Allan deviation tells the binding properties in single-molecule sensing with whispering-gallery-mode optical microcavities

We propose a scheme of exploring the kinetic properties of the receptor-ligand binding process by means of the frequency stability measurement of a whispering-gallery-mode optical microcavity. In this scheme, the probe beam is directly locked to the microcavity and the time resolution of the sensing detection is determined by the feedback loop response time 10 μs, which is much shorter than the current limit of about 20 ms. The receptor-ligand interactions are entirely mapped onto the probe light frequency and can be recovered through the analysis of its Allan deviation. As a result, one can investigate a chemical and biological reaction at the single-molecule level without the need for resolving the occurrence and duration of individual binding events. We numerically study this sensing scheme based on a practical optoplasmonic platform, where several gold nanorods are adsorbed onto the surface of a spherical microcavity that is located in the aqueous environment. The microsphere has a typical radius of 40 μm and quality factor of 106 (finesse of 2.6 × 103) at the wavelength of 785 nm. This powerful scheme may extend single-molecule sensing to the kinetic regime of the biochemical reactions.


I. INTRODUCTION
Owing to the properties of tight light confinement and high Q factor, whispering-gallery-mode (WGM) microcavities have been extensively used in chemical and biological sensing [1]. The fundamental sensing mechanism lies in the spectral shift δλ and broadening of a WGM induced by the variation in the refractive index of the surrounding environment of a passive/active microcavity [2,3]. For a WGM at the wavelength λ cav , a dielectric nanoparticle with an excess polarizability α ex and a volume V p leads to where ε 0 is the vacuum permittivity, (r) denotes the spatial distribution of the relative permittivity, and E(r) represents the WGM field. The mode volume V cav of the WGM is evaluated based on the field intensity at the nanoparticle's position r 0 . Measuring δλ allows one to estimate α ex or the nanoparticle's size. For example, a silica (refractive index of 1.45) microsphere has a typical radius R cav = 40 μm and is situated in the air. The mode volume of the fundamental TE-polarized WGM at λ cav = 785 nm is computed as V cav = 2 × 10 3 μm 3 . A spherical polystyrene nanoparticle with a radius of 50 nm and α ex /ε 0 = 1. i.e., |δλ/λ cav | ∼ 10 −7 . Besides dielectric nanoparticles, this mode-shift sensing mechanism is also applied to detect large single bioparticles, such as influenza viruses with |δλ/λ cav | ∼ 10 −8 [4]. However, detecting small single biomolecules (e.g., proteins and short DNA) demands a further suppression of V cav . Reducing R cav can strengthen the evanescent field E(r 0 ), thereby suppressing V cav and enhancing δλ [3,4]. However, the microcavity's Q factor also suffers from a large degradation due to increased radiation and surface scattering losses. An efficient measurement on δλ requires the Q factor (linewidth κ 0 = ω cav /Q with ω cav = 2π c/λ cav ) of a microcavity to be sufficiently high (narrow). Generally, the Q factor of a WGM microcavity in the air reaches over 10 8 [5][6][7], corresponding to a finesse F = 2.6 × 10 5 and linewidth κ 0 = 2π × 4 MHz of a WGM at λ cav = 785 nm with R cav = 40 μm. However, detecting single biomolecules usually requires the microcavity to be immersed in an aqueous solution. This results in a serious degradation of the microcavity's Q factor, for instance, from 10 8 in the air to 10 6 in the water [8], because of the huge absorption losses in the aqueous environment. Reducing R cav may further degrade Q, thereby deteriorating the sensor's performance.
The mode volume V cav can, alternatively, be suppressed by hybridizing the localized surface plasmon resonance (LSPR), i.e., the collective oscillation of free electrons in nanostructured metals, with WGM microcavities [ Fig. 1(a)]. The LSPR allows the subwavelength light confinement beyond the diffraction limit, boosting the local evanescent field E(r 0 ). For example, a gold nanorod, whose circular cross-section diameter and length are 10 nm and 35 nm, respectively, gives a near-field intensity enhancement factor ∼ 10 3 around two hotspots [9]. Here, the plasmonic hotspots denote the regions A silica microsphere with a radius of R cav = 40 μm is situated in the aqueous environment. Several gold nanorods are adsorbed onto the microsphere's surface. The TE-polarized cw and ccw WGMs at λ cav = 785 nm are excited by a probe laser through an optical prism. The transmitted light is fed back into the probe laser, locking its frequency to the microcavity. The Allan deviation of the probe laser is measured by the beat note with a reference laser. (b) Longitudinal LSPR spectrum of a gold nanorod and the electric-field intensity distribution of LSPR at λ cav . The circular cross section of the nanorod has a diameter of 10 nm and the nanorod's length is 35 nm. The regions around two ends of the nanorod are termed the hotspots, where the near-field intensity is strongly enhanced by a factor = 800. (c) Evanescent interaction between the microsphere and a gold nanorod. The receptor molecules, which interact with the ligand molecules, are permanently attached within the nanorod's hotspots. (d) Schematic diagrams of the time traces of the wavelength shift of a WGM. The spike events correspond to the transient receptor-ligand interactions while the step events denote the permanent binding between single receptor and ligand molecules. near sharp corners and tips of metal nanoparticles [ Fig. 1(b)]. In addition, Ohmic losses of metal nanoparticles introduce an extra decay channel of intracavity photons, degrading the Q factor of microcavities. For instance, binding five gold nanorods, which are all aligned perpendicularly to the equatorial WGM, onto a microsphere immersed in the water may reduce Q from 1 × 10 6 to 5 × 10 5 . Nevertheless, the large intensity enhancement factor still strongly enhances the sensor's sensitivity.
Moreover, modifying the surface of metal nanoparticles with receptor (R) molecules enables a label-free detection of specific ligand (L) molecules [ Fig. 1(c)] through the relevant binding reaction, much simplifying the sensing procedures. Here, k on (in units of M −1 s −1 , where M denotes molar) and k off (in units of s −1 ) are the on-rate and off-rate constants, respectively. The singlemolecule detection based on the optoplasmonic hybridization has been demonstrated for short DNA oligonucleotides [9], bovine serum albumin protein molecules [10], and even atomic ions, such as Zn 2+ and Hg 2+ [11], whose ionic radii are 0.74 and 1.02 Å, respectively. This is the highest sensitivity achieved by WGM microcavities so far. Recently, these optoplasmonic devices have been widely applied to monitor the reaction kinetics between receptor and ligand molecules in a label-free manner [12][13][14].
For conventional optoplasmonic sensors, the transmission spectrum of a WGM microcavity is measured by sweeping the wavelength of a probe laser. The single-molecule interactions are identified by the spike and step events [ Fig. 1(d)] extracted, i.e., through fitting the spectrum to a Lorentzian profile, from a time trace of the WGM wavelength λ cav . When a ligand molecule binds to a receptor molecule, λ cav experiences a shift δλ that is determined by Eq. (1). For a transient receptor-ligand interaction, δλ vanishes once the ligand molecule is removed from the receptor molecule, leaving a spike in the time trace of λ cav . By contrast, a step event occurs for a permanent receptor-ligand binding. The typical time resolution of the sensing detection is about tens of ms (e.g., 20 ms in [11][12][13][14]), which is limited by the scanning rate of the probe laser. To accurately capture the transmission spectrum of the microcavity, the probe laser frequency goes over the central frequency of the WGM with a width larger than κ 0 . Also, the laser scanning should be slow enough that the intracavity field continuously maintains a steady state, i.e., adiabatic limit.
The adjacent single-molecule events with an interval shorter than the time resolution cannot be distinguished. Thus, the sensor's applications are restricted to an aqueous solution of ligand molecules with an appropriate off-rate constant k off (e.g., k off = 47 and 23 s −1 in [13]) and also at a suitably low concentration (fM ∼ nM in [11][12][13][14]) with an appropriate onrate constant k on . This weakness may be overcome by directly locking the probe beam to the microcavity. The frequency locking time can be as short as 10 μs [15], leading to a threeorders-of-magnitude improvement in the time resolution.
Moreover, extracting the spike and step events from a time trace of the WGM wavelength is usually implemented in a manual manner [11][12][13][14]. An empirical algorithm is applied to remove the background fluctuations and drift of the probe laser [11]. However, it inevitably omits the weak spike/step events that are buried in the background noise and is thereby unable to reveal the entire picture of the chemical and biological reaction. To address this issue, the spike and step events may be treated as the extra noise to the microcavity and one performs the stability analysis on the probe beam whose frequency is locked to the microcavity.
In this paper, we numerically study a fast and sensitive single-molecule detection scheme. The probe beam is directly locked to the microcavity and the kinetic properties of the receptor-ligand binding interactions are explored by measuring the frequency stability, i.e., Allan deviation, of the probe light. This scheme is capable of capturing those single-molecule events with a duration of 10 μs, limited by the integration time of the feedback loop. The sensing method is applicable to the previously inaccessible kinetic regime with large on-rate k on (at a high ligand concentration) and off-rate k off constants.

II. PHYSICAL MODEL
The single-molecule sensing scheme is illustrated in Fig. 1(a). The light from a probe laser (amplitude A p and frequency ω p ) is evanescently coupled into a silica microsphere via the total internal reflection at the surface of an optical prism. The microsphere-prism coupling rate is κ in . The microsphere with a typical radius R cav = 40 μm is immersed in the water (refractive index of n = 1.33). The probe beam excites two degenerate fundamental clockwise (cw) and counterclockwise (ccw) WGMs (mode index l = 453) at λ cav = 785 nm. Both WGMs are TE-polarized and their quality factor in the aqueous environment approximates Q = 10 6 . The loss rate of two WGMs is κ 0 = ω cav /Q with ω cav = 2π c/λ cav . We assume a weak microsphere-prism coupling, i.e., κ in κ 0 . The mode volume of the microcavity is computed as V cav = 7.2 × 10 3 μm 3 .
The light transmitted from the microsphere is measured by a photodetector. Unlike conventional optoplasmonic sensors [11][12][13][14], the output from the photodetector is further fed back into the probe laser system through a servo, locking the probe light frequency ω p to the central frequency ω cav of the ccw WGM. The characteristic time of the feedback loop can be as short as 10 μs [15]. The frequency stability (i.e., Allan deviation) of the probe laser is assessed by the beat note between the locked probe laser and a high-Q reference laser whose stability much exceeds both probe laser and microsphere [ Fig. 1(a)]. Thus, the stability of the microcavity is indirectly measured.
The LSPR of gold nanorods (circular cross-section diameter of 10 nm and length of 35 nm) is utilized to enhance the ligand molecule-microcavity interaction (i.e., optoplasmonic sensing). These metal nanoparticles are permanently adsorbed to the microsphere's surface [9] and coupled to the evanescent field of the microcavity, thereby boosting the near-field intensity within the hotspot regions [ Fig. 1(b)]. Each nanorod has two hotspots. The surface of each nanorod is modified with many receptor molecules, enabling a label-free detection of specific ligand molecules [ Fig. 1(c)]. In the following, we assume that the receptor-ligand pair consists of partially mismatched single DNA strands [9]. Nevertheless, the sensing scheme is applicable to other receptor-ligand pairs. We compute the longitudinal LSPR spectrum of the nanorod by using the finite-difference time-domain method. Figure 1(b) illustrates that the LSPR spectrum is peaked at 780 nm. The nanorod's electric polarizability at λ cav is evaluated as α NR = 0.96 × 10 −22 + i1.73 × 10 −22 m 3 . The enhancement factor of the electric-field intensity around the nanorod's hotspots reaches = 800 [9].
The typical number of the adsorbed nanorods in experiment is N = 5. In the ideal situation, the longitudinal direction of all nanorods is perpendicular to the equatorial plane of the microsphere. The complementary single DNA strands, i.e., ligands, are injected into the solution. Since the receptor and ligand strands are partially mismatched, the transient hybridization events [i.e., the spike events in the time trace of the WGM frequency as shown in Fig. 1(c)] occur primarily. We assume the sample solution is in the equilibrium state and neglect the permanent binding events [i.e., the step events as shown in Fig. 1(c)] in the following discussion. One ligand binding to one receptor located at a nanorod's hotspot gives rise to a polarizability increment, for instance, α DNA ≈ 2.5 × 10 −25 m 3 for the short oligonucleotides [9], to the corresponding nanorod. Normally, such a small polarizability increment cannot be detected by the WGM microcavity in the absence of the LSPR.
The experimental measurement has demonstrated that two degenerate cw and ccw WGMs are indirectly coupled via the backscattering and Rayleigh scattering of the nanoparticles [16][17][18], leading to the mode shift and broadening and the degeneracy lifting. This can be described by the following equation of motion for the amplitudes A cw,ccw of two WGMs (Appendix) with κ = κ 0 + κ in . Here, φ j is the phase difference between cw and ccw optical waves at the position of the jth nanorod.
In the numerical simulation, φ j is produced according to the uniform distribution of the gold nanorods in the equatorial circumference of the microsphere. The probe beam A p is only coupled with the ccw WGM. The Rayleigh scattering rate and mode frequency shift caused by the jth nanorod are expressed as respectively. The extra time-dependent polarizability of the jth nanorod is given by where (t ) is the unit step function and t on j,k (t off j,k ) denotes the initial (end) time of the binding interaction between the kth ligand and a receptor on the jth nanorod. The coupling between cw and ccw WGMs lifts their degeneracy. We focus on the ccw WGM since the probe laser is locked to it. The transmission of the ccw wave as a function of the probe light frequency ω p can be derived from Eq. (3). The spectrum S Tran (ω p ) is single-peaked at (ω cav + δ s ) with a linewidth (κ + δ b ). That is to say, the mode splitting is unresolvable due to the relatively large cavity loss rate κ [19]. The extra spectral shift δ s (t ) and broadening δ b (t ) are caused by the nanorods and the receptor-ligand binding interactions. In what follows, we may set ω p = ω ccw with the central frequency of the ccw WGM for the reasons: (1) The frequency locking time of the feedback loop is much shorter than the time scale of interest; and (2) The extra servo and photon shot noises are sufficiently low. Figure 2(a) summaries the relations among ω cav , ω ccw , and δ s . We use δ s,0 to refer to the mode-shift component caused solely by the gold nanorods in the absence of the receptor-ligand interactions. Next, we consider the numerical simulation of the receptorligand binding events. The experimental observations [11][12][13][14] have manifested: (1) The statistics of the transient singlemolecule events follow the Poisson distribution, that is, the probability of K ∈ Z spike events occurring within a fixed period T is given by The total event rate is equal to R = N j=1 r j (in units of s −1 ) with the event rate r j of the jth nanorod. We neglect the difference between gold nanorods and have R = Nr with r = r j . The typical R in the conventional optoplasmonic sensing experiments is less than 1 s −1 for various ligand molecules [11,12]. Raising the ligand concentration enhances R. However, the higher R regime has thus far been difficult to analyze for the reason of the limited time resolution. Distinguishing the single-molecule events requires the time resolution to be much shorter than R −1 . The time resolution of the conventional optoplasmonic sensors is limited by ∼20 ms because of the slow scanning rate of the probe beam [11][12][13][14]; and (2) The duration (t off j,k − t on j,k ) of the single-molecule binding interaction follows the exponential distribution with an average duration τ ave . When τ ave < R −1 , different single-molecule events are distinguishable. By contrast, different single-molecule events are overlapped when τ ave > R −1 . Again, the current experimental detection is limited by τ ave ∼ 20 ms. As we will see below, our sensing scheme may overcome this restriction.
Indeed, R and τ ave are directly linked to the kinetic on-rate k on = R/c [11] and off-rate k off = 1/τ ave [13] constants. Here, c denotes the ligand concentration. In some literature, both k on and k off are normalized per nanorod. Measuring R and τ ave allows one to derive the kinetic properties of the receptor-ligand binding interactions. For examples, conventional optoplasmonic sensing experiments have measured k on = 0.53 × 10 3 and 3.8 × 10 3 M −1 s −1 for the gold nanorods interacting with Hg 2+ and Zn 2+ ions [11], respectively, and k off = 47 and 23 s −1 for the primer/template DNA interacting with Thermus aquaticus and Klenow fragment DNA polymerases [13], respectively.
The above two experimental manifestations allow one to numerically predict t on j,k and t off j,k in a time trace (length of T ) of ω ccw (t ). The specific numerical simulation processes are summarized as follows: (1)

III. RESULTS AND ANALYSIS
Let us focus on the stability of the frequency fluctuation signal whose power spectral density is given by Here, ω 0 = 2π × 382 THz is the constant central frequency of the ccw WGM. We first consider y(t ) in the absence of δ s (t ) and the reduced signal y(t ) = ω cav (t )/ω 0 − 1 with δ s = 0 represents a bare microsphere in the aqueous environment. Figure 2(b) shows a typical S y ( f ) of the bare microsphere measured in experiment. It is seen that y(t ) is mainly composed of the white frequency (∝ f 0 ) and random walk (∝ f −2 ) noises. The stability of y(t ) is evaluated by the Allan deviation σ y (τ ) with a sampling time τ [20,21]. The white frequency noise originates from the microsphere's thermal Brownian motion [22,23] that leads to a 1 √ τ -dependence of the short-term (τ < 0.1 s) stability [24], By contrast, the random walk noise arises from the linear frequency drift and raises σ y (τ > 0.1 s) with a scaling factor √ τ [24], Additionally, σ y (τ ) displayed in Fig. 2(c) manifests the existence of the flicker walk (∝ f −3 ) component, which gives σ y (τ ) ∝ τ for a large τ , in y(t ). As a result, S y ( f ) with δ s (t ) = 0 (i.e., the bare microsphere in the aqueous environment) takes the form with the typical values h 0 = 0.98 × 10 −19 Hz −1 , h −2 = 1.67 × 10 −19 Hz, and h −3 = 2.22 × 10 −19 Hz 2 derived from the curve fitting. We have also estimated the fundamental thermorefractive-noise-limited stability [25,26], which is much lower than √ h 0 /2τ . According to S y ( f ) with δ s (t ) = 0, one may numerically generate the noise-perturbed ω cav (t ) via the method of digitally filtering the white noise [27]. As shown in Fig. 2(c), the Allan deviation of the numerically generated ω cav (t ) well matches the experimental measurement.
After obtaining t on j,k , t off j,k , and ω cav (t ), we numerically simulate the time trace of the mode frequency ω ccw (t ) by means of calculating the transmission spectrum S Tran (ω p = ω ccw ) from Eqs. (3)-(6). In the presence of the transient spike events [i.e., δ s (t ) = 0], the Allan deviation of y(t ) exhibits a distinct behavior compared to that of δ s (t ) = 0. As shown in Fig. 2(c), σ y (τ ) for ω ccw (t ) with δ s (t ) = 0 goes up firstly at the sampling time τ = τ c1 and then degrades at τ = τ c2 . Understanding the underlying mechanism relies on extracting the spike-event noise δ s (t ) from ω ccw (t ). This is feasible in the numerical simulation but challenging in experiment.
As depicted in Fig. 3(a), multiple spikes are presented in the time trace of δ s (t ), where we have removed the constant bias δ s,0 caused by the gold nanorods. The Fourier analysis illustrates that δ s (t ) predominantly consists of the white frequency h 0 f 0 and random walk h −2 f −2 noises [ Fig. 3(b)], The cross between h 0 f 0 and h −2 f −2 occurs at The random walk component strongly raises the Allan deviation of δ s (t )/ω 0 , i.e., following the 2π 2 h −2 τ /3 line, until hitting the h 0 /2τ line that is determined by the white frequency noise [ Fig. 2(c)]. This coincides with the corresponding behavior of the Allan deviation σ y (τ ) of ω ccw (t ) with δ s (t ) = 0. Using Eq. (3), one predicts that a ligand interacting with a receptor gives rise to a 10-fm increment of δ s . That is, the single-molecule-induced mode shift is 10 fm. The spike heights in Fig. 3(a) are closely related to the single-molecule event rate R and the average event duration τ ave . The reciprocal of R measures the average interval between the initial times of two adjacent receptor-ligand binding events. When R −1 exceeds the average duration τ ave , a ligand may start to interact with a receptor before the end of another receptor-ligand binding event. This results in an overlap of different binding events, i.e., multimolecular events, and a spike higher than 10 fm. Figure 3(c) displays the statistical distribution of the spike heights extracted from Fig. 3(a). It is seen that when Rτ ave > 1 the distribution is maximized at a spike height of about Rτ ave × 10 fm. By contrast, the distribution is maximized at 10 fm for Rτ ave < 1, indicating that the single-molecule events take the predominant role. Figure 3(d) illustrates the dependence of the singlemolecule-event probability on the average duration τ ave . A short τ ave highlights the probability of the single-molecule events. Shortening τ ave relies on weakening the receptorligand interaction, for instance, increasing the number of mismatched bases between receptor and ligand single DNA strands [9]. However, a short τ ave also demands a fast feedback loop. The frequency locking time of 10 μs is feasible in experiment [15]. Thus, τ ave can be, in principle, as short as 10 μs, which is over 10 3 times smaller than the current resolution time limit ∼20 ms of the conventional optoplasmonic sensors [11][12][13][14].
We use ave to denote the average height of different spikes in a time trace of δ s (t ). It is impractical to directly derive the analytical expression of ave . Here, we introduce the empirical equation where the curve fitting gives m = 2. The validity of this expression is confirmed by Fig. 4(a). In the limit of Rτ ave 1, Eq. (18) is reduced to ave ≈ 10 fm, i.e., the single-molecule spike events predominate. By contrast, Rτ ave 1 leads to ave ≈ (Rτ ave ) × 10 fm, i.e., (Rτ ave ) individual receptor-ligand binding events overlap each other. Figures 4(b) and 4(c) plot the dependence of the white frequency parameter h 0 on τ ave and R, respectively, which is derived from the numerical simulation. It is seen that log 10 h 0 goes up almost linearly as log 10 τ ave and log 10 R are increased. We suggest the following empirical formula with the intensity of the spike-induced fluctuations and the fluctuation time scale where the curve fitting gives m = 2/3. By contrast, varying τ ave hardly changes the random walk parameter log 10 h −2 while log 10 h −2 vs log 10 R exhibits a linear behavior. One may write the following empirical formula where I 0 = (10 fm/ω 0 ) 2 and the curve fitting gives A ≈ 10.
We have also evaluated the cross times τ c1 and τ c2 from the numerical simulation [ Fig. 4(b)]. It is found that τ c1 rarely depends on τ ave , corresponding to the behavior of h −2 . By contrast, τ c1 scales as h 0 since τ c1 is determined by the cross between h 0 /2τ and 2π 2 h −2 τ /3.
Finally, we come to the question of how to derive the single-molecule event rate R and the average duration τ ave from the stability measurement of the mode frequency ω ccw (t ). In experiment, it is challenging to extract the mode shift δ s (t ) from the time trace of ω ccw (t ). Nevertheless, the values of h 0 , h 0 , τ c1 and τ c2 can be obtained directly from the Allan deviation of ω ccw (t ) in the absence and presence of the receptor-ligand binding interactions. The random walk noise in δ s (t ) is then evaluated by h −2 = h 0 f 2 c2 , where the cross frequency f c2 is given by Eq. (17). Using the empirical formulas (18)- (22), one obtains R and τ ave and further derive the kinetic rate constants k on and k off . It should be noted that the intrinsic fluctuations in ω cav (t ) need to be suppressed sufficiently so that h 0 and h −2 are lower than h 0 and h −2 , respectively. Otherwise, the spike events are deeply submerged in the background noise. This requires a high Q factor of the microsphere in the aqueous environment and a large polarizability of the ligand molecules.

IV. DISCUSSION
In summary, we have numerically studied a frequencylocking-based single-molecule sensing scheme. The kinetic properties of a specific chemical/biological reaction may be assessed by simply measuring the Allan deviation of the frequency-locked probe beam. This approach is potentially applicable to an aqueous solution with a high ligand concentration and rapid receptor-ligand interactions. The current microcavity technology is ready to implement this sensing scheme.
Conventional optoplasmonic sensing is mainly based on the resonance frequency shift that is derived from the transmission spectrum of the WGM microcavity. To capture the complete WGM spectrum, the probe laser is slowly scanned over a wide spectral range around the WGM resonance frequency, resulting in a typical time resolution of ∼20 ms. Such a relatively long time resolution of the detection restricts the relevant sensing applications. For instance, the conventional optoplasmonic sensors cannot respond to the rapid binding interactions between partially mismatched single DNA strands [9]. Also, the ligand concentration in an aqueous solution should be appropriately low (e.g., fM ∼ nM in [11][12][13][14]) so that the sensors can distinguish different single-molecule events. In comparison, the time resolution of the frequencylocking-based sensing scheme studied in this work is mainly determined by the feedback loop. The frequency locking time can be as short as 10 μs [15], potentially enabling the sensors to monitor the rapid hybridization reactions of single stranded DNA molecules and identifying the single-molecule events with a high ligand concentration.
Besides the resonance-frequency-shift-based sensing mechanism, single molecules may also be detected through the mode splitting that results from the scatterer-induced coupling between cw and ccw WGMs [17]. The modesplitting mechanism enables the self-referenced detection, where the beat note between two split modes is measured, leading to a superior noise suppression [28]. Equation (3) is also applicable to study the mode-splitting mechanism. However, observing the mode splitting requires a high enough Q (usually, Q > 10 8 ) factor of the microsphere, which is challenging in the aqueous environment [5]. In addition, the polarizability α of the ligand molecules should be large enough to give rise to a strong indirect coupling between cw and ccw WGMs. This is quantified by the condition α > V cav /Q [8]. For our specific sensor, the mode splitting resolvability demands α > 7.2 × 10 −21 m 3 . Consequently, the mode-splitting mechanism significantly restricts its applications for detecting small-polarizability single molecules.
In our sensing scheme, the detection time is shortened by locking the probe beam to the microcavity, instead of directly monitoring the transmission spectrum of the microcavity as in conventional sensing schemes. Further shortening the detection time requires a faster capture of the complete WGM spectrum. This may be implemented by measuring the ring-up signal from the WGM microcavity that is driven by a sequence of far-detuned light pulses [29]. Cavity ring-up spectroscopy allows one to acquire single-molecule-induced effects at nanosecond timescales. Combining the Allan deviation analysis with the cavity ring-up technique potentially paves the path for the study of ultrafast biochemical reactions.