## Abstract

Optical microcavities are widely used for biological and chemical sensing applications. In these devices, a sensing event is estimated by measuring the shift in the resonant wavelength, or in the quality factor of the microcavity. However, all published works to date only use one of these measures to estimate the sensing event. Here, we show that the estimation accuracy of a sensing event can be improved by employing a combination of both the quality factor and the resonant wavelength measurements in a microcavity sensor. We further demonstrate an experimental application of this model by introducing a refractive index change for a microtoroidal cavity sensor immersed in a liquid. By further using the finite element method simulations in conjunction with the estimator model, we show the existence of three distinct measurement regimes as a function of the quality factor of the microcavity. Finally, the estimator model is extended to develop a sensing metric to compare performance of optical or non-optical sensors.

© 2013 OSA

## 1. Introduction

In the pursuit of improving point-of-care applications researchers have demonstrated various forms of label-free and real-time optical sensors [1]. Among them, optical microcavities present highly sensitive diagnostic platforms for various sensing applications, e.g., detection of DNA, bacteria, virus and binding kinetics [2]. In these applications, it has been shown that a sensing event can be estimated by measuring a change in the resonant wavelength (Δ*λ*) of the micro-cavity. However, the sensing event also induces a change in quality factor (Δ*Q*) of the cavity. In some sensing applications, the quality factor has been measured by applying Lorentzian fitting algorithms to the resonant dip of the microcavity [3–5]. The challenge with such non-linear methods is that they are not suitable for real-time implementation and sensitive sensing applications, primarily because the underlying approach is not only slow but may also require additional signal processing steps. Other approaches for measuring the quality factor in micro-cavities include Cavity Ring Down Spectroscopy (CRDS) [6], and one of its variants, Phase Shift-Cavity Ring Down Spectroscopy (PS-CRDS) [7]. Recently, we have demonstrated a real-time biosensor in which both the resonant wavelength and the quality factor have been tracked as the function of a biosensing event [8]. In this scheme, we have used PS-CRDS to extract the quality factor. This technique minimizes the effect of noise on the measurement as compared to the non-linear fitting approach [8].

The existing work on the estimation of the sensing events is restricted to using the signal-to-noise ratio of only one parameter (either the change in wavelength, Δ*λ*, or the change in the quality factor, Δ*Q* as a function of the sensing event) [2]. In [8], we have shown that although both of these parameters have different signal-to-noise ratios, they carry information about the same sensing event. This observation suggests that the performance of the sensor will be improved by developing a modality in which the sensing event is estimated by utilizing the information from both of these measurements. In this paper, we show a novel concept in which the gap between the statistical estimation approaches and the microcavity sensors is bridged by estimating a sensing event with a combination of the change in the wavelength and the quality factor measurements.

In this paper, we consider *estimation* of the refractive index change. Here, the estimation refers to inferring the value of the refractive index change when the refractive index change is embedded in noisy measurements. In other words, the estimation is a process to extract a value efficiently from a noisy measurement by utilizing the probability distribution of the noise variables. Here, we show that the estimated value (of the refractive index change) is statistically more accurate when the two measurements, the wavelength shift and the quality factor shift, are combined together. Clearly, once the related values are separated from the noise (in a statistical fashion) and combined together to provide an improved estimated value of the refractive index change, the next step is to determine if the estimated refractive index change is large enough to claim that a biosensing event has occurred. This latter step where this comparison is made is referred to as *detection*. Here, we perform the comparison by employing a commercial sensor.

The rest of the paper is organized as follows. In Section 2, we present the estimator model for both uncorrelated and correlated noise in Δ*λ* and Δ*Q*, followed by a brief discussion of the estimator. In Section 3, we present the experimental setup, results, and a brief discussion of the experimental noise. In Section 4, we describe the modeling results in conjunction with the estimator model and show the existence of three measurement regimes for estimation purposes. In Section 5, we discuss the extension of the estimator model to the non-linear sensor response, and further propose a sensing metric to compare the performance of large class of sensors. Finally, Section 6 concludes the paper.

## 2. Estimation model of a sensing event

In this section, we present the mathematical modeling of our sensing system. A conceptual picture of an estimation process is shown in Fig. 1.

Mathematically, we may describe the above system as:

i.e., the change in X (Δ*X*, the sensing event) induces changes in the wavelength (Δ

*λ*) and the quality factor (Δ

*Q*) with corresponding noise,

*N*and

_{λ}*N*, in each of the measurements, respectively. Because the device physics is lumped into functions

_{Q}*f*and

_{λ}*f*, Eqs. (1)–(2) are valid for various experimental applications including bulk refractive index sensing and affinity based molecule sensing. Assuming that the sensor response is linear (valid for small Δ

_{Q}*X*), and ignoring any absorption due to the sensing event, we can simplify Eqs. (1)–(2) as follows: where

*m*and

_{λ}*m*represent the slopes of the corresponding linear response, and are determined from the calibration data. Note here that the restriction of the above setup to a linear model will be relaxed later in Section 5, where we present a non-linear sensor model and adjust the subsequent estimator.

_{Q}The objective is to estimate the sensing event,
$\widehat{\mathrm{\Delta}X}$, from the measured change in wavelength, Δ*λ*, and change in quality factor, Δ*Q*, while taking into account the experimental noise. Therefore, rewriting Eqs. (3)–(4) in the matrix form, we get:

*N*, and

_{λ}*N*, follow the normal (Gaussian) distribution:

_{Q}*σ*and

*μ*represent the standard deviation and mean for the corresponding modality as denoted by the subscript. Since the noise variables (

*N*,

_{λ}*N*) are assumed as Gaussian, the probability distributions of the change in wavelength, Δ

_{Q}*λ*, and the change in quality factor, Δ

*Q*, are also Gaussian, and are given by:

*N*and

_{λ}*N*, can be either un-correlated or correlated. For example, simultaneous measurements of Δ

_{Q}*λ*and Δ

*Q*and/or use of the same detector for each of them will result in correlated noise [8]. However, sequential measurements of Δ

*λ*and Δ

*Q*and with a separate detector for each of them will result in un-correlated noise. In the following, we provide the maximum likelihood estimate of the sensing event, Δ

*X*, for both uncorrelated and correlated noise sources.

#### 2.1. Maximum likelihood estimation: Uncorrelated noise sources

In this section, we assume that the noise variables, *N _{λ}* and

*N*, are uncorrelated. The probability density function of

_{Q}**y**(Eq. (5)), parameterized by the sensing event, Δ

*X*, is a multivariate normal with mean

**m**and covariance matrix,

_{y}*R*:

*X*, (denoted by $\widehat{\mathrm{\Delta}X}$) can be found by It can be verified that the application of Eq. (15) to Eq. (14) results in:

#### 2.2. Maximum likelihood estimation: Correlated noise sources

We now derive the maximum likelihood estimate of the sensing event, Δ*X*, when the noise variables, *N _{λ}* and

*N*, are correlated. To this end, let

_{Q}*σ*be the covariance among the two noise variables then the covariance matrix of

_{λQ}**y**is represented as

*σ*, is zero, i.e., as in the case of uncorrelated noise, Eq. (18) reduces to Eq. (16). In a sensing experiment with correlated noise,

_{λQ}*σ*can easily be evaluated by performing the numerical cross correlation of the measured noise (i.e.,

_{λQ}*N*,

_{λ}*N*).

_{Q}#### 2.3. Discussion of the estimator

The estimator in Eq. (16) and Eq. (18) provides the optimal estimate by considering both the underlying physics (*m _{λ}*,

*m*, Δ

_{Q}*λ*, Δ

*Q*) and the noise variables (

*N*,

_{λ}*N*) in each measurement. According to the estimator, the optimal estimate for the sensing event is a combination of the two measurements as long as noise in either of the measurements is neither zero nor infinite. In other words, the model provides a best estimate for the sensing event irrespective of the detection limit or signal to noise ratio of any measurement. The estimator also suggests the existence of three distinct measurement regimes in a microcavity sensor. In particular, if

_{Q}*σ*>

_{λ}/m_{λ}*σ*(i.e., change in wavelength, Δ

_{Q}/m_{Q}*λ*, is more noisy or less sensitive), then more weight is assigned to the Δ

*Q*measurement for improved estimation, and vice versa. For the case of

*σ*=

_{λ}/m_{λ}*σ*, the estimator assigns equal weights to the two measurements. These distinct regimes are explored in more detail by running finite element simulations for a microcavity in conjunction with the Eq. (16), in Section 4.

_{Q}/m_{Q}## 3. Estimation: Experimental results

In this section, we present the application of the estimator developed in the previous section by providing the experimental results for refractometric sensing. Although experiments can be performed for both uncorrelated (Eq. (16)) and correlated (Eq. (18)) noise sources, we opt for the uncorrelated case for simplicity.

The change in wavelength measurements, Δ*λ*, are conducted by using the conventional method of tracking the minimum of the resonant peak [10]. The change in quality factor measurements, Δ*Q*, are conducted by using PS-CRDS [8]. The experimental setup and the measurement procedure is similar to the one used in [8] with a few modifications: (i) To uncorrelate the noise variables, *N _{λ}*, and

*N*, measurements of Δ

_{Q}*λ*and Δ

*Q*are performed sequentially (within less than 10 seconds) on two separate detectors. This allows us to use Eq.(16); however in the case of simultaneous measurements of Δ

*λ*, and Δ

*Q*and/or a single detector we must use Eq. (18) because the noise will be correlated; (ii) The PS-CRDS measurements have been made by using the phase sensitive detection (PSD) technique. The PSD technique is implemented in lock-in amplifiers and is advantageous in terms of high noise immunity e.g., in the present work, the phase shift of the waveguide signal (w.r.t a reference signal) can be measured down to 10

^{−4}degrees, whereas, in our previous work [8] it was 0.15 degrees.

The experimental setup is shown in Fig. 2. A 1530*nm* tunable laser (Tunics T100R, Yenista Optics) is used to couple light in a microtoroidal cavity [6] via tapered optical fiber. The laser is modulated in wavelength via FG_{1} (HP 8116A function generator, triangular waveform, 7V* _{pp}*, 100Hz for the Δ

*λ*measurements, 100mHz for the Δ

*Q*measurements) and an external port (FSC) of the laser. The laser is amplitude modulated via FG

_{2}(Agilent 33250A function generator, sinusoid waveform,, 200mV

*, 40MHz), and an electro-optic modulator (OC-192, JDSU). The change in wavelength, Δ*

_{pp}*λ*, is recorded by a combination of a photodetector Det

_{1}(PDA10CS, Thorlabs), and an oscilloscope (Agilent DSO3034). The change in quality factor, Δ

*Q*, is recorded by a combination of a photodetector Det

_{2}(HP 11982), and a lock-in amplifier (Zurich Instruments HF2LI).

The microtoroidal cavity (major diameter 140*μm*, minor diameter 6*μm*) [6] is immersed in heavy water (D_{2}O); because D_{2}O has lower optical loss at 1530nm than H_{2}O, the microcavity is able to maintain a high quality factor [11]. Although 1530nm is not a preferred sensing wavelength, however, many of the components described above, such as the EO modulator and the 50/50 coupler, have been optimized for operation in the near-IR and are more commonly available in this wavelength range. As such, for the initial demonstration and verification of the estimator, it is critical to balance instrumentation availability and wavelength selection.

The refractive index change (i.e., the sensing event Δ*X*) is introduced by mixing salt in the heavy water. As a control experiment, the refractive index change is first measured by a commercial Surface Plasmon Resonance (SPR) system (GenOptics, France). The known Δ*X* is then used to measure the response of the cavity in terms of a change in wavelength, Δ*λ*, and the change in quality factor, Δ*Q*. Both results are shown in Fig. 3. These results act as calibration curves for the sensor. From the results, slopes of the curves can be extracted: *m _{λ}* = 53.6

*nm/RIU*and

*m*= 4.34 × 10

_{Q}^{6}

*RIU*

^{−1}. The linearity of the Δ

*λ*measurements for similar range of refractive index change have also been previously reported in [12].

The noise in each measurement also follows the normal distribution as assumed in Section 2. A representative noise distribution for the phase shift measurement in the waveguide signal is shown in Fig. 4. This noise, *N _{Q}*, is measured by moving the tapered fiber away from the cavity and continuously recording the phase shift of the waveguide signal w.r.t a reference signal (

*FG*

_{2}). From the noise data, standard deviation,

*σ*, of the noise in Δ

_{Q}*Q*calculation is 30 for the quality factor of 5 × 10

^{4}[13]. A similar distribution is obtained for the wavelength measurement, which gives standard deviation,

*σ*, of the noise in Δ

_{λ}*λ*measurement of 46

*fm*. The noise,

*N*, is measured with the tapered fiber in contact with the cavity and continuously recording the position of the minimum of the resonant peak. The numerically evaluated correlation coefficient of the two noise variables,

_{λ}*N*and

_{λ}*N*, is 1.163 × 10

_{Q}^{−3}i.e. the two noise sources have negligible correlation.

It should be noted that in the above noise measurements, the noise due to fluctuations in the coupling between the cavity and the tapered waveguide, also known as taper jitter, is avoided. Therefore, ideally the experiments need to be performed in a similar manner. However, if the taper and resonator are in contact, the cavity Q is extremely low (< 10^{3}) in our experiments. Hence, we have performed the measurements by maintaining the gap between the cavity and the tapered fiber. Consequently, the noise due to the taper jitter is present in our measurements and can also be seen in the linear fit to the data in Fig. 3 (*σ _{λ}* = 6.76

*pm*,

*σ*= 1572).

_{Q}The application of the estimator is demonstrated by recording the response (Δ*λ* = 91.91*pm*, Δ*Q* = 1.24 × 10^{4}) of the cavity for another refractive index change, Δ*X*, which is not included in Fig. 3. The results, obtained by using data (*m _{λ}*,

*m*,

_{Q}*σ*,

_{λ}*σ*) of Fig. 3, are shown in the Table 1. The value of Δ

_{Q}*X*is also measured from the commercial SPR system and is found to be 1.95 × 10

^{−3}.

It is clear from the Table 1 that a combination of the two measurements provides a more accurate estimate of a sensing event than using any of the two measurements (Δ*λ*, Δ*Q*) alone. The high error in the quality factor measurements is attributed to the tapered fiber jitter. We anticipate that the error in the two measurements can be reduced significantly, i.e., accuracy of the estimator further improves, by conducting the measurements with the taper in contact with the cavity. In summary, the results indicate a successful proof-of-concept of utilizing the proposed estimator for sensing applications.

## 4. Distinct estimation regimes: Modeling results

In this section, we present the application of the estimator (Eq. (16)) by providing the modeling results for refractometric sensing. In particular, we show the existence of three distinct measurement regimes for the purposes of estimation. Appropriate modifications can be easily incorporated for the correlated noise variables in the context of Eq. (18).

We use a finite element model (FEM) [14] to model the microtoriodal cavities [6] immersed in heavy water at 1530*nm*. The FEM model is used to calculate the change in the wavelength, Δ*λ*, and the change in quality factor, Δ*Q*, as a function of the intrinsic quality factor of the fundamental TM mode of the cavity, in response to the refractive index change Δ*X* (10^{−4} − 10^{−3}) in the outside medium (heavy water). The slopes (*m _{λ}*,

*m*) of the curves are then obtained for each quality factor. In simulations, the quality factor is varied by changing the major diameter of the cavity, holding the minor diameter constant (6

_{Q}*μm*).

The standard deviation, *σ _{λ}* of the wavelength noise,

*N*, is theoretically calculated by running Monte Carlo simulations at each quality factor used in the FEM model [13]. On the other hand, the standard deviation,

_{λ}*σ*, of the quality factor noise,

_{Q}*N*, is theoretically calculated by running simulations based upon the coupled mode theory of a resonator coupled with a waveguide [13]. We use the specifications (e.g., relative intensity of the laser, detector bandwidth and quantum efficiency, number of bits of the digitizer) of the components used in our experimental setup in these simulations. In short, the standard deviation,

_{Q}*σ*, and the slope,

_{λ}*m*, of the wavelength measurements decrease with the increase in quality factor [13], whereas for the quality factor measurements, the standard deviation,

_{λ}*σ*, increases with the increase in the quality factor [13] and the slope,

_{Q}*m*, has a maximum value at a particular quality factor [14].

_{λ}The modeling results are shown in Fig. 5. These results clearly indicate the existence of three distinct regimes, as mentioned in Section 2.3. The modeling results provide insight into the different parameters which can be used to optimize a microcavity measurement system; for example, the balance between the microcavity geometry, its quality factor and the noise (*N _{λ}*,

*N*) in the measurement system.

_{Q}It should be noted that the curves obtained in Fig. 5 are based upon the specifications of the equipment used in our experimental setup. It is also possible that for a different testing set-up, the two curves (*σ _{λ}/m_{λ}*,

*σ*) may never intersect, resulting in a single regime. However, even in this case, the estimator (Eq. 16) will still give more accurate result as compared to using one of the measurements (but the contribution from the more noisy measurement will be lower).

_{Q}/m_{Q}## 5. Discussion

We now provide a few remarks and some discussion on the procedure.

- We use microtoroidal cavities at 1530
*nm*and employ PS-CRDS for change in quality factor, Δ*Q*, measurement. However, the analysis can be generalized to to any wavelength. In fact, use of visible wavelengths will improve the sensitivity of the sensor due to negligible liquid absorption. Moreover, the current analysis is also valid when Full Width Half Maximum (FWHM) measurements are used to calculate the quality factor,*Q*(*Q*=*λ/δλ*, where*δλ*is FWHM of the resonant peak), rather than PS-CRDS. However,*σ*will be higher for FWHM measurements as compared to the PS-CRDS system [8, 13], and the setup will be more challenging for conducting real-time measurements. Furthermore, the present work is also applicable when other peak tracking approaches are employed for wavelength shift measurements such as the interferometric technique presented in [15]._{Q} - The use of calibration curves to determine the slope of the linear response of the change in wavelength (
*m*) and the quality factor (_{λ}*m*) of microtoroidal cavities is not necessary for all cavity geometries. For example, analytical expressions for microsphere cavities have been thoroughly established enabling the calculation of these values [12, 16, 17]._{Q} - The proposed approach assumes linearity of the change in wavelength, Δ
*λ*, and the change in quality factor, Δ*Q*, for small change Δ*X*(i.e., sensing event). This assumption may not be valid for large range of Δ*X*[16]. The analysis presented in Section 2 can be extended to the case of general dependance of the change in wavelength, Δ*λ*and the change in quality factor, Δ*Q*, on the sensing event, Δ*X*. Application of the maximum likelihood principle to the general nonlinear model, Eq. (1) and Eq. (2), results in the following equation (assuming uncorrelated noise variables,*N*,_{λ}*N*, and after setting the natural logarithm to 0):_{Q}$$\frac{\left(\mathrm{\Delta}\lambda -{f}_{\lambda}\left(\mathrm{\Delta}X\right)-{\mu}_{\lambda}\right){f}_{\lambda}^{\prime}\left(\mathrm{\Delta}X\right)}{{\sigma}_{\lambda}^{2}}+\frac{\left(\mathrm{\Delta}Q-{f}_{Q}\left(\mathrm{\Delta}X\right)-{\mu}_{Q}\right){f}_{Q}^{\prime}\left(\mathrm{\Delta}X\right)}{{\sigma}_{Q}^{2}}=0,$$where*f′*(Δ*X*) represents the derivative of*f*w.r.t Δ*X*. Solution of Eq. (19) for Δ*X*gives the optimal estimate, $\widehat{\mathrm{\Delta}X}$, for the sensing event.The simulations confirm that as Δ

*X*(refractive index change in heavy water) increases, quadratic or cubic dependance of the change in wavelength, Δ*λ*, and the change in quality factor, Δ*Q*, on Δ*X*lead to the fitting noise that is less than the experimental noise. For example, assuming a quadratic dependance of Δ*λ*, and Δ*Q*in the following form:$$\mathrm{\Delta}\lambda ={m}_{\lambda N}\mathrm{\Delta}{X}^{2}+{m}_{\lambda L}\mathrm{\Delta}X+{N}_{\lambda}={f}_{\lambda}\left(\mathrm{\Delta}X\right)+{N}_{\lambda},$$$$\mathrm{\Delta}Q={m}_{QN}\mathrm{\Delta}{X}^{2}+{m}_{QL}\mathrm{\Delta}X+{N}_{Q}={f}_{Q}\left(\mathrm{\Delta}X\right)+{N}_{Q},$$where*m*,_{λN}*m*,_{λL}*m*, and_{QN}*m*are constants then solution of Eq. (19) provides the best estimate for the sensing event._{QL} - The development of a common metric to enable the comparison of different types of label-free sensors can involve various elements [18, 19]. The present work is also applicable to compare the performance of not only microcavity sensors but a large class of sensors (either optical or non-optical). It is shown in Appendix A that a sensing metric for comparing any linear sensor may be given by where
*m*is the slope of linear response of the sensor, Δ*A*is the measured quantity (e.g., change in quality factor in microcavity sensor or change in incidence angle/wavelength in surface plasmon sensors [20]), the variables (*μ*,*σ*) are the mean and standard deviation of normal noise distribution. Although Eq. (22) is applicable to the linear sensors, the sensing metric,*SM*, can also be derived for non-linear sensors. The result presented in Eq. (22) assumes that the response of the sensor is the same regardless of the location of a sensing event on the sensor surface. Depending upon the sensing application and the sensor platform, this assumption is not always true. For example, in case of microcavity sensors, detection can only occur when a molecule binds within the evanescent field of the device, which occupies a very small portion of the sensor surface [16]. For the case of smaller (or single) sensing entities, the response of the microcavity sensor can range from a maximum (sensing event happens at the equator for the fundamental mode) value to zero (sensing event occurs where the evanescent field is absent). Therefore, a correction to Eq. (22) is necessary.We define the corrected sensing metric as follows:

where*γ*is a unit less quantity with values between 0 – 1 and depends upon the sensor geometry and underlying physics. For example, in case of microcavity sensors for bulk refractive index change applications,*γ*= 1. For the case of sensing of single or smaller sensing entities,*γ*can be estimated from simulations [14] by taking the ratio of modal overlap with the cavity surface, where the binding event occurs, and total surface area of the cavity. For microspheres,*γ*can be calculated analytically for a binding of a single protein at a random location of the sphere [17]. It should be noted, regardless of the sensing application or the sensor platform, that if microfluidics ensure that the sensing event always occur at the place of maximum sensor response, then*γ*will also be maximum. For example, the liquid core optical ring resonators [21] have high*γ*as compared to the conventional microspheres or microtoriodal cavities, but at the same time they suffer from the low quality factor. In short, for achieving optimal performance of a sensor, we may have to consider multiple factors such as*m*, Δ*A*,*γ*(i.e., the device physics and the geometry), and*μ*,*σ*(i.e., the system noise parameters).

## 6. Conclusions

A microcavity sensor can provide information about a sensing event from two sources, i.e., the resonant wavelength and the quality factor. Instead of discarding one of the information sources, we present a simple, yet statistically optimal, method for combining the two information sources in order to improve the estimation accuracy of the underlying sensing event. The optimal estimate is based on the maximum likelihood principle. We show application of the estimator by conducting proof-of-concept experiments for a refractometric sensor based on the microtoroidal cavity. We further provide insight into the estimation process by running FEM simulations in conjunction with the estimator. Finally, we postulate a sensing metric that is valuable for comparing multiple types of sensors. We believe that the present work has potential for use in a wide-range of applications employing microcavity sensors.

## Appendix A: Sensing metric

Suppose that we have *n* different sensors which produce a change in their output (Δ*A*) in response to the same sensing event (Δ*X*). We estimate Δ*X* by combining the results from all of the sensors while considering the corresponding noise. In this way, the sensor with maximum weight in the estimation process performs better and leads us to the sensing metric. Assuming that the response of each sensor is linear, we can write:

*N*are the experimental noise (assuming normal distribution) measured in the response of a sensor corresponding to the sensing event (Δ

_{n}*X*) and

*m*is the slope of each sensor’s linear response. Application of the maximum likelihood principle to the above system of equations results in the following equation:

_{n}*Z*is multiplied to all of the terms. Therefore, the individual terms on the right hand side of Eq. (26) labeled as Sensor 1, Sensor 2, Sensor 3 dictate which term receives a higher weight, justifying the sensing metric proposed in Eq. (22).

## Acknowledgments

The authors would like to thank Amir Foudeh in Prof. Tabrizian’s lab at McGill University for providing help in conducting SPR measurements. This work is supported by the NSERC-CREATE training program in Integrated Sensor Systems, McGill Institute of Advanced Materials, Montreal, Canada.

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