1. Introduction

Recently, there has been great progress in the field of biosensors with tremendous sensitivity down to single molecules or single particles of micron- or even nanometer scales [1–6]. The enhancement mechanism responsible for this high sensitivity is the use of whispering gallery modes (WGM) in optical micro-cavities such as microrings, microspheres and microdisks, in which WGM resonance results in multiple interactions between the guided, re-circulating light and the sensing object (SO). In such miniature sensors, which usually include a micro-cavity coupled to a waveguide (tapered fiber, for example), a single-shot measurement to detect the SO is feasible. A light beam is launched from a remote location, propagates in the waveguide, and then couples to the micro-cavity and thereby to a SO (single molecule or particle), adjacent to the cavity. Due to WGM resonances, the light undergoes recirculation in the cavity, and as a result, the light field interacts with the sensing object multiple times, which can yield detailed information of the SO in the output transmission. The combination of low-loss confinement of the light in the WGM cavity and recirculation of the light can provide sensitivity down to single molecule or small particles of micron- and even nanometers-scales. In 2002, Vollmer et. al. experimentally demonstrated enhancement of the sensitivity using a silica microsphere with a radius ~100 micron [2]. However, even with this enhanced sensitivity the resulting shift in the resonant wavelength induced by a single molecule was very difficult to detect. Since then, progress has been made both experimentally and theoretically to reach a sensitivity level that is high enough to enable detection of a single molecule or a nanoparticle with R ~30nm [1–4]. In 2008, using a silica micro-toroid instead of a microsphere, scientists achieved extremely high sensitivity resulting in detection of an individual IL-2 protein molecule [3]. In 2010, nanoparticles with sizes down to 30nm have been detected and analyzed using the mode splitting effect [4]. So far, most theoretical analysis and calculations of WGM microrings and microsensors have been based on coupled mode theory (CMT), which is straightforward and fast, based on approximations for simple ideal structures which may not directly apply for real systems [1–7]. Usually, a simplified theoretical analysis gives the shift of the WGM wavelength, assuming that there is a change of the optical path in the cavity when it couples with a sensing object, where the change of the optical path depends on the refractive index and the size of the sensing object. However, as pointed out in Ref. [4], the shift is very small and susceptible to noise: intensity and frequency noise of laser, thermal noise, detector noise and environmental disturbance. The shift also depends on the strength of the coupling between the object and the microring WGM. As a result a small particle with large coupling to the WGM can result in the same shift as a larger object with smaller overlap. Furthermore, these analytical approaches fail to encompass many significant and complex factors, such as the actual shape and eigenmodes of the SO, determining the radiation losses of the light field in the tapered fiber and in the microcavity, as well as the scattering loss in the system. All of these factors can be addressed through FDTD simulation as presented in this paper. It is worth stressing that it is very difficult to use CMT to describe the effect of multiple interactions between the light field with the SO. More importantly, the FDTD can yield the resonant modes of the sensing object, which are easier to observe than the small resonance wavelength shift predicted by CMT. The observation of the eigenmodes of the SO has several important implications: (i) it is easy to detect the eigenmodes of the SO by observing the transmission; (ii) once the eigenmodes are detected, it is easy to determine the size and index of the SO, and (iii) the SO eigenmodes wavelengths do not depend on the coupling strength between the WGM resonator and the SO which is an important parameter in CMT, but is difficult to determine and control in experiment, and (iv) the eigenmodes are almost immune to experimental noise sources, since it is the location of the modes that is the most important, not their strength, which can be easily perturbed.

In this paper we present a finite-difference time domain (FDTD) simulation of microring-based sensing. Single-shot sensing of a label-free, single-object sensor using a microring is simulated. A broad bandwidth pulse first propagates in a tapered fiber, couples to a resonant microring and then to a single SO adjacent to the ring. Although the FDTD method can deal effectively with different shaped objects with complex refractive indices that are applicable for biosensing, for the sake of simplicity we consider in this paper only the basic cases in which SOs are 2D particles (or microdisks) with radii r_SO and real index n_SO. The indices of the SOs are chosen in the simulation as general examples to demonstrate the eigenmodes of the objects with sizes of 1-2 micron within the frequency spectrum of 1.55 micron light source. However, we would like to stress further that particles at the nano- and micro-scales have increasingly become important objects in biosensing. The 2D SOs still represent some aspects of the real particles in terms of the sensing properties, such as optical pathlength, and the eigenmodes of the SO in the 2D plane. The results of FDTD simulation show the resonant propagation in WGM microring, the multiple interactions between the recirculating light and the SO, and the resonant light inside the SO. The microring transmission without a sensing object (reference) is compared to that with sensing objects having different size and refractive indices. As presented in the next section, our FDTD code allows us to simulate the microring sensor with a high degree of flexibility, from designing a waveguide with minimum reflection, to extracting information from inside microring and the sensing object as well. The simulation method is not only suitable for a single object (single molecule, nano-, micro-scale particle), but can also be extended to the problem of multiple objects as well. Our paper is organized as follows: a general description of the system and the simulation method is presented in Section 2, simulation results including animations of the sensing will be presented in Section 3, and finally discussions of the advantages and disadvantages of the FDTD method for modeling WGM sensing will be presented in Section 4.

2. General Description

Let us consider a typical microring sensor in which a waveguide (tapered fiber) couples to a resonant microring adjacent to a sensing object (SO) as shown in Fig. 1 . A light beam is launched into the fiber from the left. The light beam propagates in the waveguide and then couples to the microring, where WGM resonances enable multiple recirculation of the light, resulting in multiple interactions between the light and the SO.

 

Fig. 1 Schematic of a microring sensor: a ring resonator (radius R, width d_R and index n_R) coupled to a waveguide (width d_W, index n_W). The gap between the ring and the waveguide is g. A sensing object (radius r_SO and index n_SO) is adjacent to the ring (light blue microdisk).

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Note that, once the light beam is coupled to the microring its components having frequencies that are close to WGM resonances of the cavity will be re-circulated multiple times depending on the quality of the cavity Q. For example, in a planar microcavity with a quality Q factor of 10⁸, the resonant light can interact with the sensing object about 100,000 times, instead of only one interaction possible in a simple optical waveguide sensor [1–3]. By monitoring the WGM optical resonances excited in the microcavity and/or the eigenmodes of the sensing object, label-free, single-object sensing can be achieved with a single laser shot as described below in our FDTD simulation.

In this paper we investigate numerically the general problem of microring sensing for a label-free single object sensing (SO) using the finite-difference time-domain (FDTD) method. We simulate wideband pulse propagation in a waveguide coupled to a microring adjacent to a sensing object. The FDTD method has several features that are advantageous for simulation of such systems as described above. Furthermore, the FDTD method can completely describe light recirculation in the WGM cavity and the multiple interactions between the light and the SO. This feature is very unique to WGM sensing, and is almost impossible to describe accurately by other simplified modeling methods. As presented below, FDTD simulation can also yield the shifts of the resonant modes that are estimated through other simplified modeling analysis [1–4]. More importantly, the FDTD simulation can extract the resonant modes of the sensing object, which are in fact easier to detect than the small wavelength shift of the microring modes. It is worth noting that FDTD method has been extensively applied to simulate and analyze WGM of isolated microdisks, microdisks and microrings [8–10]. The accuracy of the FDTD method for problems in linear optics was first demonstrated for the directional coupler [11]. Since then, the perfectly matched layer (PML) absorbing boundary condition (ABC) were introduced, providing the means to terminate the calculated grid space with extremely low reflection [12].

Let us generally describe the FDTD simulation method for a typical label-free single-SO microring sensor. We consider a two-dimensional (2D) problem where the z-directed electric field is normal to the x-y plane of the grid. We employ the PML ABC in our simulation of the light propagation in the waveguide-microring-SO system. It is worthwhile to stress here that accurate FDTD simulations in resonant cavities in general, and especially in WGM cavities have several challenges: (i) The computation time required for accurate simulation of light recirculation in WGM cavities is much longer than that required for typical scattering or propagation problems, and (ii) even if the reflection due to the numerical boundary conditions is very small within PML ABC, its effects can adversely affect the simulation results. This is especially the case when the light is reflected at the end of the waveguide, counter-propagates, and then couples back to the microring, where it undergoes recirculation in the cavity. To avoid boundary reflection during light recirculation in the microring, we introduce a long waveguide around the sensing area, which includes the microring and the SO. The goal is to make light keep propagating or trapped after it passes the microring so as to avoiding reflection in the waveguide. The optimization challenge in this simulation is to design an extended waveguide that can keep light propagating or get trapped as long as possible, while at the same time minimizing the computational space needed.

In this configuration, after traversing the coupling region between the waveguide and the microring, the light keeps propagating in the extended waveguide, and at the end it trapped by recirculation in the extended part of the waveguide.

Figure 2 shows the intensity of the light with frequency f = 192 THz, which is resonant with the microring, whose resonant modes are shown in Fig. 3 . In the simulation, a fsec pulse is launched from the left of waveguide and its ultrawide frequency band enables a single-shot simulation that spans over several resonant modes of the microring and also some eigenmodes of the sensing object. As shown in Fig. 2(b), in the case with a sensing object adjacent to the ring, the light then couples to the object and excites its eigenmodes. The complex propagation dynamics including multiple recirculation in the microring, and multiple interactions with the sensing object will be simulated and shown in the animation discussed in next section.

 

Fig. 2 FDTD simulation of light field with frequency of 192 THz in a microring with R = 20μm, n_R = n_W = 1.46, d_R = d_W = 1μm . (a): without sensing object, and (b) with sensing object r_SO = 2 μm, n_SO = 3.6. The long waveguide around the sensing system acts to minimize the boundary reflection that can couple back into to the cavity during the resonant recirculation of the light.

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Fig. 3 A general result from FDTD simulation of the pulse propagating in the waveguide and coupling to the microring. (a) E-field of the input pulse in time domain (TD), and (b) normalized intensity in frequency domain (FD). (c) E-field inside the cavity measured at position 2 in TD, and (d) relative intensity in FD showing the resonant modes of the ring without the sensing object. (e) E-field in TD and (f) transmission in FD (at position 3). Central: intensity of the light with frequency f = 192 THz which is closest to a resonant mode of the ring shown in (d).

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Upon inspection of Fig. 2, we see that the effect of scattering due to the sensing object is clearly shown (Fig. 2(b)) as compared to the microring without the object. More importantly, the eigenmodes of the sensing object are clearly shown; the SO eigenmodes also appear in the transmission spectrum and are distinct from those without a sensing object. The eigenmodes are not only easier to detect compared with the shift of resonant modes in the measurement, but furthermore are less influenced by noise and environmental disturbances which are difficult to control or reduce in a micron-scale sensing system.

3. Numerical results

We consider in our simulation a microring with R = 20μm, d_R = d_W = 1μm and index n_R = n_W = 1.46. Figure 3 shows typical results of a single-shot simulation.

The results shown in Fig. 3 represents a generalized example, in which the FDTD simulation yields the input field at point #1, the field inside the cavity at point #2 and the output field at point #3 in the time domain (TD) and frequency domain (FD). Later, in this Section we will show the spectrum for each case under consideration, and discuss in details the results. In the simulation, a femtosecond pulse with a cosine-modulated Gaussian waveform is launched from the left into the waveguide as shown in Figs. 3(a) and 3(b) for the field E₁(t) in the time domain (TD) and normalized intensity I₁(f) = |E₁(f)|² in the frequency domain (FD), respectively. The light field inside the microring (measured at position 2 of the central figure) is shown in Figs. 3(c) and 3(d) for the field E₂(t) in TD and relative intensity I₂(f) = |E₂(f)|²/|E₁(f)|² in FD, respectively. The field measured at position 3 is shown in Figs. 3(e) and 3(f) for field E₃(t) in TD and the transmission T(f) = |E₃(f)|²/|E₁(f)|² in FD, respectively. The central figure is the intensity of light with frequency f = 192 THz, which is close to a resonant mode of the cavity as shown in Fig. 3(d) for the light intensity inside cavity. Here E_j(t) is the electric field in the time domain at point j, and the E_j(f) are the Fourier transform in frequency domain.

Now, let us investigate numerically how the light propagates in the sensing system with a sensing object adjacent to the microring, and in particular determine what kind of information the FDTD simulation can extract from sensing object. We performed FDTD simulations for the system without SO, and with SOs having different sizes and indices as shown in Fig. 4 in which the spectrum of a microring R = 20μm without an SO is compared with ones of the same microring with adjacent SO r = 2, 3 μm (n_SO = 3.6).

 

Fig. 4 Intensity of the light with f = 192 THz in the microring (R = 20, d_R = 1 μm, n = 1.46) with and without a sensing object. (a) without SO, (b) with SO r = 2 μm, n_SO = 3.6, and (c) with SO r = 3 μm, n_SO = 3.6, (d) relative intensity inside cavity, (e) transmission: without SO (red-curve), with SO r = 2 μm (blue), and with SO r = 3 μm (green); n_SO = 3.6. Media 1 (or in large format, Media 2) is a short animation of light propagation in the microring adjacent to SO = 2micron described in Fig. 4(b).

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Let us first discuss some main features of the results shown in Fig. 4. As shown in Fig. 4(a), light with frequency f = 192 THz is strongly resonant with the ring (R = 20μm, n = 1.46). The frequency 192 THz actually is one of the resonant modes of the ring as shown by the red curves in Figs. 4(d) and 4(e) for relative intensity inside the cavity and the transmission, respectively, of the ring without SO. When a sensing object is included, a disk with diameter r and index n_SO, interesting effects occur including the following: (i) scattering by the SO, with scattering intensity increasing with the size of the SO (see, Figs. 4(b), and 4(c)); the scattering decreases intensity at f = 192 THz in the ring (Figs. 4(b) and 4(c)), as is also shown in Fig. 4(d) for the relative intensity inside the ring; (ii) Figs. 4(d) and 4(e) show the shifts of the resonant modes with an SO included, and the shift depends on the size SO (and, of course, the refractive index); and (iii) it is clear that there are new modes that appear in the spectrum of the system with the SO, and again the new modes depend on the physical properties of the SO, such as size and refractive index.

To find out whether there is any relationship between the new modes of the sensing spectra, e.g. Figures 4(d) and 4(e), we simulate the same light pulse coupling directly to the SO and calculate the relative intensity inside the SO to see the eigenmodes of the SO within the bandwidth of the light beam. Figure 5 shows the eigenmodes of several SOs with r = 1 and 2 μm with different indexes.

 

Fig. 5 Eigenmodes of several SOs with different size and indices within the bandwidth of the light pulse that was used to simulate the microring sensing above. (a) r = 1μm, n = 3.6, (b) r = 2μm, n = 3.6, (c) r = 1μm, n = 4, and (d) r = 2μm, n = 2.0. The two SOs in (a) and (b) are used for the simulation of microring sensing shown in Figs. 4(b) and 4(c) above. Media 3 (or in large format, Media 4) is a short animation of light propagation in SO with r=2 m, n=3.6 (e.g., Fig. 5(b)).

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The animations Media 1 and Media 3 below illustrate the time-domain simulation of the sensing process in the entire system and the light in the SO, respectively. In Media 1, the microring sensor was described in Fig. 4(c) and the resulting spectra are in Fig. 4(d) and 4(c). The Media 3 shows the light coupling to the SO with r_SO = 2μm and n_SO = 3.6. Note that, due to limit of space, we show here only short versions of the animations with only 1 or two recirculation.

We have simulated different microring sensing systems with different ring sizes, and sensing objects with different sizes and indices as some further examples shown in Fig. 6 . The results show that FDTD simulation yields very consistent results for the resonant shifts and eigenmodes of the sensing objects. Figure 6(a) shows transmission T of a microring (R = 20 μm) without (red) and with SO r = 2μm (blue). It can be seen in Fig. 6 that there is a small shift of the cavity resonant modes when the cavity is adjacent to an SO. In this case the cavity light can couple to the SO. As a result, the optical pathlength increases and therefore shifts the resonant modes in the cavity. More importantly, there are two new modes in the transmission of the ring with an SO (indicated by two green arrows). Figure 6(b) shows the relative intensity I₂ inside the microring without (red) and with SO (blue), and the eigenmodes of the SO itself (green). Clearly, the two new modes in the transmission of the microring with SO (blue curve in Fig. 6(a)) originated from the two eigenmodes of the SO.

 

Fig. 6 (a): Transmission T of the micro-sensing without (red) and without sensing object (blue). The microring (R = 20 μm, d_R = 1μm, n_R = 1.46); SO r_SO = 2 μm and n_SO = 3.6., and (b): Relative intensity inside microring without (red) and with (blue) sensing object, and inside the sensing object (green) which show the resonant modes in the systems. The SO has r_SO = 2 μm and n_SO = 3.6. The two green arrows indicate new modes that come from SO and can be detected in the transmission.

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Note that, the SO has two eigenmodes as shown in Fig. 6(b) (green curve): the first strong mode is at ~192.8 THz and the second, but relatively weak mode is at ~189.9 THz. However, the two corresponding modes in transmission spectrum (Fig. 6(a)) have quite similar strengths. The main reason for this disproportion is that the second eigenmode at ~189.9 THz has an overlap with a cavity mode at ~190.1 THz as shown in Fig. 6. Although the overlap is weak, it enhances the apparent strength of the eigenmode in transmission as seen in Fig. 6(a).

3. Discussion

As discussed above, the FDTD method has some advantages for simulation of WGM sensing systems. The FDTD simulation can describe many complex processes in the WGM sensing system by using the real shape and eigenmodes of the SO, thereby enabling determination of the radiation losses of the light field in the tapered fiber and in the micro-cavity, as well as the scattering loss in the system. More importantly, FDTD can yield both the shift of the resonant modes and the eigenmodes of the sensing object, which are easier to detect compared with the small shift of the resonant modes. However, there are some disadvantages of the FDTD method for simulation of WGM resonators in general and in particular for the problem of WGM sensing. By far the most difficult challenge is managing the long computing time required for simulation of the light recirculation in WGM resonators. It is widely accepted that errors in FDTD simulations are negligibly small if the grid size of the computing space is on the order of λ/10 to λ/20 [13]. However, this requirement is valid only for simulation of simple systems such as scattering problems in which there is no recirculation of resonant light. In resonant cavities, the resonant frequency light undergoes recirculation multiple times, and small numerical errors are accumulated during the long recirculation time, potentially causing large errors in the final result. We have found that accurate simulations of the microring require a grid size of λ/75 (λ = 1500nm, and grid size = 20nm). A combination of small grid size and recirculation of resonant light in the WGM cavity requires large memory and computing time. Boundary reflections make the situation even worse as longer waveguides and larger computing space are necessary to avoid reflection. A simulation of a typical microring sensing system with 16GB of RAM takes tens of hours.

Even though FDTD can accurately describe light recirculation in WGM microcavities, accurate simulations of high Q-cavities in general and WGM sensing in particular are still very challenging. The main reason for that is the huge number of times that the resonant light recirculates in the WGM cavity. As mentioned in the introduction, an WGM microcavity with Q of 10⁸ has ~100,000 recicrulation cycles. Our simulations of typical microring sensors R = 20μm and about 10 recirculations in the cavity takes ~40 hours using an i7 PC with 16GB RAM. Therefore, it is almost impossible to truly simulate the whole sensing process in a microring sensing system since it would require unrealistic amounts of PC memory and is also time consuming. It is important to stress that those limitations are difficult to overcome for a single workstation. However, with the parallel computing using cluster PCs or workstations, that problem can be significantly improved over what is presented in this paper. Our FDTD simulations also confirm the physics of WGM resonances, i.e. that the higher the number of light recirculation cycles, the narrower the width of the modes (the mode spacing is not changed) and the stronger the resonances. In other words, the higher Q factor is proportional to the number of recirculations or equivalently to the lifetime of the photon in the cavity (or ~energy storage of the cavity). Figure 7 above shows the resonant modes in a microring with R = 20μm and n = 1.46 for two simulations, one with 5 recirculation cycles and the other with 10 recirculation cycles.

 

Fig. 7 Resonant modes of a microring with R = 20 μm and n = 1.46. (a) E field in time domain, 10 recirculations, (b) E field in time domain, 5 recirculations, and (c) relative intensity inside the microring in frequency domain: 5 recirculation (red) and 10 recirculation (blue).

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It is important to note from the results in Fig. 7 that while the width of the modes changes with the number of recirculations in the microring or essentially the computing time, the central frequencies of the resonant modes do not change, and are clearly a fundamental property of the microring. This important property allows us to use the resonant shifts from the FDTD simulation to estimate the change in optical path due to the presence of the sensing object. It is worthwhile to stress here that FDTD simulation can tackle a sensing object with an arbitrary shape, a task much more difficult in other analyses. Furthermore, as presented above, the eigenmodes of the sensing object from FDTD simulation provides additional important information about the sensing object. It is also important to stress here that “numerical” reflection from boundary can cause noisy spectra and even inaccurate results. Therefore, the extended waveguide that traps the light after it traverses the microring is an important feature. Good design can trap the light for a very long time, and at the same time minimize the required computing space. Finally, we would like to stress that the conditions for accurate simulation depend on the specific problems under consideration. For example, if the SO size is smaller, the grid size should also be smaller so that the simulation would be able to describe the eigenmodes of those SOs. Therefore, it is important to establish the relationship between grid spacing and microring parameters, and the relationship between the quality factor and the number of recirculations, a task that is the subject of ongoing work.

Finally, we would like to briefly discuss the results from Ref [4]. in which the authors used the “mode splitting effect” to explain their experiments. In that paper, the new mode in the spectrum of a microtoroid with a nanoparticle has been explained as the splitting of two degenerate modes due to the appearance of the particle, and the scattering due to the nanoparticle causes two standing waves in the cavity thereby lifting the degeneracy. First, it is important to point out that the condition for degenerate modes in the microtoroid only hold for an isolated microtoroid in which the system has perfect 2D symmetry. Under these conditions, the appearance of the particle breaks the symmetry and lifts the degeneracy. In real experiments, the microtoroid couples with a waveguide, the perfect 2D symmetry is broken and therefore there are no longer any degenerate modes. Second, in the experiments, the microtoroid was 5μm wide and the nanoparticle has a size of 30nm. For such a small particle, scattered light escapes in many directions and an extremely small amount of light could be coupled back to the microtoroid, therefore, its effect is very small even for the (perfect 2D symmetry) of an isolated microtoroid. Even though standing waves can form in the microtoroid due to scattering from one nanoparticle, the same is harder to realize if there are many randomly placed particles in the microtoroid as described in [4].

Our FDTD simulation with much larger size SOs (micron-scale) cannot confirm that standing waves occur due to scattering from the SO. It is simple to check in the FDTD simulation whether there is counter-propagating light in the microring. During recirculation in the microring, each time the light beam passes the coupling region (between the ring and the waveguide), there is some light coupled to the waveguide with the rest continuing to circulate as shown in Fig. 8 .

 

Fig. 8 During the recirculation of light in the ring, each time it passes the coupling region, light can couple to and subsequently co-propagates in the waveguide. Left: first time, (central) second time, and (right) third time light passes the coupling region. (See the propagation of light in the system for the whole time in the Media 1).

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Our simulation of the ring with more than 10 cycles did not evidence counter-propagating light forming a standing wave due to scattering from the sensing object. We will further investigate the WGM microring sensing system to see if the “splitting mode” or “eigenmode” are dominant in actual experiments.

In conclusion, we have numerically investigated microring sensing using the FDTD method. We considered the general problem of the FDTD simulation for WGM sensing, in terms of a generalized sensing object. The results are for a single sensing object, but can be extended to the case of many particles with different shapes. Our simulations show that the FDTD method provides additional insight into the operation of WGM based sensors, particularly with respect to the emergence of eigenmodes associated directly with sensing objects, the shift of resonant microring modes, and other important information regarding the eigenmodes of the sensing object.