Abstract

We discuss the peculiar properties of binary waveguide arrays composed of identical photonic crystal waveguides with nonuniform spacing. Coupled-mode theory is used in order to demonstrate that, as a result of the excitation of the individual waveguides’s modes, two different array modes characterized by opposite transverse velocity can propagate. Approximated expressions for the evaluation of the diffraction curves and the percentage of energy carried by each array mode are calculated and tested through numerical examples.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817-823 (2003).
    [CrossRef] [PubMed]
  2. D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90, 053902 (2003).
    [CrossRef] [PubMed]
  3. H. S. Eisenberg and Y. Silberberg, “Diffraction management,” Phys. Rev. Lett. 85, 1863-1866 (2000).
    [CrossRef] [PubMed]
  4. A. A. Sukhorukov and Y. S. Kivshar, “Discrete gap solitons in modulated waveguide arrays,” Opt. Lett. 27, 2112-2114 (2002).
    [CrossRef]
  5. R. Morandotti, D. Mandelik, Y. Silberberg, J. S. Aitchison, M. Sorel, D. N. Christodoulides, A. A. Sukhorukov, and Y. S. Kivshar, “Observation of discrete gap solitons in binary waveguide arrays,” Opt. Lett. 29, 2890-2892 (2004).
    [CrossRef]
  6. A. A. Sukhorukov and Y. S. Kivshar, “Soliton control and Bloch-wave filtering in periodic photonic lattices,” Opt. Lett. 30, 1849-1851 (2005).
    [CrossRef] [PubMed]
  7. S. Longhi, “Multiband diffraction and refraction control in binary arrays of periodically curved waveguides,” Opt. Lett. 31, 1857-1859 (2006).
    [CrossRef] [PubMed]
  8. U. Peschel and F. Lederer, “Oscillation and decay of discrete solitons in modulated waveguide arrays,” J. Opt. Soc. Am. B 19, 544-549 (2002).
    [CrossRef]
  9. R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Controlled switching of discrete solitons in waveguide arrays,” Opt. Lett. 28, 1942-1944 (2003).
    [CrossRef] [PubMed]
  10. A. A. Sukhorukov and Y. S. Kivshar, “Multigap discrete vector solitons,” Phys. Rev. Lett. 91, 113902 (2003).
    [CrossRef] [PubMed]
  11. C. M. de Sterke, L. C. Botten, A. A. Asatryan, T. P. White, and R. C. McPhedran, “Modes of coupled photonic crystal waveguides,” Opt. Lett. 29, 1384-1386 (2004).
    [CrossRef]
  12. A. Locatelli, M. Conforti, D. Modotto, and C. De Angelis, “Diffraction engineering in arrays of photonic crystal waveguides,” Opt. Lett. 30, 2894-2896 (2005).
    [CrossRef] [PubMed]
  13. A. Locatelli, M. Conforti, D. Modotto, and C. De Angelis, “Discrete negative refraction in photonic crystal waveguide arrays,” Opt. Lett. 31, 1343-1345 (2006).
    [CrossRef] [PubMed]
  14. N. Malkova and C. Z. Ning, “Interplay between Tamm-like and Shockley-like surface states in photonic crystals,” Phys. Rev. B 76, 045305 (2007).
    [CrossRef]
  15. A. A. Sukhorukov, D. Neshev, W. Krolikowski, and Y. S. Kivshar, “Nonlinear Bloch-wave interaction and Bragg scattering in optically induced lattices,” Phys. Rev. Lett. 92, 093901 (2004).
    [CrossRef] [PubMed]
  16. N. G. Broderick, C. M. de Sterke, and B. J. Eggleton, “Soliton solutions in Rowland ghost gaps,” Phys. Rev. E 52, R5788-R5791 (1995).
    [CrossRef]
  17. S. Longhi, “Image reconstruction in segmented waveguide arrays,” Opt. Lett. 33, 473-475 (2008).
    [CrossRef] [PubMed]
  18. 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] [PubMed]

2008 (1)

2007 (1)

N. Malkova and C. Z. Ning, “Interplay between Tamm-like and Shockley-like surface states in photonic crystals,” Phys. Rev. B 76, 045305 (2007).
[CrossRef]

2006 (2)

2005 (2)

2004 (3)

2003 (4)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817-823 (2003).
[CrossRef] [PubMed]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Controlled switching of discrete solitons in waveguide arrays,” Opt. Lett. 28, 1942-1944 (2003).
[CrossRef] [PubMed]

A. A. Sukhorukov and Y. S. Kivshar, “Multigap discrete vector solitons,” Phys. Rev. Lett. 91, 113902 (2003).
[CrossRef] [PubMed]

2002 (2)

2001 (1)

2000 (1)

H. S. Eisenberg and Y. Silberberg, “Diffraction management,” Phys. Rev. Lett. 85, 1863-1866 (2000).
[CrossRef] [PubMed]

1995 (1)

N. G. Broderick, C. M. de Sterke, and B. J. Eggleton, “Soliton solutions in Rowland ghost gaps,” Phys. Rev. E 52, R5788-R5791 (1995).
[CrossRef]

Aitchison, J. S.

R. Morandotti, D. Mandelik, Y. Silberberg, J. S. Aitchison, M. Sorel, D. N. Christodoulides, A. A. Sukhorukov, and Y. S. Kivshar, “Observation of discrete gap solitons in binary waveguide arrays,” Opt. Lett. 29, 2890-2892 (2004).
[CrossRef]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

Asatryan, A. A.

Botten, L. C.

Broderick, N. G.

N. G. Broderick, C. M. de Sterke, and B. J. Eggleton, “Soliton solutions in Rowland ghost gaps,” Phys. Rev. E 52, R5788-R5791 (1995).
[CrossRef]

Christodoulides, D. N.

Conforti, M.

De Angelis, C.

de Sterke, C. M.

C. M. de Sterke, L. C. Botten, A. A. Asatryan, T. P. White, and R. C. McPhedran, “Modes of coupled photonic crystal waveguides,” Opt. Lett. 29, 1384-1386 (2004).
[CrossRef]

N. G. Broderick, C. M. de Sterke, and B. J. Eggleton, “Soliton solutions in Rowland ghost gaps,” Phys. Rev. E 52, R5788-R5791 (1995).
[CrossRef]

Eggleton, B. J.

N. G. Broderick, C. M. de Sterke, and B. J. Eggleton, “Soliton solutions in Rowland ghost gaps,” Phys. Rev. E 52, R5788-R5791 (1995).
[CrossRef]

Eisenberg, H. S.

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

H. S. Eisenberg and Y. Silberberg, “Diffraction management,” Phys. Rev. Lett. 85, 1863-1866 (2000).
[CrossRef] [PubMed]

Joannopoulos, J. D.

Johnson, S. G.

Kivshar, Y. S.

Krolikowski, W.

A. A. Sukhorukov, D. Neshev, W. Krolikowski, and Y. S. Kivshar, “Nonlinear Bloch-wave interaction and Bragg scattering in optically induced lattices,” Phys. Rev. Lett. 92, 093901 (2004).
[CrossRef] [PubMed]

Lederer, F.

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817-823 (2003).
[CrossRef] [PubMed]

U. Peschel and F. Lederer, “Oscillation and decay of discrete solitons in modulated waveguide arrays,” J. Opt. Soc. Am. B 19, 544-549 (2002).
[CrossRef]

Locatelli, A.

Longhi, S.

Malkova, N.

N. Malkova and C. Z. Ning, “Interplay between Tamm-like and Shockley-like surface states in photonic crystals,” Phys. Rev. B 76, 045305 (2007).
[CrossRef]

Mandelik, D.

R. Morandotti, D. Mandelik, Y. Silberberg, J. S. Aitchison, M. Sorel, D. N. Christodoulides, A. A. Sukhorukov, and Y. S. Kivshar, “Observation of discrete gap solitons in binary waveguide arrays,” Opt. Lett. 29, 2890-2892 (2004).
[CrossRef]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

McPhedran, R. C.

Modotto, D.

Molina, M. I.

Morandotti, R.

R. Morandotti, D. Mandelik, Y. Silberberg, J. S. Aitchison, M. Sorel, D. N. Christodoulides, A. A. Sukhorukov, and Y. S. Kivshar, “Observation of discrete gap solitons in binary waveguide arrays,” Opt. Lett. 29, 2890-2892 (2004).
[CrossRef]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

Neshev, D.

A. A. Sukhorukov, D. Neshev, W. Krolikowski, and Y. S. Kivshar, “Nonlinear Bloch-wave interaction and Bragg scattering in optically induced lattices,” Phys. Rev. Lett. 92, 093901 (2004).
[CrossRef] [PubMed]

Ning, C. Z.

N. Malkova and C. Z. Ning, “Interplay between Tamm-like and Shockley-like surface states in photonic crystals,” Phys. Rev. B 76, 045305 (2007).
[CrossRef]

Peschel, U.

Silberberg, Y.

R. Morandotti, D. Mandelik, Y. Silberberg, J. S. Aitchison, M. Sorel, D. N. Christodoulides, A. A. Sukhorukov, and Y. S. Kivshar, “Observation of discrete gap solitons in binary waveguide arrays,” Opt. Lett. 29, 2890-2892 (2004).
[CrossRef]

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817-823 (2003).
[CrossRef] [PubMed]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

H. S. Eisenberg and Y. Silberberg, “Diffraction management,” Phys. Rev. Lett. 85, 1863-1866 (2000).
[CrossRef] [PubMed]

Sorel, M.

Sukhorukov, A. A.

Vicencio, R. A.

White, T. P.

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

Nature (1)

D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in linear and nonlinear waveguide lattices,” Nature 424, 817-823 (2003).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (9)

S. Longhi, “Image reconstruction in segmented waveguide arrays,” Opt. Lett. 33, 473-475 (2008).
[CrossRef] [PubMed]

C. M. de Sterke, L. C. Botten, A. A. Asatryan, T. P. White, and R. C. McPhedran, “Modes of coupled photonic crystal waveguides,” Opt. Lett. 29, 1384-1386 (2004).
[CrossRef]

A. Locatelli, M. Conforti, D. Modotto, and C. De Angelis, “Diffraction engineering in arrays of photonic crystal waveguides,” Opt. Lett. 30, 2894-2896 (2005).
[CrossRef] [PubMed]

A. Locatelli, M. Conforti, D. Modotto, and C. De Angelis, “Discrete negative refraction in photonic crystal waveguide arrays,” Opt. Lett. 31, 1343-1345 (2006).
[CrossRef] [PubMed]

R. A. Vicencio, M. I. Molina, and Y. S. Kivshar, “Controlled switching of discrete solitons in waveguide arrays,” Opt. Lett. 28, 1942-1944 (2003).
[CrossRef] [PubMed]

A. A. Sukhorukov and Y. S. Kivshar, “Discrete gap solitons in modulated waveguide arrays,” Opt. Lett. 27, 2112-2114 (2002).
[CrossRef]

R. Morandotti, D. Mandelik, Y. Silberberg, J. S. Aitchison, M. Sorel, D. N. Christodoulides, A. A. Sukhorukov, and Y. S. Kivshar, “Observation of discrete gap solitons in binary waveguide arrays,” Opt. Lett. 29, 2890-2892 (2004).
[CrossRef]

A. A. Sukhorukov and Y. S. Kivshar, “Soliton control and Bloch-wave filtering in periodic photonic lattices,” Opt. Lett. 30, 1849-1851 (2005).
[CrossRef] [PubMed]

S. Longhi, “Multiband diffraction and refraction control in binary arrays of periodically curved waveguides,” Opt. Lett. 31, 1857-1859 (2006).
[CrossRef] [PubMed]

Phys. Rev. B (1)

N. Malkova and C. Z. Ning, “Interplay between Tamm-like and Shockley-like surface states in photonic crystals,” Phys. Rev. B 76, 045305 (2007).
[CrossRef]

Phys. Rev. E (1)

N. G. Broderick, C. M. de Sterke, and B. J. Eggleton, “Soliton solutions in Rowland ghost gaps,” Phys. Rev. E 52, R5788-R5791 (1995).
[CrossRef]

Phys. Rev. Lett. (4)

A. A. Sukhorukov, D. Neshev, W. Krolikowski, and Y. S. Kivshar, “Nonlinear Bloch-wave interaction and Bragg scattering in optically induced lattices,” Phys. Rev. Lett. 92, 093901 (2004).
[CrossRef] [PubMed]

A. A. Sukhorukov and Y. S. Kivshar, “Multigap discrete vector solitons,” Phys. Rev. Lett. 91, 113902 (2003).
[CrossRef] [PubMed]

D. Mandelik, H. S. Eisenberg, Y. Silberberg, R. Morandotti, and J. S. Aitchison, “Band-gap structure of waveguide arrays and excitation of Floquet-Bloch solitons,” Phys. Rev. Lett. 90, 053902 (2003).
[CrossRef] [PubMed]

H. S. Eisenberg and Y. Silberberg, “Diffraction management,” Phys. Rev. Lett. 85, 1863-1866 (2000).
[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 (8)

Fig. 1
Fig. 1

Schematic view of the analyzed binary waveguide arrays. Two array periods are depicted. The black areas indicate the identical waveguides, Λ is the array period, Λ 1 is the distance between the n th and ( n + 1 ) th waveguide. C 1 and C 2 are the coupling coefficients between the ( n 1 ) th and n th , and the n th and the ( n + 1 ) th waveguides, respectively.

Fig. 2
Fig. 2

Diffraction curves for modes 1 and 2. C 1 and C s are fixed to 1 and 1 16 , respectively. (a) C 2 = 0.9 (solid curves), C 2 = 0.5 (dashed curves), C 2 = 0.25 (dashed-dotted curves). (b) C 2 = 0.9 (solid curves), C 2 = 0.5 (dashed curves), C 2 = 0.25 (dashed-dotted curves).

Fig. 3
Fig. 3

Group velocity curves for modes 1 and 2. C 1 and C s are fixed to 1 and 1 16 , respectively. (a) C 2 = 0.9 (solid curves), C 2 = 0.5 (dashed curves), C 2 = 0.25 (dashed-dotted curves). (b) C 2 = 0.9 (solid curves), C 2 = 0.5 (dashed curves), C 2 = 0.25 (dashed-dotted curves).

Fig. 4
Fig. 4

Energy ratio R. C 1 and C s are fixed to 1 and 1 16 , respectively. (a) C 2 = 0.9 (solid curves), C 2 = 0.5 (dashed curves), C 2 = 0.25 (dashed-dotted curves). (b) C 2 = 0.9 (solid curves), C 2 = 0.5 (dashed curves), C 2 = 0.25 (dashed-dotted curves).

Fig. 5
Fig. 5

Sketch of a PC binary waveguide array composed of identical waveguides separated by three (negative coupling coefficient) and four rods (positive coupling coefficient). Two periods of the binary array are depicted, as in Fig. 1.

Fig. 6
Fig. 6

Normalized diffraction curves calculated from the exact solution of Maxwell’s equations (solid curves) and from CMT (dashed curves). (a) PC binary waveguide array with two- and three-rod spacing. (b) PC binary waveguide array with three- and four-rod spacing.

Fig. 7
Fig. 7

Example of light propagation in a PC binary waveguide array with two- and three-rod spacing. (a) Field intensity calculated through a 2D FDTD simulation (the units of the axes are micrometers). (b) Input (light) and output field intensity (dark) from the 2D FDTD simulation. (c) Input (empty circles) and output mode intensity (filled circles) calculated through CMT.

Fig. 8
Fig. 8

Example of light propagation in a PC binary waveguide array with three- and four-rod spacing. (a) Field intensity calculated through a 2D FDTD simulation (the units of the axes are micrometers). (b) Input (light) and output field intensity (dark) from the 2D FDTD simulation. (c) Input (empty circles) and output mode intensity (filled circles) calculated through CMT.

Equations (20)

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

d A ( n , z ) d z i β A ( n , z ) i C n 1 A ( n 1 , z ) i C n + 1 A ( n + 1 , z ) = 0 ,
d X ( n , z ) d z = i C 1 Y ( n 1 , z ) + i C 2 Y ( n , z ) ,
d Y ( n , z ) d z = i C 2 X ( n , z ) + i C 1 X ( n + 1 , z ) .
X ( n , z ) = [ L 1 exp ( i k z 1 z ) + L 2 exp ( i k z 2 z ) ] exp ( i k n ) ,
Y ( n , z ) = ρ ( k ) [ L 1 exp ( i k z 1 z ) L 2 exp ( i k z 2 z ) ] exp ( i k n ) ,
k z 1 , 2 ( k ) = ± ( C 1 2 + C 2 2 + 2 C 1 C 2 cos ( k ) ) 1 2 ,
ρ ( k ) = C 2 + C 1 exp ( i k ) ( C 1 2 + C 2 2 + 2 C 1 C 2 cos ( k ) ) 1 2 ,
d A ( n , z ) d z i β A ( n , z ) i C n 1 A ( n 1 , z ) i C n + 1 A ( n + 1 , z ) i C s [ A ( n 2 , z ) + A ( n + 2 , z ) ] = 0 ,
v g 1 , 2 = d k z 1 , 2 d k , D 1 , 2 = d 2 k z 1 , 2 d k 2 ,
E ( x , z ) = n [ X ( n , z ) M 2 n ( x ) + Y ( n , z ) M 2 n + 1 ( x ) ] ,
= L 1 m 1 k ( x ) exp ( i k z 1 z ) + L 2 m 2 k ( x ) exp ( i k z 2 z ) ,
m 1 , 2 k ( x ) = n [ M 2 n ( x ) ± ρ ( k ) M 2 n + 1 ( x ) ] exp ( i k n ) .
m 1 , 2 k ( x ) = M 0 ( x ) [ δ Λ ( x ) exp ( i k x Λ ) ± δ Λ ( x Λ 1 ) ρ ( k ) exp ( i k ( x Λ 1 ) Λ ) ] ,
m 1 , 2 k ( x ) = [ u 1 , 2 k ( x ) δ Λ ( x ) ] exp ( i k x Λ ) = u p 1 , 2 k ( x ) exp ( i k x Λ ) ,
u 1 , 2 k ( x ) = [ M 0 ( x ) ± ρ ( k ) M 0 ( x Λ 1 ) ] exp ( i k x Λ ) .
E ( x , z ) a 1 ( k x 0 ) u p 1 , k x 0 ( x ) k x I ̂ ( k x ) exp ( i k x x ) exp ( i k z 1 z ) d k x + a 2 ( k x 0 ) u p 2 , k x 0 ( x ) k x I ̂ ( k x ) exp ( i k x x ) exp ( i k z 2 z ) d k x ,
E 1 , 2 = M ̂ 0 ( k x 0 ) 2 E M 0 2 1 ± exp [ i ( k x 0 Λ 1 α ( k x 0 ) ) ] 2 x I ( x ) 2 u p 1 , 2 k x 0 ( x ) 2 d x ,
R = E 1 E 1 + E 2 cos ( k x 0 Λ 1 α ( k x 0 ) 2 ) 2 .
I ( x ) [ 1 + exp ( i ( k 0 α ( 2 k 0 ) ) ) 2 ] m 1 2 k 0 ( x ) + [ 1 exp ( i ( k 0 α ( 2 k 0 ) ) ) 2 ] m 2 2 k 0 ( x ) .
R = cos ( k 0 α ( 2 k 0 ) 2 ) 2 ,

Metrics