Abstract

We propose a scattering matrix approach to the modeling of large-scale photonic crystal circuits and show that the transmission properties of complex circuits can be accurately calculated on the basis of scattering matrices of individual photonic crystal devices and waveguides that connect them. In addition, we show that functional devices such as waveguide bends generally exhibit a discontinuous frequency dependence in the phases associated with their complex reflection and transmission coefficients and emphasize its importance for the adequate modeling of photonic crystal circuits.

© 2003 Optical Society of America

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References

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  1. T. Krauss and T. Baba, eds., feature on Photonic Crystal Structures and Applications, IEEE J. Quantum Electron. 38(7), 724 (2002).
    [CrossRef]
  2. A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, Phys. Rev. Lett. 77, 3787 (1996).
    [CrossRef] [PubMed]
  3. S. F. Mingaleev and Yu. S. Kivshar, Opt. Lett. 27, 231 (2002).
    [CrossRef]
  4. S. F. Mingaleev and Yu. S. Kivshar, J. Opt. Soc. Am. B 19, 2241 (2002).
    [CrossRef]
  5. A. Garcia-Martin, D. Hermann, K. Busch, and P. Wölfle, Mater. Res. Soc. Proc. 722, L1.1 (2002).
  6. L. Li, J. Opt. Soc. Am. A 13, 1024 (1996).
    [CrossRef]
  7. E. Silberstein, Ph. Lalanne, J.-P. Hugonin, and Q. Cao, J. Opt. Soc. Am. A 18, 2865 (2001).
    [CrossRef]
  8. M. Palamaru and Ph. Lalanne, Appl. Phys. Lett. 78, 1466 (2001).
    [CrossRef]
  9. E. Chow, S. Y. Lin, J. R. Wendt, S. G. Johnson, and J. D. Joannopoulos, Opt. Lett. 26, 286 (2001).
    [CrossRef]
  10. A. Chutinan and S. Noda, Phys. Rev. B 62, 4488 (2000).
    [CrossRef]

2002 (4)

A. Garcia-Martin, D. Hermann, K. Busch, and P. Wölfle, Mater. Res. Soc. Proc. 722, L1.1 (2002).

T. Krauss and T. Baba, eds., feature on Photonic Crystal Structures and Applications, IEEE J. Quantum Electron. 38(7), 724 (2002).
[CrossRef]

S. F. Mingaleev and Yu. S. Kivshar, Opt. Lett. 27, 231 (2002).
[CrossRef]

S. F. Mingaleev and Yu. S. Kivshar, J. Opt. Soc. Am. B 19, 2241 (2002).
[CrossRef]

2001 (3)

2000 (1)

A. Chutinan and S. Noda, Phys. Rev. B 62, 4488 (2000).
[CrossRef]

1996 (2)

L. Li, J. Opt. Soc. Am. A 13, 1024 (1996).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, Phys. Rev. Lett. 77, 3787 (1996).
[CrossRef] [PubMed]

Busch, K.

A. Garcia-Martin, D. Hermann, K. Busch, and P. Wölfle, Mater. Res. Soc. Proc. 722, L1.1 (2002).

Cao, Q.

Chen, J. C.

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, Phys. Rev. Lett. 77, 3787 (1996).
[CrossRef] [PubMed]

Chow, E.

Chutinan, A.

A. Chutinan and S. Noda, Phys. Rev. B 62, 4488 (2000).
[CrossRef]

Fan, S. H.

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, Phys. Rev. Lett. 77, 3787 (1996).
[CrossRef] [PubMed]

Garcia-Martin, A.

A. Garcia-Martin, D. Hermann, K. Busch, and P. Wölfle, Mater. Res. Soc. Proc. 722, L1.1 (2002).

Hermann, D.

A. Garcia-Martin, D. Hermann, K. Busch, and P. Wölfle, Mater. Res. Soc. Proc. 722, L1.1 (2002).

Hugonin, J.-P.

Joannopoulos, J. D.

E. Chow, S. Y. Lin, J. R. Wendt, S. G. Johnson, and J. D. Joannopoulos, Opt. Lett. 26, 286 (2001).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, Phys. Rev. Lett. 77, 3787 (1996).
[CrossRef] [PubMed]

Johnson, S. G.

Kivshar, Yu. S.

Kurland, I.

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, Phys. Rev. Lett. 77, 3787 (1996).
[CrossRef] [PubMed]

Lalanne, Ph.

Li, L.

Lin, S. Y.

Mekis, A.

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, Phys. Rev. Lett. 77, 3787 (1996).
[CrossRef] [PubMed]

Mingaleev, S. F.

Noda, S.

A. Chutinan and S. Noda, Phys. Rev. B 62, 4488 (2000).
[CrossRef]

Palamaru, M.

M. Palamaru and Ph. Lalanne, Appl. Phys. Lett. 78, 1466 (2001).
[CrossRef]

Silberstein, E.

Villeneuve, P. R.

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, Phys. Rev. Lett. 77, 3787 (1996).
[CrossRef] [PubMed]

Wendt, J. R.

Wölfle, P.

A. Garcia-Martin, D. Hermann, K. Busch, and P. Wölfle, Mater. Res. Soc. Proc. 722, L1.1 (2002).

Appl. Phys. Lett. (1)

M. Palamaru and Ph. Lalanne, Appl. Phys. Lett. 78, 1466 (2001).
[CrossRef]

IEEE J. Quantum Electron. (1)

T. Krauss and T. Baba, eds., feature on Photonic Crystal Structures and Applications, IEEE J. Quantum Electron. 38(7), 724 (2002).
[CrossRef]

J. Opt. Soc. Am. A (2)

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

Mater. Res. Soc. Proc. (1)

A. Garcia-Martin, D. Hermann, K. Busch, and P. Wölfle, Mater. Res. Soc. Proc. 722, L1.1 (2002).

Opt. Lett. (2)

Phys. Rev. B (1)

A. Chutinan and S. Noda, Phys. Rev. B 62, 4488 (2000).
[CrossRef]

Phys. Rev. Lett. (1)

A. Mekis, J. C. Chen, I. Kurland, S. H. Fan, P. R. Villeneuve, and J. D. Joannopoulos, Phys. Rev. Lett. 77, 3787 (1996).
[CrossRef] [PubMed]

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Figures (4)

Fig. 1
Fig. 1

(a), (b) Geometries of two different waveguide bends, whose complex reflection coefficients, Rω=ρ expiϕ, are characterized by the (c) absolute value, ρω, and (d) phase, ϕω, and have been calculated with effective discrete equations.3 In (c) and (d), the solid lines correspond to the bend in (a) and the dashed curves, to the bend in (b).

Fig. 2
Fig. 2

(a) Double-bend system with bends of the type shown in Fig. 1(a) connected by a waveguide of length L=10a and (b) frequency dependence of the corresponding transmission T (solid curve). For reference, the transmission through a single bend is shown (dashed curve).

Fig. 3
Fig. 3

(a) Double-bend system with bends of the type shown in Fig. 1(b) connected by a waveguide of length L=20a and (b) frequency dependence of the corresponding transmission T (solid curve). For reference, the transmission through a single bend is shown (dashed curve). The importance of the phase ϕω [see Fig. 1(d)] is illustrated by evaluation of Eq. (4) for ϕω=0 (dotted curve; values for T have been shifted downward by 0.25).

Fig. 4
Fig. 4

Frequency dependence of the transmission T for double-bend structures similar to Fig. 2(a) with connecting waveguides of lengths (a) L=a and (b) L=3a. Results of direct numerical simulations (solid curves) and results of Eq. (4) (dashed curve) are shown.

Equations (4)

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uj+1dj=Sˆjujdj+1,    Sˆj=TˆjuuRˆjudRˆjduTˆjdd.
Tˆwguu=Tˆwgdd*=expik1ωL000000expikpωL.
T^cuu=T^buu1-R^audR^bdu-1T^auu, R^cdu=R^adu+T^addR^bdu1-R^audR^bdu-1T^auu, R^cud=R^bud+T^buuR^aud1-R^bduR^aud-1T^bdd, T^cdd=T^add1-R^bduR^aud-1T^bdd.
T2=1-ρ2ω21+ρ4ω-2ρ2ωcos2kωL+2ϕω,

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