We use the finite element method (FEM) to numerically calculate the hybrid plasmonic waveguide mode. Gold is chosen as the metal layer, as it is biocompatible and not easily oxidized compared to silver. The refractive indices of the constitutive materials at 1,550 nm wavelength are nox
= 1.444, nsi
= 3.477, and nAu
= 0.55 + 11.5i
for oxide, silicon, and gold, respectively. The device upper-cladding sensing medium is assumed to be water solution with its complex refractive index nc
= 1.33 +1.2e − 4i
]. The water absorption loss is relatively small (~0.0042 dB/μm) compared to the metal absorption loss and the hybrid waveguide propagation loss , and hence its influence on the device performance can be neglected. We fix the waveguide height at Hsi
= 250 nm, a typical dimension for silicon ridge waveguides. The bending radius is set at R
= 1 μm.
(a) Effective refractive index (real part) and (b) propagation loss of the hybrid microring waveguide versus slot width.
Distinct from the straight waveguide, the bent waveguide in the hybrid microring resonator has a different electromagnetic field distribution with its mode profile pushed more outwards. To calculate the bent waveguide mode, we employ the conformal mapping to convert the bent waveguide to a straight waveguide [27
]. The refractive index is no longer uniform in the converted waveguide, but instead, it increases exponentially towards the radial direction. In our FEM simulations, the subdomains in the waveguide cross section were partitioned into triangular mesh elements with quartic Lagrange functions. The grid size is set to be 10 times smaller than the dimensions of the corresponding regions and the maximum mesh element size is set to be 20 nm, such that accurate simulation results can be achieved within the allowed time and computational resources. Perfect matched layers (PMLs) were also added at the outer edges of the FEM simulation window to approximate an open geometry [28
]. The PML was positioned 0.6 μm away from the waveguide edge. The PML width is 1 μm and the grid size inside the PML is 50 nm. The PMLs have uniform, anisotropic, and complex dielectric constants so that the reflections at the edges were eliminated and radiated light was attenuated inside the PMLs. The dielectric constants of the PMLs were chosen to match those at the edge of the equivalent straight waveguide, and the absorption constant and width of the PML were chosen to ensure light only propagates several mesh elements before being completely absorbed. Perfect electric conductors were used at all boundaries. The calculated bent waveguide mode is composed of a guided part and a leaky part, and hence, both the waveguide propagation constant and the loss can be obtained simultaneously from the FEM simulations.
shows the optical power flow density contour plots in the cross section of the bent hybrid plasmonic waveguide for two slot widths of 10 nm and 30 nm. The silicon strip width is 200 nm. The corresponding optical power flow density profiles along a lateral line in the middle of the waveguide are shown in . It can be seen that the central metal block can attract the optical power to the narrow slot region. The closer the silicon strip approaches the metal block, the more the optical power is confined near the metal surface. Note that the evanescent field outside the microring outer rim is weaker for the 10 nm slot case than that for the 30 nm slot case. Therefore, the addition of the metal block reduces the mode power radiation leakage, making the low loss propagation possible even for a highly curved waveguide.
Figure 2. (a) and (b) Average optical power flow in the propagation direction for (a) Wslot = 10 nm and (b) Wslot = 30 nm. (c) and (d) are the corresponding optical power flow density along a lateral line in the middle of the waveguide. Total power flow is assumed (more ...)
shows the bent waveguide effective refractive index and the propagation loss change as function of slot width for various silicon strip widths. The effective refractive index and the propagation loss values are referred to the microring resonator outer rim. The slot width has a large effect on the hybrid mode effective index and the propagation loss, especially for narrow silicon strips. When the slot width decreases, the optical power outside the waveguide is attracted to the silicon core, causing the effective index to increase. For the propagation loss, however, when the slot width reduces, it first decreases and then increases if the silicon strip is not too wide. There are two sources for the bent waveguide loss: the radiation loss and the metal absorption loss, with the former increasing and the latter decreasing with the slot width. Their competition ultimately determines the bent waveguide loss. Therefore, there is an optimum slot width where the bent waveguide loss reaches the minimum. For example, when the silicon strip width is 200 nm, the optimum slot width is ~16 nm. The excitation of surface plasmon waves can reduce the otherwise large bending loss suffered in a regular dielectric waveguide, which is more obvious for narrow silicon waveguides. The implication is that the low-loss propagation can be extended to a highly-curved waveguide, which makes very compact microring resonators possible.
As the overlap between the optical field and the analyte solution determines the sensitivity of the sensor, we define an upper-cladding confinement factor ηclad
to describe the overlap level. The upper-cladding confinement factor can be expressed as:
where E⃗.gif" border="0" alt="[E w/ right arrow above]" title=""/>
and H⃗.gif" border="0" alt="H" title=""/>
are the electric and magnetic fields, Sclad
indicates the upper-cladding layer region (including the slot), and S
is the hybrid plasmonic waveguide cross-section. The dependence of the upper-cladding confinement factor on the slot and the silicon strip widths is shown in . A large slot gap and a narrow silicon width push the optical energy into the upper-cladding layer, which increases its overlap with the analyte solution. For example, when the silicon width is 200 nm and the gap is >16 nm, more than half of the optical energy is distributed in the upper-cladding layer.
Upper-cladding layer confinement factor changes as a function of slot size for various silicon widths.
The sensitivity S
is defined as the ratio between the resonance wavelength shift and the cladding refractive index change, i.e.
, S dλ/dnc
, which is a key parameter to describe the sensor performance. The resonance wavelength λ
satisfies the resonance condition mλ
is an integer), where L
is the circumference of the microring resonator. Taking into account the wavelength dispersion of neff
, we can express the sensitivity as:
is the group refractive index given by:
Hence, to get a high sensitivity, the change rate of the effective refractive index to the cladding refractive index should be large, or in other words, the cladding confinement factor should be large. shows the sensitivity changes as a function of slot width for various silicon widths. The change trend is similar to that of the cladding confinement factor in . To obtain a high sensitivity, the slot should be large and the silicon strip should be thin.
Sensitivity changes as a function of slot size for various silicon widths.
The sensitivity is used to measure the resonance wavelength shift in response to the cladding index change. It is independent of the resonance spectral profile. However, the detection limit DL
, which is defined as the minimum refractive index change in the sensing medium that can be detected by the sensor system, is proportional to the resonance linewidth Δλ
or inversely proportional to the resonance Q-factor. The Q-factor is determined by the resonator loss (intrinsic Q-factor) as well as the coupling loss to the straight waveguide (external Q-factor). For the microring resonator working at the critical coupling regime, the Q-factor can be expressed as [29
is the electric-field round-trip transmission coefficient in the resonator. plots the Q-factor variation with the slot width. It has an inverse change trend with respect to the loss curves in . Although the Q-factor for this hybrid microring resonator is around 102
, it is much larger than that of microring resonators composed solely by dielectric or plasmonic waveguides with the same cavity size. When the slot is very wide, the hybrid microring resonator regresses to a dielectric microring resonator; and on the contrary, when the slot becomes very narrow, it resembles a plasmonic microring resonator. The combination of the dielectric and plasmonic guiding future of the hybrid waveguide increases the resonance Q-factor.
Resonance Q-factor of the hybrid microring resonator changes as a function of slot size for various silicon widths.
As can be seen from and , the sensitivity and Q-factor have a distinct change trend and they cannot be obtained simultaneously. A high sensitivity features a large resonance wavelength shift for a given index change, but it is more difficult to detect a small refractive index change. On the contrary, a high Q-factor improves the detection limit but reduces the resonance wavelength shift. To properly evaluate the sensing performance of the proposed hybrid microring resonator sensor, a figure of merit (FOM) can be defined as [8
which equals the number of resonance linewidth shift in response to a unit cladding refractive index change. shows the FOM changes with the slot width at various silicon strip widths. The maximum achievable FOM is ~17, which occurs at Wslot
= 10 to 16 nm depending on the silicon strip width. Note that, as the slot serves as a host for the biochemical analyte, only those with dimensions smaller than the slot can be infiltrated into the slot to induce a resonance wavelength shift.
Figure of merit (FOM) of the microring sensor changes as a function of slot size for various silicon widths.