## Orders of magnitude enhancement of mode splitting by plasmonic intracavity resonance |

Optics Express, Vol. 20, Issue 20, pp. 22172-22180 (2012)

http://dx.doi.org/10.1364/OE.20.022172

Acrobat PDF (1388 KB)

### Abstract

we report on significant mode splitting in plasmonic resonators induced by intracavity resonance. In contrast to traditional dielectric resonators where only picometer range of splitting was achieved, splitting over several hundred nanometers can be obtained without using ultrahigh quality resonators. We show that by appropriately choosing the coupling length, minute reflection is sufficient to establish intracavity resonance, which effectively lifts the degeneracy of the counterpropagating modes in the resonator. The mode splitting provides two self-referenced channels enabling simultaneous monitoring of the position and the polarizability of nano-scatterers in the resonator.

© 2012 OSA

## 1. Introduction

1. V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “Very low threshold whispering-gallery-mode microsphere laser,” Phys. Rev. A **54**(3), R1777–R1780 (1996). [CrossRef] [PubMed]

3. T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. **93**(8), 083904 (2004). [CrossRef] [PubMed]

4. S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A **71**(1), 013817 (2005). [CrossRef]

5. A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science **317**(5839), 783–787 (2007). [CrossRef] [PubMed]

7. A. Weller, F. C. Liu, R. Dahint, and M. Himmelhaus, “Whispering gallery mode biosensors in the low Q limit,” Appl. Phys. B **90**(3-4), 561–567 (2008). [CrossRef]

8. S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature **440**(7083), 508–511 (2006). [CrossRef] [PubMed]

9. G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. **87**(13), 131102 (2005). [CrossRef]

10. D. F. P. Pile and D. K. Gramotnev, “Plasmonic subwavelength waveguides: next to zero losses at sharp bends,” Opt. Lett. **30**(10), 1186–1188 (2005). [CrossRef] [PubMed]

11. A. Pannipitiya, I. D. Rukhlenko, M. Premaratne, H. T. Hattori, and G. P. Agrawal, “Improved transmission model for metal-dielectric-metal plasmonic waveguides with stub structure,” Opt. Express **18**(6), 6191–6204 (2010). [CrossRef] [PubMed]

14. J. Liu, G. Fang, H. Zhao, Y. Zhang, and S. Liu, “Plasmon flow control at gap waveguide junctions using square ring resonators,” J. Phys. D Appl. Phys. **43**(5), 055103 (2010). [CrossRef]

15. T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Modal coupling in traveling-wave resonators,” Opt. Lett. **27**(19), 1669–1671 (2002). [CrossRef] [PubMed]

16. M. L. Gorodetsky, A. D. Pryamikov, and V. S. Ilchenko, “Rayleigh scattering in high-Q microspheres,” J. Opt. Soc. Am. B **17**(6), 1051–1057 (2000). [CrossRef]

5. A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science **317**(5839), 783–787 (2007). [CrossRef] [PubMed]

6. F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods **5**(7), 591–596 (2008). [CrossRef] [PubMed]

17. T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Natl. Acad. Sci. U.S.A. **108**(15), 5976–5979 (2011). [CrossRef] [PubMed]

18. J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics **4**(1), 46–49 (2010). [CrossRef]

## 2. Modeling

_{l}and G

_{p}denote the coupling length and the gap, respectively. The dielectric permittivity of the silver is modeled by Drude model ε(ω) = 1-ω

_{p}

^{2}/ω

^{2}+ iων

_{p}, where ω

_{p}stands for the plasma frequency and ν

_{p}represents for the collision frequency. These values are taken from reference [19

19. M. A. Ordal, R. J. Bell, R. W. Alexander Jr, L. L. Long, and M. R. Querry, “Optical properties of fourteen metals in the infrared and far infrared: Al, Co, Cu, Au, Fe, Pb, Mo, Ni, Pd, Pt, Ag, Ti, V, and W,” Appl. Opt. **24**(24), 4493–4499 (1985). [CrossRef] [PubMed]

_{p}= 1.38 × 10

^{16}rad/s and ν

_{p}= 2.73 × 10

^{13}rad/s at the wavelength λ = 1550 nm. Following Haus’s approach [20], critical coupling occurs when the intrinsic quality factor Q

_{in}equals to the external quality factor Q

_{ex}[16

16. M. L. Gorodetsky, A. D. Pryamikov, and V. S. Ilchenko, “Rayleigh scattering in high-Q microspheres,” J. Opt. Soc. Am. B **17**(6), 1051–1057 (2000). [CrossRef]

21. M. L. Gorodetsky and V. S. Ilchenko, “Optical microsphere resonators: optimal coupling to high-Q whispering gallery modes,” J. Opt. Soc. Am. B **16**(1), 147–154 (1999). [CrossRef]

_{0}is the resonant frequency of the unperturbed resonator. The intrinsic quality factor of the resonator was dominated by the propagation loss of the plasmonic waveguide and can be expressed as Q

_{in}= n

_{R}/2n

_{I}, where n

_{R}and n

_{I}are the real and imaginary part of the modal index, respectively. These parameters can be obtained by solving the characteristic equation of the plasmonic waveguide for the TM mode. On the other hand, the external quality factor can be represented by Q

_{ex}= 2πN/η, where N denotes the order of the resonant mode, and η represents the external coupling efficiency. At the wavelength of 1550nm, the modal index of the fundamental mode of the Ag/Air(w = 100nm)/Ag waveguide is calculated to be n = 1.200 + i0.00212, corresponding to Q

_{in}= 283. To maintain the resonance at λ~1550 nm, the total length of the resonator is kept at 3864 nm (for N = 3 modes) in all cases. Therefore, the length in x-direction has to be adjusted correspondingly for different coupling lengths. To find the critical coupling condition, resonators with different coupling length C

_{l}and gap G

_{p}are considered. Similar to the case of directional couplers, the external coupling efficiency η in the coupling zone was calculated by the finite-different time-domain (FDTD) method. The resulting Q

_{ex}can therefore be determined. In the simulation, an otherwise infinitely extended space is truncated by fifteen-layer convolution perfectly matched layer (CPML) [22

22. G.-X. Fan and Q. H. Liu, “An FDTD algorithm with perfectly matched layers for general dispersive media,” IEEE Trans. Antenn. Propag. **48**(5), 637–646 (2000). [CrossRef]

23. J. A. Roden and S. D. Gedney, “Convolution PML (CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media,” Microw. Opt. Technol. Lett. **27**(5), 334–339 (2000). [CrossRef]

^{−18}sec, respectively. To reach convergence, a total time steps of 3 × 10

^{5}is acquired.

_{ex}decreases monotonically with the increase of the perturbation, for instance, the coupling length. Normally, the larger the gap, the weaker the perturbation, and a longer coupling length is required to transfer the energy completely into the resonator.

_{ex}= Q

_{in}is satisfied, according to Eq. (1), the spectral response should exhibit a singlet anti-Lorentzian shape. However, our FDTD result shows spectral splitting under certain conditions, as shown in Fig. 3 . Although similar phenomena were observed by researchers in the case of dielectric resonators, it is widely attributed to the effect of scattering due to roughness or adsorbate on the surface of the resonator, and the maximization of the splitting has rarely been addressed. In the present study, we found that the splitting may arise from the intrinsic feedback of the resonator upon coupling with the waveguide. In particular, it is found that when the intra-cavity resonance C

_{l}= mλ

_{0}/n

_{R}= mλ

_{eff}(m = integer) is established in the coupling zone, the splitting is maximized. As shown in Fig. 3(a)-3(h), the spectral response evolved from singlet to doublet and back to singlet again. To quantify the degree of splitting, we define a modal splitting factor M

_{s}= cos

^{2}(2πC

_{l}/λ

_{eff}), corresponding to a measure of the detuning from the resonant condition. With the increase of the C

_{l}, the FP resonance is gradually established. When C

_{l}= 1300 nm (~λ

_{eff}), the splitting is maximized as predicted by the splitting factor M

_{s}. In general, the closer to the resonant condition, the larger the splitting results. It should be also noted that to achieve a mode splitting as large as 140nm, only minute backward reflection as low as 5% is needed. This is analyzed in section 5.

## 3. Correspondence with coupled mode theory

15. T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Modal coupling in traveling-wave resonators,” Opt. Lett. **27**(19), 1669–1671 (2002). [CrossRef] [PubMed]

16. M. L. Gorodetsky, A. D. Pryamikov, and V. S. Ilchenko, “Rayleigh scattering in high-Q microspheres,” J. Opt. Soc. Am. B **17**(6), 1051–1057 (2000). [CrossRef]

_{in}and 1/τ

_{ex}are the decay rate due to the intrinsic and external loss associated with the quality factor Q = ω

_{0}τ/2 at the resonant frequency ω

_{0}. β denotes the backward coupling coefficient, and κ represents the coupling strength between the incident wave and the cavity modes. In steady state, the transmittance and reflectance subject to critical coupling condition can be represented by Eq. (3). If β<<1/τ

_{ex}, Eq. (3) can be reduced to Eq. (1), corresponding to the off-resonance condition in the coupling zone and traveling wave resonance occurs in the resonator. On the contrary, if β>>1/τ

_{ex}, intracavity resonance occurs in the coupling zone, and standing wave resonance results in the resonator. Due to the strong coupling between the CW and CCW waves, the unperturbed singlet resonant frequency ω

_{0}splits into doublets and shifts to ω

_{0}+ β and ω

_{0}-β. The results are compared with our full vectorial FDTD simulation in Fig. 3.

## 4. Phase front acceleration

_{c}per corner. In contrast to this, the coupling between the plasmonic waveguide and the resonator slightly raises the modal index, corresponding to an increased optical path. This leads to a slowed down phase front δφ

_{p}in the coupling zone which counteracts the accelerated phase front around corners. The net phase shift per round trip

_{c}- δφ

_{p}, resulting in the blue shift of the resonant wavelength calculated by FDTD. Following the calculation of the modal index of directional coupler, δφ

_{p}can be determined and discriminated from the total phase shift

_{c}= 2π × 0.016 rad and δφ

_{p}= 2π × 0.006 rad are obtained, as shown in Fig. 5(b). In the calculation, structural parameters (C

_{l,}G

_{p}) = (966, 24) subject to critical coupling condition of the traveling wave resonance was applied.

## 5. Angle dependent reflectivity and mode splitting

_{120}≅0.001, R

_{90}≅0.012 and R

_{60}≅0.051 respectively, as shown in Fig. 6(b)-6(e). As in our illustration, significant mode splitting ≅141nm was achieved in triangular-shaped resonator. Compared to the rectangular-shaped resonator, the mode splitting is twice larger which is attributed to the doubled ratio of the reflected amplitude (R

_{60}/ R

_{90})

^{0.5}≅2. While for parallelogram-shaped resonator with uneven reflectivity at both ends of the coupling zone, the mode splitting valued in between the two extremes.

_{y}|

^{2}of the plasmonic resonator is shown in Fig. 7 . It is found that the energy distributions between the two standing waves are orthogonal, i.e., the nodes of energy density for the symmetric mode lie at the antinodes of antisymmetric mode.

## 6. Illustration of practical applications

## 7. Conclusion

## Acknowledgments

## References and links

1. | V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “Very low threshold whispering-gallery-mode microsphere laser,” Phys. Rev. A |

2. | T. Lu, L. Yang, R. V. A. van Loon, A. Polman, and K. J. Vahala, “On-chip green silica upconversion microlaser,” Opt. Lett. |

3. | T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. |

4. | S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A |

5. | A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science |

6. | F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods |

7. | A. Weller, F. C. Liu, R. Dahint, and M. Himmelhaus, “Whispering gallery mode biosensors in the low Q limit,” Appl. Phys. B |

8. | S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature |

9. | G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. |

10. | D. F. P. Pile and D. K. Gramotnev, “Plasmonic subwavelength waveguides: next to zero losses at sharp bends,” Opt. Lett. |

11. | A. Pannipitiya, I. D. Rukhlenko, M. Premaratne, H. T. Hattori, and G. P. Agrawal, “Improved transmission model for metal-dielectric-metal plasmonic waveguides with stub structure,” Opt. Express |

12. | J. Liu, G. Fang, H. Zhao, Y. Zhang, and S. Liu, “Surface plasmon reflector based on serial stub structure,” Opt. Express |

13. | A. Hosseini and Y. Massoud, “Nanoscale surface plasmon based resonator using rectangular geometry,” Appl. Phys. Lett. |

14. | J. Liu, G. Fang, H. Zhao, Y. Zhang, and S. Liu, “Plasmon flow control at gap waveguide junctions using square ring resonators,” J. Phys. D Appl. Phys. |

15. | T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Modal coupling in traveling-wave resonators,” Opt. Lett. |

16. | M. L. Gorodetsky, A. D. Pryamikov, and V. S. Ilchenko, “Rayleigh scattering in high-Q microspheres,” J. Opt. Soc. Am. B |

17. | T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Natl. Acad. Sci. U.S.A. |

18. | J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics |

19. | M. A. Ordal, R. J. Bell, R. W. Alexander Jr, L. L. Long, and M. R. Querry, “Optical properties of fourteen metals in the infrared and far infrared: Al, Co, Cu, Au, Fe, Pb, Mo, Ni, Pd, Pt, Ag, Ti, V, and W,” Appl. Opt. |

20. | A. Haus, |

21. | M. L. Gorodetsky and V. S. Ilchenko, “Optical microsphere resonators: optimal coupling to high-Q whispering gallery modes,” J. Opt. Soc. Am. B |

22. | G.-X. Fan and Q. H. Liu, “An FDTD algorithm with perfectly matched layers for general dispersive media,” IEEE Trans. Antenn. Propag. |

23. | J. A. Roden and S. D. Gedney, “Convolution PML (CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media,” Microw. Opt. Technol. Lett. |

24. | J. Avelin, R. Sharma, I. Hänninen, and A. H. Sihvola, “Polarizability analysis of cubical and square-shaped dielectric scatterers,” IEEE Trans. Antenn. Propag. |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(250.5403) Optoelectronics : Plasmonics

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: July 17, 2012

Revised Manuscript: September 8, 2012

Manuscript Accepted: September 9, 2012

Published: September 12, 2012

**Citation**

Chao-Yi Tai and Wen-Hsiang Yu, "Orders of magnitude enhancement of mode splitting by plasmonic intracavity resonance," Opt. Express **20**, 22172-22180 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-20-22172

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### References

- V. Sandoghdar, F. Treussart, J. Hare, V. Lefèvre-Seguin, J.-M. Raimond, and S. Haroche, “Very low threshold whispering-gallery-mode microsphere laser,” Phys. Rev. A 54(3), R1777–R1780 (1996). [CrossRef] [PubMed]
- T. Lu, L. Yang, R. V. A. van Loon, A. Polman, and K. J. Vahala, “On-chip green silica upconversion microlaser,” Opt. Lett. 34(4), 482–484 (2009). [CrossRef] [PubMed]
- T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Kerr-nonlinearity optical parametric oscillation in an ultrahigh-Q toroid microcavity,” Phys. Rev. Lett. 93(8), 083904 (2004). [CrossRef] [PubMed]
- S. M. Spillane, T. J. Kippenberg, K. J. Vahala, K. W. Goh, E. Wilcut, and H. J. Kimble, “Ultrahigh-Q toroidal microresonators for cavity quantum electrodynamics,” Phys. Rev. A 71(1), 013817 (2005). [CrossRef]
- A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science 317(5839), 783–787 (2007). [CrossRef] [PubMed]
- F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods 5(7), 591–596 (2008). [CrossRef] [PubMed]
- A. Weller, F. C. Liu, R. Dahint, and M. Himmelhaus, “Whispering gallery mode biosensors in the low Q limit,” Appl. Phys. B 90(3-4), 561–567 (2008). [CrossRef]
- S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440(7083), 508–511 (2006). [CrossRef] [PubMed]
- G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. 87(13), 131102 (2005). [CrossRef]
- D. F. P. Pile and D. K. Gramotnev, “Plasmonic subwavelength waveguides: next to zero losses at sharp bends,” Opt. Lett. 30(10), 1186–1188 (2005). [CrossRef] [PubMed]
- A. Pannipitiya, I. D. Rukhlenko, M. Premaratne, H. T. Hattori, and G. P. Agrawal, “Improved transmission model for metal-dielectric-metal plasmonic waveguides with stub structure,” Opt. Express 18(6), 6191–6204 (2010). [CrossRef] [PubMed]
- J. Liu, G. Fang, H. Zhao, Y. Zhang, and S. Liu, “Surface plasmon reflector based on serial stub structure,” Opt. Express 17(22), 20134–20139 (2009). [CrossRef] [PubMed]
- A. Hosseini and Y. Massoud, “Nanoscale surface plasmon based resonator using rectangular geometry,” Appl. Phys. Lett. 90(18), 181102 (2007). [CrossRef]
- J. Liu, G. Fang, H. Zhao, Y. Zhang, and S. Liu, “Plasmon flow control at gap waveguide junctions using square ring resonators,” J. Phys. D Appl. Phys. 43(5), 055103 (2010). [CrossRef]
- T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Modal coupling in traveling-wave resonators,” Opt. Lett. 27(19), 1669–1671 (2002). [CrossRef] [PubMed]
- M. L. Gorodetsky, A. D. Pryamikov, and V. S. Ilchenko, “Rayleigh scattering in high-Q microspheres,” J. Opt. Soc. Am. B 17(6), 1051–1057 (2000). [CrossRef]
- T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” Proc. Natl. Acad. Sci. U.S.A. 108(15), 5976–5979 (2011). [CrossRef] [PubMed]
- J. Zhu, S. K. Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Photonics 4(1), 46–49 (2010). [CrossRef]
- M. A. Ordal, R. J. Bell, R. W. Alexander, L. L. Long, and M. R. Querry, “Optical properties of fourteen metals in the infrared and far infrared: Al, Co, Cu, Au, Fe, Pb, Mo, Ni, Pd, Pt, Ag, Ti, V, and W,” Appl. Opt. 24(24), 4493–4499 (1985). [CrossRef] [PubMed]
- A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, 1984).
- M. L. Gorodetsky and V. S. Ilchenko, “Optical microsphere resonators: optimal coupling to high-Q whispering gallery modes,” J. Opt. Soc. Am. B 16(1), 147–154 (1999). [CrossRef]
- G.-X. Fan and Q. H. Liu, “An FDTD algorithm with perfectly matched layers for general dispersive media,” IEEE Trans. Antenn. Propag. 48(5), 637–646 (2000). [CrossRef]
- J. A. Roden and S. D. Gedney, “Convolution PML (CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media,” Microw. Opt. Technol. Lett. 27(5), 334–339 (2000). [CrossRef]
- J. Avelin, R. Sharma, I. Hänninen, and A. H. Sihvola, “Polarizability analysis of cubical and square-shaped dielectric scatterers,” IEEE Trans. Antenn. Propag. 49(3), 451–457 (2001). [CrossRef]

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