Abstract

We introduce the theoretical criterion for achieving three-dimensional self-collimation of light in a photonic crystal. Based on this criterion, we numerically demonstrate a body-center-cubic structure that supports wide-angle self-collimation and is directly compatible with the recently developed holographic fabrication technique. We further show that both bends and beam splitters can be introduced into this structure by the use of interfaces.

© 2005 Optical Society of America

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  1. P. V. Braun and P. Wiltzius, Nature 402, 603 (1999).
    [CrossRef]
  2. Y. A. Vlasov, X. Z. Bo, J. C. Sturm, and D. J. Norris, Nature 414, 289 (2001).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  4. L. Z. Cai, X. L. Yang, and Y. R. Wang, Opt. Lett. 27, 900 (2002).
    [CrossRef]
  5. O. Toader, T. Y.M. Chan, and S. John, Phys. Rev. Lett. 92, 043905 (2004).
    [CrossRef]
  6. C. K. Ullal, M. Maldovan, E. L. Thomas, G. Chen, Y. Han, and S. Yang, Appl. Phys. Lett. 84, 5434 (2004).
    [CrossRef]
  7. W. Lee, S. A. Pruzinsky, and P. V. Braun, Adv. Mater. (Weinheim, Ger.) 14, 271 (2002).
    [CrossRef]
  8. A. Chutinan and S. Noda, Appl. Phys. Lett. 75, 3739 (1999).
    [CrossRef]
  9. M. Povinelli, S. G. Johnson, S. Fan, and J. D. Joannopoulos, Phys. Rev. B 64, 075313 (2003).
    [CrossRef]
  10. A. Chutinan, S. John, and O. Toader, Phys. Rev. Lett. 90, 123901 (2003).
    [CrossRef]
  11. R. Zengerle, J. Mod. Opt. 34, 1589 (1987).
    [CrossRef]
  12. H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, Appl. Phys. Lett. 74, 1212 (1999).
    [CrossRef]
  13. M. Notomi, Phys. Rev. B 62, 10696 (2000).
    [CrossRef]
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    [CrossRef]
  15. J. Witzens and A. Scherer, J. Opt. Soc. Am. A 20, 935 (2003).
    [CrossRef]
  16. X. Yu and S. Fan, Appl. Phys. Lett. 83, 3251 (2003).
    [CrossRef]
  17. L. Wu, M. Mazilu, and T. F. Krauss, J. Lightwave Technol. 21, 561 (2003).
    [CrossRef]
  18. C. Chen, A. Sharkawy, D. M. Pustai, S. Shi, and D. W. Prather, Opt. Express 11, 3153 (2003).
    [CrossRef] [PubMed]
  19. S. G. Johnson and J. D. Joannopoulos, Opt. Express 8, 173 (2001).
    [CrossRef] [PubMed]
  20. K. S. Kunz and R. J. Luebbers, The Finite-Difference Time-Domain Methods for Electromagnetics (CRC Press, 1993).

2004 (2)

O. Toader, T. Y.M. Chan, and S. John, Phys. Rev. Lett. 92, 043905 (2004).
[CrossRef]

C. K. Ullal, M. Maldovan, E. L. Thomas, G. Chen, Y. Han, and S. Yang, Appl. Phys. Lett. 84, 5434 (2004).
[CrossRef]

2003 (6)

M. Povinelli, S. G. Johnson, S. Fan, and J. D. Joannopoulos, Phys. Rev. B 64, 075313 (2003).
[CrossRef]

A. Chutinan, S. John, and O. Toader, Phys. Rev. Lett. 90, 123901 (2003).
[CrossRef]

J. Witzens and A. Scherer, J. Opt. Soc. Am. A 20, 935 (2003).
[CrossRef]

X. Yu and S. Fan, Appl. Phys. Lett. 83, 3251 (2003).
[CrossRef]

L. Wu, M. Mazilu, and T. F. Krauss, J. Lightwave Technol. 21, 561 (2003).
[CrossRef]

C. Chen, A. Sharkawy, D. M. Pustai, S. Shi, and D. W. Prather, Opt. Express 11, 3153 (2003).
[CrossRef] [PubMed]

2002 (3)

W. Lee, S. A. Pruzinsky, and P. V. Braun, Adv. Mater. (Weinheim, Ger.) 14, 271 (2002).
[CrossRef]

L. Z. Cai, X. L. Yang, and Y. R. Wang, Opt. Lett. 27, 900 (2002).
[CrossRef]

J. Witzens, M. Loncar, and A. Scherer, IEEE J. Sel. Top. Quantum Electron. 8, 1246 (2002).
[CrossRef]

2001 (2)

Y. A. Vlasov, X. Z. Bo, J. C. Sturm, and D. J. Norris, Nature 414, 289 (2001).
[CrossRef] [PubMed]

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

2000 (2)

M. Notomi, Phys. Rev. B 62, 10696 (2000).
[CrossRef]

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, Nature 404, 53 (2000).
[CrossRef] [PubMed]

1999 (3)

P. V. Braun and P. Wiltzius, Nature 402, 603 (1999).
[CrossRef]

A. Chutinan and S. Noda, Appl. Phys. Lett. 75, 3739 (1999).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, Appl. Phys. Lett. 74, 1212 (1999).
[CrossRef]

1987 (1)

R. Zengerle, J. Mod. Opt. 34, 1589 (1987).
[CrossRef]

Bo, X. Z.

Y. A. Vlasov, X. Z. Bo, J. C. Sturm, and D. J. Norris, Nature 414, 289 (2001).
[CrossRef] [PubMed]

Braun, P. V.

W. Lee, S. A. Pruzinsky, and P. V. Braun, Adv. Mater. (Weinheim, Ger.) 14, 271 (2002).
[CrossRef]

P. V. Braun and P. Wiltzius, Nature 402, 603 (1999).
[CrossRef]

Cai, L. Z.

Campbell, M.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, Nature 404, 53 (2000).
[CrossRef] [PubMed]

Chan, T. Y.M.

O. Toader, T. Y.M. Chan, and S. John, Phys. Rev. Lett. 92, 043905 (2004).
[CrossRef]

Chen, C.

Chen, G.

C. K. Ullal, M. Maldovan, E. L. Thomas, G. Chen, Y. Han, and S. Yang, Appl. Phys. Lett. 84, 5434 (2004).
[CrossRef]

Chutinan, A.

A. Chutinan, S. John, and O. Toader, Phys. Rev. Lett. 90, 123901 (2003).
[CrossRef]

A. Chutinan and S. Noda, Appl. Phys. Lett. 75, 3739 (1999).
[CrossRef]

Denning, R. G.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, Nature 404, 53 (2000).
[CrossRef] [PubMed]

Fan, S.

X. Yu and S. Fan, Appl. Phys. Lett. 83, 3251 (2003).
[CrossRef]

M. Povinelli, S. G. Johnson, S. Fan, and J. D. Joannopoulos, Phys. Rev. B 64, 075313 (2003).
[CrossRef]

Han, Y.

C. K. Ullal, M. Maldovan, E. L. Thomas, G. Chen, Y. Han, and S. Yang, Appl. Phys. Lett. 84, 5434 (2004).
[CrossRef]

Harrison, M. T.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, Nature 404, 53 (2000).
[CrossRef] [PubMed]

Joannopoulos, J. D.

M. Povinelli, S. G. Johnson, S. Fan, and J. D. Joannopoulos, Phys. Rev. B 64, 075313 (2003).
[CrossRef]

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

John, S.

O. Toader, T. Y.M. Chan, and S. John, Phys. Rev. Lett. 92, 043905 (2004).
[CrossRef]

A. Chutinan, S. John, and O. Toader, Phys. Rev. Lett. 90, 123901 (2003).
[CrossRef]

Johnson, S. G.

M. Povinelli, S. G. Johnson, S. Fan, and J. D. Joannopoulos, Phys. Rev. B 64, 075313 (2003).
[CrossRef]

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

Kawakami, S.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, Appl. Phys. Lett. 74, 1212 (1999).
[CrossRef]

Kawashima, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, Appl. Phys. Lett. 74, 1212 (1999).
[CrossRef]

Kosaka, H.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, Appl. Phys. Lett. 74, 1212 (1999).
[CrossRef]

Krauss, T. F.

Kunz, K. S.

K. S. Kunz and R. J. Luebbers, The Finite-Difference Time-Domain Methods for Electromagnetics (CRC Press, 1993).

Lee, W.

W. Lee, S. A. Pruzinsky, and P. V. Braun, Adv. Mater. (Weinheim, Ger.) 14, 271 (2002).
[CrossRef]

Loncar, M.

J. Witzens, M. Loncar, and A. Scherer, IEEE J. Sel. Top. Quantum Electron. 8, 1246 (2002).
[CrossRef]

Luebbers, R. J.

K. S. Kunz and R. J. Luebbers, The Finite-Difference Time-Domain Methods for Electromagnetics (CRC Press, 1993).

Maldovan, M.

C. K. Ullal, M. Maldovan, E. L. Thomas, G. Chen, Y. Han, and S. Yang, Appl. Phys. Lett. 84, 5434 (2004).
[CrossRef]

Mazilu, M.

Noda, S.

A. Chutinan and S. Noda, Appl. Phys. Lett. 75, 3739 (1999).
[CrossRef]

Norris, D. J.

Y. A. Vlasov, X. Z. Bo, J. C. Sturm, and D. J. Norris, Nature 414, 289 (2001).
[CrossRef] [PubMed]

Notomi, M.

M. Notomi, Phys. Rev. B 62, 10696 (2000).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, Appl. Phys. Lett. 74, 1212 (1999).
[CrossRef]

Povinelli, M.

M. Povinelli, S. G. Johnson, S. Fan, and J. D. Joannopoulos, Phys. Rev. B 64, 075313 (2003).
[CrossRef]

Prather, D. W.

Pruzinsky, S. A.

W. Lee, S. A. Pruzinsky, and P. V. Braun, Adv. Mater. (Weinheim, Ger.) 14, 271 (2002).
[CrossRef]

Pustai, D. M.

Sato, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, Appl. Phys. Lett. 74, 1212 (1999).
[CrossRef]

Scherer, A.

J. Witzens and A. Scherer, J. Opt. Soc. Am. A 20, 935 (2003).
[CrossRef]

J. Witzens, M. Loncar, and A. Scherer, IEEE J. Sel. Top. Quantum Electron. 8, 1246 (2002).
[CrossRef]

Sharkawy, A.

Sharp, D. N.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, Nature 404, 53 (2000).
[CrossRef] [PubMed]

Shi, S.

Sturm, J. C.

Y. A. Vlasov, X. Z. Bo, J. C. Sturm, and D. J. Norris, Nature 414, 289 (2001).
[CrossRef] [PubMed]

Tamamura, T.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, Appl. Phys. Lett. 74, 1212 (1999).
[CrossRef]

Thomas, E. L.

C. K. Ullal, M. Maldovan, E. L. Thomas, G. Chen, Y. Han, and S. Yang, Appl. Phys. Lett. 84, 5434 (2004).
[CrossRef]

Toader, O.

O. Toader, T. Y.M. Chan, and S. John, Phys. Rev. Lett. 92, 043905 (2004).
[CrossRef]

A. Chutinan, S. John, and O. Toader, Phys. Rev. Lett. 90, 123901 (2003).
[CrossRef]

Tomita, A.

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, Appl. Phys. Lett. 74, 1212 (1999).
[CrossRef]

Turberfield, A. J.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, Nature 404, 53 (2000).
[CrossRef] [PubMed]

Ullal, C. K.

C. K. Ullal, M. Maldovan, E. L. Thomas, G. Chen, Y. Han, and S. Yang, Appl. Phys. Lett. 84, 5434 (2004).
[CrossRef]

Vlasov, Y. A.

Y. A. Vlasov, X. Z. Bo, J. C. Sturm, and D. J. Norris, Nature 414, 289 (2001).
[CrossRef] [PubMed]

Wang, Y. R.

Wiltzius, P.

P. V. Braun and P. Wiltzius, Nature 402, 603 (1999).
[CrossRef]

Witzens, J.

J. Witzens and A. Scherer, J. Opt. Soc. Am. A 20, 935 (2003).
[CrossRef]

J. Witzens, M. Loncar, and A. Scherer, IEEE J. Sel. Top. Quantum Electron. 8, 1246 (2002).
[CrossRef]

Wu, L.

Yang, S.

C. K. Ullal, M. Maldovan, E. L. Thomas, G. Chen, Y. Han, and S. Yang, Appl. Phys. Lett. 84, 5434 (2004).
[CrossRef]

Yang, X. L.

Yu, X.

X. Yu and S. Fan, Appl. Phys. Lett. 83, 3251 (2003).
[CrossRef]

Zengerle, R.

R. Zengerle, J. Mod. Opt. 34, 1589 (1987).
[CrossRef]

Adv. Mater. (Weinheim, Ger.) (1)

W. Lee, S. A. Pruzinsky, and P. V. Braun, Adv. Mater. (Weinheim, Ger.) 14, 271 (2002).
[CrossRef]

Appl. Phys. Lett. (4)

A. Chutinan and S. Noda, Appl. Phys. Lett. 75, 3739 (1999).
[CrossRef]

C. K. Ullal, M. Maldovan, E. L. Thomas, G. Chen, Y. Han, and S. Yang, Appl. Phys. Lett. 84, 5434 (2004).
[CrossRef]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, Appl. Phys. Lett. 74, 1212 (1999).
[CrossRef]

X. Yu and S. Fan, Appl. Phys. Lett. 83, 3251 (2003).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

J. Witzens, M. Loncar, and A. Scherer, IEEE J. Sel. Top. Quantum Electron. 8, 1246 (2002).
[CrossRef]

J. Lightwave Technol. (1)

J. Mod. Opt. (1)

R. Zengerle, J. Mod. Opt. 34, 1589 (1987).
[CrossRef]

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

Nature (3)

P. V. Braun and P. Wiltzius, Nature 402, 603 (1999).
[CrossRef]

Y. A. Vlasov, X. Z. Bo, J. C. Sturm, and D. J. Norris, Nature 414, 289 (2001).
[CrossRef] [PubMed]

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, and A. J. Turberfield, Nature 404, 53 (2000).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. B (2)

M. Povinelli, S. G. Johnson, S. Fan, and J. D. Joannopoulos, Phys. Rev. B 64, 075313 (2003).
[CrossRef]

M. Notomi, Phys. Rev. B 62, 10696 (2000).
[CrossRef]

Phys. Rev. Lett. (2)

A. Chutinan, S. John, and O. Toader, Phys. Rev. Lett. 90, 123901 (2003).
[CrossRef]

O. Toader, T. Y.M. Chan, and S. John, Phys. Rev. Lett. 92, 043905 (2004).
[CrossRef]

Other (1)

K. S. Kunz and R. J. Luebbers, The Finite-Difference Time-Domain Methods for Electromagnetics (CRC Press, 1993).

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Figures (4)

Fig. 1
Fig. 1

Constant-frequency surface (CSF) around the H point for an empty BCC lattice. The inset shows the first Brillouin zone.

Fig. 2
Fig. 2

(a) Band structure of a BCC gyroid crystal. The crystal is plotted in the inset. The horizontal line indicates the position of the frequency of 0.41 ( 2 π c a ) . (b), (c) Calculated constant frequency surface around the H point at frequencies (b) 0.40 ( 2 π c a ) , (c) 0.41 ( 2 π c a ) .

Fig. 3
Fig. 3

(a) Propagation of a self-collimated beam, with an initial beam radius of 2 a as calculated in a finite-difference time-domain simulation. Shown here is the intensity of the electric field. (b), (c) Beam width as a function of propagation distance for an initial beam radius of (b) 2 a and (c) 8 a .

Fig. 4
Fig. 4

Intensity plot as a light beam goes through (a) a 90° bend, (b) a splitter and then a bend. Filled arrows, normal direction of interfaces; open arrows, propagation direction.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

k z = f ( k x , k y ) = f ( k x 0 , k y 0 ) + f k x k x 0 , k y 0 ( k x k x 0 ) + f k y k x 0 , k y 0 ( k y k y 0 ) + 1 2 2 f k x 2 k x 0 , k y 0 ( k x k x 0 ) 2 + 2 f k x k y k x 0 , k y 0 ( k x k x 0 ) ( k y k y 0 ) + 1 2 2 f k y 2 k x 0 , k y 0 ( k y k y 0 ) 2 + .
R u ̂ = 1 2 f u 2 = 2 [ ( A + C ) + ( A C ) cos ( 2 θ ) + 2 B sin ( 2 θ ) ] ,
A = 2 f k x 2 , B = 2 f k x k y , C = 2 f k y 2 .
ϵ ( x , y , z ) = 12.06 if sin ( 2 π ( x + y ) a ) + sin ( 2 π ( x y ) a ) + sin ( 2 π ( y + z ) a ) + sin ( 2 π ( y z ) a ) + sin ( 2 π ( z + x ) a ) + sin ( 2 π ( z x ) a ) > 2.0 ,
ϵ ( x , y , z ) = 1 otherwise ,

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