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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Metamaterials are artificial media structured on a size scale smaller than the wavelength of external stimuli, that may provide novel tools to significantly enhance the sensitivity and resolution of the sensors. In this paper, we derive the dispersion relation of hollow cylindrical dielectric waveguide, and compute the resonant frequencies and Q factors of the corresponding Whispering-Gallery-Modes (WGM). A metamaterial sensor based on microring resonator operating in WGM is proposed, and the resonance intensity spectrum curves in the frequency range from 185 to 212 THz were studied under different sensing conditions. Full-wave simulations, considering the frequency shift sensitivity influenced by the change of core media permittivity, the thickness and permittivity of the adsorbed substance, prove that the sensitivity of the metamaterial sensor is more than 7 times that of the traditional microring resonator sensor, and the metamaterial layer loaded in the inner side of the microring doesn’t affect the high Q performance of the microring resonator.

Metamaterials, which are artificial materials whose permittivity and/or permeability can be designed to continuously change from negative to positive values, have attracted considerable attention for their intriguing electromagnetic properties. Many novel applications have been proposed based on metamaterials, such as perfect lenses, cloaks, concentrators, directive antennae, superscatterers, superabsorbers, transparent devices and so on [

WGM is a morphology-dependent resonance, which occurs when an electromagnetic wave travels in a dielectric medium of circular geometry. After repeated total internal reflections at the curved boundary, the electromagnetic field can close on itself and give rise to resonances [

The purpose of this paper is to investigate the performance of the metamaterial microring resonator sensor and to illustrate how it is different from the traditional microring resonator sensor. We derive the dispersion relation of the hollow cylindrical dielectric waveguide, and compute the resonant frequencies and Q factors of the corresponding WGMs. We then make full wave simulation on the performance of the metamaterial sensor. This study shows that the metamaterial sensor possesses much higher sensitivity than traditional microring resonator sensor, and the mechanism behind this phenomenon is the amplification of evanescent wave.

The geometry of a hollow cylindrical dielectric waveguide is shown in _{1}_{2}_{3}_{m}_{m}_{m}_{m}

The functions _{m}_{m}_{m}_{1}_{1}_{2}_{2}_{3}_{3}_{1}_{2}_{3}_{m}^{im}^{π}^{/2}_{m}_{m}_{m}(−iz)^{im}^{π}^{/2}

Finally, the radial and azimuthal electric fields are derived from the axial electric field using Maxwell’s equations. For TM mode in the cylindrical dielectric waveguide, transverse magnetic fields can be obtained as:

For WGM in the cylindrical dielectric waveguide, Equation (3) becomes:

The field matching equations at the boundary surface _{m}_{m}_{m}_{m}

In the lossless case, the resonant frequencies at mode number _{1} = _{3} = 1, _{2} = 3.2, by solving the lowest order radial WGMs of a hollow cylindrical dielectric waveguide. The results are shown in

In order to estimate the radiation-loss-limited Q factors of WGMs [_{2}2πb/λ. To solve for the complex roots of the dispersion equation |_{2} /n_{1}). The family of nearly vertical lines corresponds to WGM resonances, each characterized by an azimuthal mode number

Simulation models of the microring sensor are illustrated in _{r}_{r}

Firstly, the electric field distribution inside the traditional microring sensor (Model A) at the frequency of 198.257 THz is simulated, as shown in

From left to right, the spectral lines represent mode 25, 26, 27, 28 and 29 of the microring resonator. The inset portrays the amplification in the frequency range of 198.23–198.28 THz. The resonant frequency of the microring is calculated and compared with the theoretical results, as shown in _{s}_{t}

Next, a layer of metamaterials with thickness of 0.09 μm is loaded into the inner side of an air-core microring to build the simulation model of the metamaterial sensor (Model C). Simulation results of the electric field pattern, power flow and resonant frequency spectrum of the metamaterial sensor are illustrated in panels (a), (b) and (c) of

_{s}_{s}

Suppose that a layer of substance with thickness of t = 0.075 μm is adsorbed to the inner side of the microring (_{s}

To understand the impact of the metamaterials on the sensitivity of the microring sensor, we assume that the thickness (_{m}_{s}_{r}_{r}

For the traditional microring sensor (_{m}

To reveal the mechanism behind these phenomena, we plotted the electric field distribution along the x axis from −3 μm to −1.5 μm for mode 27, as shown in _{s}_{m}_{m}

We have theoretically analyzed here the WGMs of a hollow cylindrical dielectric waveguide by solving the Maxwell equations with the corresponding boundary conditions. The resonant frequencies in the frequency range of 185 to 212 THz are obtained analytically, and the results reveal good agreement with full wave simulation. The metamaterial sensor constituted by a microring resonator loaded with a layer of double negative metamaterial is proposed, and considering the excitation of WGMs, its performance is simulated numerically and compared with the traditional microring sensor. We show that the metamaterial sensor possesses a higher sensitivity than the traditional microring sensor, due to the amplification of evanescent waves. Moreover, the sensitivity will be further improved by increasing the thickness of the metamaterial layer. It is expected that with the development of metamaterial technology, metamaterial sensors based on the microring resonator can be realized, and will contribute to more applications in dielectric sensing in fields where high sensitivity is required.

The authors thank the training Program of Yunnan Province for Middle-aged and Young Leaders of Disciplines in Science and Technology (Grant No. 2008PY031), the Research Foundation from Ministry of Education of China (Grant No. 208133), the National Natural Science Foundation of China (Grant No. 60861002), and NSFC-YN (Grant No.U1037603) for financial support.

_{1}, ɛ_{2} and ɛ_{3}.

Radiation-loss-limited finesse (F) of the lowest order radial WGMs of a hollow cylindrical dielectric waveguide of index n_{2} in a medium of index n_{1} = n_{3} = 1 plotted against normalized radius (X).

Simulation model of the microring sensor.

_{r}

_{r}

Resonant frequency spectrum of mode 27 with respect to the change of core medium permittivity _{s}_{s}

Resonant frequency spectrum of mode 27 with respect to the change of adsorbed substance permittivity _{s}_{s}

Resonant frequency spectrum of mode 27 with respect to the increase of substance layer thickness

Resonant frequency spectrum of mode 27 with respect to the change of adsorbed substance permittivity ɛ_{s}_{s}

Electric field distribution along x axis from −3 μm to −1.5 μm for the metamaterial sensor operating in mode 27. The inset shows the electric field distribution of the sensor of _{m}

Mode number and resonant frequency.

_{r} |
186.274 | 192.332 | 198.388 | 204.441 | 210.493 |

Comparison of the theoretical resonant frequency (_{t}_{s}

_{t} |
186.274 | 192.332 | 198.388 | 204.441 | 210.493 |

_{s} |
186.156 | 192.208 | 198.257 | 204.305 | 210.351 |

Simulated Q factor and resonance frequency for Model A and C.

Mode | 25 | 26 | 27 | 28 | 29 | |
---|---|---|---|---|---|---|

Model A | Q | 9279 | 12539 | 16970 | 22600 | 30145 |

_{r} |
186.156 | 192.208 | 198.257 | 204.305 | 210.351 | |

| ||||||

Model C | Q | 9338 | 12594 | 16952 | 22740 | 30341 |

_{r} |
186.360 | 192.412 | 198.462 | 204.511 | 210.559 |

Q factor, resonant frequency and frequency shift calculated from

_{s} |
||||||
---|---|---|---|---|---|---|

_{m}(μm) | ||||||

0 | Q | 16896 | 16914 | 16870 | 16868 | 16955 |

_{r} |
198.250 | 198.244 | 198.238 | 198.231 | 198.225 | |

Δ_{r} |
6.0 | 6.0 | 7.0 | 6.0 | ||

0.06 | Q | 15502 | 15503 | 15502 | 15501 | 16513 |

_{r} |
198.320 | 198.295 | 198.270 | 198.246 | 198.222 | |

Δ_{r} |
25.0 | 25.0 | 24.0 | 24.0 | ||

0.09 | Q | 16758 | 16885 | 16901 | 16890 | 16841 |

_{r} |
198.413 | 198.363 | 198.314 | 198.265 | 198.215 | |

Δ_{r} |
50.0 | 49.0 | 49.0 | 50.0 | ||

0.12 | Q | 17006 | 17049 | 16932 | 16915 | 16867 |

_{r} |
198.600 | 198.501 | 198.401 | 198.301 | 198.201 | |

Δ_{r} |
99.0 | 100.0 | 100.0 | 100.0 | ||

0.15 | Q | 17151 | 17067 | 17022 | 16914 | 16830 |

_{r} |
198.983 | 198.781 | 198.577 | 198.373 | 198.168 | |

Δ_{r} |
202.0 | 204.0 | 204.0 | 205.0 | ||

0.18 | Q | 17470 | 17393 | 17129 | 16948 | 16746 |

_{r} |
199.772 | 199.357 | 198.939 | 198.517 | 198.092 | |

Δ_{r} |
415.0 | 418.0 | 422.0 | 425.0 |