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• Abstract

SECTION I

## INTRODUCTION

WHISPERING gallery mode (WGM) based resonators have been used widely for biosensor applications [1], [2], [3], [4]. In most cases, the presence of the analyte changes the effective refractive index of the resonator, which in turn changes its resonance wavelength. As such resonance wavelength shifts can be monitored very precisely, a very high sensitivity can be achieved, right down to single-molecule levels [2].

For most cases, two key factors that determine the overall sensing performance are the Q factor of the resonator and mode overlap, which is defined to be the fraction of the optical energy within the region affected by the analyte. Increasing the Q-factor increases the precision of the resonance peak determination. However, while Q-factors that exceed $10^{6}$ or more have been reported [1], [2], such high values are difficult to achieve using standard fabrication processes, and can also result in limited dynamic range and sensitivity to environmental noise. Increasing the mode overlap improves the overall sensitivity and detection limit regardless of such vagaries of fabrication or sensing processes. However, the need to confine the optical mode in the resonator has led most WGM based biosensors to rely on evanescent fields for sensing, which can reduce the practically achievable mode overlap.

Of the many structures that have been proposed to solve this problem of low mode overlap [4], [5], [6], [7], the slot structure, which consists of a low-refractive index region sandwiched between regions of high refractive index, has attracted a particular attention. The presence of slot not only opens up the access to the interior of optical mode, where the optical field is the highest, but can further enhance the optical field inside the slot due to the continuity requirement for the D-field [8], making it ideal for applications such as non-linear optical devices and biosensing [9], [10]. Furthermore, high Q-factors are possible with a slot structure due to its all-dielectric construction. Recently, we have demonstrated fabrication of microdisks with a 40-nm thin horizontal air slot with an intrinsic Q-factor of 34,000 [11]. Consequently, such a structure can provide more than a ten-fold increase in sensitivity over conventional, evanescent-field based WGM biosensors [3], [4]. However, the presence of air-slot complicates the optical properties of the resonator in a non-intuitive way. Thus, while the potential for highly sensitive biosensing using horizontal slot resonator has been demonstrated, the limits of its performance and the optimum design for biosensor have not yet been investigated.

In this letter, we report on the results of numerically investigating the effects of slot geometry on the performance of such horizontal slot microdisk resonators based biosensors. We find that the mode overlap is the limiting factor for the performance. Based on calculated results, we suggest that slot resonators offer advantage over conventional, evanescent-field based resonators primarily for the surface sensing that detects the modification of the sensing surface by the analyte, typically in units of nm/nm (change in resonant wavelength per analyte thickness) or nm/pg (change in resonant wavelength per analyte mass). This requires strong optical field near the surface. At the same time, possible mode mixing needs to be avoided by proper design of the resonator structure. We further propose an optimized triple-slot structure to double the possible detector sensitivity.

SECTION II

## THEORY

While there have been previous reports on optimization of waveguide structures with vertical slots [12], [13], such results cannot be applied directly to horizontal air slot resonator structures. In addition to the obvious difference in the structure, the physical quantities that affect the sensitivity are different as well. Using variation methods, the change in the effective index of a waveguide upon a small change in the refractive index can be given by [14]. TeX Source $$\Delta n_{eff}={{\int_{M}{d\nu\varepsilon E^{2}}}\over{\int_{\infty}{d\nu\varepsilon E^{2}}}}\times n_{g}(n_{2}^{2}-n_{1}^{2})/2n_{1}^{2}\eqno{\hbox{(1)}}$$ where M in the integral in the numerator denotes the region whose index has changed, and ${\rm n}_{{\rm g}}$ is group index. ${\rm n}_{1}$ and ${\rm n}_{2}$ are the refractive indices of region M before and after the change, respectively. Note that the $\Delta {\rm n}_{\rm eff}$ is directly proportional to the group index. Thus, the effects of dispersion such as the slow light can greatly affect the performance of waveguide-based biosensors such as Mach-Zehnder interferometers [15] or photonic crystals. However, in case of resonators, the resonance wavelength is a function of the refractive index, and we must consider the effect of dispersion. As a result, group index does not appear in resonator peak shift, and we obtain TeX Source $$\Delta\lambda=Overlap\times{{\lambda (n_{2}^{2}-n_{1}^{2})}\over{2n_{1}^{2}}},\;Overlap={{\int_{M}{d\nu\varepsilon E^{2}}}\over{\int_{\infty}{d\nu\varepsilon E^{2}}}}\eqno{\hbox{(2)}}$$ In fact, (2) is equivalent to the expression $\Delta\omega=-\omega\left\langle E\right\vert\!\Delta\varepsilon\!\left\vert E\right\rangle/2\!\left\langle E\right\vert\varepsilon\!\left\vert E\right\rangle$ that was reported in [16], since $\Delta\omega/\omega=\;-\Delta\lambda/\lambda$ and $\varepsilon={\rm n}_{1}^{2}$ at region of M.

In short, for a resonator-based biosensor, the only external factor that determines the shift in the resonance peak is the mode overlap, defined to be the fraction of optical energy in the region containing the analytes, as the difference between ${\rm n}_{2}$ and ${\rm n}_{1}$ is determined by sensing material. Thus we conclude that for whispering gallery mode based biosensors, the mode overlap optimization equals to sensor optimization. Henceforth, we will focus on design rules for optimizing the overlap.

SECTION III

## SIMULATION

The optical modes of horizontal slot disk structures were simulated using finite difference time-domain (FDTD) method. In all cases, a horizontal slot microdisk that consists of two 8 $\mu{\rm m}$ diameter Si disks, one on top of the other separated by a gap that forms the slot, and suspended in a low index matrix of either air or water, was used unless otherwise specified. Si was chosen for its high refractive index compatibility with standard microfabrication techniques, and possible ultra-high Q-factors [17]. The diameter was fixed at 8 $\mu{\rm m}$ to be small enough for efficient simulations, but large enough for low radiation losses. For optimization, the thickness of the slot was varied from 10 to 100 nm, while the thickness of the Si disks was varied from 100 to 200 nm. A minimum slot thickness of 10 nm was chosen to be larger than typical protein molecules [18]. The Si disks were chosen to be thick enough for reasonable optical confinements, but thin enough that for the TM modes only the zeroth order vertical modes are supported. A schematic description of the structure is given below in Fig. 1. Clearly, such a suspended structure is not physically realizable. However, we have previously shown, both theoretically and experimentally via evanescent coupling using tapered fibers, that a pedestal type resonator with a concentric ${\rm SiO}_{2}$ spacer layer between the two disks can accurately mimic such a suspended structure [11].

Fig. 1. Simulation structure. The diameter was fixed at 8 $\mu{\rm m}$. Thickness of the Si disks was varied from 100 to 200 nm and thickness of the slot was varied from 10 to 100 nm. Disks are surrounded by air or water. Perfectly matched layer is used for absorbing boundary condition.

Fig. 2(a) and (b) show the slot-mode overlap, defined to be the fraction of the optical energy in the slot region, of a disk in air and water, respectively. Fig. 2(c) and (d) show the homogeneous-mode overlap, defined to be the fraction of the optical energy in the entire environment surrounding the resonator, of a disk in air and water, respectively. For accurate comparisons between different resonators, we calculated every ${\rm TM}_{1,0}$ (first-order radial and zeroth order vertical TM mode) slot-modes in the 1510 to 1570 nm range for a given resonator structure, and used the largest overlap value. Shaded regions indicate regions where the Q-factor became too low (below 700) to be reliable for calculation of mode overlap, and were discarded from further investigations. We note that a Q-factor of only 330 was shown to be sufficient for biosensing using slot-type ring resonators [19]. Thus, low Q-factors are not expected to be a problem for any mode we investigate here. The values of Q-factors, corresponding propagation loss, and effective refractive indices of the modes that were investigated ranged $10^{3}\hbox{--}10^{7}$, 0.05–200 db/cm, and 1.2–2.5, respectively.

Fig. 2. (a) and (b): Slot-mode overlap in air and water. Slot-mode overlap remains similar for a wide range of combination of disk and slot thicknesses except for the presence of periodic dips. (c) and (d): Homogeneous-mode overlap in air and water. Homogeneous-mode overlap decreases monotonically with increasing Si disk thickness.

Somewhat surprisingly, Fig. 2(a) and (b) show that, except for the presence of periodic dips that will be discussed later, the slot-mode overlap remains largely the same for a wide range of combination of disk and slot thicknesses. Still, the largest slot-mode overlap is obtained when the Si disks are relatively thick. This is due to pulling of the evanescent fields from above and below the microdisk into the disk by the increased effective refractive index. But if the Si disks are too thick, then the fraction of the optical mode confined inside the disks becomes too large, reducing the slot-mode overlap. Conversely, this pulling-in of the evanescent field reduces the homogeneous-mode overlap such that it decreases monotonically with increasing Si disk thickness. A similar phenomenon has been reported for waveguide sensors as well [12].

The largest values for the slot-mode overlap in air and water are 30.4 and 32.3%, respectively. However, they are smaller than the largest values obtained for homogeneous-mode overlap, which were 60.9 and 68.1%. It is smaller than even the homogeneous-mode overlap calculated for a conventional, 200 nm thick Si disk resonator in water, which is 47.8% for a mode with the same azimuthal number, similar resonant frequency, and Q-factor. This indicates that for homogeneous sensing methods which probe the entire environment, a slot disk resonator offers only a little advantage in sensitivity, and none if only the slot region is used for sensing, over a conventional disk resonator.

On the other hand, the slot structure provides a significant advantage for surface sensing methods, which probe only the nm-thin layer at the surface. As shown in Fig. 3(a) and (b), the energy density in the slot, obtained by dividing the slot overlap by the slot thickness, increases monotonically with decreasing the slot thickness (except for the presence of shallow, periodic dips). As a result, a slot resonator with 10 nm thick slot and Si disks that are thicker than 150 nm will have as much as 19.5% of its optical mode within the 5 nm thin layer at the slot surface. In contrast, a conventional, 200 nm thick Si disk resonator in water will have only 1.87% of its optical mode within the 5 nm thin layer at the surface, which would result in a surface sensitivity that is 10 times weaker.

Fig. 3. Energy density in the slot in air (a) and water (b), obtained by dividing the slot overlap by the slot thickness. Energy density increases monotonically with decreasing the slot thickness except for the presence of periodic dips.

We now investigate the cause of periodic dips in the overlap. To do so, we calculate the slot-thickness dependence of resonance wavelength of other modes whose resonance frequency lies near that of the ${\rm TM}_{1,0}$, and has the same azimuthal number. The Si disk thickness was set to be 200 nm, and the resonator is taken to be immersed in water. As shown in Fig. 4, we see that at the slot thickness of 42.5 nm, where the dip in the slot-mode overlap occurs as shown in Fig. 2(b), we have anti-crossing of the ${\rm TM}_{1,0}$ mode with the ${\rm TE}_{1,1}$ mode, indicating mode-mixing. As TE modes contain more of the E-field inside the dielectric, such mixing reduces both the slot- and homogeneous-overlap, producing the observed dips.

Fig. 4. Slot-thickness dependence of resonant wavelength WGM modes whose resonance frequency lies near that of the fundamental ${\rm TM}_{1,0}$, and has the same azimuthal number. The Si disk thickness was set to be 200 nm, and the resonator is taken to be immersed in water.

However, no such mixing is observed at the slot thickness of 33 nm, where the ${\rm TM}_{1,0}$ mode is degenerate with ${\rm TE}_{3,0}$ mode. This difference is ascribed to the symmetries of the field distribution of ${\rm TM}_{1,0}$, ${\rm TE}_{1,1}$, and ${\rm TE}_{3,0}$ modes. As shown in Fig. 5, ${\rm TM}_{1,0}$ and ${\rm TE}_{1,1}$ modes have the same symmetry of axial and radial field in the vertical direction even when far away from the crossing. Consequently, ${\rm TE}_{1,1}$ mode mixes readily with the ${\rm TM}_{1,0}$ mode when the k-vector matching condition is satisfied. On the other hand, ${\rm TE}_{3,0}$ mode has a completely different symmetry, and thus does not mix with the even when the k-vector is matched. Such similar symmetry exists between ${\rm TM}_{1,0}$ and ${\rm TE}_{2,1}$ mode as well, leading to mode mixing and formation of the second dip at slot thickness of 85 nm for 200 nm thick Si slot disks in the water.

Fig. 5. Axial and radial field distribution of ${\rm TM}_{1,0}$, ${\rm TE}_{1,1}$ and ${\rm TE}_{3,0}$ modes. Note that the ${\rm TM}_{1,0}$ and ${\rm TE}_{1,1}$ modes have the same symmetry of axial and radial field distribution in vertical direction.

Practically, this means that when designing a slot disk resonator for bio-sensing applications, care should be taken to avoid the conditions under which such mode mixing can occur. One way to reduce the possibility of such mode mixing is lowering of the refractive index contrast, as such reduction in index contrast increases the spacing between the dips. This is already demonstrated in Fig. 2(a) and (b) that show 3 dips for Si slot disks in air, but only 2 for Si disks in water. However, that reducing the refractive index contrast will lead to reduction of the optical confinement in the slot and consequent reduction in the sensitivity of the resonator.

SECTION IV

## SIMULATION OF MULTI SLOT DISK

Finally, we note that by adding more slots to the disk, we can increase the sensing surfaces inside the slot that can utilize the confined optical field. Previous investigations into multiple slot ring resonators have demonstrated that presence of additional slot does not significantly affect either the coupling or the surface scattering losses [20]. On the other hand, doing so in effect increases the slot thickness, which reduces the optical field density. Fig. 6(a) shows the calculated enhancement of surface sensitivity of double and triple-slot structure over a single-slot structure, as a function of separation between the slots. In this simulation, the slot thickness was fixed at 10 nm which gave the best single-slot performance and total Si thickness are fixed at 280 nm and 370 nm for disks in air and water. Fig. 6(b) shows typical ${\rm E}^{2}$ distribution of double and triple-slot disks. We find that increasing the number of slots does help. Interestingly, the maximum enhancement is obtained not when the slots are close together in the middle, where the E-field intensity is expected to be the highest, but when the slots are separated by a well-defined optimum. The effect of additional slot is larger when the resonator is in water. The maximum value of enhancement observed is 2.0 for a triple slot in water, and 1.6 for triple slots in the air. In fact, for a resonator in water, a simple double-slot can increase the surface sensitivity by a factor of 1.6.

Fig. 6. (a) Enhancement of surface sensitivity of 10 nm double and triple-slot structure over a single-slot structure, as a function of slot interval. The maximum value of enhancement observed is 2.0 for a triple slot in water. (b) ${\rm E}^{2}$ distribution of double- and triple-slot disks.
SECTION V

## CONCLUSION

In conclusion, we have investigated sensitivity of slot disks through the overlap. For homogeneous sensing, a slot structure provides little advantage over conventional, non-slotted disk resonator with similar Q factor. On the other hand, the slot structure provides a significant advantage for surface sensing methods, with the surface sensitivity increasing continuously with decreasing slot thickness for an order-of magnitude enhancement over a conventional, non-slotted disk. Possible mode mixing needs to be avoided by proper design of the resonator structure. Finally, adapting multislots we further enhance the surface sensitivity. Maximum value of enhancement observed is 2.0 for a triple slot in water.

## Footnotes

This work was supported in part by the National Research Foundation of Korea Grant Funded by the Korea Government under Grant 2010-0029255 and in part by the Basic Science Research Program under Grant 2009-0087691.

S. C. Eom is with the Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea.

J. H. Shin is with the Department of Physics, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea, and also with the Graduate School of Nanoscience and Technology, Korea Advanced Institute of Science and Technology, Daejeon 305-701, Korea (e-mail: jhs@kaist.ac.kr).

Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

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