J. Y. Chang, C. C. Lee, G. T. Chen, and J. I. Chyi, "Laser emission from GaN photonic crystals," Appl. Phys. Lett. 89, 071116 (2006).
[CrossRef]

C. C. Chen, Y. L. Tsai, C. L. Hsu, and J. Y. Chang, "Propagation loss reduction of photonic crystal slab waveguides by microspheres," Opt. Express 12, 3934-3939 (2004).
[CrossRef] [PubMed]

Joannopoulos, J. D.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104 (2002).
[CrossRef]

Johnson, S. G.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104 (2002).
[CrossRef]

Khoo, E. H.

Krokhin, A. A.

P. Halevi, A. A. Krokhin, and J. Arriaga, "Photonic crystals as optical components," Appl. Phys. Lett. 75, 2725-2727 (1999).
[CrossRef]

Kuo, C. H.

H. T. Chien, H. T. Tang, C. H. Kuo, C.C. Chien, and Z. Ye, "Directed diffraction without negative refraction," Phys. Rev. B 70, 113101 (2004).
[CrossRef]

Lee, C. C.

L. M. Chang, C. H. Hou, Y. C. Ting, C. C. Chen, C. L. Hsu, J. Y. Chang, C. C. Lee, G. T. Chen, and J. I. Chyi, "Laser emission from GaN photonic crystals," Appl. Phys. Lett. 89, 071116 (2006).
[CrossRef]

Li, J.

Li, Z. Y.

Z. Y. Li and L. L. Lin, "Evaluation of lensing in photonic crystal slabs exhibiting negative refraction," Phys. Rev. B 68, 245110 (2003).
[CrossRef]

Lin, L. L.

Z. Y. Li and L. L. Lin, "Evaluation of lensing in photonic crystal slabs exhibiting negative refraction," Phys. Rev. B 68, 245110 (2003).
[CrossRef]

Liu, A. Q.

Lo, S. S.

Luan, P. G.

Luo, C.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104 (2002).
[CrossRef]

Martí, J.

A. Martínez and J. Martí, "Negative refraction in two-dimensional photonic crystals: Role of lattice orientation and interface termination," Phys. Rev. B 71, 235115 (2005).
[CrossRef]

Martínez, A.

A. Martínez and J. Martí, "Negative refraction in two-dimensional photonic crystals: Role of lattice orientation and interface termination," Phys. Rev. B 71, 235115 (2005).
[CrossRef]

Pendry, J. B.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104 (2002).
[CrossRef]

Pertsch, T.

E. Pshenay-Severin, C. C. Chen, T. Pertsch, M. Augustin, A. Chipouline, A. Tünnermann, "Photonic Crystal Lens for Photonic Crystal Waveguide Coupling," Proceeding of CLEO2006,  CTHK3, 2006.

Pinjala, D.

Pshenay-Severin, E.

E. Pshenay-Severin, C. C. Chen, T. Pertsch, M. Augustin, A. Chipouline, A. Tünnermann, "Photonic Crystal Lens for Photonic Crystal Waveguide Coupling," Proceeding of CLEO2006,  CTHK3, 2006.

Tang, H. T.

H. T. Chien, H. T. Tang, C. H. Kuo, C.C. Chien, and Z. Ye, "Directed diffraction without negative refraction," Phys. Rev. B 70, 113101 (2004).
[CrossRef]

Ting, Y. C.

L. M. Chang, C. H. Hou, Y. C. Ting, C. C. Chen, C. L. Hsu, J. Y. Chang, C. C. Lee, G. T. Chen, and J. I. Chyi, "Laser emission from GaN photonic crystals," Appl. Phys. Lett. 89, 071116 (2006).
[CrossRef]

Tsai, C. H.

Tsai, Y. L.

Tünnermann, A.

E. Pshenay-Severin, C. C. Chen, T. Pertsch, M. Augustin, A. Chipouline, A. Tünnermann, "Photonic Crystal Lens for Photonic Crystal Waveguide Coupling," Proceeding of CLEO2006,  CTHK3, 2006.

Wu, J. H.

Ye, Z.

H. T. Chien, H. T. Tang, C. H. Kuo, C.C. Chien, and Z. Ye, "Directed diffraction without negative refraction," Phys. Rev. B 70, 113101 (2004).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (2)

L. M. Chang, C. H. Hou, Y. C. Ting, C. C. Chen, C. L. Hsu, J. Y. Chang, C. C. Lee, G. T. Chen, and J. I. Chyi, "Laser emission from GaN photonic crystals," Appl. Phys. Lett. 89, 071116 (2006).
[CrossRef]

P. Halevi, A. A. Krokhin, and J. Arriaga, "Photonic crystals as optical components," Appl. Phys. Lett. 75, 2725-2727 (1999).
[CrossRef]

J. Lightwave Technol. (1)

Opt. Commun. (1)

H. T. Chien, C. C. Chen, and P. G. Luan, "Photonic crystal beam splitters," Opt. Commun. 259, 873-875 (2006).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. B (4)

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, "All-angle negative refraction without negative effective index," Phys. Rev. B 65, 201104 (2002).
[CrossRef]

A. Martínez and J. Martí, "Negative refraction in two-dimensional photonic crystals: Role of lattice orientation and interface termination," Phys. Rev. B 71, 235115 (2005).
[CrossRef]

H. T. Chien, H. T. Tang, C. H. Kuo, C.C. Chien, and Z. Ye, "Directed diffraction without negative refraction," Phys. Rev. B 70, 113101 (2004).
[CrossRef]

Z. Y. Li and L. L. Lin, "Evaluation of lensing in photonic crystal slabs exhibiting negative refraction," Phys. Rev. B 68, 245110 (2003).
[CrossRef]

Proceeding of CLEO (1)

E. Pshenay-Severin, C. C. Chen, T. Pertsch, M. Augustin, A. Chipouline, A. Tünnermann, "Photonic Crystal Lens for Photonic Crystal Waveguide Coupling," Proceeding of CLEO2006,  CTHK3, 2006.

Other (1)

L. Sanchis, A. Håkansson, D. López-Zanón, J. Bravo-Abad, and José Sánchez-Dehesa, " Integrated optical devices design by genetic algorithm," Appl. Phys. Lett. 84, 4460-4462 (2004).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

The graded PC consists of m × n air holes. The refractive index in the air holes and the background is 1and 3.46, respectively. The radius of the air holes in each column is identical. The radius of the air holes for different columns is linearly increased from the center (r1) to the edge (r13) of the array. The coordinate of the center of the structure is defined as (0,0).

Fig. 2.
Fig. 2.

Intensity spectra at the position along Z-axis. (a)~(m) show the results for the structures with m from 1 to 14, respectively. The upper edge of the graded PC is at the position of Z=(m/2)×a in each figure.

Fig. 3.
Fig. 3.

Photonic bandgap of the uniform (non-graded) PC structures formed by air holes in the Γ-X direction. The calculation is performed for the structure with the radius from 0.27a to 0.39a.

Fig. 4.
Fig. 4.

(a). Intensity and spot size at focus for the structures with different layer number, m. The intensity at focus reaches maximum as the value of m is 10. The corresponding spot size is around 1.94a. (b). Focal length and NA of the graded PC with different layer number, m.

Fig. 5.
Fig. 5.

Intensity distribution of magnetic field in steady state for the structures (a) m=10 and (b) m=14.

Fig. 6.
Fig. 6.

Intensity distributions of magnetic field in steady state (a) for the linear taper and (b) for the graded PC. (c) for the graded PC with the width of 9a.

Fig. 7.
Fig. 7.

Intensity distributions of magnetic field in steady state while the incident beams are oblique. The oblique angle are (a) 5, (b) 10, and (c) 15 degrees, respectively.

Equations (1)

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NA = n S i F F 2 R 2

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