## The transmission characteristics of surface plasmon polaritons in ring resonator

Optics Express, Vol. 17, Issue 26, pp. 24096-24101 (2009)

http://dx.doi.org/10.1364/OE.17.024096

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### Abstract

A two-dimensional nanoscale structure which consists of two metal-insulator-metal (MIM) waveguides coupled to each other by a ring resonator is designed. The transmission characteristics of surface plasmon polaritons are studied in this structure. There are several types of modes in the transmission spectrum. These modes exhibit red shift when the radius of the ring increases. The transmission properties of such structure are simulated by the Finite-Difference Time-Domain (FDTD) method, and the eignwavelengths of the ring resonator are calculated theoretically. Results obtained by the theory of the ring resonator are consistent with those from the FDTD simulations.

© 2009 OSA

## 1. Introduction

1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature **424**(6950), 824–830 (2003). [CrossRef] [PubMed]

3. 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]

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

6. T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyi, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. **85**(24), 5833–5835 (2004). [CrossRef]

7. Y. Chu, E. Schonbrun, T. Yang, and K. B. Crozier, “Experimental observation of narrow surface plasmon resonances in gold nanoparticle arrays,” Appl. Phys. Lett. **93**(18), 181108 (2008). [CrossRef]

8. E. Verhagen, A. Polman, and L. K. Kuipers, “Nanofocusing in laterally tapered plasmonic waveguides,” Opt. Express **16**(1), 45–57 (2008). [CrossRef] [PubMed]

9. Z. Han, E. Forsberg, and S. He, “Surface plasmon Bragg gratings formed in metal-insulator-metal waveguides,” IEEE Photon. Technol. Lett. **19**(2), 91–93 (2007). [CrossRef]

10. B. Wang and G. P. Wang, “Plasmon Bragg reflectors and nanocavities on flat metallic surfaces,” Appl. Phys. Lett. **87**(1), 013107 (2005). [CrossRef]

11. J. Q. Liu, L. L. Wang, M. D. He, W. Q. Huang, D. Wang, B. S. Zou, and S. Wen, “A wide bandgap plasmonic Bragg reflector,” Opt. Express **16**(7), 4888–4894 (2008). [CrossRef] [PubMed]

12. A. Hosseini and Y. Massoud, “Nanoscale surface Plasmon based resonator using rectangular geometry,” Appl. Phys. Lett. **90**(18), 181102 (2007). [CrossRef]

13. X. S. Lin and X. G. Huang, “Tooth-shaped plasmonic waveguide filters with nanometeric sizes,” Opt. Lett. **33**(23), 2874–2876 (2008). [CrossRef] [PubMed]

## 2. Theory model

*r*, and inner radius is

_{a}*r*. The radius of the ring is

_{i}*r*, which is the average of the inner and the outer radius,

*r = (r*, as depicted by the dashed circle in Fig. 1(a).

_{a}+ r_{i})/2*d*are the widths of the waveguides and the ring.

*w*are the coupling lengths between the waveguides and the ring. As is well known, the dispersion equation of SPPs in the MIM structure can be written as [14

14. I. P. Kaminow, W. L. Mammel, and H. P. Weber, “Metal-Clad optical waveguides: analytical and experimental study,” Appl. Opt. **13**(2), 396–405 (1974). [CrossRef] [PubMed]

*β*is the propagation constant of gap SPPs [15

_{gspp}15. E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. **182**(2), 539–554 (1969). [CrossRef]

*k*

_{0}

*= 2π/λ*is the wave number of light in the air,

*λ*is the wavelength of incident light.

*ε*

_{0}and

*ε*are the dielectric functions of air and silver, respectively. For more accurately match the experimental optical constant of silver,

_{m}*ε*can be characterized by the Lorentz-Drude model [16

_{m}16. A. D. Rakic, A. B. Djurisic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. **37**(22), 5271–5283 (1998). [CrossRef] [PubMed]

*ε*is the relative permittivity in the infinity frequency.

_{∞}*G*is the oscillator strengths, Ω

_{m}*is the plasma frequency,*

_{m}*ω*is the resonant frequency, Γ

_{m}*is the damping factor, and*

_{m}*ω*is the angular frequency of incident light. All the parameters of the Lorentz-Drude model can be found in Ref [16

16. A. D. Rakic, A. B. Djurisic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. **37**(22), 5271–5283 (1998). [CrossRef] [PubMed]

*n*

_{eff}= β/k_{0}, which can be calculated by Eq. (1). Figure 1(b) shows the real part of

*n*as a function of

_{eff}*d*and

*λ*. As seen from the figure, the

*n*decreases as the width

_{eff}*d*increases with the same wavelength

*λ*, while

*n*varies very slowly when the value of

_{eff}*λ*changes with

*d*fixed.

## 3. Results and discussions

*A*

_{1}and the transmitted amplitude

*A*

_{2}. The transmittance is defined to be

*T = A*

_{2}

*/A*

_{1}. Only those wavelengths satisfy the resonance condition can be transported efficiently, while others are stopped. In Fig. 2 , we plot the transmission spectrum and the field distributions of the propagation of SPPs in the structure. The parameters of the structure are set to be

*d =*50 nm,

*w*= 10 nm, and

*r =*170 nm in calculation. One can see in Fig. 2(a), that there are three transmission peaks corresponding to the wavelength

*λ =*1599.2,

*λ =*817.92, and

*λ =*579.84 nm, respectively. Figures 2(b), 2(c), and 2(d) depict the contour profiles of fields

*|H*for different wavelengths. The field distributions in Figs. 2(b) and 2(d) correspond to the Modes I and II in Fig. 2(a), respectively. One can see that the SPPs can propagate through the ring, and transmit from the waveguide II for these two cases. The standing waves form in the ring at resonance. There are one and two modes in the rings of Fig. 2(b) and 2(d), respectively. It must be noted here that the wavelengths of the transmission peaks do not satisfy the simple relation

_{y}|^{2}*λ = Ln*, where

_{eff}/N*N*is the mode number in the ring,

*L = 2πr*is the perimeter of the ring. The resonance condition will be discussed later. In Fig. 2(c), the wavelength of the SPPs is

*λ =*1200 nm. As can be seen in Fig. 2(c), the field in the ring is very weak, and almost no Spps exist in the right-side waveguide, the transmission of SPPs is forbidden in this case.

*w*= 10 nm and

*w*= 20 nm are depicted in blue and red scattering lines, respectively. We can see that the transmission modes exhibit blue shift as the coupling length increase. Three modes are separated from each other, and intervals between them are large enough. These results will provide the theoretical basis for designing band-pass filters at the given wavelength. We will theoretically study the relationship between the wavelengths of the transmission peaks and the radii of the rings in the following.

17. I. Wolff and N. Knoppik, “Microstrip ring resonator and dispersion measurement on microstrip lines,” Electron. Lett. **7**(26), 779–781 (1971). [CrossRef]

*k = ω(ε*

_{0}

*ε*

_{r}μ_{0}

*)*

^{½}.

*μ*

_{0}is the permeability in the air.

*J*is a Bessel function of the first kind and order

_{n}*n*, and

*N*is a Bessel function of second kind and order

_{n}*n*.

*kr*). Equation (3) is a transcendent equation which can be numerically solved. In principle, the ERI is wavelength dependent. However, we can see that, from Fig. 1(b), the wavelength has little influence on ERI when the width of the ring is fixed. Therefore, we can treat ERI as a constant in calculation for convenience. The real part of ERI is 1.4325 for a thickness of the waveguide of

*d*= 50 nm in the calculation, which can be obtained from Fig. 1(b). The imaginary part can be neglected because it is small and only affects the propagation length of SPPs. The calculation results are shown in Fig. 4 with black solid lines. Modes I, II, and III correspond to the one, two, and three orders of Bessel functions. One can see that there is a little difference between the theory result and the FDTD ones. In the calculation, we assume that the effective relative permittivity is the same in the whole ring, but this assumption is not exact. Since the coupling length between the ring and the waveguides are finite, the effective relative permittivity in the coupling areas is different from that in other areas of the ring. Only when the coupling length is infinite, the effective relative permittivity will be the same with that of the standard MIM model. Consequently, we can see that the FDTD results for

*w*= 20 nm are closer to the theory results than those for

*w*= 10 nm. When the radius of the ring gets larger, the effects of the coupling areas on the effective relative permittivity is smaller or even can be neglected. In addition, the curvature of the ring also have an effect on the effective permittivity, which means that for larger radius (with smaller curvature), every part of arc in the ring is closer to the straight MIM structure. Therefore, the theory results for larger radius are closer to FDTD results than that for smaller radius, which can be seen in Fig. 4. From the analysis above, we can see that the theory results match the FDTD results quite well.

## 4. Conclusion

## Acknowledgments

## References and links

1. | W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature |

2. | H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science |

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

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

5. | Z. Han, L. Liu, and E. Forsberg, “Ultra-compact directional couplers and Mach-Zehnder interferometers employing surface plasmon polaritons,” Opt. Commun. |

6. | T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyi, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. |

7. | Y. Chu, E. Schonbrun, T. Yang, and K. B. Crozier, “Experimental observation of narrow surface plasmon resonances in gold nanoparticle arrays,” Appl. Phys. Lett. |

8. | E. Verhagen, A. Polman, and L. K. Kuipers, “Nanofocusing in laterally tapered plasmonic waveguides,” Opt. Express |

9. | Z. Han, E. Forsberg, and S. He, “Surface plasmon Bragg gratings formed in metal-insulator-metal waveguides,” IEEE Photon. Technol. Lett. |

10. | B. Wang and G. P. Wang, “Plasmon Bragg reflectors and nanocavities on flat metallic surfaces,” Appl. Phys. Lett. |

11. | J. Q. Liu, L. L. Wang, M. D. He, W. Q. Huang, D. Wang, B. S. Zou, and S. Wen, “A wide bandgap plasmonic Bragg reflector,” Opt. Express |

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

13. | X. S. Lin and X. G. Huang, “Tooth-shaped plasmonic waveguide filters with nanometeric sizes,” Opt. Lett. |

14. | I. P. Kaminow, W. L. Mammel, and H. P. Weber, “Metal-Clad optical waveguides: analytical and experimental study,” Appl. Opt. |

15. | E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. |

16. | A. D. Rakic, A. B. Djurisic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. |

17. | I. Wolff and N. Knoppik, “Microstrip ring resonator and dispersion measurement on microstrip lines,” Electron. Lett. |

**OCIS Codes**

(140.4780) Lasers and laser optics : Optical resonators

(240.6680) Optics at surfaces : Surface plasmons

(130.7408) Integrated optics : Wavelength filtering devices

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: October 6, 2009

Revised Manuscript: November 17, 2009

Manuscript Accepted: December 7, 2009

Published: December 17, 2009

**Citation**

Tong-Biao Wang, Xie-Wen Wen, Cheng-Ping Yin, and He-Zhou Wang, "The transmission characteristics of surface plasmon polaritons in ring resonator," Opt. Express **17**, 24096-24101 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-26-24096

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### References

- W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef] [PubMed]
- H. J. Lezec, A. Degiron, E. Devaux, R. A. Linke, L. Martin-Moreno, F. J. Garcia-Vidal, and T. W. Ebbesen, “Beaming light from a subwavelength aperture,” Science 297(5582), 820–822 (2002). [CrossRef] [PubMed]
- 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]
- G. Veronis and S. Fan, “Bends and splitters in metal-dielectric-metal subwavelength plasmonic waveguides,” Appl. Phys. Lett. 87(13), 131102 (2005). [CrossRef]
- Z. Han, L. Liu, and E. Forsberg, “Ultra-compact directional couplers and Mach-Zehnder interferometers employing surface plasmon polaritons,” Opt. Commun. 259(2), 690–695 (2006). [CrossRef]
- T. Nikolajsen, K. Leosson, and S. I. Bozhevolnyi, “Surface plasmon polariton based modulators and switches operating at telecom wavelengths,” Appl. Phys. Lett. 85(24), 5833–5835 (2004). [CrossRef]
- Y. Chu, E. Schonbrun, T. Yang, and K. B. Crozier, “Experimental observation of narrow surface plasmon resonances in gold nanoparticle arrays,” Appl. Phys. Lett. 93(18), 181108 (2008). [CrossRef]
- E. Verhagen, A. Polman, and L. K. Kuipers, “Nanofocusing in laterally tapered plasmonic waveguides,” Opt. Express 16(1), 45–57 (2008). [CrossRef] [PubMed]
- Z. Han, E. Forsberg, and S. He, “Surface plasmon Bragg gratings formed in metal-insulator-metal waveguides,” IEEE Photon. Technol. Lett. 19(2), 91–93 (2007). [CrossRef]
- B. Wang and G. P. Wang, “Plasmon Bragg reflectors and nanocavities on flat metallic surfaces,” Appl. Phys. Lett. 87(1), 013107 (2005). [CrossRef]
- J. Q. Liu, L. L. Wang, M. D. He, W. Q. Huang, D. Wang, B. S. Zou, and S. Wen, “A wide bandgap plasmonic Bragg reflector,” Opt. Express 16(7), 4888–4894 (2008). [CrossRef] [PubMed]
- A. Hosseini and Y. Massoud, “Nanoscale surface Plasmon based resonator using rectangular geometry,” Appl. Phys. Lett. 90(18), 181102 (2007). [CrossRef]
- X. S. Lin and X. G. Huang, “Tooth-shaped plasmonic waveguide filters with nanometeric sizes,” Opt. Lett. 33(23), 2874–2876 (2008). [CrossRef] [PubMed]
- I. P. Kaminow, W. L. Mammel, and H. P. Weber, “Metal-Clad optical waveguides: analytical and experimental study,” Appl. Opt. 13(2), 396–405 (1974). [CrossRef] [PubMed]
- E. N. Economou, “Surface plasmons in thin films,” Phys. Rev. 182(2), 539–554 (1969). [CrossRef]
- A. D. Rakic, A. B. Djurisic, J. M. Elazar, and M. L. Majewski, “Optical properties of metallic films for vertical-cavity optoelectronic devices,” Appl. Opt. 37(22), 5271–5283 (1998). [CrossRef] [PubMed]
- I. Wolff and N. Knoppik, “Microstrip ring resonator and dispersion measurement on microstrip lines,” Electron. Lett. 7(26), 779–781 (1971). [CrossRef]

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