Development and analysis of whispering gallery mode model for refractive index sensing

A refractive index sensor based on a whispering gallery mode resonator is numerically investigated in this paper. Whispering gallery mode resonators confine light and can be used in sensing applications, and this work presents the model built using COMSOL software to analyze their sensing performance. This work has investigated the WGM resonators’ response to refractive index changes numerically using 2D models in the refractive index ranges 1.000 and 1.333. The resonator showed a sensitivity of 167 nm/RIU in the refractive index range of 1.3331.4 and a sensitivity of 95 nm/RIU in the refractive index range of 1.000-1.010. The sensing resonator has potential applications for biological and chemical analytes.


Introduction
The Micro-bottle resonators (MBR) are bottle-like structures fabricated by varying the radius of an optical fiber [1].MBRs can support whispering gallery modes (WGM) due to the total internal reflection of the light coupled into the MBRs [2][3][4].MBRs have a significant advantage in sensing and other advanced optical applications due to their light confinement abilities in a relatively small mode volume and high Q factor [5].They can be used for various applications including environmental, chemical, or biological sensors [6].For instance, hollow core silica capillary MBRs are used for magnetic field sensing and showed an 8.45 pm/Gs sensitivity with a Q factor of 1.28 × 10 6 [7].A packaged optofluidic MBR for fluid flow rate sensing showed a maximum sensitivity of 0.079 pm/(μL/min) within 0-200 μL [8].However, hollow MBRs sensors can become fragile and challenging.
Many works have been reported on WGM based sensors [9,10].The numerical analysis of WGM resonators showed a 3.14 mV/Pa sensitivity as an acoustic sensor.The broadband sensor was siliconbased [11].A WGM gyroscope was shown to have a 3.002 pm/(°.s - ) [12].Dual measurements of refractive index changes and pressure were conducted using a WGM resonator.The sensor used two whispering gallery modes to achieve an RI sensitivity of 29.07 nm/RIU and a pressure sensitivity of 0.576 pm/kPa for the first mode, and an RI sensitivity of 38.68 nm/RIU and a pressure sensitivity of 0.589 pm/kPa for the other mode [13].Dual temperature and refractive index measurements have also shown refractive index sensitivities of 45.8821 and 72.9402 nm/RIU and 0.0730 and 0.0703 nm/K temperature sensitivity [14].Plasmonic whispering gallery modes have been shown to reach a 180 nm/RIU sensitivity [15].These numerical investigations show the potential whispering gallery mode resonators have as chemical, environmental, and biomedical sensors.Solid MBR sensors can have the analyte on the outer medium while a hollow MBR can have the analyte inside it or outside the MBR.
The sensing principle of MBR is to measure changes in the resonance wavelength shift or power output variations in the presence of the analyte.MBRs can be used in many sensing and measurement applications [2,16,17].This model presents MBRs' azimuthal plane simulation for refractive index sensing.In MBRs, two projections exist, one in the (r, ) plane and the other in the (r, z) plane.The (r, ) plane for MBRs confines light like a whispering gallery.The light propagation and interactions with the changing surrounding medium are modelled and studied as a WGM resonator coupled with a straight waveguide.The effect of temperature on the transmission spectrum is also investigated to highlight the potential of MBR sensor to detect temperature change.Due to the complexity of the calculation, a 2D model is used to study the WGM resonator.Using a 3D model to simulate a WGM resonator coupled with a straight waveguide or fibre is computationally intensive and challenging to maintain due to their large size.
In this paper, the objective was to build a model in the (r, ) plane to study the MBR's sensing response.The model was used to investigate the resonator's response to refractive index changes in the ranges 1.333 and 1.000.The 2D model was also used to study the effects of temperature variations on the resonance wavelength shift.

Methodology and model development
The This section provides a detailed method for developing the COMSOL model for the WGM resonator.COMSOL 6.0 was used in this model.COMSOL solvers decompose the system into subdomains.The partial differential equations for each of the subdomains are solved with boundary conditions.The solutions are combined to finalize the system calculation.COMSOL can mesh and solve curved structures, making the software suitable for studying our WGM resonator.Other COMSOL wave optics modules simulate electromagnetic wave propagation and resonance phenomena.The module calculates electromagnetic field distributions and transmission coefficients for a photonics sensor design.The module's model formulas are based on Maxwell's equations combined with material properties for varied media propagation.The module solves Maxwell's equations in differential forms with initial/boundary conditions.
In this work, a model for a WGM resonator coupled with a waveguide was built using COMSOL 6.0.The simulation was conducted using electromagnetic waves and frequency domain analysis.A 2D structure was used to minimize the running time.The WGM resonator was tested with multiple radii from 3-11 µm.The waveguide was tested at radii 0.55-1.25 µm.The combination of WGM with a 3 µm radius and a 0.65 µm waveguide was chosen for further analysis.This combination was suitable for our study since it is optimum in terms of run time and mesh complexity.Larger resonator radii resulted in file sizes larger than 18 gigabytes for the model and computation times up to 12 hours for a single run of wavelength sweep with one value of the refractive index.The waveguide diameter was the value at which the transmission of the fiber has the minimum transmission at resonance dip.
The first step is using the model wizard.The dimension of the model is chosen as 2D.The physics chosen is the electromagnetics wave frequency domain (ewfd).The electric field of this model became (Ex, Ey, Ez).The WGM resonator radius and the waveguide diameter were used in the geometry definition.The refractive indices of the core, cladding, and analyte were used in the material definitions.A circle with a 7 µm radius was built in the geometry window.The circle represents the WGM resonator.A rectangle with a diameter of 0.65 µm was built.The rectangle represented the waveguide.The design has two elements: WGM resonator, and straight waveguide, and they were enclosed in a larger rectangle to form a model.The model needed a perfectly matching layer to operate correctly.A perfectly matching layer mimics an open, non-reflecting, infinite domain.It is an alternative to non-reflecting boundary conditions.The layers work for all wave types.The perfectly matching layer was added to the boundary of the model to truncate the computations.The perfectly matching domain was used to avoid reflections.The layer absorbs the outgoing waves from the components inside the model without reflecting those waves into the elements of the model.The PML is applied to domains, not boundaries.Thus, domains need to be included in the model to use the PML.This model included the PML surrounding the encasement rectangle.The material of the PML was the same as the material of the encasement rectangle.Both materials had a 1.00 refractive index.The PML with cartesian coordinates as defined in the definition's tap in the components window.PML domains effectively weaken reflections.However, PML domains require high computational performance, as shown in Figure 1.

Results and discussion
This section presents the output performance of the proposed model.The input light was set within the 1550 nm range.The transmission spectra of the straight waveguide and the whispering gallery mode (WGM)-coupled straight waveguide (or tapered fiber) were investigated by sweeping the frequency.The effect of coupling gap distance on the performance of Q factor and extinction ratio was also calculated.The study was conducted within two refractive index ranges: 1.00-1.01and 1.333-1.34.The resonance dip shifts were calculated and presented to quantify the potential of the sensor.The temperature response of the sensor was calculated to show the sensor's potential.The light was launched through port 1 and collected from port 2.
Figure 2 shows the transmission spectra of the straight waveguide (or fiber) and the WGM coupled fiber.It is observed that a resonance occurs in a WGM-coupled fiber.Far from the resonance range, most of the waveguide power remains in the waveguide.When the light propagating inside the fiber has a frequency and phase match with the resonator, the light gets coupled into the resonator instead of propagating through the fiber.Figure 3 shows the light field propagation inside the proposed WGMcoupled fiber at two resonance wavelengths of 1.545 and 1.564 µm.At the resonance range of 1.545 µm, the transmission spectrum show that the energy is coupled into the WGM resonator, see Figure 3.

WGM-coupled fiber
The resonator can sense biochemical analytes by measuring the surrounding medium refractive index changes.These biological and chemical concentration changes induce refractive index changes in the resonator's surroundings.The fields of the resonator interact with the analytes, and a subsequent resonance wavelength shift occurs.This shifting is the change in the resonant wavelength of MBR.This wavelength shift can be tracked and used to detect and measure analytes.The resonant wavelength shift can be measured directly or indirectly.These resonance shifts can occur when the surrounding medium has a refractive index change or if the resonator's size or refractive index changes.When the MBR has a particle with a higher refractive index attached to its surface, part of the electric field is outwardly pulled.Thus, this particle causes the optical path to be longer, and a redshift happens to the resonance mode.
The sensitivity of this reaction is the ratio of the resonance wavelength shift to the refractive index change.The refractive index of the surrounding media directly affects the coupling conditions of the resonator and its transmission spectrum, as shown in Figure 5.With higher refractive indices, a red shift occurs in the WGM transmission spectra.The refractive index was changed within a range from 1.0001.010with a step size of 0.0025.Figure 4 shows the resonance wavelength shifting towards longer wavelengths with increasing refractive index.
The model was then used to measure the sensitivity of the proposed WGM setup as the surrounding refractive index was changed in a range of 1.000-1.010for gas sensing.The resonance wavelength experiences a redshift when the test gas is applied to the resonator surface.Figure 5 shows the sensing response, which indicates the sensitivity of 95 nm/RIU with a linearity of 100%.The shift in WGM resonance wavelength (or frequency) is not only caused by the change in the resonator size and refractive index of the resonator material, but also attributed to the change in the refractive index of surrounding medium [18].Here, the refractive index range of 1.333-1.34represents the range of various liquid solutions.Sodium nitrate, formalin, and ethanol solution have been shown to have refractive index values in this range.A test layer was added to the model to represent the liquid solution.Then, frequency sweep is applied to obtain the transmission spectra of the resonator.Figure 6 shows the transmission spectra at various liquid refractive indexes.As the surrounding media's refractive index is increased, the resonance dip experiences a redshift.The resonance dip moves to a longer wavelength due to the change in the coupling condition.It experienced a shift from 1.629 to 1.641 µm with the increase of refractive index from 1.333 to 1.400.Figure 7 shows a plot of resonance wavelength as a function of the surrounding refractive index.It shows a linear response with a sensitivity of 167.8 nm/RIU.The modelling results confirm the WGM resonance shift induced by the surrounding medium.WGM resonators are sensitive to the change of surrounding refractive index and could be useful for sensing applications.The sensitivity of the proposed WGM resonator with the change of surrounding refractive index is comparable with the previous work on the gold-coated plasmonic WGM resonator [15,20,21].WGM resonators are sensitive to the change of surrounding refractive index and could be useful for sensing applications.Table 1 compares the performance of this WGM resonator model with the previous WGM resonators, which were reported for various applications.The sensitivity of the proposed WGM resonator with the change of surrounding refractive index is comparable with the previous work on the gold-coated plasmonic WGM resonator.Gold-coated plasmonic WGM 180 nm/RIU [15] This model 167.8 nm/RIU This work

Conclusion
MBRs confine light as a whispering gallery in the (r, ) plane.This work has investigated the WGM resonators' response to refractive index changes in the ranges 1.333 and 1.000.A COMSOL 2D model for the WGM resonator was used for refractive index detection.The 3 µm resonator showed a sensitivity of 167 nm/RIU in the refractive index range of 1.333-1.40and showed a sensitivity of 95 nm/RIU in the refractive index range of 95 nm/RIU.The temperature effects on the resonator are shown to be 24 pm/°K in the temperature change range of 0-50 ΔK from room temperature.The sensing resonator has potential applications for biological and chemical analytes.

Figure 1 .
Figure 1.The PML configuration and geometry.

4 A
WGM of the second radial order occurs at this wavelength.At the resonance range of 1.545 µm.The power coupling between the WGM and the waveguide is apparent in both figures.

Figure 2 .
Figure 2. The transmission Figure 3.The resonance spectra of the fiber and the response at 1.545 µm.WGM-coupled fiber

Figure 4 .
Figure 4.The transmission spectra at different refractive indices.

Figure 5 :
Figure 5 : The sensitivity of the resonator at the refractive index range 1-1.01.

Figure 6 .
Figure 6.The effect of the surrounding refractive index change on the transmission spectrum.

Figure 7 :
Figure 7: The model's response to refractive index change within a range of 1.333-1.400.