Abstract

We designed and analyzed a “mesh-stack” three-dimensional photonic crystal of a 12.4% bandgap with a dielectric constant ratio of 121. The mesh-stack consists of four offset identical square-lattice air-hole patterned membranes in each vertical period that is equal to the in-plane period of the square lattice. This design is fully compatible with the membrane-stacking fabrication method, which is based on alignment and stacking of large-area single-crystal membranes containing engineered defects. A bandgap greater than 10% is preserved as long as the membranes are subjected to in-plane misalignment less than 3% of the square period. By introducing a linear defect with a nonsymmorphic symmetry into the mesh-stack, we achieved a single-mode waveguide over a wide bandwidth.

© 2012 Optical Society of America

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  1. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).
  2. E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
    [CrossRef]
  3. S. John, Phys. Rev. Lett. 58, 2486 (1987).
    [CrossRef]
  4. S. Ghadarghadr, C. P. Fucetola, L. L. Cheong, E. E. Moon, and H. I. Smith, J. Vac. Sci. Technol. B 29, 06F401 (2011).
    [CrossRef]
  5. A. A. Patel and H. I. Smith, J. Vac. Sci. Technol. B 25, 2662 (2007).
    [CrossRef]
  6. K. Aoki, H. T. Miyazaki, H. Hirayama, K. Inoshita, T. Baba, K. Sakoda, N. Shinya, and Y. Aoyagi, Nat. Mater. 2, 117 (2003).
    [CrossRef]
  7. K. Ho, C. Chan, C. Soukoulis, R. Biswas, and M. Sigalas, Solid State Commun. 89, 413 (1994).
    [CrossRef]
  8. M. Maldovan and E. L. Thomas, Nat. Mater. 3, 593 (2004).
    [CrossRef]
  9. m is an arbitrary real number. When m=2, the equation defines a circle. When m=1, it defines a diamond. When m=∞, it defines a square. This is illustrated on the right side of Fig. 1(c).
  10. The bandstructure calculations were done using MIT Photonic-Bands package (MPB, http://ab-initio.mit.edu/mpb ) and the optimizations were done using the nonlinear optimization package (NLopt, http://ab-initio.mit.edu/nlopt ).
  11. A. Mock, L. Lu, and J. O’Brien, Phys. Rev. B 81, 155115 (2010).
    [CrossRef]
  12. This degeneracy, protected by the glide spatial symmetry, ensures the continuity of the bands across the Brillouin zone boundary, while the continuities of the dispersion curves inside the Brillouin zone are protected by translational spatial symmetry.
  13. K. Aoki, Appl. Phys. Lett. 95, 191910 (2009).
    [CrossRef]
  14. E. Yablonovitch, T. J. Gmitter, and K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
    [CrossRef]
  15. C. C. Cheng and A. Scherer, J. Vac. Sci. Technol. B 13, 2696 (1995).
    [CrossRef]
  16. S. Takahashi, K. Suzuki, M. Okano, M. Imada, T. Nakamori, Y. Ota, K. Ishizaki, and S. Noda, Nat. Mater. 8, 721 (2009).
    [CrossRef]
  17. L. Tang and T. Yoshie, J. Vac. Sci. Technol. B 28, 301 (2010).
    [CrossRef]
  18. O. Toader and S. John, Phys. Rev. E 71, 036605 (2005).
    [CrossRef]
  19. S. G. Johnson and J. D. Joannopoulos, Opt. Express 8, 173 (2001).
    [CrossRef]
  20. E. E. Moon, M. K. Mondol, P. N. Everett, and H. I. Smith, J. Vac. Sci. Technol. B 23, 2607 (2005).

2011

S. Ghadarghadr, C. P. Fucetola, L. L. Cheong, E. E. Moon, and H. I. Smith, J. Vac. Sci. Technol. B 29, 06F401 (2011).
[CrossRef]

2010

A. Mock, L. Lu, and J. O’Brien, Phys. Rev. B 81, 155115 (2010).
[CrossRef]

L. Tang and T. Yoshie, J. Vac. Sci. Technol. B 28, 301 (2010).
[CrossRef]

2009

S. Takahashi, K. Suzuki, M. Okano, M. Imada, T. Nakamori, Y. Ota, K. Ishizaki, and S. Noda, Nat. Mater. 8, 721 (2009).
[CrossRef]

K. Aoki, Appl. Phys. Lett. 95, 191910 (2009).
[CrossRef]

2007

A. A. Patel and H. I. Smith, J. Vac. Sci. Technol. B 25, 2662 (2007).
[CrossRef]

2005

E. E. Moon, M. K. Mondol, P. N. Everett, and H. I. Smith, J. Vac. Sci. Technol. B 23, 2607 (2005).

O. Toader and S. John, Phys. Rev. E 71, 036605 (2005).
[CrossRef]

2004

M. Maldovan and E. L. Thomas, Nat. Mater. 3, 593 (2004).
[CrossRef]

2003

K. Aoki, H. T. Miyazaki, H. Hirayama, K. Inoshita, T. Baba, K. Sakoda, N. Shinya, and Y. Aoyagi, Nat. Mater. 2, 117 (2003).
[CrossRef]

2001

1995

C. C. Cheng and A. Scherer, J. Vac. Sci. Technol. B 13, 2696 (1995).
[CrossRef]

1994

K. Ho, C. Chan, C. Soukoulis, R. Biswas, and M. Sigalas, Solid State Commun. 89, 413 (1994).
[CrossRef]

1991

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
[CrossRef]

1987

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef]

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef]

Aoki, K.

K. Aoki, Appl. Phys. Lett. 95, 191910 (2009).
[CrossRef]

K. Aoki, H. T. Miyazaki, H. Hirayama, K. Inoshita, T. Baba, K. Sakoda, N. Shinya, and Y. Aoyagi, Nat. Mater. 2, 117 (2003).
[CrossRef]

Aoyagi, Y.

K. Aoki, H. T. Miyazaki, H. Hirayama, K. Inoshita, T. Baba, K. Sakoda, N. Shinya, and Y. Aoyagi, Nat. Mater. 2, 117 (2003).
[CrossRef]

Baba, T.

K. Aoki, H. T. Miyazaki, H. Hirayama, K. Inoshita, T. Baba, K. Sakoda, N. Shinya, and Y. Aoyagi, Nat. Mater. 2, 117 (2003).
[CrossRef]

Biswas, R.

K. Ho, C. Chan, C. Soukoulis, R. Biswas, and M. Sigalas, Solid State Commun. 89, 413 (1994).
[CrossRef]

Chan, C.

K. Ho, C. Chan, C. Soukoulis, R. Biswas, and M. Sigalas, Solid State Commun. 89, 413 (1994).
[CrossRef]

Cheng, C. C.

C. C. Cheng and A. Scherer, J. Vac. Sci. Technol. B 13, 2696 (1995).
[CrossRef]

Cheong, L. L.

S. Ghadarghadr, C. P. Fucetola, L. L. Cheong, E. E. Moon, and H. I. Smith, J. Vac. Sci. Technol. B 29, 06F401 (2011).
[CrossRef]

Everett, P. N.

E. E. Moon, M. K. Mondol, P. N. Everett, and H. I. Smith, J. Vac. Sci. Technol. B 23, 2607 (2005).

Fucetola, C. P.

S. Ghadarghadr, C. P. Fucetola, L. L. Cheong, E. E. Moon, and H. I. Smith, J. Vac. Sci. Technol. B 29, 06F401 (2011).
[CrossRef]

Ghadarghadr, S.

S. Ghadarghadr, C. P. Fucetola, L. L. Cheong, E. E. Moon, and H. I. Smith, J. Vac. Sci. Technol. B 29, 06F401 (2011).
[CrossRef]

Gmitter, T. J.

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
[CrossRef]

Hirayama, H.

K. Aoki, H. T. Miyazaki, H. Hirayama, K. Inoshita, T. Baba, K. Sakoda, N. Shinya, and Y. Aoyagi, Nat. Mater. 2, 117 (2003).
[CrossRef]

Ho, K.

K. Ho, C. Chan, C. Soukoulis, R. Biswas, and M. Sigalas, Solid State Commun. 89, 413 (1994).
[CrossRef]

Imada, M.

S. Takahashi, K. Suzuki, M. Okano, M. Imada, T. Nakamori, Y. Ota, K. Ishizaki, and S. Noda, Nat. Mater. 8, 721 (2009).
[CrossRef]

Inoshita, K.

K. Aoki, H. T. Miyazaki, H. Hirayama, K. Inoshita, T. Baba, K. Sakoda, N. Shinya, and Y. Aoyagi, Nat. Mater. 2, 117 (2003).
[CrossRef]

Ishizaki, K.

S. Takahashi, K. Suzuki, M. Okano, M. Imada, T. Nakamori, Y. Ota, K. Ishizaki, and S. Noda, Nat. Mater. 8, 721 (2009).
[CrossRef]

Joannopoulos, J. D.

S. G. Johnson and J. D. Joannopoulos, Opt. Express 8, 173 (2001).
[CrossRef]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

John, S.

O. Toader and S. John, Phys. Rev. E 71, 036605 (2005).
[CrossRef]

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef]

Johnson, S. G.

S. G. Johnson and J. D. Joannopoulos, Opt. Express 8, 173 (2001).
[CrossRef]

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Leung, K. M.

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
[CrossRef]

Lu, L.

A. Mock, L. Lu, and J. O’Brien, Phys. Rev. B 81, 155115 (2010).
[CrossRef]

Maldovan, M.

M. Maldovan and E. L. Thomas, Nat. Mater. 3, 593 (2004).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Miyazaki, H. T.

K. Aoki, H. T. Miyazaki, H. Hirayama, K. Inoshita, T. Baba, K. Sakoda, N. Shinya, and Y. Aoyagi, Nat. Mater. 2, 117 (2003).
[CrossRef]

Mock, A.

A. Mock, L. Lu, and J. O’Brien, Phys. Rev. B 81, 155115 (2010).
[CrossRef]

Mondol, M. K.

E. E. Moon, M. K. Mondol, P. N. Everett, and H. I. Smith, J. Vac. Sci. Technol. B 23, 2607 (2005).

Moon, E. E.

S. Ghadarghadr, C. P. Fucetola, L. L. Cheong, E. E. Moon, and H. I. Smith, J. Vac. Sci. Technol. B 29, 06F401 (2011).
[CrossRef]

E. E. Moon, M. K. Mondol, P. N. Everett, and H. I. Smith, J. Vac. Sci. Technol. B 23, 2607 (2005).

Nakamori, T.

S. Takahashi, K. Suzuki, M. Okano, M. Imada, T. Nakamori, Y. Ota, K. Ishizaki, and S. Noda, Nat. Mater. 8, 721 (2009).
[CrossRef]

Noda, S.

S. Takahashi, K. Suzuki, M. Okano, M. Imada, T. Nakamori, Y. Ota, K. Ishizaki, and S. Noda, Nat. Mater. 8, 721 (2009).
[CrossRef]

O’Brien, J.

A. Mock, L. Lu, and J. O’Brien, Phys. Rev. B 81, 155115 (2010).
[CrossRef]

Okano, M.

S. Takahashi, K. Suzuki, M. Okano, M. Imada, T. Nakamori, Y. Ota, K. Ishizaki, and S. Noda, Nat. Mater. 8, 721 (2009).
[CrossRef]

Ota, Y.

S. Takahashi, K. Suzuki, M. Okano, M. Imada, T. Nakamori, Y. Ota, K. Ishizaki, and S. Noda, Nat. Mater. 8, 721 (2009).
[CrossRef]

Patel, A. A.

A. A. Patel and H. I. Smith, J. Vac. Sci. Technol. B 25, 2662 (2007).
[CrossRef]

Sakoda, K.

K. Aoki, H. T. Miyazaki, H. Hirayama, K. Inoshita, T. Baba, K. Sakoda, N. Shinya, and Y. Aoyagi, Nat. Mater. 2, 117 (2003).
[CrossRef]

Scherer, A.

C. C. Cheng and A. Scherer, J. Vac. Sci. Technol. B 13, 2696 (1995).
[CrossRef]

Shinya, N.

K. Aoki, H. T. Miyazaki, H. Hirayama, K. Inoshita, T. Baba, K. Sakoda, N. Shinya, and Y. Aoyagi, Nat. Mater. 2, 117 (2003).
[CrossRef]

Sigalas, M.

K. Ho, C. Chan, C. Soukoulis, R. Biswas, and M. Sigalas, Solid State Commun. 89, 413 (1994).
[CrossRef]

Smith, H. I.

S. Ghadarghadr, C. P. Fucetola, L. L. Cheong, E. E. Moon, and H. I. Smith, J. Vac. Sci. Technol. B 29, 06F401 (2011).
[CrossRef]

A. A. Patel and H. I. Smith, J. Vac. Sci. Technol. B 25, 2662 (2007).
[CrossRef]

E. E. Moon, M. K. Mondol, P. N. Everett, and H. I. Smith, J. Vac. Sci. Technol. B 23, 2607 (2005).

Soukoulis, C.

K. Ho, C. Chan, C. Soukoulis, R. Biswas, and M. Sigalas, Solid State Commun. 89, 413 (1994).
[CrossRef]

Suzuki, K.

S. Takahashi, K. Suzuki, M. Okano, M. Imada, T. Nakamori, Y. Ota, K. Ishizaki, and S. Noda, Nat. Mater. 8, 721 (2009).
[CrossRef]

Takahashi, S.

S. Takahashi, K. Suzuki, M. Okano, M. Imada, T. Nakamori, Y. Ota, K. Ishizaki, and S. Noda, Nat. Mater. 8, 721 (2009).
[CrossRef]

Tang, L.

L. Tang and T. Yoshie, J. Vac. Sci. Technol. B 28, 301 (2010).
[CrossRef]

Thomas, E. L.

M. Maldovan and E. L. Thomas, Nat. Mater. 3, 593 (2004).
[CrossRef]

Toader, O.

O. Toader and S. John, Phys. Rev. E 71, 036605 (2005).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Yablonovitch, E.

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
[CrossRef]

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef]

Yoshie, T.

L. Tang and T. Yoshie, J. Vac. Sci. Technol. B 28, 301 (2010).
[CrossRef]

Appl. Phys. Lett.

K. Aoki, Appl. Phys. Lett. 95, 191910 (2009).
[CrossRef]

J. Vac. Sci. Technol. B

C. C. Cheng and A. Scherer, J. Vac. Sci. Technol. B 13, 2696 (1995).
[CrossRef]

L. Tang and T. Yoshie, J. Vac. Sci. Technol. B 28, 301 (2010).
[CrossRef]

S. Ghadarghadr, C. P. Fucetola, L. L. Cheong, E. E. Moon, and H. I. Smith, J. Vac. Sci. Technol. B 29, 06F401 (2011).
[CrossRef]

A. A. Patel and H. I. Smith, J. Vac. Sci. Technol. B 25, 2662 (2007).
[CrossRef]

E. E. Moon, M. K. Mondol, P. N. Everett, and H. I. Smith, J. Vac. Sci. Technol. B 23, 2607 (2005).

Nat. Mater.

K. Aoki, H. T. Miyazaki, H. Hirayama, K. Inoshita, T. Baba, K. Sakoda, N. Shinya, and Y. Aoyagi, Nat. Mater. 2, 117 (2003).
[CrossRef]

M. Maldovan and E. L. Thomas, Nat. Mater. 3, 593 (2004).
[CrossRef]

S. Takahashi, K. Suzuki, M. Okano, M. Imada, T. Nakamori, Y. Ota, K. Ishizaki, and S. Noda, Nat. Mater. 8, 721 (2009).
[CrossRef]

Opt. Express

Phys. Rev. B

A. Mock, L. Lu, and J. O’Brien, Phys. Rev. B 81, 155115 (2010).
[CrossRef]

Phys. Rev. E

O. Toader and S. John, Phys. Rev. E 71, 036605 (2005).
[CrossRef]

Phys. Rev. Lett.

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, Phys. Rev. Lett. 67, 2295 (1991).
[CrossRef]

E. Yablonovitch, Phys. Rev. Lett. 58, 2059 (1987).
[CrossRef]

S. John, Phys. Rev. Lett. 58, 2486 (1987).
[CrossRef]

Solid State Commun.

K. Ho, C. Chan, C. Soukoulis, R. Biswas, and M. Sigalas, Solid State Commun. 89, 413 (1994).
[CrossRef]

Other

m is an arbitrary real number. When m=2, the equation defines a circle. When m=1, it defines a diamond. When m=∞, it defines a square. This is illustrated on the right side of Fig. 1(c).

The bandstructure calculations were done using MIT Photonic-Bands package (MPB, http://ab-initio.mit.edu/mpb ) and the optimizations were done using the nonlinear optimization package (NLopt, http://ab-initio.mit.edu/nlopt ).

This degeneracy, protected by the glide spatial symmetry, ensures the continuity of the bands across the Brillouin zone boundary, while the continuities of the dispersion curves inside the Brillouin zone are protected by translational spatial symmetry.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

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Figures (3)

Fig. 1.
Fig. 1.

(a) To-scale illustration of the mesh-stack 3D PhC for five layers, where the first and fifth red membranes are periodically equivalent. The r=0.45(a/2), m=2, and c=a/2. A portion of the top red layer is peeled back to show the relative displacements of the four layers that constitute one period of the structure. The left bottom inset is a top view illustration of the in-plane offsets between the four membranes. (b) Illustration of the fct lattice with a diamond-like structure. It becomes the fcc/diamond lattice when c=a. The colors of the spheres match the membrane colors in (a). (c) Bandgap sizes with optimized edge curvatures (m) for different minimum vein widths (v). (d) Band diagram of the optimized PhC at v/(a/2)=0.1.

Fig. 2.
Fig. 2.

(a) Two consecutive membranes containing identical defects as waveguide cores. (b) Dispersion relations of the nonsymmorphic waveguide. (c) Field profiles of one of the degenerate modes at k=π/(a/2) plotted in a linear scale overlaid on the dielectric distribution in transparent gray color.

Fig. 3.
Fig. 3.

Mesh-stack bandgap sizes and central frequencies as a function of the alignment offsets of the membranes. σ is the standard deviation of the random offsets along both in-plane directions.

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