## Control of dispersion in photonic crystal waveguides using group symmetry theory |

Optics Express, Vol. 20, Issue 12, pp. 13108-13114 (2012)

http://dx.doi.org/10.1364/OE.20.013108

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### Abstract

We demonstrate dispersion tailoring by coupling the even and the odd modes in a line-defect photonic crystal waveguide. Coupling is determined ab-initio using group theory analysis, rather than by trial-error optimisation of the design parameters. A family of dispersion curves is generated by controlling a single geometrical parameter. This concept is demonstrated experimentally with very good agreement with theory.

© 2012 OSA

3. L. Yin, Q. Lin, and G. P. Agrawal, “Dispersion tailoring and soliton propagation in silicon waveguides,” Opt. Lett. **31**, 1295–1297 (2006). [CrossRef] [PubMed]

3. L. Yin, Q. Lin, and G. P. Agrawal, “Dispersion tailoring and soliton propagation in silicon waveguides,” Opt. Lett. **31**, 1295–1297 (2006). [CrossRef] [PubMed]

*dispersion engineering*[4

4. J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express **16**, 6227–6232 (2008). [CrossRef] [PubMed]

5. O. Khayam and H. Benisty, “General recipe for flatbands in photonic crystalwaveguides,” Opt. Express **17**, 14634–14648 (2009). [CrossRef] [PubMed]

6. J. C. Knight, “Photonic crystal fibres,” Nature **424**, 847–851 (2003). [CrossRef] [PubMed]

4. J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express **16**, 6227–6232 (2008). [CrossRef] [PubMed]

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

10. Y. Hamachi, S. Kubo, and T. Baba, “Slow light with low dispersion and nonlinear enhancement in a lattice-shifted photonic crystal waveguide,” Opt. Lett. **34**, 1072–1074 (2009). [CrossRef] [PubMed]

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

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

8. A. Petrov and M. Eich, “Dispersion compensation with photonic crystal line-defect waveguides,” IEEE J. Sel. Area. Commun. **23**, 1396–1401 (2005). [CrossRef]

12. M. Patterson, S. Hughes, D. Dalacu, and R. L. Williams, “Broadband purcell factor enhancements in photonic-crystal ridge waveguides,” Phys. Rev. B **80**, 125307 (2009). [CrossRef]

13. S. Lü, J. Zhao, and D. Zhang, “Flat band slow light in asymmetric photonic crystal waveguide based on microfluidic infiltration,” Appl. Opt. **49**, 3930–3934 (2010). [CrossRef] [PubMed]

16. N. Gutman, W. Dupree, Y. Sun, A. Sukhorukov, and C. de Sterke, “Frozen and broadband slow light in coupled periodic nanowire waveguides,” Opt. Express **20**, 3519–3528 (2012). [CrossRef] [PubMed]

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

*γ*(

*k*) of the modes of the individual waveguides, with dispersion

*ω*

_{wg}_{{1,2}}(

*k*), results from the diagonalisation of: Introducing the coupling results into a modification of the dispersion of the original modes, which can be controlled within some extent by playing with some parameters, such as the radius of the holes or the separation between the two waveguides. We point out that

*γ*(

*k*) could be modified, in principle, almost independently from

*ω*

_{wg}_{{i}}

*i*= {1, 2}.

*x*–

*z*, where

*x*is the direction of the propagation and

*z*is perpendicular to the 2D PhC lattice.

17. S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express **8**, 173–190 (2001). [CrossRef] [PubMed]

*r/a*= 0.26, slab thickness

*h/a*= 0.41, width of the line defect

*W/a*= 0.95 √3, outward displacement of the first row of holes

*s/a*= 0.14, with radius reduced to

*r*

_{1}

*/a*= 0.23. The refractive index of the material is assumed to be 3.17. Here, the parameters have been chosen in order to lower the edge of the odd mode (at

*k*= 0.5

**K**,

**K**being the reciprocal lattice vector); indeed, here the frequency spacing is only

*δfa/c*= 610

^{−3}. If a suitable coupling between these two modes, which are now closer in frequency, is introduced, then the anti-crossing will result into a change in the dispersion. As a consequence, controlling the coupling strength would enable, for example, the flattening of the lower frequency branch.

*ε*(

*r⃗*) such that

*γ*(

_{eo}*k*) is non-zero, but also dominates also substantially over the other terms. Let us first look at the integrand

*C*

_{2v}, that statement is equivalent to requiring the integrand to have at least the element of symmetry

*A*

_{1}(namely, the fully symmetric function). Conversely, if that condition is not respected, then integrand will have both positive and negative parts which would mutually cancel out.

18. O. Painter and K. Srinivasan, “Localized defect states in two-dimensional photonic crystal slab waveguides: a simple model based upon symmetry analysis,” Phys. Rev. B **68**, 035110 (2003). [CrossRef]

*B*

_{1}and

*B*

_{2}respectively. The symmetry condition on the integrand in

*γ*(

_{oe}*k*) is then formulated as: where

*X*is the symmetry of Δ

*ε*(

*r⃗*), still to be determined. It is found that

*X*must have the symmetry

*A*

_{2}. This implies that the dielectric perturbation is anti-symmetric with respect both to the

*x*= 0 and the

*y*= 0 symmetry planes. This also implies that

*γ*(

_{oo}*k*) =

*γ*(

_{ee}*k*) =

*γ*(

_{ss}*k*) =

*γ*(

_{se}*k*) = 0, i.e., the only non-zero elements are

*γ*(

_{eo}*k*) and

*γ*(

_{so}*k*). Indeed, since the even and the substrate modes have the same symmetry, they are not coupled together by a such perturbation. It can be observed that the coupling

*γ*(

_{so}*k*) between the odd and the substrate modes is minimized if the initial frequency spacing between the two modes is large and if the changes of the dielectric function Δ

*ε*(

*r⃗*) is localized where the field of the even and odd modes is much stronger than that of the substrate mode. This is in general the case in the region close to the line defect (see Fig. 2(a)).

*B*

_{1}perturbation, as used in ref. [10

10. Y. Hamachi, S. Kubo, and T. Baba, “Slow light with low dispersion and nonlinear enhancement in a lattice-shifted photonic crystal waveguide,” Opt. Lett. **34**, 1072–1074 (2009). [CrossRef] [PubMed]

*B*

_{1}readily produces a strong change in the dispersion.

*T*is changed (Fig. 3(c)). Indeed, while a flat-band over more then 10

*nm*is obtained with T=0.05 (delay-bandwidth product of 0.15), the other values generate a dispersion where the group dispersion changes sign twice and a local maximum of the group index increasing sharply as T is further increased.

19. S. Combrié, Q. V. Tran, A. D. Rossi, C. Husko, and P. Colman, “High quality gainp nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett. **95**, 221108 (2009). [CrossRef]

20. A. Parini, P. Hamel, A. De Rossi, S. Combrie, N.-V.-Q. Tran, Y. Gottesman, R. Gabet, A. Talneau, Y. Jaouen, and G. Vadala, “Time-wavelength reflectance maps of photonic crystal waveguides: a new view on disorder-induced scattering,” J. Lightwave Technol . **26**, 3794–3802 (2008). [CrossRef]

21. Q. V. Tran, S. Combrié, P. Colman, and A. D. Rossi, “Photonic crystal membrane waveguides with low insertion losses,” Appl. Phys. Lett. **95**, 061105 (2009). [CrossRef]

*T/a*= 0.05) can be attributed to the fabrication tolerances (e.g. the radius of the holes). The important point is that the transition from one of these dispersion profiles to another follows continuously the change in the parameter T; which can be further adjusted in order to reach the desired dispersion profile.

## Acknowledgments

## References and links

1. | G. P. Agrawal, |

2. | R. W. Boyd, |

3. | L. Yin, Q. Lin, and G. P. Agrawal, “Dispersion tailoring and soliton propagation in silicon waveguides,” Opt. Lett. |

4. | J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express |

5. | O. Khayam and H. Benisty, “General recipe for flatbands in photonic crystalwaveguides,” Opt. Express |

6. | J. C. Knight, “Photonic crystal fibres,” Nature |

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

8. | A. Petrov and M. Eich, “Dispersion compensation with photonic crystal line-defect waveguides,” IEEE J. Sel. Area. Commun. |

9. | L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, and P. I. Borel, “Photonic crystal waveguides with semi-slow light and tailored dispersion properties,” Opt. Express |

10. | Y. Hamachi, S. Kubo, and T. Baba, “Slow light with low dispersion and nonlinear enhancement in a lattice-shifted photonic crystal waveguide,” Opt. Lett. |

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

12. | M. Patterson, S. Hughes, D. Dalacu, and R. L. Williams, “Broadband purcell factor enhancements in photonic-crystal ridge waveguides,” Phys. Rev. B |

13. | S. Lü, J. Zhao, and D. Zhang, “Flat band slow light in asymmetric photonic crystal waveguide based on microfluidic infiltration,” Appl. Opt. |

14. | X. Mao, Y. Huang, W. Zhang, and J. Peng, “Coupling between even- and oddlike modes in a single asymmetric photonic crystal waveguide,” Appl. Phys. Lett. |

15. | J. Ma and C. Jiang, “Demonstration of ultraslow modes in asymmetric line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett. |

16. | N. Gutman, W. Dupree, Y. Sun, A. Sukhorukov, and C. de Sterke, “Frozen and broadband slow light in coupled periodic nanowire waveguides,” Opt. Express |

17. | S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express |

18. | O. Painter and K. Srinivasan, “Localized defect states in two-dimensional photonic crystal slab waveguides: a simple model based upon symmetry analysis,” Phys. Rev. B |

19. | S. Combrié, Q. V. Tran, A. D. Rossi, C. Husko, and P. Colman, “High quality gainp nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett. |

20. | A. Parini, P. Hamel, A. De Rossi, S. Combrie, N.-V.-Q. Tran, Y. Gottesman, R. Gabet, A. Talneau, Y. Jaouen, and G. Vadala, “Time-wavelength reflectance maps of photonic crystal waveguides: a new view on disorder-induced scattering,” J. Lightwave Technol . |

21. | Q. V. Tran, S. Combrié, P. Colman, and A. D. Rossi, “Photonic crystal membrane waveguides with low insertion losses,” Appl. Phys. Lett. |

22. | Y. Vlasov, W. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photonics |

23. | A. Shinya, S. Matsuo, Yosia, T. Tanabe, E. Kuramochi, T. Sato, T. Kakitsuka, and M. Notomi, “All-optical on-chip bit memory based on ultra high Q InGaAsP photonic crystal,” Opt. Express |

24. | A. Melloni, A. Canciamilla, C. Ferrari, F. Morichetti, L. Ó Faolain, T. F. Krauss, R. De La Rue, A. Samarelli, and M. Sorel, “On-chip tunable delay lines in silicon photonics,” IEE Photon. J. |

25. | S. Schultz, L. O’Faolain, D. Beggs, T. P. White, A. Melloni, and T. F. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt. |

26. | P. Colman, S. Combrié, I. Cestier, A. Willinger, G. Eisenstein, A. de Rossi, and G. Lehoucq, “Observation of gain due to four-wave-mixing in dispersion engineered GaInP photonic crystal waveguides,” Opt. Lett. |

**OCIS Codes**

(130.3120) Integrated optics : Integrated optics devices

(260.2030) Physical optics : Dispersion

(130.5296) Integrated optics : Photonic crystal waveguides

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: March 7, 2012

Revised Manuscript: April 12, 2012

Manuscript Accepted: April 12, 2012

Published: May 25, 2012

**Citation**

Pierre Colman, Sylvain Combrié, Gaëlle Lehoucq, and Alfredo De Rossi, "Control of dispersion in photonic crystal waveguides using group symmetry theory," Opt. Express **20**, 13108-13114 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-12-13108

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### References

- G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 2007).
- R. W. Boyd, Nonlinear Optics (Academic Press, 2003).
- L. Yin, Q. Lin, and G. P. Agrawal, “Dispersion tailoring and soliton propagation in silicon waveguides,” Opt. Lett.31, 1295–1297 (2006). [CrossRef] [PubMed]
- J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express16, 6227–6232 (2008). [CrossRef] [PubMed]
- O. Khayam and H. Benisty, “General recipe for flatbands in photonic crystalwaveguides,” Opt. Express17, 14634–14648 (2009). [CrossRef] [PubMed]
- J. C. Knight, “Photonic crystal fibres,” Nature424, 847–851 (2003). [CrossRef] [PubMed]
- 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. Petrov and M. Eich, “Dispersion compensation with photonic crystal line-defect waveguides,” IEEE J. Sel. Area. Commun.23, 1396–1401 (2005). [CrossRef]
- L. H. Frandsen, A. V. Lavrinenko, J. Fage-Pedersen, and P. I. Borel, “Photonic crystal waveguides with semi-slow light and tailored dispersion properties,” Opt. Express14, 9444–9450 (2006). [CrossRef] [PubMed]
- Y. Hamachi, S. Kubo, and T. Baba, “Slow light with low dispersion and nonlinear enhancement in a lattice-shifted photonic crystal waveguide,” Opt. Lett.34, 1072–1074 (2009). [CrossRef] [PubMed]
- A. Y. Petrov and M. Eich, “Zero dispersion at small group velocities in photonic crystal waveguides,” Appl. Phys. Lett.85, 4866–4868 (2004). [CrossRef]
- M. Patterson, S. Hughes, D. Dalacu, and R. L. Williams, “Broadband purcell factor enhancements in photonic-crystal ridge waveguides,” Phys. Rev. B80, 125307 (2009). [CrossRef]
- S. Lü, J. Zhao, and D. Zhang, “Flat band slow light in asymmetric photonic crystal waveguide based on microfluidic infiltration,” Appl. Opt.49, 3930–3934 (2010). [CrossRef] [PubMed]
- X. Mao, Y. Huang, W. Zhang, and J. Peng, “Coupling between even- and oddlike modes in a single asymmetric photonic crystal waveguide,” Appl. Phys. Lett.95, 183106 (2009). [CrossRef]
- J. Ma and C. Jiang, “Demonstration of ultraslow modes in asymmetric line-defect photonic crystal waveguides,” IEEE Photon. Technol. Lett.20, 1237–1239 (2008). [CrossRef]
- N. Gutman, W. Dupree, Y. Sun, A. Sukhorukov, and C. de Sterke, “Frozen and broadband slow light in coupled periodic nanowire waveguides,” Opt. Express20, 3519–3528 (2012). [CrossRef] [PubMed]
- S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express8, 173–190 (2001). [CrossRef] [PubMed]
- O. Painter and K. Srinivasan, “Localized defect states in two-dimensional photonic crystal slab waveguides: a simple model based upon symmetry analysis,” Phys. Rev. B68, 035110 (2003). [CrossRef]
- S. Combrié, Q. V. Tran, A. D. Rossi, C. Husko, and P. Colman, “High quality gainp nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett.95, 221108 (2009). [CrossRef]
- A. Parini, P. Hamel, A. De Rossi, S. Combrie, N.-V.-Q. Tran, Y. Gottesman, R. Gabet, A. Talneau, Y. Jaouen, and G. Vadala, “Time-wavelength reflectance maps of photonic crystal waveguides: a new view on disorder-induced scattering,” J. Lightwave Technol. 26, 3794–3802 (2008). [CrossRef]
- Q. V. Tran, S. Combrié, P. Colman, and A. D. Rossi, “Photonic crystal membrane waveguides with low insertion losses,” Appl. Phys. Lett.95, 061105 (2009). [CrossRef]
- Y. Vlasov, W. Green, and F. Xia, “High-throughput silicon nanophotonic wavelength-insensitive switch for on-chip optical networks,” Nat. Photonics2, 242–246 (2008). [CrossRef]
- A. Shinya, S. Matsuo, Yosia, T. Tanabe, E. Kuramochi, T. Sato, T. Kakitsuka, and M. Notomi, “All-optical on-chip bit memory based on ultra high Q InGaAsP photonic crystal,” Opt. Express16, 19382–19387 (2008). [CrossRef]
- A. Melloni, A. Canciamilla, C. Ferrari, F. Morichetti, L. Ó Faolain, T. F. Krauss, R. De La Rue, A. Samarelli, and M. Sorel, “On-chip tunable delay lines in silicon photonics,” IEE Photon. J.2, 181–194 (2010). [CrossRef]
- S. Schultz, L. O’Faolain, D. Beggs, T. P. White, A. Melloni, and T. F. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt.12, 104004 (2010). [CrossRef]
- P. Colman, S. Combrié, I. Cestier, A. Willinger, G. Eisenstein, A. de Rossi, and G. Lehoucq, “Observation of gain due to four-wave-mixing in dispersion engineered GaInP photonic crystal waveguides,” Opt. Lett.36, 2629–2631 (2011). [CrossRef] [PubMed]

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