Due to their intriguing electromagnetic properties, a great deal of attention has been focused recently on metamaterials. The permittivity and permeability of metamaterials can be designed to continuously change from negative to positive values. Many novel metamaterial-based applications have been proposed, such as perfect lenses, cloaks, concentrators, directive antennas, superscatterers, superabsorbers, transparent devices, etc. [1–6]. Recently, great interest has been devoted to the sensing applications of metamaterials. For example, Jakšić et al. [7] investigated some peculiarities of electromagnetic metamaterials convenient for plasmon-based chemical sensing with enhanced sensitivity, and they envisioned practical applications of metamaterial-based sensors in biosensing, chemical sensing, environmental sensing, homeland security, etc. He et al. [8], studied the resonant modes of a 2D subwavelength open resonator, and showed it was suitable for biosensing. Melik et al. [9] presented telemetric sensing of surface strains on different industrial materials using split-ring-resonator based metamaterials, and desirable properties were obtained. Lee et al. [10] demonstrated experimentally the effectiveness of a split-ring resonator (SRR) array as a biosensing device at microwave frequencies. Cubukcu et al. [11] reported a surface enhanced molecular detection technique with zeptomole sensitivity that relies on the resonant electromagnetic coupling between a split ring resonator and the infrared vibrational modes of molecules. Alù et al. [12] proposed a method of dielectric sensing using ε near-zero narrow waveguide channels. Shreiber et al. [13] developed a novel microwave nondestructive evaluation sensor using a metamaterial lens for detection of material defects small relative to a wavelength. Zheludev [14] reviewed the road ahead for metamaterials, and pointed out that sensor applications are another growth area in metamaterials research. Our team has studied the performance of metamaterial sensors, and shown that the sensitivity and resolution of sensors can be greatly enhanced by using metamaterials [15–17].

WGM is a morphology-dependent resonance, which occurs when light within a dielectric microsphere, microdisk, or microring has a higher refractive index than its surroundings. In a ring resonator, WGMs form due to the total internal reflection of the light along the curved boundary surface [18]. The WGM resonance phenomenon has attracted increasing attention due to its high potential for the realization of microcavity lasers [19], quantum computers [20], sensing applications [21–29], etc. Examples of the applications of WGM sensors include biosensing [24], nanoparticle detection [25], single-molecule detection [26], temperature measurement [27], ammonia detection [28], and TNT detection [29]. However, to the best of our knowledge, there are no reports about any NPM sensors based on microring resonators operating in WGM.

In this paper, we derive the dispersion relation of a cylindrical dielectric waveguide loaded on a NPM layer, and compute the resonant frequencies and electric field distributions of the corresponding WGMs. We perform a full wave simulation of the performance of the NPM sensor, and compared it with the theoretical results. We show that the NPM sensor possesses much higher sensitivity than a traditional microring sensor, and the mechanism behind these phenomena is verified by theoretical analysis and simulation.

Figure 1 shows the geometry of a cylindrical dielectric waveguide loaded with a layer of metamaterials. The inner side of the e cylindrical dielectric waveguide (ε₃,μ₃) is loaded on a metamaterial layer (ε₂,μ₂). The waveguide has a four-layer structure. The material parameters of regions 1, 2, 3, 4 are denoted as (ε₁,μ₁), (ε₂,μ₂), (ε₃,μ₃), (ε₄,μ₄), respectively. The axial fields in corresponding regions for TM mode [30] are: (1a) E z ( 1 ) ( r , θ ) = A m J m ( p 1 r ) e ± jm θ (1b) E z ( 2 ) ( r , θ ) = ( B m J m ( p 2 r ) + B m ′ Y m ( p 2 r ) ) e ± jm θ (1c) E z ( 3 ) ( r , θ ) = ( C m J m ( p 3 r ) + C m ′ Y m ( p 3 r ) ) e ± j mθ (1d) E z ( 4 ) ( r , θ ) = D m K m ( q r ) ) e ± jm θwhere A_m, B_m, C_m, D_m, B m ′ and C ′ m are chosen here to weight the field, but they are interdependent. 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. The terms p 1 = ω 2 ε 1 μ 1 − β 2, p 2 = ω 2 ε 2 μ 2 − β 2, p 3 = ω 2 ε 3 μ 3 − β 2, q = β 2 − ω 2 ε 4 μ 4. β is the propagation constant, and m is the angular order. For an infinite cylindrical dielectric waveguide with negligible absorption and no axial component of the propagation constant (β = 0), TM mode degenerates to WGM [31], and Equation (1) becomes: (2a) E z ( 1 ) ( r , θ ) = A m J m ( p 1 r ) e ± jm θ (2b) E z ( 2 ) ( r , θ ) = ( B m J m ( p 2 r ) + B m ′ Y m ( p 2 r ) ) e ± jm θ (2c) E z ( 3 ) ( r , θ ) = ( C m J m ( p 3 r ) + C m ′ Y m ( p 3 r ) ) e ± jm θ (2d) E z ( 4 ) ( r , θ ) = D m ′ H m ( 1 ) ( p 4 r ) ) e ± jm θwhere p 1 = ω ε 1 μ 1, p 2 = ω ε 2 μ 2, p 3 = ω ε 3 μ 3, p 4 = − q 2 = ω ε 4 μ 4, D m ′ = ( i π / 2 ) e im π / 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 ( − i z ) = ( i π / 2 ) e im π / 2 H m ( 1 ) ( z ). For TM mode in an infinite cylindrical dielectric waveguide, transverse magnetic fields can be obtained as: (3a) H r ( r , θ ) = 1 p 2 ( j ω ε r ∂ E z ( r , θ ) ∂ θ ) (3b) H θ ( r , θ ) = 1 p 2 ( − j ω ε ∂ E z ( r , θ ) ∂ r )

The tangential fields matching equations at the boundary surfaces r=r₁, r=r₂ and r=r₃ are expressed as: E z ( 1 ) ( r 1 , θ ) = E z ( 2 ) ( r 1 , θ ) ,         H θ ( 1 ) ( r 1 , θ ) = H θ ( 2 ) ( r 1 , θ ) ,         E z ( 2 ) ( r 2 , θ ) = E z ( 3 ) ( r 2 , θ ) , H θ ( 2 ) ( r 2 , θ ) = H θ ( 3 ) ( r 2 , θ ) ,         E z ( 3 ) ( r 3 , θ ) = E z ( 4 ) ( r 3 , θ ) ,         H θ ( 3 ) ( r 4 , θ ) = H θ ( 4 ) ( r 4 , θ )

Satisfying these conditions gives: (4) [ M ] [ A m , B m , B m ′ , C m , C m ′ , D m ′ ] T = 0where: (5) [ M ] = [ J m ( p 1 r 1 ) − J m ( p 2 r 1 ) − Y m ( p 2 r 1 ) 0 0 0 − ε 1 J m ′ ( p 1 r 1 ) p 1 ε 2 J m ′ ( p 2 r 1 ) p 2 ε 2 Y m ′ ( p 2 r 1 ) p 2 0 0 0 0 J m ( p 2 r 2 ) Y m ( p 2 r 2 ) − J m ( p 3 r 2 ) − Y m ( p 3 r 2 ) 0 0 − ε 2 J m ′ ( p 2 r 2 ) p 2 − ε 2 Y m ′ ( p 2 r 2 ) p 2 ε 3 J m ′ ( p 3 r 2 ) p 3 ε 3 Y m ′ ( p 3 r 2 ) p 3 0 0 0 0 J m ( p 3 r 3 ) Y m ( p 3 r 3 ) − H m ( 1 ) ( p 4 r 3 ) 0 0 0 − ε 3 J m ′ ( p 3 r 3 ) p 3 − ε 3 Y m ′ ( p 3 r 3 ) p 3 ε 4 H m ′ ( 1 ) ( p 4 r 3 ) p 4 ]

The dispersion equation can be obtained by setting | M |=0. The resonant frequency for different modes can be calculated using the software Mathematica (Wolfram Research Inc., Champaign, IL, USA). Coefficients B_m, B m ′, C_m, C m ′ and D m ′ can be expressed in terms of the arbitrary coefficient A_m, and B m = f m ( 1 ) A m, B ′ m = f m ( 2 ) A m, C m = f m ( 3 ) A m, C m ′ = f m ( 4 ) A m, D m ′ = f m ( 5 ) A m. Parameters f n ( 1 ), f n ( 2 ), f n ( 3 ), f n ( 4 ) and f n ( 5 ) may be found from Equation (4). Electric field distribution for different mode can be obtained by substituting these coefficients in to Equation (2): (6) f m ( 1 ) = ( p 2 ε 1 J m ′ ( p 1 r 1 ) Y m ( p 2 r 1 ) − p 1 ε 2 J m ( p 1 r 1 ) Y m ′ ( p 2 r 1 ) ) / ( p 1 ε 2 J m ′ ( p 2 r 1 ) Y m ( p 2 r 1 ) − p 1 ε 2 J m ( p 2 r 1 ) Y m ′ ( p 2 r 1 ) ) (7) f m ( 2 ) = − ( p 2 ε 1 J m ′ ( p 1 r 1 ) J m ( p 2 r 1 ) − p 1 ε 2 J m ( p 1 r 1 ) J m ′ ( p 2 r 1 ) ) / ( p 1 ε 2 J m ′ ( p 2 r 1 ) Y m ( p 2 r 1 ) − p 1 ε 2 J m ( p 2 r 1 ) Y m ′ ( p 2 r 1 ) ) (8) f m ( 3 ) = p 2 ɛ 3 Y m ′ ( p 3 r 2 ) J m ( p 2 r 2 ) ( p 1 ɛ 2 J m ( p 1 r 1 ) Y m ′ ( p 2 r 1 ) − p 2 ɛ 1 J m ′ ( p 1 r 1 ) Y m ( p 2 r 1 ) ) + Y m ( p 2 r 2 ) ⋅ ( p 2 ɛ 1 J m ′ ( p 1 r 1 ) J m ( p 2 r 1 ) − p 1 ɛ 2 J m ( p 1 r 1 ) J m ′ ( p 2 r 1 ) ) ) + p 3 ɛ 2 Y m ( p 3 r 2 ) ( J m ′ ( p 2 r 2 ) ( p 2 ɛ 1 J m ′ ( p 1 r 1 ) ⋅ Y m ( p 2 r 1 ) − p 1 ɛ 2 J m ( p 1 r 1 ) Y m ′ ( p 2 r 1 ) ) + Y m ′ ( p 2 r 2 ) ( p 1 ɛ 2 J m ( p 1 r 1 ) J m ′ ( p 2 r 1 ) − p 2 ɛ 1 J m ′ ( p 1 r 1 ) J m ( p 2 r 1 ) ) ) ) / ( p 1 p 2 ɛ 2 ɛ 3 ( J m ′ ( p 2 r 1 ) Y m ( p 2 r 1 ) − J m ( p 2 r 1 ) Y m ′ ( p 2 r 1 ) ) J ′ m ( p 3 r 2 ) Y m ( p 3 r 2 ) − J m ( p 3 r 2 ) Y m ′ ( p 3 r 2 ) ) ) (9) f m ( 4 ) = p 2 ɛ 3 J m ′ ( p 3 r 2 ) J m ( p 2 r 2 ) ( p 2 ɛ 1 J m ′ ( p 1 r 1 ) Y m ( p 2 r 1 ) − p 1 ɛ 2 J m ( p 1 r 1 ) Y m ′ ( p 2 r 1 ) ) + Y m ( p 2 r 2 ) ⋅ ( p 1 ɛ 2 J m ( p 1 r 1 ) J m ′ ( p 2 r 1 ) − p 2 ɛ 1 J m ′ ( p 1 r 1 ) J m ( p 2 r 1 ) ) ) + p 3 ɛ 2 J m ( p 3 r 2 ) ( J m ′ ( p 2 r 2 ) ( p 1 ɛ 2 J m ( p 1 r 1 ) ⋅ Y m ′ ( p 2 r 1 ) − p 2 ɛ 1 J m ′ ( p 1 r 1 ) Y m ( p 2 r 1 ) ) + Y m ′ ( p 2 r 2 ) ( p 2 ɛ 1 J m ′ ( p 1 r 1 ) J m ( p 2 r 1 ) − p 1 ɛ 2 J m ( p 1 r 1 ) J m ′ ( p 2 r 1 ) ) ) ) / ( p 1 p 2 ɛ 2 ɛ 3 ( J m ′ ( p 2 r 1 ) Y m ( p 2 r 1 ) − J m ( p 2 r 1 ) Y m ′ ( p 2 r 1 ) ) J m ′ ( p 3 r 2 ) Y m ( p 3 r 2 ) − J m ( p 3 r 2 ) Y m ′ ( p 3 r 2 ) ) ) (10) f m ( 5 ) = ( Y m ( p 3 r 3 ) ( p 2 ɛ 3 J m ′ ( p 3 r 2 ) ( J m ( p 2 r 2 ) ( p 2 ɛ 1 J m ′ ( p 1 r 1 ) Y m ( p 2 r 1 ) − p 1 ɛ 2 J m ( p 1 r 1 ) Y m ′ ( p 2 r 1 ) ) + Y m ( p 2 r 2 ) ( p 1 ɛ 2 J m ( p 1 r 1 ) J m ′ ( p 2 r 1 ) − p 2 ɛ 1 J m ′ ( p 1 r 1 ) J m ( p 2 r 1 ) ) ) + p 3 ɛ 2 J m ( p 3 r 2 ) ( J m ′ ( p 2 r 2 ) ( p 1 ɛ 2 J m ( p 1 r 1 ) ⋅ Y m ′ ( p 2 r 1 ) − p 2 ɛ 1 J m ′ ( p 1 r 1 ) Y m ( p 2 r 1 ) ) + Y m ′ ( p 2 r 2 ) ( p 2 ɛ 1 J m ′ ( p 1 r 1 ) J m ( p 2 r 1 ) − p 1 ɛ 2 J m ( p 1 r 1 ) J m ′ ( p 2 r 1 ) ) ) ) + J m ( p 3 r 3 ) ( p 2 ɛ 3 Y m ′ ( p 3 r 2 ) ( J m ( p 2 r 2 ) ( p 1 ɛ 2 J m ( p 1 r 1 ) Y m ′ ( p 2 r 1 ) − p 2 ɛ 1 J m ′ ( p 1 r 1 ) Y m ( p 2 r 1 ) ) + Y m ( p 2 r 2 ) ⋅ ( p 2 ɛ 1 J m ′ ( p 1 r 1 ) J m ( p 2 r 1 ) ) − p 1 ɛ 2 J m ( p 1 r 1 ) J m ′ ( p 2 r 1 ) ) ) + p 3 ɛ 2 Y m ( p 3 r 2 ) ( J m ′ ( p 2 r 2 ) ( p 2 ɛ 1 J m ′ ( p 1 r 1 ) Y m ( p 2 r 1 ) − p 1 ɛ 2 J m ( p 1 r 1 ) Y m ′ ( p 2 r 1 ) ) + Y m ′ ( p 2 r 2 ) ( p 1 ɛ 2 J m ( p 1 r 1 ) J m ′ ( p 2 r 1 ) − p 2 ɛ 1 J m ′ ( p 1 r 1 ) J m ( p 2 r 1 ) ) ) ) ) / ( p 1 p 2 ɛ 2 ɛ 3 H m ( 1 ) ( p 4 r 3 ) ( J m ′ ( p 2 r 1 ) Y m ( p 2 r 1 ) − J m ( p 2 r 1 ) Y m ′ ( p 2 r 1 ) ) ( J m ′ ( p 3 r 2 ) Y m ( p 3 r 2 ) − J m ( p 3 r 2 ) Y m ′ ( p 3 r 2 ) ) )

Simulation models of the NPM sensor based on a microring resonator are shown in Figure 2. A layer of NPM with thickness t is located on the inner side of the microring. Permittivity and permeability of the NPM are ε₂ = ε₀, μ₂ = –μ₀. Width of the microring and the waveguide is w = 0.3 μm. The outer diameter of the microring is d = 5 μm. The distance from outer microring to the waveguide is g = 0.232 μm. The permittivity of the microring and the waveguide is ε₃ = 10.24ε₀. Figure 2(a) is the simulation model for homogeneous sensing. The dielectric core with permittivity ε₁ = ε_rε₀ is colored in light blue. Figure 2(b) is the simulation model for surface sensing. The dielectric substance with thickness t_s and permittivity ε₁ = ε_rε₀ is attached to the NPM layer.

The frequency spectrum of the NPM sensor for homogeneous sensing is simulated by the finite element software COMSOL Multiphysics (COMSOL Inc., Burlington, MA, USA), as shown in Figure 3. In the simulation, the computational space is surrounded by a scattering boundary. The excitation is set at port A of the waveguide. The spectrum is obtained by frequency sweep. From left to right, the spectral lines represent modes 25, 26, 27, 28 and 29 of the NPM sensor. The inset shows the amplification in the 191.83–191.87 THz frequency range. Table 1 shows the comparison of the analytical and simulated resonant frequency for the microring sensor and the NPM sensor. Therefore, WGMs (m = 25, 26, 27, 28, 29) in the cross section of the waveguide correspond to the modes of the microring sensor and the NPM sensor. The analytical resonant frequency of the sensor can be obtained by setting | M | = 0 (details may be found in next Section). The maximum deviation between simulation results and analytical results is 0.011 THz. The analytical results are in good agreement with the simulation results.

Supposed that material parameters of the waveguide in region 1, 2, 3, 4 are ε₁ = ε_rε₀, μ₁ = μ₀, ε₂ = ε₀, μ₂ = –μ₀, ε₃ = 10.24ε₀, μ₃ = μ₀, ε₄ = ε₀, μ₄ = μ₀, respectively. The resonant frequency of WGM (m = 27) in the cross section of the waveguide can be calculated by setting | M |=0. The coefficients B m = f m ( 1 ) A m, B m ′ = f m ( 2 ) A m, C m = f m ( 3 ) A m, C m ′ = f m ( 4 ) A m, D m ′ = f m ( 5 ) A m can be easily obtained according to Equations (6)–(10). The electric field distribution of the WGM can be calculated according to Equation (2), and are shown in Figure 4(a,c). To confirm the WGM in the cross section of the waveguide corresponds to the mode of the microring resonator, we simulate the electric field distribution of the microring resonator, as shown in Figure 4(b,d). From Figure 4, we can observe that the theoretical results are in good agreement with the simulation results. Interestingly, we find that the maximum electric field is located at the interface of the NPM layer and core medium. This implies that a microring resonator loaded on an NPM layer has higher sensitivity than a traditional microring resonator without loading on the NPM layer.

To confirm the above idea, we simulated the performance of the microring sensor and the NPM sensor for homogeneous sensing, as shown in Figure 5. Permittivity (ε_r) of the dielectric core varies from 1 to 1.1 with an interval of 0.02. From Figure 5(a,b), we can observe that the spectra red shift with the increase of ε_r. Sensitivity for the microring sensor and the NPM sensor is 5.9 nm/RIU and 64.2 nm/RIU, respectively. Here, sensitivity is defined as Δ λ / Δ n = [ λ ( ε r , t ) − λ ( 1 , t ) ] / ( ε r − 1 ). Figure 6(a,b) show the analytical and simulating resonant frequency for the NPM sensor and microring sensor, respectively. Simulating frequencies are calculated from Figure 5, while the theoretical frequencies are obtained by setting | M |=0. From Figure 6, we find that the simulation results are in good agreement with the theoretical results. With an increase of 0.02 in core medium permittivity, average frequency shift for the NPM sensor is very large [Figure 6(a)], but the average frequency shift of the microring sensor is quite small [Figure 6(b)]. Therefore, the NPM sensor possesses much higher sensitivity than the traditional microring sensor.

To reveal the mechanism behind these phenomena, we plotted the electric field distribution of the NPM sensor along the x axis from −3 μm to −1.5 μm for mode 27, as shown in Figure 7. Permittivity of the core medium is set to be ε_r = 1. It is seen that the electric field intensity increases with NPM layer thickness (t). The inset shows the electric field distribution of the NPM sensor. From Figure 7, we can clearly observe that the stronger electric field of evanescent wave penetrates into the detecting region when the thickness of NPM layer increases. Figure 8 shows the relation between core medium permittivity and wavelength shift for different NPM layer thickness. Permittivity of the core medium increases from 1 to 1.1 with an interval of 0.02. Resonant wavelength shift is calculated by Δλ = λ(ε_r,t) – λ(1,t). For the microring sensor (t = 0), the sensitivity is only 5.9 nm/RIU. For the NPM sensor, the sensitivity increases with NPM layer thickness. When the thickness of the NPM layer is 0.06 μm, 0.09 μm, 0.12 μm, and 0.15 μm, the corresponding sensitivity will be 28.4 nm/RIU, 64.2 nm/RIU, 136.8 nm/RIU, and 240.7 nm/RIU, respectively. Therefore, the essence for the enhancement of sensitivity is the evanescent wave amplified by the metamaterial. Interestingly, we find that the sensitivity of the NPM sensor can be up to 327.3 nm/RIU when NPM thickness is 0.174 μm. But when the thickness is larger than 0.174 μm, WGM with m = 27 will be transferred to the WGM with m = 26 or 28. Details are not shown here for brevity.

Surface sensing performance of the NPM sensor can also be analyzed according to the above procedures, and it is not shown here for brevity. Figure 9 shows the simulation results for surface sensing. Similarly, the sensitivity increases with NPM layer thickness. When the thickness of the NPM layer is 0.06 μm, 0.09 μm, 0.12 μm, and 0.15 μm, the sensitivity of the NPM sensor will be 24.1 nm/RIU, 54.9 nm/RIU, 117.7 nm/RIU, 208.9 nm/RIU, respectively. Therefore, sensitivity of the NPM sensor can be greatly improved by increasing the thickness of the NPM layer attached to its inner side. This is a novel method for sensor design with specified sensitivity.

WGMs of a dielectric waveguide with a layer of negative permeability metamaterial are theoretically analyzed, and the dispersion relation is derived. Analytical results of the resonant frequency shift and electric field distribution of the sensor are in good agreement with the simulation results. We show that the NPM sensor possesses a higher sensitivity than the traditional microring sensor, due to the amplification of the evanescent wave. Moreover, the sensitivity will be further improved by increasing the thickness of the metamaterial layer, opening a door for the design of novel sensors with desired sensitivity.