Abstract

Self-collimation in photonic crystals has received a lot of attention in the literature, partly due to recent interest in silicon photonics, yet no performance metrics have been proposed. This paper proposes a figure of merit (FOM) for self-collimation and outlines a methodical approach for calculating it. Performance metrics include bandwidth, angular acceptance, strength, and an overall FOM. Two key contributions of this work include the performance metrics and identifying that the optimum frequency for self-collimation is not at the inflection point. The FOM is used to optimize a planar photonic crystal composed of a square array of cylinders. Conclusions are drawn about how the refractive indices and fill fraction of the lattice impact each of the performance metrics. The optimization is demonstrated by simulating two spatially variant self-collimating photonic crystals, where one has a high FOM and the other has a low FOM. This work gives optical designers tremendous insight into how to design and optimize robust self-collimating photonic crystals, which promises many applications in silicon photonics and integrated optics.

© 2013 Optical Society of America

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  1. S. Feng, Z.-Y. Li, Z.-F. Feng, K. Ren, B.-Y. Cheng, and D.-Z. Zhang, “Focusing properties of a rectangular-rod photonic-crystal slab,” J. Appl. Phys. 98, 063102 (2005).
    [CrossRef]
  2. R. Iliew, C. Etrich, and F. Lederer, “Self-collimation of light in three-dimensional photonic crystals,” Opt. Express 13, 7076–7085 (2005).
    [CrossRef]
  3. R. Iliew, C. Etrich, U. Peschel, F. Lederer, M. Augustin, H.-J. Fuchs, D. Schelle, E.-B. Kley, S. Nolte, and A. Tunnermann, “Diffractionless propagation of light in a low-index photonic-crystal film,” Appl. Phys. Lett. 85, 5854–5856 (2004).
    [CrossRef]
  4. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
    [CrossRef]
  5. Z. Lu, S. Shi, J. A. Murakowski, G. J. Schneider, C. A. Schuetz, and D. W. Prather, “Experimental demonstration of self-collimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett. 96, 173902 (2006).
    [CrossRef]
  6. J. Shin and S. Fan, “Conditions for self-collimation in three-dimensional photonic crystals,” Opt. Lett. 30, 2397 (2005).
    [CrossRef]
  7. J. Witzens, M. Loncar, and A. Scherer, “Self-collimation in planar photonic crystals,” IEEE J. Sel. Top. Quantum Electron. 8, 1246–1257 (2002).
    [CrossRef]
  8. T.-T. Kim, S.-G. Lee, H. Y. Park, J.-E. Kim, and C.-S. Kee, “Asymmetric Mach-Zehnder filter based on self-collimation phenomenon in two-dimensional photonic crystals,” Opt. Express 18, 5384–5389 (2010).
    [CrossRef]
  9. A. F. Matthews, S. K. Morrison, and Y. S. Kivshar, “Self-collimation and beam splitting in low-index photonic crystals,” Opt. Commun. 279, 313–319 (2007).
    [CrossRef]
  10. D. W. Prather, S. Shi, D. M. Pustai, C. Chen, S. Venkataraman, A. Sharkawy, G. Schneider, and J. Murakowski, “Dispersion-based optical routing in photonic crystals,” Opt. Lett. 29, 50–52 (2004).
    [CrossRef]
  11. Y. Zhang, Y. Zhang, and B. Li, “Optical switches and logic gates based on self-collimated beams in two-dimensional photonic crystals,” Opt. Express 15, 9287–9292 (2007).
    [CrossRef]
  12. D. Zhao, C. Zhou, Q. Gong, and X. Jiang, “Lasing cavities and ultra-fast switch based on self-collimation of photonic crystal,” J. Phys. D 41, 115108 (2008).
    [CrossRef]
  13. X. Chen, Z. Qiang, D. Zhao, H. Li, Y. Qiu, W. Yang, and W. Zhou, “Polarization-independent drop filters based on photonic crystal self-collimation ring resonators,” Opt. Express 17, 19808–19813 (2009).
    [CrossRef]
  14. Y. Wang, H. Wang, Q. Xue, and W. Zheng, “Photonic crystal self-collimation sensor,” Opt. Express 20, 12111–12118 (2012).
    [CrossRef]
  15. D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. Miao, and R. Martin, “Self-collimation in photonic crystal structures: a new paradigm for applications and device development,” J. Phys. D 40, 2635 (2007).
    [CrossRef]
  16. Z. Fang and C. Z. Zhao, “Recent progress in silicon photonics: a review,” ISRN Opt. 2012, 1–27 (2012).
  17. R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 1678–1687 (2006).
    [CrossRef]
  18. R. C. Rumpf and J. Pazos, “Synthesis of spatially variant lattices,” Opt. Express 20, 15263–15274 (2012).
    [CrossRef]
  19. M. I. Hussein, “Reduced Bloch mode expansion for periodic media band structure calculations,” Proc. Roy. Soc. Lond. Ser. A 465, 2825–2848 (2009).
    [CrossRef]
  20. S. Guo and S. Albin, “Simple plane wave implementation for photonic crystal calculations,” Opt. Express 11, 167–175 (2003).
    [CrossRef]
  21. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8, 173–190 (2001).
    [CrossRef]
  22. M. J. Lax, Symmetry Principles in Solid State and Molecular Physics (Wiley, 1974), p. 449.
  23. H. Anton, Elementary Linear Algebra (Wiley, 2010), pp. 303–305.
  24. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999), pp. 673–690.
  25. S. C. Chapra and R. P. Canale, Numerical Methods for Engineers, 6th ed. (McGraw-Hill, 2010), Vol. 2, pp. 852–887.

2012

2010

2009

X. Chen, Z. Qiang, D. Zhao, H. Li, Y. Qiu, W. Yang, and W. Zhou, “Polarization-independent drop filters based on photonic crystal self-collimation ring resonators,” Opt. Express 17, 19808–19813 (2009).
[CrossRef]

M. I. Hussein, “Reduced Bloch mode expansion for periodic media band structure calculations,” Proc. Roy. Soc. Lond. Ser. A 465, 2825–2848 (2009).
[CrossRef]

2008

D. Zhao, C. Zhou, Q. Gong, and X. Jiang, “Lasing cavities and ultra-fast switch based on self-collimation of photonic crystal,” J. Phys. D 41, 115108 (2008).
[CrossRef]

2007

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. Miao, and R. Martin, “Self-collimation in photonic crystal structures: a new paradigm for applications and device development,” J. Phys. D 40, 2635 (2007).
[CrossRef]

A. F. Matthews, S. K. Morrison, and Y. S. Kivshar, “Self-collimation and beam splitting in low-index photonic crystals,” Opt. Commun. 279, 313–319 (2007).
[CrossRef]

Y. Zhang, Y. Zhang, and B. Li, “Optical switches and logic gates based on self-collimated beams in two-dimensional photonic crystals,” Opt. Express 15, 9287–9292 (2007).
[CrossRef]

2006

R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 1678–1687 (2006).
[CrossRef]

Z. Lu, S. Shi, J. A. Murakowski, G. J. Schneider, C. A. Schuetz, and D. W. Prather, “Experimental demonstration of self-collimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett. 96, 173902 (2006).
[CrossRef]

2005

2004

D. W. Prather, S. Shi, D. M. Pustai, C. Chen, S. Venkataraman, A. Sharkawy, G. Schneider, and J. Murakowski, “Dispersion-based optical routing in photonic crystals,” Opt. Lett. 29, 50–52 (2004).
[CrossRef]

R. Iliew, C. Etrich, U. Peschel, F. Lederer, M. Augustin, H.-J. Fuchs, D. Schelle, E.-B. Kley, S. Nolte, and A. Tunnermann, “Diffractionless propagation of light in a low-index photonic-crystal film,” Appl. Phys. Lett. 85, 5854–5856 (2004).
[CrossRef]

2003

2002

J. Witzens, M. Loncar, and A. Scherer, “Self-collimation in planar photonic crystals,” IEEE J. Sel. Top. Quantum Electron. 8, 1246–1257 (2002).
[CrossRef]

2001

1999

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[CrossRef]

Albin, S.

Anton, H.

H. Anton, Elementary Linear Algebra (Wiley, 2010), pp. 303–305.

Augustin, M.

R. Iliew, C. Etrich, U. Peschel, F. Lederer, M. Augustin, H.-J. Fuchs, D. Schelle, E.-B. Kley, S. Nolte, and A. Tunnermann, “Diffractionless propagation of light in a low-index photonic-crystal film,” Appl. Phys. Lett. 85, 5854–5856 (2004).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999), pp. 673–690.

Canale, R. P.

S. C. Chapra and R. P. Canale, Numerical Methods for Engineers, 6th ed. (McGraw-Hill, 2010), Vol. 2, pp. 852–887.

Chapra, S. C.

S. C. Chapra and R. P. Canale, Numerical Methods for Engineers, 6th ed. (McGraw-Hill, 2010), Vol. 2, pp. 852–887.

Chen, C.

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. Miao, and R. Martin, “Self-collimation in photonic crystal structures: a new paradigm for applications and device development,” J. Phys. D 40, 2635 (2007).
[CrossRef]

D. W. Prather, S. Shi, D. M. Pustai, C. Chen, S. Venkataraman, A. Sharkawy, G. Schneider, and J. Murakowski, “Dispersion-based optical routing in photonic crystals,” Opt. Lett. 29, 50–52 (2004).
[CrossRef]

Chen, X.

Cheng, B.-Y.

S. Feng, Z.-Y. Li, Z.-F. Feng, K. Ren, B.-Y. Cheng, and D.-Z. Zhang, “Focusing properties of a rectangular-rod photonic-crystal slab,” J. Appl. Phys. 98, 063102 (2005).
[CrossRef]

Etrich, C.

R. Iliew, C. Etrich, and F. Lederer, “Self-collimation of light in three-dimensional photonic crystals,” Opt. Express 13, 7076–7085 (2005).
[CrossRef]

R. Iliew, C. Etrich, U. Peschel, F. Lederer, M. Augustin, H.-J. Fuchs, D. Schelle, E.-B. Kley, S. Nolte, and A. Tunnermann, “Diffractionless propagation of light in a low-index photonic-crystal film,” Appl. Phys. Lett. 85, 5854–5856 (2004).
[CrossRef]

Fan, S.

Fang, Z.

Z. Fang and C. Z. Zhao, “Recent progress in silicon photonics: a review,” ISRN Opt. 2012, 1–27 (2012).

Feng, S.

S. Feng, Z.-Y. Li, Z.-F. Feng, K. Ren, B.-Y. Cheng, and D.-Z. Zhang, “Focusing properties of a rectangular-rod photonic-crystal slab,” J. Appl. Phys. 98, 063102 (2005).
[CrossRef]

Feng, Z.-F.

S. Feng, Z.-Y. Li, Z.-F. Feng, K. Ren, B.-Y. Cheng, and D.-Z. Zhang, “Focusing properties of a rectangular-rod photonic-crystal slab,” J. Appl. Phys. 98, 063102 (2005).
[CrossRef]

Fuchs, H.-J.

R. Iliew, C. Etrich, U. Peschel, F. Lederer, M. Augustin, H.-J. Fuchs, D. Schelle, E.-B. Kley, S. Nolte, and A. Tunnermann, “Diffractionless propagation of light in a low-index photonic-crystal film,” Appl. Phys. Lett. 85, 5854–5856 (2004).
[CrossRef]

Gong, Q.

D. Zhao, C. Zhou, Q. Gong, and X. Jiang, “Lasing cavities and ultra-fast switch based on self-collimation of photonic crystal,” J. Phys. D 41, 115108 (2008).
[CrossRef]

Guo, S.

Hussein, M. I.

M. I. Hussein, “Reduced Bloch mode expansion for periodic media band structure calculations,” Proc. Roy. Soc. Lond. Ser. A 465, 2825–2848 (2009).
[CrossRef]

Iliew, R.

R. Iliew, C. Etrich, and F. Lederer, “Self-collimation of light in three-dimensional photonic crystals,” Opt. Express 13, 7076–7085 (2005).
[CrossRef]

R. Iliew, C. Etrich, U. Peschel, F. Lederer, M. Augustin, H.-J. Fuchs, D. Schelle, E.-B. Kley, S. Nolte, and A. Tunnermann, “Diffractionless propagation of light in a low-index photonic-crystal film,” Appl. Phys. Lett. 85, 5854–5856 (2004).
[CrossRef]

Jiang, X.

D. Zhao, C. Zhou, Q. Gong, and X. Jiang, “Lasing cavities and ultra-fast switch based on self-collimation of photonic crystal,” J. Phys. D 41, 115108 (2008).
[CrossRef]

Joannopoulos, J. D.

Johnson, S. G.

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[CrossRef]

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[CrossRef]

Kee, C.-S.

Kim, J.-E.

Kim, T.-T.

Kivshar, Y. S.

A. F. Matthews, S. K. Morrison, and Y. S. Kivshar, “Self-collimation and beam splitting in low-index photonic crystals,” Opt. Commun. 279, 313–319 (2007).
[CrossRef]

Kley, E.-B.

R. Iliew, C. Etrich, U. Peschel, F. Lederer, M. Augustin, H.-J. Fuchs, D. Schelle, E.-B. Kley, S. Nolte, and A. Tunnermann, “Diffractionless propagation of light in a low-index photonic-crystal film,” Appl. Phys. Lett. 85, 5854–5856 (2004).
[CrossRef]

Kosaka, H.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[CrossRef]

Lax, M. J.

M. J. Lax, Symmetry Principles in Solid State and Molecular Physics (Wiley, 1974), p. 449.

Lederer, F.

R. Iliew, C. Etrich, and F. Lederer, “Self-collimation of light in three-dimensional photonic crystals,” Opt. Express 13, 7076–7085 (2005).
[CrossRef]

R. Iliew, C. Etrich, U. Peschel, F. Lederer, M. Augustin, H.-J. Fuchs, D. Schelle, E.-B. Kley, S. Nolte, and A. Tunnermann, “Diffractionless propagation of light in a low-index photonic-crystal film,” Appl. Phys. Lett. 85, 5854–5856 (2004).
[CrossRef]

Lee, S.-G.

Li, B.

Li, H.

Li, Z.-Y.

S. Feng, Z.-Y. Li, Z.-F. Feng, K. Ren, B.-Y. Cheng, and D.-Z. Zhang, “Focusing properties of a rectangular-rod photonic-crystal slab,” J. Appl. Phys. 98, 063102 (2005).
[CrossRef]

Loncar, M.

J. Witzens, M. Loncar, and A. Scherer, “Self-collimation in planar photonic crystals,” IEEE J. Sel. Top. Quantum Electron. 8, 1246–1257 (2002).
[CrossRef]

Lu, Z.

Z. Lu, S. Shi, J. A. Murakowski, G. J. Schneider, C. A. Schuetz, and D. W. Prather, “Experimental demonstration of self-collimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett. 96, 173902 (2006).
[CrossRef]

Martin, R.

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. Miao, and R. Martin, “Self-collimation in photonic crystal structures: a new paradigm for applications and device development,” J. Phys. D 40, 2635 (2007).
[CrossRef]

Matthews, A. F.

A. F. Matthews, S. K. Morrison, and Y. S. Kivshar, “Self-collimation and beam splitting in low-index photonic crystals,” Opt. Commun. 279, 313–319 (2007).
[CrossRef]

Miao, B.

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. Miao, and R. Martin, “Self-collimation in photonic crystal structures: a new paradigm for applications and device development,” J. Phys. D 40, 2635 (2007).
[CrossRef]

Morrison, S. K.

A. F. Matthews, S. K. Morrison, and Y. S. Kivshar, “Self-collimation and beam splitting in low-index photonic crystals,” Opt. Commun. 279, 313–319 (2007).
[CrossRef]

Murakowski, J.

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. Miao, and R. Martin, “Self-collimation in photonic crystal structures: a new paradigm for applications and device development,” J. Phys. D 40, 2635 (2007).
[CrossRef]

D. W. Prather, S. Shi, D. M. Pustai, C. Chen, S. Venkataraman, A. Sharkawy, G. Schneider, and J. Murakowski, “Dispersion-based optical routing in photonic crystals,” Opt. Lett. 29, 50–52 (2004).
[CrossRef]

Murakowski, J. A.

Z. Lu, S. Shi, J. A. Murakowski, G. J. Schneider, C. A. Schuetz, and D. W. Prather, “Experimental demonstration of self-collimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett. 96, 173902 (2006).
[CrossRef]

Nolte, S.

R. Iliew, C. Etrich, U. Peschel, F. Lederer, M. Augustin, H.-J. Fuchs, D. Schelle, E.-B. Kley, S. Nolte, and A. Tunnermann, “Diffractionless propagation of light in a low-index photonic-crystal film,” Appl. Phys. Lett. 85, 5854–5856 (2004).
[CrossRef]

Notomi, M.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[CrossRef]

Park, H. Y.

Pazos, J.

Peschel, U.

R. Iliew, C. Etrich, U. Peschel, F. Lederer, M. Augustin, H.-J. Fuchs, D. Schelle, E.-B. Kley, S. Nolte, and A. Tunnermann, “Diffractionless propagation of light in a low-index photonic-crystal film,” Appl. Phys. Lett. 85, 5854–5856 (2004).
[CrossRef]

Prather, D. W.

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. Miao, and R. Martin, “Self-collimation in photonic crystal structures: a new paradigm for applications and device development,” J. Phys. D 40, 2635 (2007).
[CrossRef]

Z. Lu, S. Shi, J. A. Murakowski, G. J. Schneider, C. A. Schuetz, and D. W. Prather, “Experimental demonstration of self-collimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett. 96, 173902 (2006).
[CrossRef]

D. W. Prather, S. Shi, D. M. Pustai, C. Chen, S. Venkataraman, A. Sharkawy, G. Schneider, and J. Murakowski, “Dispersion-based optical routing in photonic crystals,” Opt. Lett. 29, 50–52 (2004).
[CrossRef]

Pustai, D. M.

Qiang, Z.

Qiu, Y.

Ren, K.

S. Feng, Z.-Y. Li, Z.-F. Feng, K. Ren, B.-Y. Cheng, and D.-Z. Zhang, “Focusing properties of a rectangular-rod photonic-crystal slab,” J. Appl. Phys. 98, 063102 (2005).
[CrossRef]

Rumpf, R. C.

Sato, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[CrossRef]

Schelle, D.

R. Iliew, C. Etrich, U. Peschel, F. Lederer, M. Augustin, H.-J. Fuchs, D. Schelle, E.-B. Kley, S. Nolte, and A. Tunnermann, “Diffractionless propagation of light in a low-index photonic-crystal film,” Appl. Phys. Lett. 85, 5854–5856 (2004).
[CrossRef]

Scherer, A.

J. Witzens, M. Loncar, and A. Scherer, “Self-collimation in planar photonic crystals,” IEEE J. Sel. Top. Quantum Electron. 8, 1246–1257 (2002).
[CrossRef]

Schneider, G.

Schneider, G. J.

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. Miao, and R. Martin, “Self-collimation in photonic crystal structures: a new paradigm for applications and device development,” J. Phys. D 40, 2635 (2007).
[CrossRef]

Z. Lu, S. Shi, J. A. Murakowski, G. J. Schneider, C. A. Schuetz, and D. W. Prather, “Experimental demonstration of self-collimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett. 96, 173902 (2006).
[CrossRef]

Schuetz, C. A.

Z. Lu, S. Shi, J. A. Murakowski, G. J. Schneider, C. A. Schuetz, and D. W. Prather, “Experimental demonstration of self-collimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett. 96, 173902 (2006).
[CrossRef]

Sharkawy, A.

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. Miao, and R. Martin, “Self-collimation in photonic crystal structures: a new paradigm for applications and device development,” J. Phys. D 40, 2635 (2007).
[CrossRef]

D. W. Prather, S. Shi, D. M. Pustai, C. Chen, S. Venkataraman, A. Sharkawy, G. Schneider, and J. Murakowski, “Dispersion-based optical routing in photonic crystals,” Opt. Lett. 29, 50–52 (2004).
[CrossRef]

Shi, S.

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. Miao, and R. Martin, “Self-collimation in photonic crystal structures: a new paradigm for applications and device development,” J. Phys. D 40, 2635 (2007).
[CrossRef]

Z. Lu, S. Shi, J. A. Murakowski, G. J. Schneider, C. A. Schuetz, and D. W. Prather, “Experimental demonstration of self-collimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett. 96, 173902 (2006).
[CrossRef]

D. W. Prather, S. Shi, D. M. Pustai, C. Chen, S. Venkataraman, A. Sharkawy, G. Schneider, and J. Murakowski, “Dispersion-based optical routing in photonic crystals,” Opt. Lett. 29, 50–52 (2004).
[CrossRef]

Shin, J.

Soref, R.

R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 1678–1687 (2006).
[CrossRef]

Tamamura, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[CrossRef]

Tomita, A.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[CrossRef]

Tunnermann, A.

R. Iliew, C. Etrich, U. Peschel, F. Lederer, M. Augustin, H.-J. Fuchs, D. Schelle, E.-B. Kley, S. Nolte, and A. Tunnermann, “Diffractionless propagation of light in a low-index photonic-crystal film,” Appl. Phys. Lett. 85, 5854–5856 (2004).
[CrossRef]

Venkataraman, S.

Wang, H.

Wang, Y.

Witzens, J.

J. Witzens, M. Loncar, and A. Scherer, “Self-collimation in planar photonic crystals,” IEEE J. Sel. Top. Quantum Electron. 8, 1246–1257 (2002).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999), pp. 673–690.

Xue, Q.

Yang, W.

Zhang, D.-Z.

S. Feng, Z.-Y. Li, Z.-F. Feng, K. Ren, B.-Y. Cheng, and D.-Z. Zhang, “Focusing properties of a rectangular-rod photonic-crystal slab,” J. Appl. Phys. 98, 063102 (2005).
[CrossRef]

Zhang, Y.

Zhao, C. Z.

Z. Fang and C. Z. Zhao, “Recent progress in silicon photonics: a review,” ISRN Opt. 2012, 1–27 (2012).

Zhao, D.

X. Chen, Z. Qiang, D. Zhao, H. Li, Y. Qiu, W. Yang, and W. Zhou, “Polarization-independent drop filters based on photonic crystal self-collimation ring resonators,” Opt. Express 17, 19808–19813 (2009).
[CrossRef]

D. Zhao, C. Zhou, Q. Gong, and X. Jiang, “Lasing cavities and ultra-fast switch based on self-collimation of photonic crystal,” J. Phys. D 41, 115108 (2008).
[CrossRef]

Zheng, W.

Zhou, C.

D. Zhao, C. Zhou, Q. Gong, and X. Jiang, “Lasing cavities and ultra-fast switch based on self-collimation of photonic crystal,” J. Phys. D 41, 115108 (2008).
[CrossRef]

Zhou, W.

Appl. Phys. Lett.

R. Iliew, C. Etrich, U. Peschel, F. Lederer, M. Augustin, H.-J. Fuchs, D. Schelle, E.-B. Kley, S. Nolte, and A. Tunnermann, “Diffractionless propagation of light in a low-index photonic-crystal film,” Appl. Phys. Lett. 85, 5854–5856 (2004).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Self-collimating phenomena in photonic crystals,” Appl. Phys. Lett. 74, 1212–1214 (1999).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

J. Witzens, M. Loncar, and A. Scherer, “Self-collimation in planar photonic crystals,” IEEE J. Sel. Top. Quantum Electron. 8, 1246–1257 (2002).
[CrossRef]

R. Soref, “The past, present, and future of silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 12, 1678–1687 (2006).
[CrossRef]

ISRN Opt.

Z. Fang and C. Z. Zhao, “Recent progress in silicon photonics: a review,” ISRN Opt. 2012, 1–27 (2012).

J. Appl. Phys.

S. Feng, Z.-Y. Li, Z.-F. Feng, K. Ren, B.-Y. Cheng, and D.-Z. Zhang, “Focusing properties of a rectangular-rod photonic-crystal slab,” J. Appl. Phys. 98, 063102 (2005).
[CrossRef]

J. Phys. D

D. Zhao, C. Zhou, Q. Gong, and X. Jiang, “Lasing cavities and ultra-fast switch based on self-collimation of photonic crystal,” J. Phys. D 41, 115108 (2008).
[CrossRef]

D. W. Prather, S. Shi, J. Murakowski, G. J. Schneider, A. Sharkawy, C. Chen, B. Miao, and R. Martin, “Self-collimation in photonic crystal structures: a new paradigm for applications and device development,” J. Phys. D 40, 2635 (2007).
[CrossRef]

Opt. Commun.

A. F. Matthews, S. K. Morrison, and Y. S. Kivshar, “Self-collimation and beam splitting in low-index photonic crystals,” Opt. Commun. 279, 313–319 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

Z. Lu, S. Shi, J. A. Murakowski, G. J. Schneider, C. A. Schuetz, and D. W. Prather, “Experimental demonstration of self-collimation inside a three-dimensional photonic crystal,” Phys. Rev. Lett. 96, 173902 (2006).
[CrossRef]

Proc. Roy. Soc. Lond. Ser. A

M. I. Hussein, “Reduced Bloch mode expansion for periodic media band structure calculations,” Proc. Roy. Soc. Lond. Ser. A 465, 2825–2848 (2009).
[CrossRef]

Other

M. J. Lax, Symmetry Principles in Solid State and Molecular Physics (Wiley, 1974), p. 449.

H. Anton, Elementary Linear Algebra (Wiley, 2010), pp. 303–305.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999), pp. 673–690.

S. C. Chapra and R. P. Canale, Numerical Methods for Engineers, 6th ed. (McGraw-Hill, 2010), Vol. 2, pp. 852–887.

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Figures (10)

Fig. 1.
Fig. 1.

Construction of the IFCs using RBME/PWEM.

Fig. 2.
Fig. 2.

Determining the performance metrics for self-collimation from the IFCs. Only the upper right quadrant of the BZ is shown here.

Fig. 3.
Fig. 3.

Illustration of the self-collimation effect and the strength metric.

Fig. 4.
Fig. 4.

Visualization of self-collimation over a range of frequencies. The optimum frequency for self-collimation is slightly offset from the inflection point.

Fig. 5.
Fig. 5.

Illustration of the fast L algorithm for estimating the performance metrics.

Fig. 6.
Fig. 6.

Unit cells for optimization.

Fig. 7.
Fig. 7.

Performance metrics as a function of dielectric constant for the TE mode.

Fig. 8.
Fig. 8.

Performance metrics as a function of dielectric constant for the TM mode.

Fig. 9.
Fig. 9.

Performance metrics as a function of fill fraction.

Fig. 10.
Fig. 10.

Demonstration of the performance enhancement in a spatially variant self-collimating photonic crystal for the TM mode. (Left) Lattice with high FOM. (Right) Lattice with lower FOM.

Equations (36)

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EzyEyz=k0μrH˜x,
ExzEzx=k0μrH˜y,
EyxExy=k0μrH˜z,
H˜zyH˜yz=k0εrEx,
H˜xzH˜zx=k0εrEy,
H˜yxH˜xy=k0εrEz.
H˜yxH˜xy=k0εrEz,
Ezy=k0μrH˜x,
Ezx=k0μrH˜y.
EyxExy=k0μrH˜z,
H˜zy=k0εrEx,
H˜zx=k0εrEy.
E⃗(r⃗)=p=q=S⃗pqej(kx,pqx+ky,pqy),
H˜(r⃗)=p=q=U⃗pqej(kx,pqx+ky,pqy),
εr(r⃗)=p=q=apqej(2πpΛx+2πqΛy),
μr(r⃗)=p=q=bpqej(2πpΛx+2πqΛy).
kx,pq=βx2πpΛx,
ky,pq=βy2πqΛy.
jkx,pqUy,pqjky,pqUx,pq=k0p=q=app,qqSz,pq,
jky,pqSz,pq=k0p=q=bpp,qqUx,pq,
jkx,pqSz,pq=k0p=q=bpp,qqUy,pq.
KxuyKyux=jk0εrsz,
Kysz=jk0μrux,
Kxsz=jk0μruy.
ATMsz=k02BTMszATM=Kxμr1Kx+Kyμr1KyBTM=εr,
ATEuz=k02BTEuzATE=Kxεr1Kx+Kyεr1KyBTE=μr,
R=Gram Schmidt([VΓ|1NVX|1NVM|1N]).
ATMsz=k02BTMszATM=RHATMRBTM=RHBTMR,
ATEuz=k02BTEuzATE=RHATERBTE=RHBTER.
ϕ=tan1(ω/βyω/βx).
FBW=2ω(βx2)ω(βx1)ω(βx2)+ω(βx1),
θA=tan1(βy2/βx2)90°.
A=FBW·θA.
S=1|2Λxπβxi1|.
FOM=FBW·θA·S3.
FBWωβxω(βxi+Δβ/2)ω(βxiΔβ/2)Δβ.

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