## Paired modes of heterostructure cavities in photonic crystal waveguides with split band edges |

Optics Express, Vol. 18, Issue 25, pp. 25693-25701 (2010)

http://dx.doi.org/10.1364/OE.18.025693

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### Abstract

We investigate the modes of double heterostructure cavities where the underlying photonic crystal waveguide has been dispersion engineered to have two band-edges inside the Brillouin zone. By deriving and using a perturbative method, we show that these structures possess two modes. For unapodized cavities, the relative detuning of the two modes can be controlled by changing the cavity length, and for particular lengths, a resonant-like effect makes the modes degenerate. For apodized cavities no such resonances exist and the modes are always non-degenerate.

© 2010 Optical Society of America

## 1. Introduction

1. K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics **4**, 477–483 (2010). [CrossRef]

2. J. Vuĉković and Y. Yamamoto, “Photonic crystal microcavities for cavity quantum electrodynamics with a single quantum dot,” Appl. Phys. Lett. **82**, 2374–2376 (2003). [CrossRef]

3. M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature **462**, 78–82 (2009). [CrossRef] [PubMed]

4. O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science **284**, 1819–1821 (1999). [CrossRef] [PubMed]

*∼*10

^{6}[5

5. B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. **4**, 207–210 (2005). [CrossRef]

6. E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. **88**, 041112 (2006). [CrossRef]

^{8}[7

7. Y. Tanaka, T. Asano, and S. Noda, “Design of photonic crystal nanocavity with *Q*-factor of ∼10^{9},” J. Lightwave Technol. **26**, 1532–1539 (2008). [CrossRef]

*λ*

_{0}/

*n*)

^{3}. As shown in Fig. 1(a), DHCs are constructed by weakly perturbing a photonic crystal waveguide (PCW). This can be done, for example, by changing the period of the PCW [5

5. B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. **4**, 207–210 (2005). [CrossRef]

6. E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. **88**, 041112 (2006). [CrossRef]

8. S. Tomljenovic Hanic, M. J. Steel, C. M. de Sterke, and D. J. Moss, “High-Q cavities in photosensitive photonic crystals,” Opt. Lett. **32**, 542–544 (2007). [CrossRef] [PubMed]

9. S. Gardin, F. Bordas, X. Letartre, C. Seassal, A. Rahmani, R. Bozio, and P. Viktorovitch, “Microlasers based on effective index confined slow light modes in photonic crystal waveguides,” Opt. Express **16**, 6331–6339 (2008). [CrossRef] [PubMed]

7. Y. Tanaka, T. Asano, and S. Noda, “Design of photonic crystal nanocavity with *Q*-factor of ∼10^{9},” J. Lightwave Technol. **26**, 1532–1539 (2008). [CrossRef]

10. T. Asano, B. S. Song, Y. Akahane, and S. Noda, “Ultrahigh-Q nanocavities in two-dimensional photonic crystal slabs,” IEEE J. Sel. Top. Quantum Electron. **12**, 1123–1134 (2006). [CrossRef]

11. D. Englund, I. Fushman, and J. Vuĉković, “General recipe for designing photonic crystal cavities,” Opt. Express **13**, 5961–5975 (2005). [CrossRef] [PubMed]

10. T. Asano, B. S. Song, Y. Akahane, and S. Noda, “Ultrahigh-Q nanocavities in two-dimensional photonic crystal slabs,” IEEE J. Sel. Top. Quantum Electron. **12**, 1123–1134 (2006). [CrossRef]

11. D. Englund, I. Fushman, and J. Vuĉković, “General recipe for designing photonic crystal cavities,” Opt. Express **13**, 5961–5975 (2005). [CrossRef] [PubMed]

12. M. Ibanescu, S. G. Johnson, D. Roundy, Y. Fink, and J. D. Joannopoulos, “Microcavity confinement based on an anomalous zero group-velocity waveguide mode,” Opt. Lett. **30**, 552–554 (2005). [CrossRef] [PubMed]

13. A. Figotin and I. Vitebskiy, “Slow-wave resonance in periodic stacks of anisotropic layers,” Phys. Rev. A **76**, 053839 (2007). [CrossRef]

15. K. Y. Jung and F. L. Teixeira, “Numerical study of photonic crystals with a split band edge: Polarization dependence and sensitivity analysis,” Phys. Rev. A **78**, 043826 (2008). [CrossRef]

16. S. Ha, A. A. Sukhorukov, A. V. Lavrinenko, and Yu. S. Kivshar, “Cavity mode control in side-coupled periodic waveguides: theory and experiment,” Photonics Nanostruct.: Fundam. Appl. **8**, 310–317 (2010). [CrossRef]

16. S. Ha, A. A. Sukhorukov, A. V. Lavrinenko, and Yu. S. Kivshar, “Cavity mode control in side-coupled periodic waveguides: theory and experiment,” Photonics Nanostruct.: Fundam. Appl. **8**, 310–317 (2010). [CrossRef]

10. T. Asano, B. S. Song, Y. Akahane, and S. Noda, “Ultrahigh-Q nanocavities in two-dimensional photonic crystal slabs,” IEEE J. Sel. Top. Quantum Electron. **12**, 1123–1134 (2006). [CrossRef]

## 2. Formulation of theory

17. S. Mahmoodian, C. G. Poulton, K. B. Dossou, R. C. McPhedran, L. C. Botten, and C. M. de Sterke, “Modes of Shallow Photonic Crystal Waveguides: Semi-Analytic Treatment,” Opt. Express **17**, 19629–19643 (2009). [CrossRef] [PubMed]

18. J. M. Luttinger and W. Kohn, “Motion of electrons and holes in perturbed periodic fields,” Phys. Rev. **97**, 869–883 (1955). [CrossRef]

*x*-direction (see Fig. 1) and is dispersion engineered in a fashion such that the minima of its dispersion relation are not at the BZ edge or centre [Fig. 1(d)]. Throughout this paper we look at

*H*polarised modes of a PCW with a background index

_{z}*n*= 3.0, air holes of radius 0.3

_{b}*d*, where

*d*is the period, and the radius of the holes adjacent to the waveguide are changed to 0.4

*d*. The equation governing the modes of the PCW is where

*ε*(

**r**) is the permittivity of the PCW and the subscript

*k*indicates the dependence on the Bloch wavevector. The cavity mode then satisfies where we have constructed the cavity by adding the perturbation

_{x}*δε*(

**r**) as shown in Fig. 1. We choose to expand the cavity modes using the Bloch basis set, that is, the modes of the PCW. Here we can choose our basis set such that we can use the wavevector

*k*, spanning all reciprocal space, to index our modes. The expansion takes the form Substituting Eq. (3) in Eq. (2), using Eq. (1) and taking the inner product with

_{x}*x*-direction. Thus the typical width of a DHC mode is larger than a period of the PCW.

*C*(

*k*) is then expected to be narrow in reciprocal space. Since there are two inequivalent band minima,

_{x}*C*(

*k*) is then composed of two narrow peaks centred on each minimum. We can thus write

_{x}21. S. W. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, A. V. Lavrinenko, D. N. Chigrin, and Yu. S. Kivshar, “Dispersionless tunneling of slow light in antisymmetric photonic crystal couplers,” Opt. Express **16**, 1104–1114 (2008). [CrossRef] [PubMed]

*k*′

*has values where the function*

_{x}*k*′

*. On the left hand side this gives where*

_{x}*id/dx*and arises as a result of inverse Fourier transforming with respect to

*k*′

*. It is given by*

_{x}*ω*is the frequency of the band-edge and

_{L}*D*is the curvature of the band-edge. We choose to keep only the leading order derivative as the band-edge is quadratic to first order and the envelope function along

_{L}*x*is slowly varying compared to the period of the PCW. We now turn to the RHS of Eq. (6). Since

*F*

_{1}and

*F*

_{2}are narrow functions, the main Bloch mode contribution is from the Bloch mode corresponding to the Bloch wavevector upon which the minimum is centred, that is, modes corresponding to the Bloch wavevectors

*j*denotes either 1 or 2. After some manipulation, the final equation is where

*f*

_{1}(

*x*) and

*f*

_{2}(

*x*) are envelope functions and Due to the symmetry relations formulated in Eq. (5),

*δℰ*

_{11}=

*δℰ*

_{22}and

*M*in terms of that over a unit cell, that is,

*M*= (2

*π*/

*d*)

*ℰ*with

*f*=

_{R}*f*

_{1}+

*f*

_{2}and

*f*=

_{I}*if*

_{1}–

*if*

_{2}. Accordingly, the eigenmode of Eq. (10) can be reformulated for the real envelope functions

*f*(

_{R}*x*) and

*f*(

_{I}*x*),

## 3. Results and Discussion

*n*= 0.005, as well as apodized cavities where the index of the background is altered according to a Gaussian profile with a maximum index change of Δ

*n*= 0.005. Creating the cavity in this way is similar to the photosensitive cavities previously investigated numerically in [8

8. S. Tomljenovic Hanic, M. J. Steel, C. M. de Sterke, and D. J. Moss, “High-Q cavities in photosensitive photonic crystals,” Opt. Lett. **32**, 542–544 (2007). [CrossRef] [PubMed]

22. M. W. Lee, C. Grillet, S. Tomljenovic Hanic, E. C. Magi, D. J. Moss, B. J. Eggleton, X. Gai, S. Madden, D. Y. Choi, D. A. P. Bulla, and B. Luther-Davies, “Photowritten high-Q cavities in two-dimensional chalcogenide glass photonic crystals,” Opt. Lett. **34**, 3671–3673 (2009). [CrossRef] [PubMed]

*k*-space distribution of the field, in turn leading to a slight shift in the positions of the zero-crossings in Fig. 2(b). To correct this discrepancy a higher order theory would be necessary. As shown in Figs. 2(c) and 2(f), the average frequency of the two modes is in excellent agreement with the numerics, which indicates that the diagonal terms in Eq. (10) accurately describe how deep the modes move into the bandgap. Importantly, our theory highlights the key difference between the apodized and unapodized cavities, that is, the unapodized cavities have modes whose detuning oscillates with width, while the apodized cavity’s modes have a detuning which decreases with cavity length.

*d*computed with our perturbation theory. Here, the odd mode is now the fundamental mode as, from Fig. 2(a), it is evident that the modes have crossed. Figure 3(c) shows a comparison between a full numerical calculation of the fields and our perturbation theory. The good agreement indicates that the theory correctly computes both the decay rate of the envelope functions as well as their phases. From these figures it is clear that the perturbation theory contains the underlying physics governing the behaviour of these modes. The theory thus has the potential to be used as a first-principles guide to designing such structures.

*δℰ*(

_{ij}*x*) terms depend on

*x*and do not completely vanish unless the perturbation is zero. At the degenerate points however, they evidently have no net contribution to the frequency of the DHC modes. From the symmetry relations in Eq. (5), it is easy to see that the off-diagonal terms are oscillating complex valued functions. When these terms are such that they average to zero along the length of the cavity, the contributions of the off-diagonal terms vanish and the modes become degenerate.

**12**, 1123–1134 (2006). [CrossRef]

*d*and has a zero at

*d*are almost degenerate [c.f. Fig. 2(a)]. The broken green line in Fig. 4(a) is for a cavity with length 8

*d*. Its sidelobes do not have a zero near

## 4. Conclusion

## Acknowledgments

## References and links

1. | K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, “Sub-femtojoule all-optical switching using a photonic-crystal nanocavity,” Nat. Photonics |

2. | J. Vuĉković and Y. Yamamoto, “Photonic crystal microcavities for cavity quantum electrodynamics with a single quantum dot,” Appl. Phys. Lett. |

3. | M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, “Optomechanical crystals,” Nature |

4. | O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science |

5. | B. S. Song, S. Noda, T. Asano, and Y. Akahane, “Ultra-high-Q photonic double-heterostructure nanocavity,” Nat. Mater. |

6. | E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, “Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect,” Appl. Phys. Lett. |

7. | Y. Tanaka, T. Asano, and S. Noda, “Design of photonic crystal nanocavity with |

8. | S. Tomljenovic Hanic, M. J. Steel, C. M. de Sterke, and D. J. Moss, “High-Q cavities in photosensitive photonic crystals,” Opt. Lett. |

9. | S. Gardin, F. Bordas, X. Letartre, C. Seassal, A. Rahmani, R. Bozio, and P. Viktorovitch, “Microlasers based on effective index confined slow light modes in photonic crystal waveguides,” Opt. Express |

10. | T. Asano, B. S. Song, Y. Akahane, and S. Noda, “Ultrahigh-Q nanocavities in two-dimensional photonic crystal slabs,” IEEE J. Sel. Top. Quantum Electron. |

11. | D. Englund, I. Fushman, and J. Vuĉković, “General recipe for designing photonic crystal cavities,” Opt. Express |

12. | M. Ibanescu, S. G. Johnson, D. Roundy, Y. Fink, and J. D. Joannopoulos, “Microcavity confinement based on an anomalous zero group-velocity waveguide mode,” Opt. Lett. |

13. | A. Figotin and I. Vitebskiy, “Slow-wave resonance in periodic stacks of anisotropic layers,” Phys. Rev. A |

14. | A. A. Chabanov, “Strongly resonant transmission of electromagnetic radiation in periodic anisotropic layered media,” Phys. Rev. A |

15. | K. Y. Jung and F. L. Teixeira, “Numerical study of photonic crystals with a split band edge: Polarization dependence and sensitivity analysis,” Phys. Rev. A |

16. | S. Ha, A. A. Sukhorukov, A. V. Lavrinenko, and Yu. S. Kivshar, “Cavity mode control in side-coupled periodic waveguides: theory and experiment,” Photonics Nanostruct.: Fundam. Appl. |

17. | S. Mahmoodian, C. G. Poulton, K. B. Dossou, R. C. McPhedran, L. C. Botten, and C. M. de Sterke, “Modes of Shallow Photonic Crystal Waveguides: Semi-Analytic Treatment,” Opt. Express |

18. | J. M. Luttinger and W. Kohn, “Motion of electrons and holes in perturbed periodic fields,” Phys. Rev. |

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

20. | P. St. J. Russell, T. A. Birks, and F. D. Lloyd Lucas, “Photonic Bloch waves and photonic band gaps,” in |

21. | S. W. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, A. V. Lavrinenko, D. N. Chigrin, and Yu. S. Kivshar, “Dispersionless tunneling of slow light in antisymmetric photonic crystal couplers,” Opt. Express |

22. | M. W. Lee, C. Grillet, S. Tomljenovic Hanic, E. C. Magi, D. J. Moss, B. J. Eggleton, X. Gai, S. Madden, D. Y. Choi, D. A. P. Bulla, and B. Luther-Davies, “Photowritten high-Q cavities in two-dimensional chalcogenide glass photonic crystals,” Opt. Lett. |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(140.3948) Lasers and laser optics : Microcavity devices

(350.4238) Other areas of optics : Nanophotonics and photonic crystals

(130.5296) Integrated optics : Photonic crystal waveguides

(050.5298) Diffraction and gratings : Photonic crystals

(230.5298) Optical devices : Photonic crystals

**ToC Category:**

Integrated Optics

**History**

Original Manuscript: September 21, 2010

Revised Manuscript: November 18, 2010

Manuscript Accepted: November 20, 2010

Published: November 23, 2010

**Citation**

Sahand Mahmoodian, Andrey A. Sukhorukov, Sangwoo Ha, Andrei V. Lavrinenko, Christopher G. Poulton, Kokou B. Dossou, Lindsay C. Botten, Ross C. McPhedran, and C. M. de Sterke, "Paired modes of heterostructure cavities in photonic crystal waveguides with split band edges," Opt. Express **18**, 25693-25701 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-25-25693

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### References

- K. Nozaki, T. Tanabe, A. Shinya, S. Matsuo, T. Sato, H. Taniyama, and M. Notomi, "Sub-femtojoule all-optical switching using a photonic-crystal nanocavity," Nat. Photonics 4, 477-483 (2010). [CrossRef]
- J. Vuckovic, and Y. Yamamoto, "Photonic crystal microcavities for cavity quantum electrodynamics with a single quantum dot," Appl. Phys. Lett. 82, 2374-2376 (2003). [CrossRef]
- M. Eichenfield, J. Chan, R. M. Camacho, K. J. Vahala, and O. Painter, "Optomechanical crystals," Nature 462, 78-82 (2009). [CrossRef] [PubMed]
- O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O’Brien, P. D. Dapkus, and I. Kim, "Two-dimensional photonic band-gap defect mode laser," Science 284, 1819-1821 (1999). [CrossRef] [PubMed]
- B. S. Song, S. Noda, T. Asano, and Y. Akahane, "Ultra-high-Q photonic double-heterostructure nanocavity," Nat. Mater. 4, 207-210 (2005). [CrossRef]
- E. Kuramochi, M. Notomi, S. Mitsugi, A. Shinya, T. Tanabe, and T. Watanabe, "Ultrahigh-Q photonic crystal nanocavities realized by the local width modulation of a line defect," Appl. Phys. Lett. 88, 041112 (2006). [CrossRef]
- Y. Tanaka, T. Asano, and S. Noda, "Design of photonic crystal nanocavity with Q-factor of ~109," J. Lightwave Technol. 26, 1532-1539 (2008). [CrossRef]
- S. Tomljenovic Hanic, M. J. Steel, C. M. de Sterke, and D. J. Moss, "High-Q cavities in photosensitive photonic crystals," Opt. Lett. 32, 542-544 (2007). [CrossRef] [PubMed]
- S. Gardin, F. Bordas, X. Letartre, C. Seassal, A. Rahmani, R. Bozio, and P. Viktorovitch, "Microlasers based on effective index confined slow light modes in photonic crystal waveguides," Opt. Express 16, 6331-6339 (2008). [CrossRef] [PubMed]
- T. Asano, B. S. Song, Y. Akahane, and S. Noda, "Ultrahigh-Q nanocavities in two-dimensional photonic crystal slabs," IEEE J. Sel. Top. Quantum Electron. 12, 1123-1134 (2006). [CrossRef]
- D. Englund, I. Fushman, and J. Vuckovic, "General recipe for designing photonic crystal cavities," Opt. Express 13, 5961-5975 (2005). [CrossRef] [PubMed]
- M. Ibanescu, S. G. Johnson, D. Roundy, Y. Fink, and J. D. Joannopoulos, "Microcavity confinement based on an anomalous zero group-velocity waveguide mode," Opt. Lett. 30, 552-554 (2005). [CrossRef] [PubMed]
- A. Figotin, and I. Vitebskiy, "Slow-wave resonance in periodic stacks of anisotropic layers," Phys. Rev. A 76, 053839 (2007). [CrossRef]
- A. A. Chabanov, "Strongly resonant transmission of electromagnetic radiation in periodic anisotropic layered media," Phys. Rev. A 77, 033811 (2008). [CrossRef]
- K. Y. Jung, and F. L. Teixeira, "Numerical study of photonic crystals with a split band edge: Polarization dependence and sensitivity analysis," Phys. Rev. A 78, 043826 (2008). [CrossRef]
- S. Ha, A. A. Sukhorukov, A. V. Lavrinenko, and Yu. S. Kivshar, "Cavity mode control in side-coupled periodic waveguides: theory and experiment," Photonics Nanostruct.: Fundam. Appl. 8, 310-317 (2010). [CrossRef]
- S. Mahmoodian, C. G. Poulton, K. B. Dossou, R. C. McPhedran, L. C. Botten, and C. M. de Sterke, "Modes of Shallow Photonic Crystal Waveguides: Semi-Analytic Treatment," Opt. Express 17, 19629-19643 (2009). [CrossRef] [PubMed]
- J. M. Luttinger, and W. Kohn, "Motion of electrons and holes in perturbed periodic fields," Phys. Rev. 97, 869-883 (1955). [CrossRef]
- J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, 1995).
- P. St, J. Russell, T. A. Birks, and F. D. Lloyd Lucas, "Photonic Bloch waves and photonic band gaps," in Confined Electrons and Photons, E. Burstein and C. Weisbuch, eds., (1995), pp. 585-633.
- S. W. Ha, A. A. Sukhorukov, K. B. Dossou, L. C. Botten, A. V. Lavrinenko, D. N. Chigrin, and Yu. S. Kivshar, "Dispersionless tunneling of slow light in antisymmetric photonic crystal couplers," Opt. Express 16, 1104-1114 (2008). [CrossRef] [PubMed]
- M. W. Lee, C. Grillet, S. Tomljenovic Hanic, E. C. Magi, D. J. Moss, B. J. Eggleton, X. Gai, S. Madden, D. Y. Choi, D. A. P. Bulla, and B. Luther-Davies, "Photowritten high-Q cavities in two-dimensional chalcogenide glass photonic crystals," Opt. Lett. 34, 3671-3673 (2009). [CrossRef] [PubMed]

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