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 [1–5]. Recently, a great deal of interest has been devoted to the potential sensing applications of metamaterials. For example, He et al. [6], studied the resonant modes of a 2D subwavelength open resonator, and showed it was suitable for biosensing. Jakšić et al. [7] investigated some peculiarities of electromagnetic metamaterials convenient for plasmon-based chemical sensing with enhanced sensitivity, and it was believed that metamaterial sensor may have potential applications in environmental sensing, homeland security and biosensing. Alù et al. [8] proposed a method of dielectric sensing using ɛ near-zero narrow waveguide channels. Shreiber et al. [9] developed a novel microwave nondestructive evaluation sensor using metamaterial lens for detection of material defects small relative to a wavelength. Zheludev [10] analysed the road ahead for metamaterials, and pointed out that sensor applications are another growth area in metamaterials research. Our team [11–13] has studied the performance of metamaterial sensors, and shown that the sensitivity and resolution of the sensors can be greatly enhanced by metamaterials.

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 [14]. The WGM resonance phenomenon has received increasing attention due to its high potential for the realization of microcavity lasers [15,16], quantum computers [17], sensing applications [18–26], etc. Examples of the applications of WGM sensors include biosensing [21], nanoparticle detection [22], single-molecule detection [23], temperature measurement [24], ammonia detection [25], and TNT detection [26]. However, to the best of our knowledge, there is no report about the metamaterial sensor based on microring resonator operating in WGM.

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 Figure 1. A hollow core of relative permittivity ɛ₁ = 1 is surrounded by a concentric layer of material having relative permittivity ɛ₂. The region exterior to the layer is free-space (ɛ₃ = 1). The axial fields in regions 1, 2, and 3 for TM mode [27] are: (1a) E z ( 1 )   ( r ,   θ )   =   A m J m ( p 1 r ) e ± j m θ (1b) E z ( 2 )   ( r ,   θ )   =   ( B m J m ( p 2 r )   +   C m Y m   ( p 2   r ) ) e ± j m θ (1c) E z ( 3 )   ( r ,   θ )   =   D m K m   ( q r ) e ± j m θwhere A_m, B_m, C_m, and D_m are arbitrary constants.

The functions J_m, Y_m, and K_m are, respectively, the Bessel functions of the first kind, of the second kind, and the modified Bessel function of the second kind. p 1 2   =   ω 2 ɛ 1 μ 0   −   β 2   =   k 1 2   −   β 2, p 2 2   =   ω 2 ɛ 2 μ 0   −   β 2   =   k 2 2   −   β 2, and q 2   =   β 2   −   ω 2 ɛ 3 μ 0   =   β 2   −   k 3 2. k₁ = n₁ ω/c, k₂ = n₂ ω/c, k₃ = n₃ ω/c are respectively the wave number of region 1, 2, 3, while n₁, n₂, n₃ are the corresponding reflective index. β is the propagation constant, and m is the angular order. For an infinite hollow cylindrical dielectric waveguide with negligible absorption and no axial component of the propagation constant (β = 0), TM mode degenerates to WGM [28], Equation (1) becomes: (2a) E z ( 1 )   ( r ,   θ )   =   A m J m ( k 1 r ) e ± j m θ (2b) E z ( 2 )   ( r ,   θ )   =   ( B m J m ( k 2 r )   +   C m Y m   ( k 2   r ) ) e ± j m θ (2c) E z ( 3 )   ( r ,   θ )   =   D m ′ H m ( 1 )   ( k 3 r ) e ± j m θwhere D′_m = (iπ/2)e^imπ/2 D_m, H m ( 1 ) is the Hankel function of the first kind. The relation between H m ( 1 ) and K_m is K_m(−iz) = (iπ/2)e^imπ/2 H m ( 1 )   ( z ).

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: (3a) H r   ( r ,   θ )   =   1 p 2   ( i ω ɛ r ⋅ ∂ E z   ( r ,   θ ) ∂ θ ) (3b) H θ   ( r ,   θ )   =   1 p 2 ( − i ω ɛ ⋅ ∂ E z   ( r ,   θ ) ∂ r )

For WGM in the cylindrical dielectric waveguide, Equation (3) becomes: (4a) H r   ( r ,   θ )   =   i Z 0 k 0 1 r ∂ ∂ θ E z   ( r ,   θ ) (4b) H θ   ( r ,   θ )   =   − i Z 0 k 0 ∂ ∂ r   E z   ( r ,   θ )

The field matching equations at the boundary surface r = a and r = b are expressed as: E z ( 1 )   ( a ,   θ )   =   E z ( 2 )   ( a ,   θ ), H θ ( 1 )   ( a ,   θ )   =   H θ ( 2 )   ( a ,   θ ), E z ( 2 )   ( b ,   θ )   =   E z ( 3 )   ( b ,   θ ), H θ ( 2 )   ( b ,   θ )   =   H θ ( 3 )   ( b ,   θ ).Satisfying these conditions gives: (5) [ M ] [ A m ,   B m ,   C m ,   D m ′ ] T   =   0where: (6) [ M ]   =   [ J m   ( k 1 a ) − J m   ( k 2 a ) − Y m   ( k 2 a ) 0 k 1 J m ′   ( k 1 a ) − k 2 J m ′   ( k 2 a ) − k 2 Y m ′   ( k 2 a ) 0 0 J m   ( k 2 b ) Y m   ( k 2 b ) − H m ( 1 )   ( k 3 b ) 0 k 2 J m ′   ( k 2 b ) k 2 Y m ′   ( k 2 b ) − k 3 H m ′ ( 1 )   ( k 3 b ) ]J′_m, Y′_m, H m ′ ( 1 ) are the derivative of J_m, Y_m and H m ( 1 ). The dispersion equation can be obtained by setting | M = 0|. (7) − k 2 J m   ( k 1 a ) J m ′   ( k 2 a ) [ − k 3 Y m   ( k 2 b ) H m ′ ( 1 )   ( k 3 b )   +   k 2 H m ( 1 )   ( k 3 b ) Y m ′   ( k 2 b ) ] + k 2 J m   ( k 1 a ) Y m ′   ( k 2 a ) [ − k 3 J m   ( k 2 b ) H m ′ ( 1 )   ( k 3 b )   +   k 2 J m ′   ( k 2 b ) H m ( 1 )   ( k 3 b ) ] + k 1 J m ′   ( k 1 a ) J m   ( k 2 a ) [ − k 3 Y m   ( k 2 b ) H m ′ ( 1 )   ( k 3 b )   +   k 2 H m ( 1 )   ( k 3 b ) Y m ′   ( k 2 b ) ] − k 1 J m ′   ( k 1 a ) Y m   ( k 2 a ) [ − k 3 J m   ( k 2 b ) H m ′ ( 1 )   ( k 3 b )   +   k 2 H m ( 1 )   ( k 3 b ) J m ′   ( k 2 b ) ]   =   0

In the lossless case, the resonant frequencies at mode number m in the frequency range of 185 to 212 THz are obtained under the condition of a = 2.2 μm, b = 2.5 μm, n₁ = n₃ = 1, n₂ = 3.2, by solving the lowest order radial WGMs of a hollow cylindrical dielectric waveguide. The results are shown in Table 1.

In order to estimate the radiation-loss-limited Q factors of WGMs [28,29], complex frequencies of the resonances ω = ω′ + jω″ were introduced into the dispersion equation. The real part ω′ determines the wavelength of the resonances and the Q factor can be estimated with the expression: Q = ω′ /2ω″. This dimensionless parameter generalizes the imaginary part of the solution space. In order to generalize the real part, a normalized radius is defined as X = n₂2πb/λ. To solve for the complex roots of the dispersion equation |M|= 0, a global optimization scheme can be used to minimize the absolute value of the equation over two variables: Q and X. Figure 2 displays the radiation-loss-limited finesse (F = Q/m) vs. normalized radius for a variety of azimuthal mode numbers and index ratios. The family of diagonal lines represents varying refractive index contrast (n = n₂ /n₁). The family of nearly vertical lines corresponds to WGM resonances, each characterized by an azimuthal mode number m. From Figure 2, we can observe that the radiation-loss-limited finesse increases with normalized radius and mode number. Besides, for a give normalized radius and mode number, the radiation-loss-limited finesse increases with the refractive index contrast. These results are beneficial for designing a microring resonator. In the following section, we will simulate the microring resonator and compare its resonant frequencies with the analytical results. Then, the metamaterial sensor constituted by the microring resonator with a layer of metamaterial loaded into its inner side and coupled to a straight waveguide is studied.

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.