Abstract

We report and analyze hybridization of s-state and p-state modes in photonic crystal one-dimensional defect cavity array. When embedding a nano-strip into a dielectric rod photonic crystal, an effective cavity array is made, where each cavity possesses two cavity modes: s-state and p-state. The two modes are laterally even versus the nano-strip direction, and interact with each other, producing defect bands, of which the group velocity becomes zero within the first Brillouin zone. We could model and describe the phenomena by using the tight-binding method, well agreeing with the plane-wave expansion method analysis. We note that the reported s- and p-state mode interaction corresponds to the hybridization of atomic orbital in solid-state physics. The concept of multiple period s-p hybridization and the proposed model can be useful for analyzing and developing novel photonic crystal waveguides and devices.

© 2005 Optical Society of America

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References

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Appl. Phys. Lett. (2)

D. Mori and T. Baba, �??Dispersion-controlled optical group delay device by chirped photonic crystal waveguides,�?? Appl. Phys. Lett. 85, 1101-1103 (2004).
[CrossRef]

A. Y. Petrov and M. Eich, �??Zero dispersion at small group velocities in photonic crystal waveguides,�?? Appl. Phys. Lett. 85, 4866-4868 (2004).
[CrossRef]

IEEE J. Quantum Electron. (1)

E. Ozbay, M. Bayindir, I. Bulu, and E. Cubukcu, �??Investigation of localized coupled-cavity modes in twodimensional photonic bandgap structures,�?? IEEE J. Quantum Electron. 38, 837-843 (2002).
[CrossRef]

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

Opt. Lett. (1)

Phys. Rev. B (4)

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

S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, �??Linear waveguides in photonic-crystal slabs,�?? Phys. Rev. B 62, 8212-8222 (2000).
[CrossRef]

N. Stefanou and A. Modinos, �??Impurity bands in photonic insulators,�?? Phys. Rev. B 57, 12127-12133 (1998).
[CrossRef]

J. P. Albert, C. Jouanin, D. Cassagne, and D. Bertho, �??Generalized Wannier function method for photonic crystals,�?? Phys. Rev. B 61, 4381-4384 (2000).
[CrossRef]

Phys. Rev. Lett. (2)

E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, �??Tight-binding parametrization for photonic band gap materials,�?? Phys. Rev. Lett. 81, 1405-1408 (1998).
[CrossRef]

M. Notomi, K. Yamada, A. Shinya, J. Takahashi, C. Takahashi, and I. Yokohama, �??Extremely large groupvelocity dispersion of line-defect waveguides in photonic crystal slabs,�?? Phys. Rev. Lett. 87, 253902 (2001).
[CrossRef] [PubMed]

Other (2)

L. Brillouin, Wave Propagation in Periodic Structures 2nd ed. (Dover Publications, New York, 1953).

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

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Figures (7)

Fig. 1.
Fig. 1.

Two-dimensional NEPC: (a) dielectric rod photonic crystal with nano-strip embedding, (b) projected band diagram, red and blue lines: NEPC defect bands, (c) conceptual interpretation of NEPC: one-dimensional defect cavity array.

Fig. 2.
Fig. 2.

Effective defect cavity of NEPC: (a) elementary effective defect cavity, (b) s-state cavity mode field distribution, (c) p-state cavity mode field distribution.

Fig. 3.
Fig. 3.

Multiple period s-p hybridization in NEPC: solid lines; PWE results, circles; tight-binding model results for (a) nearest neighbor interaction only, (b) up to third-nearest neighbor interactions, (c) up to fourth-nearest neighbor interactions.

Fig. 4.
Fig. 4.

Mode profiles for (a) k=0 (amplitude), (b) k=0.25 (intensity), (c) k=0.4 (intensity), (d) k=0.5 (amplitude), left side; tight-binding model, right side; PWE analysis, upper side; upper band mode, lower side; lower band mode.

Fig. 5.
Fig. 5.

Relative amplitudes of the s-state (solid line) and p-state (dashed line) cavity modes for (a) lower defect band, (b) upper defect band.

Fig. 6.
Fig. 6.

Effective cavity fitting for tight-binding model convergence to the PWE method: blue line; PWE method, red dashed line; tight-binding method (nano-strip with length of a), red crosses; tight-binding method (nano-strip with increased length of 1.2a), red circles; tight-binding method (nano-strip with increased length of 1.2a and increased refractive index of 3.9).

Fig. 7.
Fig. 7.

Mode field profile comparison for (a) k=0 (amplitude), (b) k=0.32 (intensity), (c) k=0.5 (amplitude), left side; tight-binding model, right side; the PWE method.

Equations (11)

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E k ( r , t ) = exp ( i ω k t ) n exp ( i n k a ) l A l E Ω l ( r n a e ̂ z ) .
× × E k = ε ( r ) ω k 2 c 2 E k
d r ε o ( r ) E Ω l ( r ) · E Ω m ( r ) = δ l , m ,
l A l Ω l 2 [ δ m , l + n 0 exp ( i n k a ) β m , l n ] = ω k 2 l A l [ δ m , l + Δ α m , l + n 0 exp ( i n k a ) α m , l n ]
α m , l n = d r ε ( r ) E Ω m ( r ) · E Ω l ( r n a e ̂ z ) ,
β m , l n = d r ε o ( r n a e ̂ z ) E Ω m ( r ) · E Ω l ( r n a e ̂ z ) ,
Δ α m , l = d r [ ε ( r ) ε o ( r ) ] E Ω m ( r ) · E Ω l ( r ) .
A m [ ω k 2 { 1 + Δ α m , m + n 0 2 α m , m n cos ( n k a ) } + Ω m 2 { 1 + n 0 2 β m , m n cos ( n k a ) } ]
i A l [ ω k 2 n 0 2 α m , l n sin ( n k a ) + Ω l 2 n 0 2 β m , l n sin ( n k a ) ] = 0 ,
κ m , l n = β m , l n α m , l n
= d r [ ε o ( r n a e ̂ z ) ε ( r n a e ̂ z ) ] E Ω m ( r ) · E Ω l ( r n a e ̂ z )

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