Abstract

Slow-wave optical structures such as coupled photonic crystal cavities, coupled microresonators, and similar coupled-resonator optical waveguides are being proposed for slowing light because of the nature of their dispersion relationship. Since the group velocity becomes small, slow light and enhanced light–matter interaction may be observed at the edges of the waveguiding band. We derive a model of the effects of disorder on slow light in such structures, obtaining a relationship between the root-mean-square variation in the coupling coefficients and how slow the light is at the band edge.

© 2007 Optical Society of America

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  1. N. Stefanou and A. Modinos, Phys. Rev. B 57, 12127 (1998).
    [CrossRef]
  2. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, Opt. Lett. 24, 711 (1999).
    [CrossRef]
  3. G. Gutroff, M. Bayer, J. P. Reithmaier, A. Forchel, P. A. Knipp, and T. L. Reinecke, Phys. Rev. B 64, 155313 (2001).
    [CrossRef]
  4. M. Ghulinyan, M. Galli, C. Toninelli, J. Bertolotti, S. Gottardo, F. Marabelli, D. S. Wiersma, L. Pavesi, and L. C. Andreani, Appl. Phys. Lett. 88, 241103 (2006).
    [CrossRef]
  5. J. C. Slater, Microwave Electronics (Dover, 1969), Chap. VIII.
  6. M. Bayindir, B. Temelkuran, and E. Ozbay, Phys. Rev. Lett. 84, 2140 (2000).
    [CrossRef] [PubMed]
  7. S. Mookherjea, J. Opt. Soc. Am. B 23, 1137 (2006).
    [CrossRef]
  8. P. Dean, Proc. Phys. Soc. London 84, 727 (1964).
    [CrossRef]
  9. H. Matsuoka and R. Grobe, Phys. Rev. E 71, 046606 (2005).
    [CrossRef]
  10. B. Z. Steinberg, A. Boag, and R. Lisitsin, J. Opt. Soc. Am. A 20, 138 (2003).
    [CrossRef]
  11. M. Bayindir, E. Cubukcu, I. Bulu, T. Tut, E. Ozbay, and C. Soukoulis, Phys. Rev. B 64, 195113 (2001).
    [CrossRef]
  12. F. J. Dyson, Phys. Rev. 92, 1331 (1953).
    [CrossRef]
  13. E. R. Smith, J. Phys. C 3, 1419 (1970).
    [CrossRef]
  14. J. K. S. Poon, L. Zhu, G. DeRose, and A. Yariv, Opt. Lett. 31, 456 (2006).
    [CrossRef] [PubMed]
  15. In Ref. , the authors derived Fig. 4 (inset) by fitting the data points by a sum-of-cosines, which forces the slope at the band edge to zero, even if this seems not to be indicated by the measured data points. We extracted Sbe≈6.5 from the data points (open circles) shown in Fig. 4.

2006

M. Ghulinyan, M. Galli, C. Toninelli, J. Bertolotti, S. Gottardo, F. Marabelli, D. S. Wiersma, L. Pavesi, and L. C. Andreani, Appl. Phys. Lett. 88, 241103 (2006).
[CrossRef]

S. Mookherjea, J. Opt. Soc. Am. B 23, 1137 (2006).
[CrossRef]

J. K. S. Poon, L. Zhu, G. DeRose, and A. Yariv, Opt. Lett. 31, 456 (2006).
[CrossRef] [PubMed]

2005

H. Matsuoka and R. Grobe, Phys. Rev. E 71, 046606 (2005).
[CrossRef]

2003

2001

M. Bayindir, E. Cubukcu, I. Bulu, T. Tut, E. Ozbay, and C. Soukoulis, Phys. Rev. B 64, 195113 (2001).
[CrossRef]

G. Gutroff, M. Bayer, J. P. Reithmaier, A. Forchel, P. A. Knipp, and T. L. Reinecke, Phys. Rev. B 64, 155313 (2001).
[CrossRef]

2000

M. Bayindir, B. Temelkuran, and E. Ozbay, Phys. Rev. Lett. 84, 2140 (2000).
[CrossRef] [PubMed]

1999

1998

N. Stefanou and A. Modinos, Phys. Rev. B 57, 12127 (1998).
[CrossRef]

1970

E. R. Smith, J. Phys. C 3, 1419 (1970).
[CrossRef]

1964

P. Dean, Proc. Phys. Soc. London 84, 727 (1964).
[CrossRef]

1953

F. J. Dyson, Phys. Rev. 92, 1331 (1953).
[CrossRef]

Andreani, L. C.

M. Ghulinyan, M. Galli, C. Toninelli, J. Bertolotti, S. Gottardo, F. Marabelli, D. S. Wiersma, L. Pavesi, and L. C. Andreani, Appl. Phys. Lett. 88, 241103 (2006).
[CrossRef]

Bayer, M.

G. Gutroff, M. Bayer, J. P. Reithmaier, A. Forchel, P. A. Knipp, and T. L. Reinecke, Phys. Rev. B 64, 155313 (2001).
[CrossRef]

Bayindir, M.

M. Bayindir, E. Cubukcu, I. Bulu, T. Tut, E. Ozbay, and C. Soukoulis, Phys. Rev. B 64, 195113 (2001).
[CrossRef]

M. Bayindir, B. Temelkuran, and E. Ozbay, Phys. Rev. Lett. 84, 2140 (2000).
[CrossRef] [PubMed]

Bertolotti, J.

M. Ghulinyan, M. Galli, C. Toninelli, J. Bertolotti, S. Gottardo, F. Marabelli, D. S. Wiersma, L. Pavesi, and L. C. Andreani, Appl. Phys. Lett. 88, 241103 (2006).
[CrossRef]

Boag, A.

Bulu, I.

M. Bayindir, E. Cubukcu, I. Bulu, T. Tut, E. Ozbay, and C. Soukoulis, Phys. Rev. B 64, 195113 (2001).
[CrossRef]

Cubukcu, E.

M. Bayindir, E. Cubukcu, I. Bulu, T. Tut, E. Ozbay, and C. Soukoulis, Phys. Rev. B 64, 195113 (2001).
[CrossRef]

Dean, P.

P. Dean, Proc. Phys. Soc. London 84, 727 (1964).
[CrossRef]

DeRose, G.

Dyson, F. J.

F. J. Dyson, Phys. Rev. 92, 1331 (1953).
[CrossRef]

Forchel, A.

G. Gutroff, M. Bayer, J. P. Reithmaier, A. Forchel, P. A. Knipp, and T. L. Reinecke, Phys. Rev. B 64, 155313 (2001).
[CrossRef]

Galli, M.

M. Ghulinyan, M. Galli, C. Toninelli, J. Bertolotti, S. Gottardo, F. Marabelli, D. S. Wiersma, L. Pavesi, and L. C. Andreani, Appl. Phys. Lett. 88, 241103 (2006).
[CrossRef]

Ghulinyan, M.

M. Ghulinyan, M. Galli, C. Toninelli, J. Bertolotti, S. Gottardo, F. Marabelli, D. S. Wiersma, L. Pavesi, and L. C. Andreani, Appl. Phys. Lett. 88, 241103 (2006).
[CrossRef]

Gottardo, S.

M. Ghulinyan, M. Galli, C. Toninelli, J. Bertolotti, S. Gottardo, F. Marabelli, D. S. Wiersma, L. Pavesi, and L. C. Andreani, Appl. Phys. Lett. 88, 241103 (2006).
[CrossRef]

Grobe, R.

H. Matsuoka and R. Grobe, Phys. Rev. E 71, 046606 (2005).
[CrossRef]

Gutroff, G.

G. Gutroff, M. Bayer, J. P. Reithmaier, A. Forchel, P. A. Knipp, and T. L. Reinecke, Phys. Rev. B 64, 155313 (2001).
[CrossRef]

Knipp, P. A.

G. Gutroff, M. Bayer, J. P. Reithmaier, A. Forchel, P. A. Knipp, and T. L. Reinecke, Phys. Rev. B 64, 155313 (2001).
[CrossRef]

Lee, R. K.

Lisitsin, R.

Marabelli, F.

M. Ghulinyan, M. Galli, C. Toninelli, J. Bertolotti, S. Gottardo, F. Marabelli, D. S. Wiersma, L. Pavesi, and L. C. Andreani, Appl. Phys. Lett. 88, 241103 (2006).
[CrossRef]

Matsuoka, H.

H. Matsuoka and R. Grobe, Phys. Rev. E 71, 046606 (2005).
[CrossRef]

Modinos, A.

N. Stefanou and A. Modinos, Phys. Rev. B 57, 12127 (1998).
[CrossRef]

Mookherjea, S.

Ozbay, E.

M. Bayindir, E. Cubukcu, I. Bulu, T. Tut, E. Ozbay, and C. Soukoulis, Phys. Rev. B 64, 195113 (2001).
[CrossRef]

M. Bayindir, B. Temelkuran, and E. Ozbay, Phys. Rev. Lett. 84, 2140 (2000).
[CrossRef] [PubMed]

Pavesi, L.

M. Ghulinyan, M. Galli, C. Toninelli, J. Bertolotti, S. Gottardo, F. Marabelli, D. S. Wiersma, L. Pavesi, and L. C. Andreani, Appl. Phys. Lett. 88, 241103 (2006).
[CrossRef]

Poon, J. K. S.

Reinecke, T. L.

G. Gutroff, M. Bayer, J. P. Reithmaier, A. Forchel, P. A. Knipp, and T. L. Reinecke, Phys. Rev. B 64, 155313 (2001).
[CrossRef]

Reithmaier, J. P.

G. Gutroff, M. Bayer, J. P. Reithmaier, A. Forchel, P. A. Knipp, and T. L. Reinecke, Phys. Rev. B 64, 155313 (2001).
[CrossRef]

Scherer, A.

Slater, J. C.

J. C. Slater, Microwave Electronics (Dover, 1969), Chap. VIII.

Smith, E. R.

E. R. Smith, J. Phys. C 3, 1419 (1970).
[CrossRef]

Soukoulis, C.

M. Bayindir, E. Cubukcu, I. Bulu, T. Tut, E. Ozbay, and C. Soukoulis, Phys. Rev. B 64, 195113 (2001).
[CrossRef]

Stefanou, N.

N. Stefanou and A. Modinos, Phys. Rev. B 57, 12127 (1998).
[CrossRef]

Steinberg, B. Z.

Temelkuran, B.

M. Bayindir, B. Temelkuran, and E. Ozbay, Phys. Rev. Lett. 84, 2140 (2000).
[CrossRef] [PubMed]

Toninelli, C.

M. Ghulinyan, M. Galli, C. Toninelli, J. Bertolotti, S. Gottardo, F. Marabelli, D. S. Wiersma, L. Pavesi, and L. C. Andreani, Appl. Phys. Lett. 88, 241103 (2006).
[CrossRef]

Tut, T.

M. Bayindir, E. Cubukcu, I. Bulu, T. Tut, E. Ozbay, and C. Soukoulis, Phys. Rev. B 64, 195113 (2001).
[CrossRef]

Wiersma, D. S.

M. Ghulinyan, M. Galli, C. Toninelli, J. Bertolotti, S. Gottardo, F. Marabelli, D. S. Wiersma, L. Pavesi, and L. C. Andreani, Appl. Phys. Lett. 88, 241103 (2006).
[CrossRef]

Xu, Y.

Yariv, A.

Zhu, L.

Appl. Phys. Lett.

M. Ghulinyan, M. Galli, C. Toninelli, J. Bertolotti, S. Gottardo, F. Marabelli, D. S. Wiersma, L. Pavesi, and L. C. Andreani, Appl. Phys. Lett. 88, 241103 (2006).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Phys. C

E. R. Smith, J. Phys. C 3, 1419 (1970).
[CrossRef]

Opt. Lett.

Phys. Rev.

F. J. Dyson, Phys. Rev. 92, 1331 (1953).
[CrossRef]

Phys. Rev. B

M. Bayindir, E. Cubukcu, I. Bulu, T. Tut, E. Ozbay, and C. Soukoulis, Phys. Rev. B 64, 195113 (2001).
[CrossRef]

G. Gutroff, M. Bayer, J. P. Reithmaier, A. Forchel, P. A. Knipp, and T. L. Reinecke, Phys. Rev. B 64, 155313 (2001).
[CrossRef]

N. Stefanou and A. Modinos, Phys. Rev. B 57, 12127 (1998).
[CrossRef]

Phys. Rev. E

H. Matsuoka and R. Grobe, Phys. Rev. E 71, 046606 (2005).
[CrossRef]

Phys. Rev. Lett.

M. Bayindir, B. Temelkuran, and E. Ozbay, Phys. Rev. Lett. 84, 2140 (2000).
[CrossRef] [PubMed]

Proc. Phys. Soc. London

P. Dean, Proc. Phys. Soc. London 84, 727 (1964).
[CrossRef]

Other

J. C. Slater, Microwave Electronics (Dover, 1969), Chap. VIII.

In Ref. , the authors derived Fig. 4 (inset) by fitting the data points by a sum-of-cosines, which forces the slope at the band edge to zero, even if this seems not to be indicated by the measured data points. We extracted Sbe≈6.5 from the data points (open circles) shown in Fig. 4.

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Figures (3)

Fig. 1
Fig. 1

Density of states, ρ ̂ ( ω ) , obtained from a numerical calculation of the eigenvalues of the matrix M in Eq. (1). Indicated values of δ κ κ represent the standard deviation in the coupling coefficients. The horizontal axis is the radian frequency detuning from the band center.

Fig. 2
Fig. 2

Dispersion relationship of a weakly disordered optical tight-binding lattice for δ κ κ = 3 % in the uniform random distribution is compared with the theoretical result (straight line). (a) The shape of the ideal dispersion curve, indicated by the solid curve, is reproduced over most of the range by the data points, except near the band edge, where as shown by (b), the slope in the disordered structure is not exactly zero and a tail extends into the bandgap region.

Fig. 3
Fig. 3

Band-edge slowing factor obtained from numerical calculations of ρ ( ω ) . The horizontal axis is the inverse strength of disorder, ( δ κ κ ) 1 , varying from 50% to 1%. U and G refer to the uniform and Gaussian random distributions in Eqs. (9a, 9b), respectively. Those equations describe the straight-line best fit to the data points. The vertical distance between the lines is approximately ( 2 3 ) log 10 12 , to equate the variances of the two distributions.

Equations (12)

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i d d t u = M u ,
k n = 1 N e i n k R E single ( r n R z ̂ ) ,
κ ̂ κ + δ κ U [ 1 2 , 1 2 ] , ( Uniform ) ,
κ ̂ κ + δ κ G [ 0 ; 1 ] , ( Gaussian ) ,
[ v g ] max = 1 [ ρ ̂ ( ω ) ] min 1 π 10 μ m = 1.0 × 10 6 m s .
ϕ ( k ) k ( k k ) L × ( number of states at k ) × ( probability of transition : k k ) = d k ( k k ) L ρ ideal ( k ) W ( k , k ) ,
ρ ( k ) = ρ ideal ( k ) ( 1 + 1 L d ϕ d k ) .
v g = 1 2 ρ ̂ ( ω ) 1 2 π R ( 1 + 1 N d ϕ R d k ) .
S be v g at band center v g at band edge ,
log 10 ( S be ) = 0.644 log 10 ( κ δ κ ) + 0.272 , ( Uniform ) ,
log 10 ( S be ) = 0.648 log 10 ( κ δ κ ) + 0.281 0.648 log 10 12 , ( Gaussian ) .
log 10 ( S be ) theory = 0.667 log 10 ( κ δ κ ) + 0.313 .

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