Abstract

We demonstrate that high transmission through sharp bends in photonic crystal waveguides can be described by a simple model of the Fano resonance where the waveguide bend plays a role of a specific localized defect. We derive effective discrete equations for two types of the waveguide bends in two-dimensional photonic crystals and obtain exact analytical solutions for the resonant transmission and reflection. This approach allows us to get a deeper insight into the physics of resonant transmission, and it is also useful for the study and design of power-dependent transmission through the waveguide bends with embedded nonlinear defects.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |

  1. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, �??Photonic crystals: putting a new twist on light,�?? Nature 386, 143-149 (1997).
    [CrossRef]
  2. A. Mekis, J.C. Chen, I. Kurland, S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, �??High transmission through sharp bends in photonic crystal waveguides,�?? Phys. Rev. Lett. 77, 3787-3790 (1996).
    [CrossRef] [PubMed]
  3. S.-Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, �??Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,�?? Science 282, 274-276 (1998).
    [CrossRef] [PubMed]
  4. A. Chutinan and S. Noda, �??Waveguides and waveguide bends in two-dimensional photonic crystal slabs,�?? Phys. Rev. B 62, 4488-4492 (2000).
    [CrossRef]
  5. R.L. Espinola, R.U. Ahmad, F. Pizzuto, M.J. Steel, and R.M. Osgood Jr., �??A study of high-index-contrast 90 degree waveguide bend structures,�?? Opt. Express 8, 517-528 (2001).
    [CrossRef] [PubMed]
  6. E. Chow, S. Y. Lin, J. R. Wendt, S. G. Johnson, and J. D. Joannopoulos, �??Quantitative analysis of bending efficiency in photonic-crystal waveguide bends at λ= 1.55 μm wavelengths,�?? Opt. Lett. 26, 286-288 (2001).
    [CrossRef]
  7. S. Olivier, H. Benisty, C. Weisbuch, C. J. M. Smith, T.F. Krauss, R. Houdré, and U. Oesterle, �??Improved 60 degree bend transmission of Ssubmicron-width waveguides defined in two-dimensional photonic crystals,�?? J. Lightwave Technol. 20, 1198-1203 (2002).
    [CrossRef]
  8. J. Smajic, C. Hafner, and D. Erni, �??Design and optimization of an achromatic photonic crystal bend,�?? Opt. Express 11, 1378-1384 (2003).
    [CrossRef] [PubMed]
  9. Z.Y. Li and K.M. Ho, �??Light propagation through photonic crystal waveguide bends by eigenmode examinations,�?? Phys. Rev. B 68, 045201 (12) (2003).
    [CrossRef]
  10. A. Talneau, M. Agio, C. M. Soukoulis, M. Mulot, S. Anand, and Ph. Lalanne, �??High-bandwidth transmission of an efficient photonic-crystal mode converter,�?? Opt. Lett. 29, 1745-1747 (2004).
    [CrossRef] [PubMed]
  11. J. S. Jensen and O. Sigmund, �??Systematic design of photonic crystal structures using topology optimization: Low-loss waveguide bends,�?? App. Phys. Lett. 84, 2022-2024 (2004).
    [CrossRef]
  12. I. Ntakis, P. Pottier, and R. M. De La Rue, �??Optimization of transmission properties of two-dimensional photonic crystal channel waveguide bends through local lattice deformation,�?? J. App. Phys. 96, 12-18 (2004).
    [CrossRef]
  13. N. Malkova, �??Tunable resonant light propagation through 90° bend waveguide based on strained photonic crystal,�?? J. Phys.: Cond. Matt. 16, 1523-1530 (2004).
    [CrossRef]
  14. S.F. Mingaleev, Yu.S. Kivshar, and R.A. Sammut, �??Long-range interaction and nonlinear localized modes in photonic crystal waveguides,�?? Phys. Rev. E 62, 5777-5782 (2000).
    [CrossRef]
  15. S.F. Mingaleev and Yu.S. Kivshar, �??Effective equations for photonic-crystal waveguides and circuits,�?? Opt. Lett. 27, 231-233 (2002).
    [CrossRef]
  16. S. F. Mingaleev and Yu. S. Kivshar, �??Nonlinear transmission and light localization in photonic-crystal waveguides,�?? J. Opt. Soc. Am. B 19, 2241-2249 (2002).
    [CrossRef]
  17. U. Fano, �??Effects of configuration interaction on intensities and phase shifts,�?? Phys. Rev. 124, 1866-1878 (1961).
    [CrossRef]
  18. A.E. Miroshnichenko, S.F. Mingaleev, S. Flach, and Yu.S. Kivshar, �??Nonlinear Fano resonance and bistable wave transmission,�?? Phys. Rev. E 71, 036626-8 (2005).
    [CrossRef]
  19. T. Zijlstra, E. van der Drift, M. J. A. de Dood, E. Snoeks, and A. Polman, �??Fabrication of two-dimensional photonic crystal waveguides for 1.5 μm in silicon by deep anisotropic dry etching,�?? J. Vac. Sci. Tech. B 17, 2734-2739 (1999).
    [CrossRef]

App. Phys. Lett.

J. S. Jensen and O. Sigmund, �??Systematic design of photonic crystal structures using topology optimization: Low-loss waveguide bends,�?? App. Phys. Lett. 84, 2022-2024 (2004).
[CrossRef]

J. App. Phys.

I. Ntakis, P. Pottier, and R. M. De La Rue, �??Optimization of transmission properties of two-dimensional photonic crystal channel waveguide bends through local lattice deformation,�?? J. App. Phys. 96, 12-18 (2004).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. B

J. Phys.: Cond. Matt.

N. Malkova, �??Tunable resonant light propagation through 90° bend waveguide based on strained photonic crystal,�?? J. Phys.: Cond. Matt. 16, 1523-1530 (2004).
[CrossRef]

J. Vac. Sci. Tech. B

T. Zijlstra, E. van der Drift, M. J. A. de Dood, E. Snoeks, and A. Polman, �??Fabrication of two-dimensional photonic crystal waveguides for 1.5 μm in silicon by deep anisotropic dry etching,�?? J. Vac. Sci. Tech. B 17, 2734-2739 (1999).
[CrossRef]

Nature

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, �??Photonic crystals: putting a new twist on light,�?? Nature 386, 143-149 (1997).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev.

U. Fano, �??Effects of configuration interaction on intensities and phase shifts,�?? Phys. Rev. 124, 1866-1878 (1961).
[CrossRef]

Phys. Rev. B

A. Chutinan and S. Noda, �??Waveguides and waveguide bends in two-dimensional photonic crystal slabs,�?? Phys. Rev. B 62, 4488-4492 (2000).
[CrossRef]

Z.Y. Li and K.M. Ho, �??Light propagation through photonic crystal waveguide bends by eigenmode examinations,�?? Phys. Rev. B 68, 045201 (12) (2003).
[CrossRef]

Phys. Rev. E

A.E. Miroshnichenko, S.F. Mingaleev, S. Flach, and Yu.S. Kivshar, �??Nonlinear Fano resonance and bistable wave transmission,�?? Phys. Rev. E 71, 036626-8 (2005).
[CrossRef]

S.F. Mingaleev, Yu.S. Kivshar, and R.A. Sammut, �??Long-range interaction and nonlinear localized modes in photonic crystal waveguides,�?? Phys. Rev. E 62, 5777-5782 (2000).
[CrossRef]

Phys. Rev. Lett.

A. Mekis, J.C. Chen, I. Kurland, S. Fan, P.R. Villeneuve, and J.D. Joannopoulos, �??High transmission through sharp bends in photonic crystal waveguides,�?? Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Science

S.-Y. Lin, E. Chow, V. Hietala, P. R. Villeneuve, and J. D. Joannopoulos, �??Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,�?? Science 282, 274-276 (1998).
[CrossRef] [PubMed]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (5)

Fig. 1.
Fig. 1.

(a) Spatial structure of the coupling coefficients Jnm (ω) of the effective discrete model (3) at ω=0.4×2πc/a. (b) Dependence of the specific coupling coefficients (marked) on the frequency ω.

Fig. 2.
Fig. 2.

Schematic view of two designs of the waveguide bends studied in the paper. Empty circles correspond to removed rods, dashed lines denote the effective coupling. Yellow circles mark the defects with different dielectric constant εd (|E|), which can be nonlinear.

Fig. 3.
Fig. 3.

Transmission coefficient of the waveguide bend for different values of the dielectric constant εd . The Fano resonance is observed when the value of the dielectric constant of the defect rod εd approaches the value of the dielectric constant of the lattice rod ε rod. The plot εd =εbg corresponds to the case when a rod is removed from the bend corner, whereas the plot εd =ε rod corresponds to the case when the lattice rod remains at the corner. For comparison, the crosses (×) show the results of the direct FDTD numerical calculations.

Fig. 4.
Fig. 4.

Transmission coefficient through the waveguide bend with a (yellow) defect rod placed outside the corner. In this case, there exist two Fano resonances, one of them is characterized by an asymmetric profile and corresponds to the perfect transmission.

Fig. 5.
Fig. 5.

Nonlinear transmission calculated for two types of the waveguide bends shown in Fig. 2. In both the cases, the Fano resonance is observed as the perfect reflection. The waveguide bend of the type B allows the perfect transmission that can be also tuned.

Equations (20)

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

[ 2 + ( ω c ) 2 ε ( x ) ] E ( x ω ) = 0 .
E ( x ω ) = ( ω c ) 2 d 2 y G ( x , y ω ) δ ε ( y ) E ( y ω ) ,
E n , m = k , l J n k , m l ( ω ) δ ε k , l E k , l
J n , m ( ω ) = ( ω c ) 2 r d d 2 y G ( x n , x m + y ω )
δ ε n , m = ε n , m ε rod ,
ε n , m = ε bg , for n 1 and m = 0 or n = 0 and m 1 .
( 1 J 0 , 0 δ ε 0 ) E n , 0 = δ ε 0 J 1 , 0 ( E n + 1 , 0 + E n 1 , 0 ) , n < 1 ,
( 1 J 0 , 0 δ ε 0 ) E 0 , m = δ ε 0 J 0 , 1 ( E 0 , m + 1 + E m , n 1 ) , m > 1 ,
( 1 J 0 , 0 δ ε 0 ) E 1 , 0 = J 1 , 0 ( δ ε 1 E 0 , 0 + δ ε 0 E 2 , 0 ) + J 1 , 1 δ ε 0 E 0 , 1 ,
( 1 J 0 , 0 δ ε 1 ) E 0 , 0 = δ ε 0 ( J 0 , 1 E 0 , 1 + J 1 , 0 E 1 , 0 ) ,
( 1 J 0 , 0 δ ε 0 ) E 0 , 1 = J 0 , 1 ( δ ε 1 E 0 , 0 + δ ε 0 E 0 , 2 ) + J 1 , 1 δ ε 0 E 1 , 0 ,
cos k = 1 δ ε 0 J 0 , 0 δ ε 0 J 0 , 1 ,
T = 4 a 2 sin 2 k b ( c b ) 2 ,
a = ( J 11 + J 0 , 1 2 δ ε 1 J 0 , 0 J 1 , 1 δ ε 1 ) J 0 , 1 δ ε 0 2 , b = ( J 0 , 0 + exp ( i k ) J 0 , 1 J 1 , 1 ) δ ε 0 1 ,
c = ( J 0 , 0 2 2 J 0 , 1 2 + J 0 , 0 J 1 , 1 + exp ( i k ) J 0 , 0 J 0 , 1 ) δ ε 0 δ ε 1 J 0 , 0 δ ε 1 .
J 0 , 0 J 1 , 1 δ ε 1 = J 11 + J 0 , 1 2 δ ε 1 .
J 0 , 0 = ω ω d , J 0 , 1 = C , J 1 , 1 = V J 0 , 0 = V ( ω ω d ) ,
δ ε 1 V ω 2 ( 2 δ ε 1 ω d + 1 ) V ω + ( V δ ε 1 ω d 2 + V ω d C 2 δ ε 1 ) = 0 ,
ω F = ω d + 1 2 δ ε 1 ± [ C 2 V + 1 4 δ ε 1 2 ] 1 2 .
ε d ( E ) = ε d + λ E 2 .

Metrics