## Analysis of hybrid plasmonic-photonic crystal structures using perturbation theory |

Optics Express, Vol. 20, Issue 15, pp. 16992-17000 (2012)

http://dx.doi.org/10.1364/OE.20.016992

Acrobat PDF (913 KB)

### Abstract

A perturbation theory approach for the analysis of hybrid plasmonic- photonic crystal structures is presented. This theory allows for accurate calculation of the resonance frequency shift and quality factor change when introducing a resonant plasmonic structure into a photonic crystal microcavity. An example calculation is shown, agreeing to within 5% with comprehensive finite difference time domain simulations but taking an order of magnitude less time. This theoretical approach overcomes the challenge of poor scaling in computations with hybrid plasmonic-photonic crystal structures, allowing for rapid design optimization in such hybrid geometries.

© 2012 OSA

## 1. Introduction

1. K. J. Vahala, “Optical microcavities,” Nature **424**(6950), 839–846 (2003). [CrossRef]

3. S. Hughes, “Coupled cavity QED using planar photonic crystals,” Phys. Rev. Lett. **98**(8), 083603 (2007). [CrossRef]

4. H. Ryu, M. Notomi, and Y. Lee, “High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,” Appl. Phys. Lett. **83**(21), 4294–4296 (2003). [CrossRef]

5. S. A. Maier, “Effective mode volume of nanoscale plasmon cavities,” Opt. Quantum Electron. **38**(1-3), 257–267 (2006). [CrossRef]

6. S. A. Maier, “Plasmonics: metal nanostructures for subwavelength photonic devices,” IEEE J. Sel. Top. Quantum Electron. **12**(6), 1214–1220 (2006). [CrossRef]

10. Y. X. Ni, D. L. Gao, Z. F. Sang, L. Gao, and C. W. Qiu, “Influence of spherical anisotropy on the optical properties of plasmon resonant metallic nanoparticles,” Appl. Phys., A Mater. Sci. Process. **102**(3), 673–679 (2011). [CrossRef]

11. M. Barth, S. Schietinger, S. Fischer, J. Becker, N. Nüsse, T. Aichele, B. Löchel, C. Sönnichsen, and O. Benson, “Nanoassembled plasmonic-photonic hybrid cavity for tailored light-matter coupling,” Nano Lett. **10**(3), 891–895 (2010). [CrossRef]

14. C. Grillet, C. Monat, C. L. Smith, B. L. Eggleton, D. J. Moss, S. Frédérick, D. Dalacu, P. J. Poole, J. Lapointe, G. Aers, and R. L. Williams, “Nanowire coupling to photonic crystal nanocavities for single photon sources,” Opt. Express **15**(3), 1267–1276 (2007). [CrossRef]

11. M. Barth, S. Schietinger, S. Fischer, J. Becker, N. Nüsse, T. Aichele, B. Löchel, C. Sönnichsen, and O. Benson, “Nanoassembled plasmonic-photonic hybrid cavity for tailored light-matter coupling,” Nano Lett. **10**(3), 891–895 (2010). [CrossRef]

15. I. Mukherjee, G. Hajisalem, and R. Gordon, “One step Integration of metal nanoparticles in photonic crystal nanobeam cavity,” Opt. Express **19**(23), 22462–22469 (2011). [CrossRef]

16. S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and E. F. Schubert, “High extraction efficiency of spontaneous emission from slabs of photonic crystals,” Phys. Rev. Lett. **78**(17), 3294–3297 (1997). [CrossRef]

17. M. Pelton, C. Santori, J. Vučković, B. Zhang, G. S. Solomon, J. Plant, and Y. Yamamoto, “Efficient sources of single photons: a single quantum dot in a micropost microcavity,” Phys. Rev. Lett. **89**(23), 233602 (2002). [CrossRef]

15. I. Mukherjee, G. Hajisalem, and R. Gordon, “One step Integration of metal nanoparticles in photonic crystal nanobeam cavity,” Opt. Express **19**(23), 22462–22469 (2011). [CrossRef]

18. M. Kim, S. H. Lee, M. Choi, B. Ahn, N. Park, Y. H. Lee, and B. Min, “Low-loss surface-plasmonic nanobeam cavities,” Opt. Express **18**(11), 11089–11096 (2010). [CrossRef]

19. M. W. McCutcheon and M. Loncar, “Design of a silicon nitride photonic crystal cavity with a quality factor of one million for coupling to a diamond nanocrystal,” Opt. Express **16**(23), 19136–19144 (2008). [CrossRef]

21. G. D. Kondylis, F. D. Flaviis, G. J. Pottie, and T. Itoh, “A memory efficient formulation of the finite difference time domain method for the solution of Maxwell’s equations,” IEEE Trans. Microw. Theory Tech. **49**(7), 1310–1320 (2001). [CrossRef]

22. S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **65**(6), 066611 (2002). [CrossRef]

28. L. Lalouat, B. Cluzel, P. Velha, E. Picard, D. Peyrade, J. P. Hugonin, P. Lalanne, E. Hadji, and F. de Fornel, “Near-field interactions between a subwavelength tip and a small-volume photonic-crystal nanocavity,” Phys. Rev. B **76**(4), 041102 (2007). [CrossRef]

## 2. Perturbation theory

_{2}and B

_{2}can be written as: From Maxwell’s differential equations, we can derive from Eqs. (1) and (5): Similarly, the expressions for magnetic fields can be obtained as: Now we use the vector identity

28. L. Lalouat, B. Cluzel, P. Velha, E. Picard, D. Peyrade, J. P. Hugonin, P. Lalanne, E. Hadji, and F. de Fornel, “Near-field interactions between a subwavelength tip and a small-volume photonic-crystal nanocavity,” Phys. Rev. B **76**(4), 041102 (2007). [CrossRef]

26. A. Morand, K. Phan-Huy, Y. Desieres, and P. Benech, “Analytical study of the microdisk’s resonant modes coupling with a waveguide based on the perturbation theory,” J. Lightwave Technol. **22**(3), 827–832 (2004). [CrossRef]

## 3. Perturbation theory applied to a photonic cavity with an Ag nanoparticle

15. I. Mukherjee, G. Hajisalem, and R. Gordon, “One step Integration of metal nanoparticles in photonic crystal nanobeam cavity,” Opt. Express **19**(23), 22462–22469 (2011). [CrossRef]

^{3}nanobeam with a mode source centered at the resonant frequency of the cavity (shown in Fig. 2(a)), we obtain the unperturbed electric and magnetic field values (

^{3}) that represents the cavity volume (Fig. 2(b)). Next, a silver nanoparticle is placed at the center of the nanobeam and the perturbed cavity fields (

31. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B **6**(12), 4370–4379 (1972). [CrossRef]

^{3}produces an extinction peak at ~600 nm as shown in Fig. 3(a) , when simulated on a silicon nitride substrate. Figure 3(b) shows a cross section of the particle on silicon nitride indicating non-uniformity of fields inside the particle due to the effect of the substrate.

## 4. Theory results and FDTD verification

^{3}, applied perturbation theory and verified the results using FDTD. Our theory predicted a frequency shift of 1.39 THz and a quality factor of 1270.17 as compared to FDTD which calculated a frequency shift of 1.42 THz and a quality factor of 1327.67. Similarly, to test the effect of our theory on other hybrid structures, we replaced the nanobeam cavity with a reduced hybrid cavity using perfect electric conductor walls as mirrors instead of the photonic crystal holes. Calculations were repeated using the ellipsoidal particle as the source of perturbation and a mesh step size of 1 nm in all directions to achieve the best possible convergence in a reasonable simulation time.

## 5. Conclusions

## Acknowledgment

## References and links

1. | K. J. Vahala, “Optical microcavities,” Nature |

2. | S. Hughes and P. Yao, “Theory of quantum light emission from strongly-coupled single quantum dot photonic-crystal cavity system,” Opt. Express |

3. | S. Hughes, “Coupled cavity QED using planar photonic crystals,” Phys. Rev. Lett. |

4. | H. Ryu, M. Notomi, and Y. Lee, “High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,” Appl. Phys. Lett. |

5. | S. A. Maier, “Effective mode volume of nanoscale plasmon cavities,” Opt. Quantum Electron. |

6. | S. A. Maier, “Plasmonics: metal nanostructures for subwavelength photonic devices,” IEEE J. Sel. Top. Quantum Electron. |

7. | S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. |

8. | E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science |

9. | J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater. |

10. | Y. X. Ni, D. L. Gao, Z. F. Sang, L. Gao, and C. W. Qiu, “Influence of spherical anisotropy on the optical properties of plasmon resonant metallic nanoparticles,” Appl. Phys., A Mater. Sci. Process. |

11. | M. Barth, S. Schietinger, S. Fischer, J. Becker, N. Nüsse, T. Aichele, B. Löchel, C. Sönnichsen, and O. Benson, “Nanoassembled plasmonic-photonic hybrid cavity for tailored light-matter coupling,” Nano Lett. |

12. | F. De Angelis, M. Patrini, G. Das, I. Maksymov, M. Galli, L. Businaro, L. C. Andreani, and E. Di Fabrizio, “A hybrid plasmonic photonic nanodevice for label free detection of a few molecules,” Nano Lett. |

13. | P. E. Barclay, K. M. Fu, C. Santori, and R. G. Beausoleil, “Hybrid photonic crystal cavity and waveguide for coupling to diamond NV centers,” Opt. Express |

14. | C. Grillet, C. Monat, C. L. Smith, B. L. Eggleton, D. J. Moss, S. Frédérick, D. Dalacu, P. J. Poole, J. Lapointe, G. Aers, and R. L. Williams, “Nanowire coupling to photonic crystal nanocavities for single photon sources,” Opt. Express |

15. | I. Mukherjee, G. Hajisalem, and R. Gordon, “One step Integration of metal nanoparticles in photonic crystal nanobeam cavity,” Opt. Express |

16. | S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and E. F. Schubert, “High extraction efficiency of spontaneous emission from slabs of photonic crystals,” Phys. Rev. Lett. |

17. | M. Pelton, C. Santori, J. Vučković, B. Zhang, G. S. Solomon, J. Plant, and Y. Yamamoto, “Efficient sources of single photons: a single quantum dot in a micropost microcavity,” Phys. Rev. Lett. |

18. | M. Kim, S. H. Lee, M. Choi, B. Ahn, N. Park, Y. H. Lee, and B. Min, “Low-loss surface-plasmonic nanobeam cavities,” Opt. Express |

19. | M. W. McCutcheon and M. Loncar, “Design of a silicon nitride photonic crystal cavity with a quality factor of one million for coupling to a diamond nanocrystal,” Opt. Express |

20. | Y. Liu, Z. Yang, Z. Liang, and L. Qi, “A memory efficient strategy for FDTD implementation applied to photonic crystal problems,” Prog. Electromagn. Res. |

21. | G. D. Kondylis, F. D. Flaviis, G. J. Pottie, and T. Itoh, “A memory efficient formulation of the finite difference time domain method for the solution of Maxwell’s equations,” IEEE Trans. Microw. Theory Tech. |

22. | S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

23. | A. Kumar, K. Thyagarajan, and A. K. Ghatak, “Analysis of rectangular-core dielectric waveguides: an accurate perturbation approach,” Opt. Lett. |

24. | A. Parkash, J. K. Vaid, and A. Mansingh, “Measurement of dielectric parameters at microwave frequencies by cavity-perturbation technique,” IEEE Trans. Microw. Theory Tech. |

25. | M. Lohmeyer, N. Bahlmann, and P. Hertel, “Geometry tolerance estimation for rectangular dielectric waveguide devices by means of perturbation theory,” Opt. Commun. |

26. | A. Morand, K. Phan-Huy, Y. Desieres, and P. Benech, “Analytical study of the microdisk’s resonant modes coupling with a waveguide based on the perturbation theory,” J. Lightwave Technol. |

27. | M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

28. | L. Lalouat, B. Cluzel, P. Velha, E. Picard, D. Peyrade, J. P. Hugonin, P. Lalanne, E. Hadji, and F. de Fornel, “Near-field interactions between a subwavelength tip and a small-volume photonic-crystal nanocavity,” Phys. Rev. B |

29. | R. A. Waldron, |

30. | H. A. Haus, |

31. | P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

32. | C. F. Bohren and D. R. Huffman, |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(230.5298) Optical devices : Photonic crystals

(250.5403) Optoelectronics : Plasmonics

**ToC Category:**

Photonic Crystals

**History**

Original Manuscript: May 15, 2012

Revised Manuscript: June 29, 2012

Manuscript Accepted: July 3, 2012

Published: July 11, 2012

**Citation**

Ishita Mukherjee and Reuven Gordon, "Analysis of hybrid plasmonic-photonic crystal structures using perturbation theory," Opt. Express **20**, 16992-17000 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-15-16992

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### References

- K. J. Vahala, “Optical microcavities,” Nature424(6950), 839–846 (2003). [CrossRef]
- S. Hughes and P. Yao, “Theory of quantum light emission from strongly-coupled single quantum dot photonic-crystal cavity system,” Opt. Express17(5), 3322–3330 (2009). [CrossRef]
- S. Hughes, “Coupled cavity QED using planar photonic crystals,” Phys. Rev. Lett.98(8), 083603 (2007). [CrossRef]
- H. Ryu, M. Notomi, and Y. Lee, “High-quality-factor and small-mode-volume hexapole modes in photonic-crystal-slab nanocavities,” Appl. Phys. Lett.83(21), 4294–4296 (2003). [CrossRef]
- S. A. Maier, “Effective mode volume of nanoscale plasmon cavities,” Opt. Quantum Electron.38(1-3), 257–267 (2006). [CrossRef]
- S. A. Maier, “Plasmonics: metal nanostructures for subwavelength photonic devices,” IEEE J. Sel. Top. Quantum Electron.12(6), 1214–1220 (2006). [CrossRef]
- S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys.98(1), 011101 (2005). [CrossRef]
- E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science311(5758), 189–193 (2006). [CrossRef]
- J. A. Schuller, E. S. Barnard, W. Cai, Y. C. Jun, J. S. White, and M. L. Brongersma, “Plasmonics for extreme light concentration and manipulation,” Nat. Mater.9(3), 193–204 (2010). [CrossRef]
- Y. X. Ni, D. L. Gao, Z. F. Sang, L. Gao, and C. W. Qiu, “Influence of spherical anisotropy on the optical properties of plasmon resonant metallic nanoparticles,” Appl. Phys., A Mater. Sci. Process.102(3), 673–679 (2011). [CrossRef]
- M. Barth, S. Schietinger, S. Fischer, J. Becker, N. Nüsse, T. Aichele, B. Löchel, C. Sönnichsen, and O. Benson, “Nanoassembled plasmonic-photonic hybrid cavity for tailored light-matter coupling,” Nano Lett.10(3), 891–895 (2010). [CrossRef]
- F. De Angelis, M. Patrini, G. Das, I. Maksymov, M. Galli, L. Businaro, L. C. Andreani, and E. Di Fabrizio, “A hybrid plasmonic photonic nanodevice for label free detection of a few molecules,” Nano Lett.8(8), 2321–2327 (2008). [CrossRef]
- P. E. Barclay, K. M. Fu, C. Santori, and R. G. Beausoleil, “Hybrid photonic crystal cavity and waveguide for coupling to diamond NV centers,” Opt. Express17(12), 9588–9601 (2009). [CrossRef]
- C. Grillet, C. Monat, C. L. Smith, B. L. Eggleton, D. J. Moss, S. Frédérick, D. Dalacu, P. J. Poole, J. Lapointe, G. Aers, and R. L. Williams, “Nanowire coupling to photonic crystal nanocavities for single photon sources,” Opt. Express15(3), 1267–1276 (2007). [CrossRef]
- I. Mukherjee, G. Hajisalem, and R. Gordon, “One step Integration of metal nanoparticles in photonic crystal nanobeam cavity,” Opt. Express19(23), 22462–22469 (2011). [CrossRef]
- S. Fan, P. R. Villeneuve, J. D. Joannopoulos, and E. F. Schubert, “High extraction efficiency of spontaneous emission from slabs of photonic crystals,” Phys. Rev. Lett.78(17), 3294–3297 (1997). [CrossRef]
- M. Pelton, C. Santori, J. Vučković, B. Zhang, G. S. Solomon, J. Plant, and Y. Yamamoto, “Efficient sources of single photons: a single quantum dot in a micropost microcavity,” Phys. Rev. Lett.89(23), 233602 (2002). [CrossRef]
- M. Kim, S. H. Lee, M. Choi, B. Ahn, N. Park, Y. H. Lee, and B. Min, “Low-loss surface-plasmonic nanobeam cavities,” Opt. Express18(11), 11089–11096 (2010). [CrossRef]
- M. W. McCutcheon and M. Loncar, “Design of a silicon nitride photonic crystal cavity with a quality factor of one million for coupling to a diamond nanocrystal,” Opt. Express16(23), 19136–19144 (2008). [CrossRef]
- Y. Liu, Z. Yang, Z. Liang, and L. Qi, “A memory efficient strategy for FDTD implementation applied to photonic crystal problems,” Prog. Electromagn. Res.3, 374–378 (2007).
- G. D. Kondylis, F. D. Flaviis, G. J. Pottie, and T. Itoh, “A memory efficient formulation of the finite difference time domain method for the solution of Maxwell’s equations,” IEEE Trans. Microw. Theory Tech.49(7), 1310–1320 (2001). [CrossRef]
- S. G. Johnson, M. Ibanescu, M. A. Skorobogatiy, O. Weisberg, J. D. Joannopoulos, and Y. Fink, “Perturbation theory for Maxwell’s equations with shifting material boundaries,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.65(6), 066611 (2002). [CrossRef]
- A. Kumar, K. Thyagarajan, and A. K. Ghatak, “Analysis of rectangular-core dielectric waveguides: an accurate perturbation approach,” Opt. Lett.8(1), 63–65 (1983). [CrossRef]
- A. Parkash, J. K. Vaid, and A. Mansingh, “Measurement of dielectric parameters at microwave frequencies by cavity-perturbation technique,” IEEE Trans. Microw. Theory Tech.27(9), 791–795 (1979). [CrossRef]
- M. Lohmeyer, N. Bahlmann, and P. Hertel, “Geometry tolerance estimation for rectangular dielectric waveguide devices by means of perturbation theory,” Opt. Commun.163(1-3), 86–94 (1999). [CrossRef]
- A. Morand, K. Phan-Huy, Y. Desieres, and P. Benech, “Analytical study of the microdisk’s resonant modes coupling with a waveguide based on the perturbation theory,” J. Lightwave Technol.22(3), 827–832 (2004). [CrossRef]
- M. Soljačić, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, “Optimal bistable switching in nonlinear photonic crystals,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.66(5), 055601 (2002). [CrossRef]
- L. Lalouat, B. Cluzel, P. Velha, E. Picard, D. Peyrade, J. P. Hugonin, P. Lalanne, E. Hadji, and F. de Fornel, “Near-field interactions between a subwavelength tip and a small-volume photonic-crystal nanocavity,” Phys. Rev. B76(4), 041102 (2007). [CrossRef]
- R. A. Waldron, Theory of Waveguides and Cavities (Maclaren & Sons, 1967).
- H. A. Haus, Waves and Fields in Optoelectronics (Prentice Hall Inc., 1984).
- P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972). [CrossRef]
- C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley & Sons, Inc., 1983).

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