## Multiple period *s-p* hybridization in nano-strip embedded photonic crystal

Optics Express, Vol. 13, Issue 7, pp. 2774-2781 (2005)

http://dx.doi.org/10.1364/OPEX.13.002774

Acrobat PDF (367 KB)

### 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

## 1. Introduction

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

2. 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]

3. N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B **57**, 12127–12133 (1998). [CrossRef]

6. E. Ozbay, M. Bayindir, I. Bulu, and E. Cubukcu, “Investigation of localized coupled-cavity modes in two-dimensional photonic bandgap structures,” IEEE J. Quantum Electron. **38**, 837–843 (2002). [CrossRef]

7. 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]

8. M. Lončar, J. Vučković, and A. Scherer, “Methods for controlling positions of guided modes of photoniccrystal waveguides,” J. Opt. Soc. Am. B **18**, 1362–1368 (2001). [CrossRef]

8. M. Lončar, J. Vučković, and A. Scherer, “Methods for controlling positions of guided modes of photoniccrystal waveguides,” J. Opt. Soc. Am. B **18**, 1362–1368 (2001). [CrossRef]

10. A. Y. Petrov and M. Eich, “Zero dispersion at small group velocities in photonic crystal waveguides,” Appl. Phys. Lett. **85**, 4866–4868 (2004). [CrossRef]

6. E. Ozbay, M. Bayindir, I. Bulu, and E. Cubukcu, “Investigation of localized coupled-cavity modes in two-dimensional photonic bandgap structures,” IEEE J. Quantum Electron. **38**, 837–843 (2002). [CrossRef]

7. 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]

10. A. Y. Petrov and M. Eich, “Zero dispersion at small group velocities in photonic crystal waveguides,” Appl. Phys. Lett. **85**, 4866–4868 (2004). [CrossRef]

*s*- and

*p*-state cavity modes are interacting with each other over the multiple periods in the propagation direction. This interaction can be described by the tight-binding method with multiples of resonating modes [3

3. N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B **57**, 12127–12133 (1998). [CrossRef]

5. Y. Xu, R. K. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” J. Opt. Soc. Am. B **17**, 387–400 (2000). [CrossRef]

13. 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]

15. D. Leuenberger, R. Ferrini, and R. Houdré, “An *initio* tight-binding approach to photonic-crystal based coupled cavity waveguides,” J. Appl. Phys. **95**, 806–809 (2004). [CrossRef]

15. D. Leuenberger, R. Ferrini, and R. Houdré, “An *initio* tight-binding approach to photonic-crystal based coupled cavity waveguides,” J. Appl. Phys. **95**, 806–809 (2004). [CrossRef]

## 2. NEPC: effective defect cavity array with s and p cavity modes

*a*and 0.1

*a*, respectively, where

*a*is the period of the photonic crystal structure. The band diagram was obtained using plane wave expansion (PWE) calculation with 8 by 1 (

*x*by

*z*direction) supercell method [22]. In the calculation, electromagnetic variational theorem [23] was used with iterative minimization techniques. The band diagram of Fig. 1(b) is for the case of TM polarization, in which only

*y*-directional electric field exists. Within the PBG, the two defect bands become waveguide bands with multi-mode characteristics for certain frequencies and zero group velocity away from the Brillouin zone edge (i.e., near normalized frequency 0.34). However, for the upper defect band (blue line in Fig. 1(b)), the low wave vector part goes into the upper air band so that large loss occurs due to interactions with the air band extended modes. Although this is an artificial structure to analyze and interpret the hybridization, actual realization of the hybridized band structure may use some lower index background even with three-dimensional slab structure [2

2. 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]

*a*. This interpretation forms the basis of the theoretical modeling in the following section.

*a*nano-strip with length a. The NEPC structure of Fig. 1(a) is considered to be composed of elementary defect nano-strips, like Fig. 2(a), forming an effective cavity array, as seen in Fig. 1(c). By using the PWE calculation with 8 by 10 (

*x*by

*z*) supercell method for the defect cavity of Fig. 2(a), we could find that there are two cavity modes with field distribution, as plotted in Figs. 2(b) and 2(c). With respect to the one-dimensional periodicity along the defect cavity array direction, we can see that each mode corresponds to the

*s*-state (Fig. 2(b), normalized resonant frequency 0.354) and

*p*-state (Fig. 2(c), normalized resonant frequency 0.385) similar to the atomic crystal orbital states [2

2. 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]

24. B. E. A. Saleh and M. C. Teich, *Fundamentals of Photonics* (John Wiley & Sons, New York, 1991). [CrossRef]

## 3. Tight-binding model: multiple period s-p hybridization

4. A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. **24**, 711–713 (1999). [CrossRef]

5. Y. Xu, R. K. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” J. Opt. Soc. Am. B **17**, 387–400 (2000). [CrossRef]

*a*is the period along the cavity array direction,

*A*

_{l}and

*l*, and its resonating mode profile (i.e.,

*s*and

*p*modes), and

*n*is an integer for multiple period interaction. Since the cavity array is periodic along its direction, the defect band mode also becomes Bloch wave, where we can consider only the first Brillouin zone of the wave vector

*k*(i.e., -

*π/a*≤

*k*≤

*π*/

*a*) in characterizing its band dispersion.

_{k}(

*t*) satisfies the Maxwell equations, as

*ε(*r →
)is the dielectric constant of the NEPC system and

*ω*

_{k}is the waveguide mode eigenfrequency. In the same way,

*ε(*r →
)is replaced by

*ε*

_{o}

*(*r →
), the effective defect cavity dielectric profile, and

*ω*

_{k}is replaced with

*Ω*

_{l}, the resonance frequency of the effective cavity. We take orthonormal condition to the cavity modes as

*m*’s) for

*s*- and

*p*-state) and spatially integrating, we can get two eigen equations for the mode amplitudes as

*α*

_{m,l}are defined as

*p*-state cavity mode along the defect cavity array direction (i.e.,

*z*-direction), the symmetry relations of the parameters of Eq. (5a) and (5b) become

*m*and

*l*denote different cavity mode states, respectively. Equation (6) can have nontrivial solution when its determinant becomes zero, which gives amplitude ratio between

*A*

_{s}and

*A*

_{p}, and the band dispersion relations.

## 4. Numerical analysis : tight-binding model and PWE analysis

15. D. Leuenberger, R. Ferrini, and R. Houdré, “An *initio* tight-binding approach to photonic-crystal based coupled cavity waveguides,” J. Appl. Phys. **95**, 806–809 (2004). [CrossRef]

*s-p*hybridization. We can see that by increasing the interaction lengths from just the nearest neighbors (Fig. 3(a)) to the fourth-nearest neighbors (Fig. 3(c)), the proposed model correctly converges to the PWE results. This means that the defect bands of the NEPC comes from the

*s*- and

*p*-state cavity mode interactions over up to the fourth-nearest neighbors. For the upper defect band (blue in Fig. 3(c)) with low wave vector, there occurs large deviation of the proposed model from the PWE calculation. This is due to the coupling between the upper

*s-p*hybridized band mode and the extended band modes of the air band (Fig. 1(b)), where the two-cavity modes assumption of the proposed model is not satisfied. Figure 4 is a contour map of the mode profiles of the two defect bands. Mode profiles in 8 by 1 (

*x*by

*z*) unit area are obtained using the proposed tight-binding model as well as the PWE calculation. We can see the results of the two methods to be agreeing well with each other, although in the vicinity of the air band, the upper defect band mode does not match the PWE result (i.e., Fig. 4(a) and (b) upper contours). In Fig. 5, the relative amplitudes of the two cavity modes in each defect band are plotted. These amplitudes were normalized as |

*A*

_{s}|

^{2}+|

*A*

_{p}|

^{2}=1. We can see that at the Brillouin zone edges and the zero wave vector, only one cavity mode vibrates and the amplitude is monotonically decreasing or increasing in the intermediate wave vector regions. This hybridization of the

*s*- and

*p*-state modes originates from the relationship of the coefficients

*Ω*

_{l}’s. From Eq. (6), we can see that at the zero wave vector and at the Brillouin zone edge, only one cavity mode vibrates for each band. Generally, when the coefficients and cavity resonance frequency relationship induces relative frequency change for the two vibrating cavity modes at these two wave vectors, the

*s*- and

*p*-state hybridization occurs over the Brillouin zone wave vectors. We also note that at the Brillouin zone edge, the two bands can support two lasing modes if we consider fabricating distributed feedback (DFB) laser. The

*p*-state mode for the lower frequency and the s-state mode for the higher frequency of the NEPC case (i.e., Fig. 4(d)), correspond to the lasing modes of the uniform corrugated DFB laser [25

25. S. Akiba, M. Usami, and K. Utaka, “1.5-µm λ/4-shifted InGaAsP/InP DFB lasers,” J. Lightwave Technol. **LT-5**, 1564–1573 (1987). [CrossRef]

26. X. Checoury, P. Boucaud, J-M. Lourtioz, F. Pommereau, C. Cuisin, E. Derouin, O. Drisse, L. Legouezigou, F. Lelarge, F. Poingt, G. H. Duan, D. Mulin, S. Bonnefont, O. Gauthier-Lafaye, J. Valentin, F. Lozes, and A. Talneau, “Distributed feedback regime of photonic crystal waveguide lasers at 1.5 *µ*m,” Appl. Phys. Lett. **85**, 5502–5504(2004). [CrossRef]

26. X. Checoury, P. Boucaud, J-M. Lourtioz, F. Pommereau, C. Cuisin, E. Derouin, O. Drisse, L. Legouezigou, F. Lelarge, F. Poingt, G. H. Duan, D. Mulin, S. Bonnefont, O. Gauthier-Lafaye, J. Valentin, F. Lozes, and A. Talneau, “Distributed feedback regime of photonic crystal waveguide lasers at 1.5 *µ*m,” Appl. Phys. Lett. **85**, 5502–5504(2004). [CrossRef]

*a*, of the nano-strip, as in Fig. 2(a). Note that there are some deviations from the PWE calculation (blue line in Fig. 6). However, when we increase the nano-strip length of the effective defect cavity to 1.2

*a*(red crosses) and then increase the refractive index of the strip to 3.9 (red circles), we can see that the tight-binding approach converges more well to the PWE calculation result. It means that for the optical field confinement along the propagation direction (i.e.,

*z*direction in Fig. 2(a)) in each cavity, the nano-strip waveguiding effect needs to be considered, to some extent, differently from the simple cavity model of Fig. 2(a), based on just the PBG reflection effect. However, with some clever cavity structures or theoretical method development for the real effective cavity, we are expecting that practical application of the multiple period

*s-p*hybridization might be possible. In relation to the weak coupling condition of the tight-binding method, we note that the coupling factors (i.e.,

^{-2}or less for all the multiple period interactions. For this cavity model, Fig. 7 shows comparison of the guiding mode profiles (for 8 by 1 (

*x*by

*z*) unit area) between the tight-binding model and PWE calculation. We can see that the proposed model’s results agree very well with the PWE field results.

## 5. Conclusion

*s-p*hybridization in the case of photonic crystal. This phenomenon is possible in NEPC structure, where the embedded nano-strip forms an effective defect cavity array along its direction. Using the tight-binding method, we could analyze and describe the characteristics of the hybridized bands that agree well with the PWE calculations. We note that the proposed multiple period hybridization is a novel concept in photonic crystal one-dimensional defects, which is distinguished from other defects like CROW (i.e., nearest neighbor interaction) and conventional waveguides (i.e., long range interaction). Although the phenomena were considered only in NEPC structure, by developing some clever structures or theoretical methods for design (e.g., effective cavity defining method), various applications will be possible from waveguides to photonic devices.

## References and links

1. | A. Chutinan and S. Noda, “Waveguides and waveguide bends in two-dimensional photonic crystal slabs,” Phys. Rev. B |

2. | S. G. Johnson, P. R. Villeneuve, S. Fan, and J. D. Joannopoulos, “Linear waveguides in photonic-crystal slabs,” Phys. Rev. B |

3. | N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B |

4. | A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, “Coupled-resonator optical waveguide: a proposal and analysis,” Opt. Lett. |

5. | Y. Xu, R. K. Lee, and A. Yariv, “Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,” J. Opt. Soc. Am. B |

6. | E. Ozbay, M. Bayindir, I. Bulu, and E. Cubukcu, “Investigation of localized coupled-cavity modes in two-dimensional photonic bandgap structures,” IEEE J. Quantum Electron. |

7. | 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. |

8. | M. Lončar, J. Vučković, and A. Scherer, “Methods for controlling positions of guided modes of photoniccrystal waveguides,” J. Opt. Soc. Am. B |

9. | D. Mori and T. Baba, “Dispersion-controlled optical group delay device by chirped photonic crystal waveguides,” Appl. Phys. Lett. |

10. | A. Y. Petrov and M. Eich, “Zero dispersion at small group velocities in photonic crystal waveguides,” Appl. Phys. Lett. |

11. | L. Brillouin, |

12. | N. W. Ashcroft and N. D. Mermin, |

13. | E. Lidorikis, M. M. Sigalas, E. N. Economou, and C. M. Soukoulis, “Tight-binding parametrization for photonic band gap materials,” Phys. Rev. Lett. |

14. | J. P. Albert, C. Jouanin, D. Cassagne, and D. Bertho, “Generalized Wannier function method for photonic crystals,” Phys. Rev. B |

15. | D. Leuenberger, R. Ferrini, and R. Houdré, “An |

16. | V. Yannopapas, A. Modinos, and N. Stefanou, “Optical properties of metallodielectric photonic crystals,” Phys. Rev. B |

17. | V. Yannopapas, A. Modinos, and N. Stefanou, “Waveguides of defect chains in photonic crystals,” Phys. Rev. B |

18. | W. A. Harrison, |

19. | S. T. Pantelides and W. A. Harrison, “Structure of the valence bands of zinc-blende-type semiconductors,” Phys. Rev. B |

20. | W. T. Lau and S. Fan, “Creating large bandwidth line defects by embedding dielectric waveguides into photonic crystal slabs,” Appl. Phys. Lett. |

21. | K. Sakoda, |

22. | The PWE simulations were carried out with BandSOLVE commercial software by RSoft Design Group. |

23. | J. D. Joannopoulos, R. D. Meade, and J. N. Winn, |

24. | B. E. A. Saleh and M. C. Teich, |

25. | S. Akiba, M. Usami, and K. Utaka, “1.5-µm λ/4-shifted InGaAsP/InP DFB lasers,” J. Lightwave Technol. |

26. | X. Checoury, P. Boucaud, J-M. Lourtioz, F. Pommereau, C. Cuisin, E. Derouin, O. Drisse, L. Legouezigou, F. Lelarge, F. Poingt, G. H. Duan, D. Mulin, S. Bonnefont, O. Gauthier-Lafaye, J. Valentin, F. Lozes, and A. Talneau, “Distributed feedback regime of photonic crystal waveguide lasers at 1.5 |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(230.5750) Optical devices : Resonators

**ToC Category:**

Research Papers

**History**

Original Manuscript: February 22, 2005

Revised Manuscript: March 26, 2005

Published: April 4, 2005

**Citation**

Seunghoon Han, Il-Min Lee, Hwi Kim, and Byoungho Lee, "Multiple period s-p hybridization in nano-strip embedded photonic crystal," Opt. Express **13**, 2774-2781 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-7-2774

Sort: Journal | Reset

### References

- 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]
- A. Yariv, Y. Xu, R. K. Lee, and A. Scherer, �??Coupled-resonator optical waveguide: a proposal and analysis,�?? Opt. Lett. 24, 711-713 (1999). [CrossRef]
- Y. Xu, R. K. Lee, and A. Yariv, �??Propagation and second-harmonic generation of electromagnetic waves in a coupled-resonator optical waveguide,�?? J. Opt. Soc. Am. B 17, 387-400 (2000). [CrossRef]
- 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]
- 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]
- M. Lon�?ar, J. Vu�?kovi�?, and A. Scherer, �??Methods for controlling positions of guided modes of photoniccrystal waveguides,�?? J. Opt. Soc. Am. B 18, 1362-1368 (2001). [CrossRef]
- 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]
- 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).
- 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]
- 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]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.