Abstract

We show that the propagation effects of optical beams in three-dimensional photonic crystal structures can be modeled using a direction-dependent effective diffractive index model. The parameters of the model (i.e., the effective diffractive indices) can be calculated using the curvatures of the band structure of the photonic crystal at the operation point. After finding these indices, the wave propagation inside the photonic crystal can be analyzed using simple geometrical optics formulas. We show that the model has good accuracy for most practical applications of photonic crystals. As an example, the application of the model for diffraction compensation in a tetragonal woodpile photonic crystal is demonstrated.

© 2008 Optical Society of America

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  1. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987).
    [CrossRef] [PubMed]
  2. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489 (1987).
    [CrossRef] [PubMed]
  3. J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).
  4. X. Ao and S. He, “Three-dimensional photonic crystal of negative refraction achieved by interference lithography,” Opt. Lett. 29, 2542-2544 (2004).
    [CrossRef] [PubMed]
  5. Z. Lu, S. Shi, C. A. Schuetz, J. A. Murakowski, and D. W. Prather, “Three-dimensional photonic crystal flat lens by full 3D negative refraction,” Opt. Express 13, 5592-5599 (2005).
    [CrossRef] [PubMed]
  6. T. Prasad, V. Colvin, and D. Mittleman, “Superprism phenomenon in three-dimensional macroporous polymer photonic crystals,” Phys. Rev. B 67, 165103 (2003).
    [CrossRef]
  7. S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251-253 (1998).
    [CrossRef]
  8. M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater. 3, 444-447 (2004).
    [CrossRef] [PubMed]
  9. R. Guo, Z. Li, Z. Jiang, D. Yuan, W. Huang, and A. Xia, “Log-pile photonic crystal fabricated by two-photon photopolymerization,” J. Opt. A, Pure Appl. Opt. 7, 396-399 (2005).
    [CrossRef]
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    [CrossRef] [PubMed]
  11. Y. C. Zhong, S. A. Zhu, H. M. Su, H. Z. Wang, J. M. Chen, Z. H. Zeng, and Y. L. Chen, “Photonic crystal with diamondlike structure fabricated by holographic lithography,” Appl. Phys. Lett. 87, 061103 (2005).
    [CrossRef]
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    [CrossRef]
  13. J. H. Moon, S. Yang, and S.-M. Yang, “Photonic band-gap structures of core-shell simple cubic crystals from holographic lithography,” Appl. Phys. Lett. 88, 121101 (2006).
    [CrossRef]
  14. J. Chen, W. Jiang, X. Chen, L. Wang, S. Zhang, and R. T. Chen, “Holographic three-dimensional polymeric photonic crystals operating in the 1550nm window,” Appl. Phys. Lett. 90, 093102 (2007).
    [CrossRef]
  15. R. Zengerle and P. C. Hoang, “Photonic crystal structures for potential dispersion management in optical telecommunication systems,” Proc. SPIE 5595, 78-91 (2004).
    [CrossRef]
  16. B. Momeni and A. Adibi, “An approximate effective index model for efficient analysis and control of beam propagation effects in photonic crystals,” J. Lightwave Technol. 23, 1522-1532 (2005).
    [CrossRef]
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    [CrossRef]
  18. B. Momeni and A. Adibi, “Preconditioned superprism-based photonic crystal demultiplexers: analysis and design,” Appl. Opt. 45, 8466-8476 (2006).
    [CrossRef] [PubMed]
  19. J. Shin and S. Fan, “Conditions for self-collimation in three-dimensional photonic crystals,” Opt. Lett. 30, 2397-2399 (2005).
    [CrossRef] [PubMed]
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    [CrossRef]
  21. M. Badieirostami, B. Momeni, and A. Adibi, are preparing a paper to be called “Polarization state for modes of low-contrast three-dimensional photonic crystal structures.”
  22. J. J. Stoker, Differential Geometry (Wiley, 1969), Chap. 4.
  23. B. Momeni, M. Badieirostami, and A. Adibi, “Accurate and efficient techniques for the analysis of reflection at the interfaces of three-dimensional photonic crystals,” J. Opt. Soc. Am. B 24, 2957-2963 (2007).
    [CrossRef]

2007 (2)

J. Chen, W. Jiang, X. Chen, L. Wang, S. Zhang, and R. T. Chen, “Holographic three-dimensional polymeric photonic crystals operating in the 1550nm window,” Appl. Phys. Lett. 90, 093102 (2007).
[CrossRef]

B. Momeni, M. Badieirostami, and A. Adibi, “Accurate and efficient techniques for the analysis of reflection at the interfaces of three-dimensional photonic crystals,” J. Opt. Soc. Am. B 24, 2957-2963 (2007).
[CrossRef]

2006 (4)

M. Deubel, M. Wegener, S. Linden, G. von Freymann, and S. John, “3D-2D-3D photonic crystal heterostructures fabricated by direct laser writing,” Opt. Lett. 31, 805-807 (2006).
[CrossRef] [PubMed]

B. Momeni and A. Adibi, “Preconditioned superprism-based photonic crystal demultiplexers: analysis and design,” Appl. Opt. 45, 8466-8476 (2006).
[CrossRef] [PubMed]

J. H. Moon, J. Ford, and S. Yang, “Fabricating three-dimensional polymeric photonic structures by multi-beam interference lithography,” Polym. Adv. Technol. 17, 83-93 (2006).
[CrossRef]

J. H. Moon, S. Yang, and S.-M. Yang, “Photonic band-gap structures of core-shell simple cubic crystals from holographic lithography,” Appl. Phys. Lett. 88, 121101 (2006).
[CrossRef]

2005 (5)

R. Guo, Z. Li, Z. Jiang, D. Yuan, W. Huang, and A. Xia, “Log-pile photonic crystal fabricated by two-photon photopolymerization,” J. Opt. A, Pure Appl. Opt. 7, 396-399 (2005).
[CrossRef]

Y. C. Zhong, S. A. Zhu, H. M. Su, H. Z. Wang, J. M. Chen, Z. H. Zeng, and Y. L. Chen, “Photonic crystal with diamondlike structure fabricated by holographic lithography,” Appl. Phys. Lett. 87, 061103 (2005).
[CrossRef]

B. Momeni and A. Adibi, “An approximate effective index model for efficient analysis and control of beam propagation effects in photonic crystals,” J. Lightwave Technol. 23, 1522-1532 (2005).
[CrossRef]

J. Shin and S. Fan, “Conditions for self-collimation in three-dimensional photonic crystals,” Opt. Lett. 30, 2397-2399 (2005).
[CrossRef] [PubMed]

Z. Lu, S. Shi, C. A. Schuetz, J. A. Murakowski, and D. W. Prather, “Three-dimensional photonic crystal flat lens by full 3D negative refraction,” Opt. Express 13, 5592-5599 (2005).
[CrossRef] [PubMed]

2004 (3)

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater. 3, 444-447 (2004).
[CrossRef] [PubMed]

X. Ao and S. He, “Three-dimensional photonic crystal of negative refraction achieved by interference lithography,” Opt. Lett. 29, 2542-2544 (2004).
[CrossRef] [PubMed]

R. Zengerle and P. C. Hoang, “Photonic crystal structures for potential dispersion management in optical telecommunication systems,” Proc. SPIE 5595, 78-91 (2004).
[CrossRef]

2003 (3)

B. Momeni and A. Adibi, “Optimization of photonic crystal demultiplexers based on superprism effect,” Appl. Phys. B 77, 556-560 (2003).
[CrossRef]

J. Mizuguchi, Y. Tanaka, S. Tamura, and M. Notomi, “Focusing of light in a three-dimensional cubic photonic crystal,” Phys. Rev. B 67, 075109 (2003).
[CrossRef]

T. Prasad, V. Colvin, and D. Mittleman, “Superprism phenomenon in three-dimensional macroporous polymer photonic crystals,” Phys. Rev. B 67, 165103 (2003).
[CrossRef]

1998 (1)

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251-253 (1998).
[CrossRef]

1987 (2)

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

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

Adibi, A.

Ao, X.

Badieirostami, M.

Biswas, R.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251-253 (1998).
[CrossRef]

Bur, J.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251-253 (1998).
[CrossRef]

Busch, K.

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater. 3, 444-447 (2004).
[CrossRef] [PubMed]

Chen, J.

J. Chen, W. Jiang, X. Chen, L. Wang, S. Zhang, and R. T. Chen, “Holographic three-dimensional polymeric photonic crystals operating in the 1550nm window,” Appl. Phys. Lett. 90, 093102 (2007).
[CrossRef]

Chen, J. M.

Y. C. Zhong, S. A. Zhu, H. M. Su, H. Z. Wang, J. M. Chen, Z. H. Zeng, and Y. L. Chen, “Photonic crystal with diamondlike structure fabricated by holographic lithography,” Appl. Phys. Lett. 87, 061103 (2005).
[CrossRef]

Chen, R. T.

J. Chen, W. Jiang, X. Chen, L. Wang, S. Zhang, and R. T. Chen, “Holographic three-dimensional polymeric photonic crystals operating in the 1550nm window,” Appl. Phys. Lett. 90, 093102 (2007).
[CrossRef]

Chen, X.

J. Chen, W. Jiang, X. Chen, L. Wang, S. Zhang, and R. T. Chen, “Holographic three-dimensional polymeric photonic crystals operating in the 1550nm window,” Appl. Phys. Lett. 90, 093102 (2007).
[CrossRef]

Chen, Y. L.

Y. C. Zhong, S. A. Zhu, H. M. Su, H. Z. Wang, J. M. Chen, Z. H. Zeng, and Y. L. Chen, “Photonic crystal with diamondlike structure fabricated by holographic lithography,” Appl. Phys. Lett. 87, 061103 (2005).
[CrossRef]

Colvin, V.

T. Prasad, V. Colvin, and D. Mittleman, “Superprism phenomenon in three-dimensional macroporous polymer photonic crystals,” Phys. Rev. B 67, 165103 (2003).
[CrossRef]

Deubel, M.

M. Deubel, M. Wegener, S. Linden, G. von Freymann, and S. John, “3D-2D-3D photonic crystal heterostructures fabricated by direct laser writing,” Opt. Lett. 31, 805-807 (2006).
[CrossRef] [PubMed]

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater. 3, 444-447 (2004).
[CrossRef] [PubMed]

Fan, S.

Fleming, J. G.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251-253 (1998).
[CrossRef]

Ford, J.

J. H. Moon, J. Ford, and S. Yang, “Fabricating three-dimensional polymeric photonic structures by multi-beam interference lithography,” Polym. Adv. Technol. 17, 83-93 (2006).
[CrossRef]

Guo, R.

R. Guo, Z. Li, Z. Jiang, D. Yuan, W. Huang, and A. Xia, “Log-pile photonic crystal fabricated by two-photon photopolymerization,” J. Opt. A, Pure Appl. Opt. 7, 396-399 (2005).
[CrossRef]

He, S.

Hetherington, D. L.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251-253 (1998).
[CrossRef]

Ho, K. M.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251-253 (1998).
[CrossRef]

Hoang, P. C.

R. Zengerle and P. C. Hoang, “Photonic crystal structures for potential dispersion management in optical telecommunication systems,” Proc. SPIE 5595, 78-91 (2004).
[CrossRef]

Huang, W.

R. Guo, Z. Li, Z. Jiang, D. Yuan, W. Huang, and A. Xia, “Log-pile photonic crystal fabricated by two-photon photopolymerization,” J. Opt. A, Pure Appl. Opt. 7, 396-399 (2005).
[CrossRef]

Jiang, W.

J. Chen, W. Jiang, X. Chen, L. Wang, S. Zhang, and R. T. Chen, “Holographic three-dimensional polymeric photonic crystals operating in the 1550nm window,” Appl. Phys. Lett. 90, 093102 (2007).
[CrossRef]

Jiang, Z.

R. Guo, Z. Li, Z. Jiang, D. Yuan, W. Huang, and A. Xia, “Log-pile photonic crystal fabricated by two-photon photopolymerization,” J. Opt. A, Pure Appl. Opt. 7, 396-399 (2005).
[CrossRef]

Joannopoulos, J.

J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

John, S.

Kurtz, S. R.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251-253 (1998).
[CrossRef]

Li, Z.

R. Guo, Z. Li, Z. Jiang, D. Yuan, W. Huang, and A. Xia, “Log-pile photonic crystal fabricated by two-photon photopolymerization,” J. Opt. A, Pure Appl. Opt. 7, 396-399 (2005).
[CrossRef]

Lin, S. Y.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251-253 (1998).
[CrossRef]

Linden, S.

Lu, Z.

Meade, R.

J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Mittleman, D.

T. Prasad, V. Colvin, and D. Mittleman, “Superprism phenomenon in three-dimensional macroporous polymer photonic crystals,” Phys. Rev. B 67, 165103 (2003).
[CrossRef]

Mizuguchi, J.

J. Mizuguchi, Y. Tanaka, S. Tamura, and M. Notomi, “Focusing of light in a three-dimensional cubic photonic crystal,” Phys. Rev. B 67, 075109 (2003).
[CrossRef]

Momeni, B.

Moon, J. H.

J. H. Moon, J. Ford, and S. Yang, “Fabricating three-dimensional polymeric photonic structures by multi-beam interference lithography,” Polym. Adv. Technol. 17, 83-93 (2006).
[CrossRef]

J. H. Moon, S. Yang, and S.-M. Yang, “Photonic band-gap structures of core-shell simple cubic crystals from holographic lithography,” Appl. Phys. Lett. 88, 121101 (2006).
[CrossRef]

Murakowski, J. A.

Notomi, M.

J. Mizuguchi, Y. Tanaka, S. Tamura, and M. Notomi, “Focusing of light in a three-dimensional cubic photonic crystal,” Phys. Rev. B 67, 075109 (2003).
[CrossRef]

Pereira, S.

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater. 3, 444-447 (2004).
[CrossRef] [PubMed]

Prasad, T.

T. Prasad, V. Colvin, and D. Mittleman, “Superprism phenomenon in three-dimensional macroporous polymer photonic crystals,” Phys. Rev. B 67, 165103 (2003).
[CrossRef]

Prather, D. W.

Schuetz, C. A.

Shi, S.

Shin, J.

Sigalas, M. M.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251-253 (1998).
[CrossRef]

Smith, B. K.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251-253 (1998).
[CrossRef]

Soukoulis, C. M.

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater. 3, 444-447 (2004).
[CrossRef] [PubMed]

Stoker, J. J.

J. J. Stoker, Differential Geometry (Wiley, 1969), Chap. 4.

Su, H. M.

Y. C. Zhong, S. A. Zhu, H. M. Su, H. Z. Wang, J. M. Chen, Z. H. Zeng, and Y. L. Chen, “Photonic crystal with diamondlike structure fabricated by holographic lithography,” Appl. Phys. Lett. 87, 061103 (2005).
[CrossRef]

Tamura, S.

J. Mizuguchi, Y. Tanaka, S. Tamura, and M. Notomi, “Focusing of light in a three-dimensional cubic photonic crystal,” Phys. Rev. B 67, 075109 (2003).
[CrossRef]

Tanaka, Y.

J. Mizuguchi, Y. Tanaka, S. Tamura, and M. Notomi, “Focusing of light in a three-dimensional cubic photonic crystal,” Phys. Rev. B 67, 075109 (2003).
[CrossRef]

von Freymann, G.

M. Deubel, M. Wegener, S. Linden, G. von Freymann, and S. John, “3D-2D-3D photonic crystal heterostructures fabricated by direct laser writing,” Opt. Lett. 31, 805-807 (2006).
[CrossRef] [PubMed]

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater. 3, 444-447 (2004).
[CrossRef] [PubMed]

Wang, H. Z.

Y. C. Zhong, S. A. Zhu, H. M. Su, H. Z. Wang, J. M. Chen, Z. H. Zeng, and Y. L. Chen, “Photonic crystal with diamondlike structure fabricated by holographic lithography,” Appl. Phys. Lett. 87, 061103 (2005).
[CrossRef]

Wang, L.

J. Chen, W. Jiang, X. Chen, L. Wang, S. Zhang, and R. T. Chen, “Holographic three-dimensional polymeric photonic crystals operating in the 1550nm window,” Appl. Phys. Lett. 90, 093102 (2007).
[CrossRef]

Wegener, M.

M. Deubel, M. Wegener, S. Linden, G. von Freymann, and S. John, “3D-2D-3D photonic crystal heterostructures fabricated by direct laser writing,” Opt. Lett. 31, 805-807 (2006).
[CrossRef] [PubMed]

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater. 3, 444-447 (2004).
[CrossRef] [PubMed]

Winn, J.

J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

Xia, A.

R. Guo, Z. Li, Z. Jiang, D. Yuan, W. Huang, and A. Xia, “Log-pile photonic crystal fabricated by two-photon photopolymerization,” J. Opt. A, Pure Appl. Opt. 7, 396-399 (2005).
[CrossRef]

Yablonovitch, E.

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

Yang, S.

J. H. Moon, S. Yang, and S.-M. Yang, “Photonic band-gap structures of core-shell simple cubic crystals from holographic lithography,” Appl. Phys. Lett. 88, 121101 (2006).
[CrossRef]

J. H. Moon, J. Ford, and S. Yang, “Fabricating three-dimensional polymeric photonic structures by multi-beam interference lithography,” Polym. Adv. Technol. 17, 83-93 (2006).
[CrossRef]

Yang, S.-M.

J. H. Moon, S. Yang, and S.-M. Yang, “Photonic band-gap structures of core-shell simple cubic crystals from holographic lithography,” Appl. Phys. Lett. 88, 121101 (2006).
[CrossRef]

Yuan, D.

R. Guo, Z. Li, Z. Jiang, D. Yuan, W. Huang, and A. Xia, “Log-pile photonic crystal fabricated by two-photon photopolymerization,” J. Opt. A, Pure Appl. Opt. 7, 396-399 (2005).
[CrossRef]

Zeng, Z. H.

Y. C. Zhong, S. A. Zhu, H. M. Su, H. Z. Wang, J. M. Chen, Z. H. Zeng, and Y. L. Chen, “Photonic crystal with diamondlike structure fabricated by holographic lithography,” Appl. Phys. Lett. 87, 061103 (2005).
[CrossRef]

Zengerle, R.

R. Zengerle and P. C. Hoang, “Photonic crystal structures for potential dispersion management in optical telecommunication systems,” Proc. SPIE 5595, 78-91 (2004).
[CrossRef]

Zhang, S.

J. Chen, W. Jiang, X. Chen, L. Wang, S. Zhang, and R. T. Chen, “Holographic three-dimensional polymeric photonic crystals operating in the 1550nm window,” Appl. Phys. Lett. 90, 093102 (2007).
[CrossRef]

Zhong, Y. C.

Y. C. Zhong, S. A. Zhu, H. M. Su, H. Z. Wang, J. M. Chen, Z. H. Zeng, and Y. L. Chen, “Photonic crystal with diamondlike structure fabricated by holographic lithography,” Appl. Phys. Lett. 87, 061103 (2005).
[CrossRef]

Zhu, S. A.

Y. C. Zhong, S. A. Zhu, H. M. Su, H. Z. Wang, J. M. Chen, Z. H. Zeng, and Y. L. Chen, “Photonic crystal with diamondlike structure fabricated by holographic lithography,” Appl. Phys. Lett. 87, 061103 (2005).
[CrossRef]

Zubrzycki, W.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251-253 (1998).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

B. Momeni and A. Adibi, “Optimization of photonic crystal demultiplexers based on superprism effect,” Appl. Phys. B 77, 556-560 (2003).
[CrossRef]

Appl. Phys. Lett. (3)

J. H. Moon, S. Yang, and S.-M. Yang, “Photonic band-gap structures of core-shell simple cubic crystals from holographic lithography,” Appl. Phys. Lett. 88, 121101 (2006).
[CrossRef]

J. Chen, W. Jiang, X. Chen, L. Wang, S. Zhang, and R. T. Chen, “Holographic three-dimensional polymeric photonic crystals operating in the 1550nm window,” Appl. Phys. Lett. 90, 093102 (2007).
[CrossRef]

Y. C. Zhong, S. A. Zhu, H. M. Su, H. Z. Wang, J. M. Chen, Z. H. Zeng, and Y. L. Chen, “Photonic crystal with diamondlike structure fabricated by holographic lithography,” Appl. Phys. Lett. 87, 061103 (2005).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. A, Pure Appl. Opt. (1)

R. Guo, Z. Li, Z. Jiang, D. Yuan, W. Huang, and A. Xia, “Log-pile photonic crystal fabricated by two-photon photopolymerization,” J. Opt. A, Pure Appl. Opt. 7, 396-399 (2005).
[CrossRef]

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

Nat. Mater. (1)

M. Deubel, G. von Freymann, M. Wegener, S. Pereira, K. Busch, and C. M. Soukoulis, “Direct laser writing of three-dimensional photonic-crystal templates for telecommunications,” Nat. Mater. 3, 444-447 (2004).
[CrossRef] [PubMed]

Nature (1)

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394, 251-253 (1998).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Phys. Rev. B (2)

J. Mizuguchi, Y. Tanaka, S. Tamura, and M. Notomi, “Focusing of light in a three-dimensional cubic photonic crystal,” Phys. Rev. B 67, 075109 (2003).
[CrossRef]

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[CrossRef]

Phys. Rev. Lett. (2)

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Other (3)

M. Badieirostami, B. Momeni, and A. Adibi, are preparing a paper to be called “Polarization state for modes of low-contrast three-dimensional photonic crystal structures.”

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J. Joannopoulos, R. Meade, and J. Winn, Photonic Crystals: Molding the Flow of Light (Princeton U. Press, 1995).

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Figures (6)

Fig. 1
Fig. 1

Portion of an isofrequency surface (at normalized frequency ω n 0 ) of a general 3D PC in the k space is shown. The directions tangent to the surface (i.e., ξ and η) and the direction normal to the surface ( ζ ) are defined in the figure.

Fig. 2
Fig. 2

(a) Schematic of the 3D tetragonal woodpile PC considered throughout this paper is shown. Lattice constants and filling factors in different directions of this lattice are marked in this figure. (b) The general direction of the incident beam is shown, with α being the angle between the incident wavevector and the normal to the interface ( z ) , and ϕ being the angle between the plane of incidence and the xz plane.

Fig. 3
Fig. 3

Calculated cross sections of an optical beam propagating through a tetragonal woodpile PC structure (with f x = f y = 0.3 , f z = 0.5 , and a = a x = a y = a z 2.4 ) are shown at different propagation lengths using (a) the direct mode-matching (brute-force approach) and (b) the ETF approximation. The three snapshots show the calculated E y field at z = a , 200 a , and 400 a , respectively. The beam has a normalized frequency of a λ = 0.45 and a symmetric beam waist of 41.2 λ , and it is incident upon the PC from a homogeneous material with relative permittivity 2.5 at α = 38 ° and ϕ = 0 ° as shown in Fig. 2.

Fig. 4
Fig. 4

Comparison of the beam widths along the x and y directions for an optical beam propagating inside a 3D PC (same parameters as defined in Fig. 3). The results obtained using the ETF (shown by markers) are in good agreement with the expected beam widths from a diffractive index model (shown by solid curves).

Fig. 5
Fig. 5

(a) Isofrequency surface of a tetragonal woodpile PC structure (with f x = f y = 0.3 , f z = 0.5 , ε r = 2.5 , and a = a x = a y = a z 2.4 ) in the 3D k space at the normalized frequency of a λ = 0.57 is shown. Only the surface corresponding to the excitation polarization (i.e., E y ) is retained. The excitation is a Gaussian beam incident from the substrate region ( ε r = 2.5 ) at an angle of α = 21.75 ° and ϕ = 0 ° , with a symmetric beam waist of 2 w 0 = 41.2 λ , and is originally broadened to a beam width of 77 λ . Cross sections of the beam inside the PC structure is shown at (b) z = a (i.e., upon entrance to the PC region) and (c) z = 500 a . (d) The evolution of the width of the beam during propagation through the PC structure is calculated using the ETF method and our simple diffractive index model, showing good agreement. Using Gaussian beam propagation formulas and by fitting the parameters into the calculated ETF beam widths, the diffractive indices are estimated to be n d x = 0.14 and n d y = 0.87 , which are in good agreement with those calculated in our simple model.

Fig. 6
Fig. 6

Effect of higher-order diffraction on the profile of a beam inside a tetragonal woodpile PC structure is shown, for the same parameters as those described in Fig. 5, but with an incident beam waist of 2 w 0 = 20.6 λ . The cross section of the incident beam at the input inside the PC is shown in (a). The beam profile at z = 650 a after propagation through the PC with negative diffractive index in the x direction is calculated and its intensity is plotted using (b) the effective index model and (c) the ETF approach. The appearance of sidelobes in the output beam profile in (c) is a result of higher-order diffraction effects that are neglected in the simple effective diffractive index calculations used to find (b).

Equations (26)

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p 1 ( x , y ) = 1 4 π 2 A ( k x , k y ) U k ¯ ( x , y , z 1 ) exp ( j k x x j k y y ) exp ( j k z z 1 ) d k x d k y ,
U k ¯ ( x , y , z ) = m n l U ̃ m n l ( k x , k y , k z ) exp [ j ( m K x x + n K y y + l K z z ) ] ,
p 1 ( x , y ) = 1 4 π 2 A ( k x , k y ) ( m n l U ̃ m n l ( k x , k y ) exp [ j ( k x + m K x ) x ] exp [ j ( k y + n K y ) y ] exp [ j ( k z + l K z ) z 1 ] ) d k x d k y ,
P 1 ( k x , k y ) = p 1 ( x , y ) exp ( j k x x ) exp ( j k y y ) d x d y = m n l A ( k x , k y ) E ̃ m n l ( k x , k y ) exp { j [ k z ( k x , k y ) + l K z ] z 1 } k x = k x m K x k y = k y n K y ,
P 1 ( k x , k y ) = m n l A ( k x m K x , k y n K y ) E ̃ m n l ( k x m K x , k y n K y ) exp { j [ k z ( k x m K x , k y n K y ) + l K z ] z 1 } .
P ¯ 1 ( k x , k y ) = A ( k x + k x 0 , k y + k y 0 ) exp [ j k z ( k x + k x 0 , k y + k y 0 ) z 1 ] × ( l E ̃ 00 l ( k x + k x 0 , k y + k y 0 ) exp ( j l K z z 1 ) ) .
P ¯ 2 ( k x , k y ) = A ( k x + k x 0 , k y + k y 0 ) exp [ j k z ( k x + k x 0 , k y + k y 0 ) z 2 ] × ( l E ̃ 00 l ( k x + k x 0 , k y + k y 0 ) exp ( j l K z z 2 ) ) .
P ¯ 2 ( k x , k y ) = P ¯ 1 ( k x , k y ) exp [ j k z ( k x + k x 0 , k y + k y 0 ) ( z 2 z 1 ) ] ,
H ( k x , k y ) = P ¯ 2 ( k x , k y ) P ¯ 1 ( k x , k y ) = exp [ j ( z 2 z 1 ) k z ] ,
H ( k ξ , k η ) = P ¯ 2 ( k ξ , k η ) P ¯ 1 ( k ξ , k η ) = exp [ j ( ζ 2 ζ 1 ) k ζ ( k ξ , k η ) ] ,
k z = k z 0 + a 2 ( k x k x 0 ) + a 3 ( k y k y 0 ) + a 4 ( k x k x 0 ) 2 + a 5 ( k x k x 0 ) ( k y k y 0 ) + a 6 ( k y k y 0 ) 2 ,
W = 1 + a 2 2 + a 3 2 .
E = ( 1 + a 2 2 ) W ,
F = a 2 a 3 W ,
G = ( 1 + a 3 2 ) W ,
L = 2 a 4 ,
M = a 5 ,
N = 2 a 6 .
K = L N M 2 E G F 2 ,
H = 1 2 E N 2 F M + G L E G F 2 .
κ 1 = H + H 2 K ,
κ 2 = H H 2 K .
( L κ i E M κ i F M κ i F N κ i G ) v i = 0 i = 1 , 2 ,
n = ( a 2 , a 3 , 1 ) .
n d ξ = 1 k 0 κ 1 ,
n d η = 1 k 0 κ 2 .

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