#### 2.1. Schematic and Simulation Method

As shown in

Figure 1a, the DSHP ring resonator consists of a Si ring located between Ag (or other plasmonic material) circular sheets. When the slots between the Ag sheets and Si ring are narrow enough (less than 500 nm), the quasi-transverse electric (TE) hybrid photonics (Si-slot materials) and plasmonic (slot material-Ag) guided mode can be supported. In the ring structure, it is propagating as the known whispering-gallery mode.

**Figure 1.**
(**a**) Schematic of a double-slot hybrid plasmonic ring resonator; (**b**) x-z plane cross-section view; (**c**) power distribution of the x-z plane cross-section simulated by the axisymmetric finite element method. IPA, 2-isopropanol.

**Figure 1.**
(**a**) Schematic of a double-slot hybrid plasmonic ring resonator; (**b**) x-z plane cross-section view; (**c**) power distribution of the x-z plane cross-section simulated by the axisymmetric finite element method. IPA, 2-isopropanol.

The x-z cross-section view is shown in

Figure 1b. SiO

_{2} is used as the buffer layer, and liquids to be tested (or electro-optic polymer) are filling the slots and covering the device (infiltrating the device). The radius of the DSHP ring

r is equal to the middle radius of the Si ring, as shown in

Figure 1b. The width of the Si ring is denoted by

w_{Si}; the heights of the Si and metal layers are identical and denoted by

h_{WG}; the widths of the slots are

w_{slot}, and

w is the total width (

w =

w_{Si} + 2

w_{slot}).

The power distribution of the simulated DSHP ring is shown in

Figure 1c using the axisymmetric finite element method (FEM) [

23], where the commercial simulation software, COMSOL Multriphysics, is used to solve the partial differential equation in cylindrical coordinates. The geometrical parameters are:

h_{WG} = 250 nm,

w_{Si} = 300 nm and

w_{slot} = 150 nm. The refractive indices of Si and SiO

_{2} are 3.45 and 1.45, respectively, for the operation wavelength of 1550 nm. The infiltrating material is pure 2-isopropanol (IPA), whose refractive index is 1.3739 [

24] at 1550 nm.

The properties of Ag are calculated by the Drude model [

25]:

where

${\epsilon}_{\infty}=3.1$,

${\omega}_{p}=140\times {10}^{14}\text{rad/s}$ and

$\gamma =0.31\times {10}^{14}\text{rad/s}$.

#### 2.2. Model Investigation

The model investigations start from the generalized analysis of the quality factors (

Q factors) of the DSHP ring with various widths of the slots, as shown in

Figure 2a. The geometrical parameters of the DSHP ring are:

w_{Si} = 350 nm and

h_{WG} = 250 nm. In the simulation process, the resonant wavelength is fixed around 1550 nm, and the azimuthal numbers (

m) are changed from 15 to 55 with a step of five. The radii of the DSHP ring are adjusted according to the resonant condition of whispering-gallery mode:

The

Q factor is calculated by:

where

f_{eig}(real) and

f_{eig}(imag) are the real and imaginary parts of the eigenvalue of the

m-th order whispering-gallery mode, which can be directly readout from the simulation software.

Without considering the coupling loss, the

Q factor of the DSHP ring is influenced by absorption and radiation losses [

26], which can be written as:

Here, Q_{abs} and Q_{rad} are the absorption and radiation quality factors of the ring, respectively.

As shown in the figure, the Q factor increases with the radius due to lower radiation loss; when the radius is large enough (larger than 6 μm), the radiation loss can be ignored, while the initial properties of the DSHP waveguide (absorption loss) play a more significant role in the Q factor, which tends to a constant value for specific geometrical parameters. In addition, the Q factor of the DSHP ring with wider slots is larger, which means that the absorption loss of the DSHP waveguide with wider slots is smaller due to lower plasmonic material influence.

Summarizing the information given above, the absorption loss (or Q_{abs} factor) is the major property of a DSHP ring with a large enough radius. In order to study the Q_{abs} factor of the DSHP ring, one can fix the radius of the DSHP ring at 6 μm (radiation loss can be ignored) and change the geometry of the DSHP waveguide. In this way, the normal FEM simulation method can be used to study the properties of a DSHP waveguide per se. Since the hybrid propagation mode is a mixture between photonic and Surface Plasmon Polariton (SPP) modes, a q-parameter (q = w_{Si}/w), representing the occupancy of a photonic mode, is used to study and optimize the performance of the DSHP ring resonator.

Figure 2b shows the

Q_{abs} factor

versus the

q-parameter, when the total width of the DSHP ring is changed from 500 nm to 1000 nm. The

Q_{abs} factor is estimated by:

where

n_{g} is the group index of the confined whispering-gallery mode, which is expressed by:

n_{eff}(real) is the real part of effective refractive index. The dispersion of n_{eff}, $d{n}_{eff}(real)/d\lambda $, is computed by re-simulating the DSHP waveguide with a wavelength of $\lambda +\Delta \lambda $, and the n_{eff} at $\lambda +\Delta \lambda $ can be obtained, which is denoted as ${n}_{eff}(\lambda +\Delta \lambda )$.

**Figure 2.**
(**a**) Q factor versus the radius of the double-slot hybrid plasmonic (DSHP) ring resonator with various widths of the slots (150 nm, 250 nm and 350 nm); the other geometrical parameters are: w_{Si} = 350 nm and h_{WG} = 250 nm; (**b**,**c**,**d**) the Q_{abs} factor, effective refractive index and sensitivity changes with the q-parameter for waveguides with a total width w of 500 nm, 600 nm, 700 nm, 800 nm, 900 nm and 1000 nm.

**Figure 2.**
(**a**) Q factor versus the radius of the double-slot hybrid plasmonic (DSHP) ring resonator with various widths of the slots (150 nm, 250 nm and 350 nm); the other geometrical parameters are: w_{Si} = 350 nm and h_{WG} = 250 nm; (**b**,**c**,**d**) the Q_{abs} factor, effective refractive index and sensitivity changes with the q-parameter for waveguides with a total width w of 500 nm, 600 nm, 700 nm, 800 nm, 900 nm and 1000 nm.

Then, the dispersion can be estimated by:

α in Equation (5) is the absorption coefficient, which can be written as:

where

n_{eff}(

imag) is the imaginary part of effective refractive index.

After simulating the DSHP waveguide, the

Q_{abs} factors

versus the

q-parameter are calculated according to Equation (5), as shown in

Figure 2b, where the total widths of the DSHP waveguide are changed from 500 nm to 1000 nm with a step of 100 nm. One can observe that the

Q_{abs} factor of the DSHP ring decreases with a smaller width due to the larger influence of the plasmonic layers in the narrower DSHP waveguide. For the DSHP rings with the same total width, there are optimal values of the

q-parameter giving the maximum

Q_{abs} factor, which expresses the hybrid propagation modes of the DSHP ring, where the propagation loss is much lower than that of a pure plasmonic slot waveguide (when

q is equal to either zero or one).

Values of

n_{eff} versus the

q-parameter for different total widths are shown in

Figure 2c. The

n_{eff} increases with the occupancy of the slot by the Si ring (increase of

q), which needs to be explained from two aspects: (1) the high refractive index of Si has greater influence and causes higher

n_{eff} of the DSHP waveguide; and (2) with the higher

q-parameter,

w_{slot} of the DSHP ring is small, and so, the optical confinement is larger. The former explanation concerns the photonic mode, and the latter one concerns the plasmonic optical enhancement. The interaction between photonic and plasmonic modes leads to the nonlinear behavior of

n_{eff}.

In addition to the basic studies of the DSHP ring, we also analyze its sensitivity to infiltrating materials. The sensitivity is given by:

where

∂λ_{res} is the resonant wavelength shift and

∂n_{cover} is the refractive index change of infiltrating materials (

∂n_{cover} is set to be 0.003 in the simulation); Δ

n_{eff} is the change of effective indices of the DSHP waveguide infiltrated with different materials.

As shown in

Figure 2d, the sensitivity of the narrower waveguide is higher due to the better optical confinement. In addition, optimal

q-parameters are also observed for the sensitivities of the DSHP ring, which means that the sensitivity is also enhanced by the hybrid modes compared to pure plasmonic modes (

q = 0 or 1). The maximum sensitivity can reach a value as high as 700 nm/RIU when

w = 500 nm and

q = 0.35, which can be further increased by decreasing the total width, however, at the expense of a lower

Q_{abs} factor. The trade-off between the

Q_{abs} factor and sensitivity is similar to the one between propagation loss and optical confinement, which is a general behavior in hybrid plasmonic and plasmonic waveguides [

21,

27]. Based on the different behaviors of the

Q_{abs} factor and the sensitivity of the DSHP ring, one needs to carefully design the geometry of the DSHP ring to satisfy the requirements for particular applications.