Abstract

We analyze the supermodes in multiple coupled photonic crystal waveguides for long-wavelengths. In the tight-binding limit we obtain analytic results that agree with fully numerical calculations. We find that when the field flips sign after a single photonic crystal period, and there is an odd number of periods between adjacent waveguides, the supermode order is reversed, compared to that in conventional coupled waveguides, generalizing earlier results obtained for two coupled waveguides.

© 2006 Optical Society of America

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References

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  1. See, e.g., Y. Sugimoto, Y. Tanaka, N. Ikeda, Y. Nakamura, K. Asakawa, and K. Inoue, "Low propagation loss of 0.76 dB/mm in GaAs-based single-line-defect two-dimensional photonic crystal slab waveguides up to 1 cm in length" Opt. Express 12, 1090-1096 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1090">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-6-1090</a>.
    [CrossRef] [PubMed]
  2. A. Martinez, F. Cuesta, and J. Marti, "Ultrashort 2-D photonic crystal directional couplers," IEEE Photonics Technol. Lett. 15, 694-696 (2003).
    [CrossRef]
  3. 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]
  4. P. Yeh, Optical Waves in Layered Media (Wiley, Hoboken, 1988).
  5. L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Non-Relativistic Theory), 3rd Ed. (Pergamon, Oxford, 1977).
  6. L.C. Botten, N.A. Nicorovici, R.C. McPhedran, C.M. de Sterke, and A.A. Asatryan, "Photonic bandstructure calculations using scattering matrices," Phys. Rev. E 64, 046603:1-18 (2001).
    [CrossRef]
  7. L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, "Electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculation. Part 1: Formulation", J. Opt. Soc. Am. 17, 2165-76 (2000).
    [CrossRef]
  8. M. Born and E. Wolf, Principles of Optics, 6th Edition (Pergamon, Oxford, 1980), p. 66.
  9. L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke and R. C. McPhedran, "Bloch mode scattering matrix methods for modelling extended photonic crystal structures. Part I: Theory," Phys. Rev. E 70, 056606, 1-13 (2004).
    [CrossRef]
  10. D. Felbacq, A. Moreau, and Rafik Smaâli, "Goos-Hänchen effect in the gaps of photonic crystals," Opt. Lett. 28, 1633-1635 (2003).
    [CrossRef] [PubMed]
  11. C.M. de Sterke, "Superstructure gratings in the tight-binding approximation," Phys. Rev. E 57 3502-3509 (1998).
    [CrossRef]
  12. M. Bayindir, B. Temelkuran, and E. Özbay, "Tight-binding description of the coupled defect modes in three-dimensional photonic crystals," Phys. Rev. Lett. 84, 2140 (2000).
    [CrossRef] [PubMed]
  13. S. Mookherjea and A. Yariv, "Optical pulse propagation in the tight-binding approximation," Opt. Express 9, 91-96 (2001) <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-2-91">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-9-2-91</a>.
    [CrossRef] [PubMed]
  14. N.W. Ashcroft and N.D. Mermin, Solid State Physics,(Saunders College, Philadelphia, 1976).

IEEE Photonics Technol. Lett.

A. Martinez, F. Cuesta, and J. Marti, "Ultrashort 2-D photonic crystal directional couplers," IEEE Photonics Technol. Lett. 15, 694-696 (2003).
[CrossRef]

J. Opt. Soc. Am.

L. C. Botten, N. A. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, "Electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculation. Part 1: Formulation", J. Opt. Soc. Am. 17, 2165-76 (2000).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. E

L.C. Botten, N.A. Nicorovici, R.C. McPhedran, C.M. de Sterke, and A.A. Asatryan, "Photonic bandstructure calculations using scattering matrices," Phys. Rev. E 64, 046603:1-18 (2001).
[CrossRef]

C.M. de Sterke, "Superstructure gratings in the tight-binding approximation," Phys. Rev. E 57 3502-3509 (1998).
[CrossRef]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke and R. C. McPhedran, "Bloch mode scattering matrix methods for modelling extended photonic crystal structures. Part I: Theory," Phys. Rev. E 70, 056606, 1-13 (2004).
[CrossRef]

Phys. Rev. Lett.

M. Bayindir, B. Temelkuran, and E. Özbay, "Tight-binding description of the coupled defect modes in three-dimensional photonic crystals," Phys. Rev. Lett. 84, 2140 (2000).
[CrossRef] [PubMed]

Other

N.W. Ashcroft and N.D. Mermin, Solid State Physics,(Saunders College, Philadelphia, 1976).

M. Born and E. Wolf, Principles of Optics, 6th Edition (Pergamon, Oxford, 1980), p. 66.

P. Yeh, Optical Waves in Layered Media (Wiley, Hoboken, 1988).

L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Non-Relativistic Theory), 3rd Ed. (Pergamon, Oxford, 1977).

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Figures (5)

Fig. 1.
Fig. 1.

Schematic of the geometry that we are considering, consisting of m coupled PCWs, separated by m-1 barriers, all sandwiched between two semi-infinite PCs. Each barrier consists of l PC periods.

Fig. 2.
Fig. 2.

Exact numerical results for β 2 s - β 0 2 with increasing layer thickness l for (a) m = 5, and (b) m = 6 identical waveguides. Parameter values are as given in the text, with the red and blue dots respectively showing values for even and odd symmetric supermodes.

Fig. 3.
Fig. 3.

Comparison of exact numerical results for a set of (a) m = 5, and (b) m = 6 identical waveguides with parameters given in the text. The horizontal lines give the limiting values of F(βs ) in the tight binding approximation, cos[/(m + 1)], whereas the dots give exact results as discussed in the text. The red dots are for even modes while the blue dots indicate odd modes.

Fig. 4.
Fig. 4.

Fields of the supermodes for m = 5 and (a) l = 6 and (b) l = 7, with the associated propagation constants βsd indicated above the figures. In each case, the fundamental mode is shown on the right, with consecutive higher order modes shown towards the left. The guides are oriented parallel to the x-axis.

Fig. 5.
Fig. 5.

As Figure 4, but for m = 6 and (a) l = 6 and (b) l = 7.

Equations (28)

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V j ( r ) = p = χ P 1 2 [ g j , p e P ( y y i ) + g j , p + e P ( y y j ) ] e i β P x ,
R l = ( R Q l R Q l ) ( I R Q l R Q l ) 1 ,
T l = ( I R 2 ) Q l ( I R Q l R Q l ) 1 ,
f 1 = R ˜ f 1 + , f 1 + = R ˜ m 1 f 1 + T ˜ m 1 f m + ,
f m + = R ˜ f m , f m = T ˜ m 1 f 1 + R ˜ m 1 f m + ,
˜ n = ( T ˜ n R ˜ n T ˜ n 1 R ˜ n R ˜ n T ˜ n 1 T ˜ n 1 R ˜ n T ˜ n 1 ) A n B n C n D n = ˜ n ( A B C D ) n ,
g m g m + = ( A n B n C n D n ) g 1 g 1 + .
[ f 1 + ± f m f 1 ± f m + ] = ˜ [ ± ( f 1 ± f m + ) ± ( f 1 + ± f m ) ] ,
( A n R ˜ + B n ) g = σg , ( C n R ˜ + D n ) g = σ R ˜ g ,
M ( k , β o ) g [ I ( R ˜ n + σ T ˜ n ) R ˜ ] g = 0 ,
A n = A u n 1 ( t ) u n 2 ( t ) , B n = B u n 1 ( t ) C n = C u n 1 ( t ), D n = D u n 1 ( t ) u n 2 ( t ) ,
t = 1 2 ( A + D ) , A = ( T l 2 R l 2 ) P T l ,
B = C = R l T l , D = 1 T l P .
R l = ρ ( 1 μ 2 l ) 1 ρ 2 μ 2 l , T l = ( 1 ρ 2 ) μ l 1 ρ 2 μ 2 l ,
= ± exp [ iv ( ξ ) ] ,
( APρ + B ) u n 1 ( t ) u n 2 ( t ) σ = 0
t = α ( 2 ρ ) + O ( ξ ) , + B = 2 t + O ( ξ ) , = 1 + O ( ξ ) .
u n ( t ) σ = 0 .
k 2 β s 2 h + arg [ ρ ( β s ) ] = 2 [ ρ ( β s ) ] cos ( ϑ s ) μ ( β s ) j + .
β s 2 β 0 2 4 [ ρ ( β s ) ] h χ s + β s 1 ( arg [ ρ ( β s ) ] β ) μ l cos ϑ s .
β s 2 β 0 2 4 χ 0 [ ρ ( β 0 ) ] h eff μ l cos ϑ s .
g j = T ˜ j 1 g 1 + R ˜ j 1 g j + , g j + = T ˜ m j g m + + R ˜ m j g j
g j + g j + = T ˜ j 1 ( 1 + R ˜ m j ) 1 g j + T ˜ m j ( 1 + R ˜ j 1 ) 1 g m + 1 R ˜ j 1 R ˜ m j .
g j + g j + sin j ϑ s + O ( ξ ) ,
F ( β s ) = ( β s 2 β 0 2 ) h eff 4 χ 0 [ ρ ( β 0 ) ] μ l ,
v 1 ρP v 1 + ( 1 ρ 2 ) ξP g 2 + , v 2 ( 1 ρ 2 ) ξP g 1 + ρP v 2 ,
v 1 ρP v 1 + ( 1 ρ 2 ) ξ 2 ρ v 2 ,
v 2 ( 1 ρ 2 ) ξ 2 ρ v 1 ρP v 2 ,

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