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The opto-fluidic ring resonator (OFRR) biosensor is numerically characterized in whispering gallery mode (WGM). The ring resonator includes a ring, a waveguide and a gap separating the ring and the waveguide. Dependence of the resonance characteristics on the resonator size parameters such as the ring diameter, the ring thickness, the waveguide width, and the gap width between the ring and the waveguide are investigated. For this purpose, we use the finite element method with COMSOL Multiphysics software to solve the Maxwell's equations. The resonance frequencies, the free spectral ranges (FSR), the full width at half-maximum (FWHM), finesse (F), and quality factor of the resonances (Q) are examined. The resonant frequencies are dominantly affected by the resonator diameter while the gap width, the ring thickness and the waveguide width have negligible effects on the resonant frequencies. FWHM, the quality factor Q and the finesse F are most strongly affected by the gap width and moderately influenced by the ring diameter, the waveguide width and the ring thickness. In addition, our simulation demonstrates that there is an optimum range of the waveguide width for a given ring resonator and this value is between ∼2.25 μm and ∼2.75 μm in our case.

The opto-fluidic ring resonator sensing platform has recently emerged as a new solution for highly sensitive detection of biological and chemical analytes [

A high quality OFRR configuration design is essential to optimize the sensitivity of specific biomolecule detection. However, because the experimental approach to this task is expensive and time-consuming, theoretical models have been utilized to describe the WGMs of the OFRR using Mie theory by considering a three-layer radial structure. Using this model, the radial distribution of the WGMs electrical field is derived and the resonant wavelength can be obtained numerically by matching the boundary conditions. Furthermore, the WGM spectral position can be obtained as a function of wall thickness, the resonator size, operating wavelength,

The binding of analytes to the resonator inner surface results in a modification of the effective refractive index and the ring thickness experienced by the WGM, leading to change in the WGM spectral position. A number of analytical studies have been conducted to understand the evanescent coupling for various resonator designs [

In this study, the finite element method is used for solving the Helmholtz equation of electromagnetic waves reduced from Maxwell's equations. The commercial COMSOL Multiphysics package (version 4.1) is applied to perform a finite element analysis in the OFRR simulation. We mainly investigate the resonance characteristics associated with the ring diameter, the ring thickness, the gap width, and the waveguide width. We also discuss in detail the effects of these parameters.

The sketch of a conceptual OFRR is shown in ^{−}^{iωt}

When the resonance ring is placed in the (x, y) plane as shown in _{o} is the free-space wave number.

For simplicity, the resonance ring is placed on the same plane as the waveguide, as shown in _{0} = 0 For laser excitation source, the scattering boundary condition is also used with input amplitude of _{0} = 1

The finite difference method (FDM) has been commonly used for numerical studies in electromagnetics. However, the finite element method (FEM) has an advantage in dealing with irregular configurations or system analysis. In this study, the FEM is used to solve the Helmholtz equation. COMSOL Multiphysics with RF module (version 4.1, COMSOL Inc., Burlington, MA, USA) is employed for numerical analysis and post processing.

A glass capillary and an optical fiber are used as the ring resonator and the waveguide, respectively. _{2}), respectively. The refractive indices for air and water are 1.0 and 1.333, respectively. A tunable laser beam for resonance excitation is directed straight from the left end of the waveguide. The frequency of the incident laser ranges between 192.31 THz (1,560 nm) and 196.80 THz (1,530 nm). It is noted that the wavelength

A mesh is a discretization of the geometry model into small simple shapes, and in this work the computational domain is meshed by triangle elements. The normal mesh size predefined in COMSOL is selected. All domains are meshed in two steps; the free triangular step followed by the refine step. The free triangular step is taken once to all domains. The refine step is taken thrice for the domains of the resonator and the waveguide. For the remaining domains of air and the sample, two refine steps are taken. The ratio of average mesh size in the domains for air and the sample to that in the domains for the resonator and the waveguide is about 58.3 to 1. Though the computational resolution of the laser wavelength is basically 0.5 nm, a 0.01 nm resolution is chosen in the vicinity of the resonance frequency for accuracy. The detailed simulation procedure using the COMSOL Multiphysics package is described elsewhere [

The distributions for electric fields and radiation energy density were examined under two WGM resonances (1st and 2nd order) and off resonance, respectively. We compared the experimental results for three operating conditions. The diameter and the thickness of the resonator ring were 100 μm and 3 μm, respectively. The waveguide thickness was 2 μm and the gap width between the resonator and the waveguide was 0 nm.

In addition, we conducted four parametric studies. In order to find the effects of each parameter, we varied only one parameter at a time with other three parameters fixed. The diameter of the resonator (d) was varied between 80 and 120 μm with 10 μm step. The thickness of resonator (t) was controlled between 2.5 and 4.0 μm with 0.5 μm step. The width of the waveguide (w) was changed between 1.5 and 3.25 μm with varied step size. The gap size between the resonator and the waveguide (g) was varied between 0 and 300 nm with 100 nm step. For accuracy, we obtained four sets of scattering spectra for each parameter in the frequency range of 192 THz to 196 THz. Then, we extracted the resonance characteristics from the scattering spectra to investigate the resonator configuration effects. Our study was mainly focused on the resonance characteristics such as the resonant frequency, the full width at half-maximum of the resonant frequency band, the quality factor defined by the resonant frequency divided by FWHM, the free spectral range, and the finesse of the resonant mode defined by FSR divided by FWHM.

The first and second order resonances are found at

In

In the actual simulation process, we solve Maxwell's equations for a given resonator structure and material properties. Therefore, our simulation results naturally include the power loss in the ring as well as the coupling power loss between the waveguide and the ring. The resonance characteristics determined from this simulation are realistic, and therefore it should be noted that the quality factor Q discussed in this work is the total Q, defined by 1/Q = 1/Q_{external} + 1/Q_{internal}.

In

In order to study dependence of resonance parameters on ring thickness, we calculated the scattering spectra for four different OFRR thicknesses (t = 2.5, 3.0, 3.5 and 4.0 μm). The ring diameter, the gap width and the waveguide width were 100 μm, 0 nm and 2 μm, respectively. Four first-order resonant frequencies (modes) were found for each of the resonator thickness in the frequency range of 192–195 THz. The resonant characteristics obtained from the scattering spectra (not shown) are displayed in _{s}/E_{d}_{s}_{d}

In order to understand how the resonance parameters are affected by the waveguide width, we simulated the scattering spectra for ten different waveguide widths (w = 1.5–3.25 μm). The ring diameter, the gap width and the ring thickness were 100 μm, 0 nm and 3 μm, respectively. The resonant characteristics obtained from the scattering spectra (not shown) are plotted in

The sharp jumps of Q, F and FWHM near 3.0 μm are quite surprising. In order to investigate this interesting behavior in more detail, we extended our study for four different ring thicknesses (t = 2.5, 3.0, 3.5, 4.0 μm). The simulated FWHM values plotted in

With increasing waveguide width, the electric field in the waveguide extends less into the ring resonator and coupling strength between them decreases. The decreased coupling strength causes Q and F to increase and FWHM to decrease in the range of w = 1.5–2.5 μm, as seen in

The sharp jump of FWHM near 3.0 μm, as displayed in

Finally, we investigated how the resonance parameters are influenced by the gap width. For this purpose, we simulated the scattering spectra for four different gap widths (g = 0, 100, 200 and 300 nm). The ring diameter, the waveguide width and the ring thickness were 100 μm, 2 μm and 3 μm, respectively. Four first-order resonant frequencies (modes) are found for each of the gap width in the frequency range between 192 THz and 195 THz. The resonant characteristics obtained from the scattering spectra (not shown) are plotted in

It is clearly seen in

These interesting results can be qualitatively explained by the coupled-mode theory (CMT) [_{1}_{2}_{0}, n_{1}, n_{2} and n_{3} are the refractive indices of air, the wave guide, the ring and water, respectively.

This resonator can be modeled by a coupled system with several physical parameters, as depicted in

In _{2} = |_{2}|^{2}/|_{1}|^{2},
_{0} = 1−(1 − ^{−2}^{αL}^{0}.
_{2} on resonance and K_{0} is the power splitting ratio on resonance. K_{0} is an intrinsic resonance parameter related to the power loss of the system. Therefore, it can be shown that
_{2}_{0} + _{av}L_{c}_{av}_{c}_{o} is the length of ring excluding the coupling region. They are related to each other by _{0} = 2_{c}_{0} is smaller [_{0} = _{0} and K generally decreases nearly exponentially with increasing the gap width, FWHM is expected to decrease with increasing gap width. Our results are in good agreement with CMT, as observed in _{0}

WGM resonators with a capillary ring-waveguide coupling arrangement were numerically characterized by FEM. By solving Maxwell's equations, the electric fields and radiation energy distributions in the rings were determined. As WGM resonance occurs, a very shining loop with a strong electric field and high radiation intensity exists inside the periphery of the ring resonator under first-order resonance. There are two glittering loops inside the ring under the second-order resonance and the light intensity of the inner loop is higher than that of the outer loop. Thus, the second order resonances may be preferred in sensing applications because interesting interactions with analytes occur mainly in the vicinity of the inner boundary of the ring through the evanescent field.

The WGM resonant frequencies are predominantly determined by the resonator diameter. Contrastingly, the gap width, the ring thickness and the waveguide width have negligible effects on the resonant frequencies.

FWHM, the quality factor Q and the finesse F are most substantially affected by the gap width and moderately influenced by the waveguide width and the ring thickness. For example, with increasing the gap width from 0 to 300 nm, Q and F increase tenfold while FWHM decreases by one tenth. Contrastingly, FWHM, Q and F vary by a factor of 2 as the ring diameter, the ring thickness and the waveguide width change.

In addition, our simulation demonstrates that there is an optimum range of the waveguide width for a given ring resonator and this value is between ∼2.25 μm and ∼2.75 μm in our case.

This work was supported by the research grant of the Chungbuk National University in 2010.

The schematic of (

A photograph of a fabricated OFRR.

Electric fields distributions for the ring-waveguide coupling part (d = 100 μm, t = 3 μm, w = 2 μm, g = 0 μm); (

Resonance spectra of the ring for the OFRR (d = 100 μm, t = 3 μm, w = 2 μm, g = 0 nm). The large peaks are for the first-order resonances while the small peaks are for the second-order resonances.

Effects of the ring diameter on the resonance characteristics (t = 3 μm, w = 2 μm, g = 0 μm). Displayed four data sets are obtained from adjacent resonance peaks, respectively.

Effects of the ring thickness on the resonance characteristics (d = 100 μm, w = 2 μm, g = 0 μm). Displayed four data sets are obtained from adjacent resonance peaks, respectively.

Effects of the waveguide width on the resonance characteristics (d = 100 μm, t = 3 μm, g = 0 μm). Displayed four data sets are obtained from adjacent resonance peaks, respectively.

Effect of the waveguide width on FWHM (d = 100 μm, g = 0 μm).

Effects of the gap width on the resonance characteristics (d = 100 μm, t = 3 μm, w = 2 μm). Displayed four data sets are obtained from adjacent resonance peaks, respectively.

Schematic diagram of the ring resonator coupled to a waveguide.

Model of wave coupling.