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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 19 — Sep. 23, 2013
  • pp: 22817–22828
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Towards more accurate microcavity sensors: maximum likelihood estimation applied to a combination of quality factor and wavelength shifts

M. Imran Cheema, Usman A. Khan, Andrea M. Armani, and Andrew G. Kirk  »View Author Affiliations


Optics Express, Vol. 21, Issue 19, pp. 22817-22828 (2013)
http://dx.doi.org/10.1364/OE.21.022817


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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

1. H. K. Hunt and A. M. Armani, “Label-free biological and chemical sensors,” Nanoscale 2, 1544–1559 (2010). [CrossRef] [PubMed]

]. Among them, optical microcavities present highly sensitive diagnostic platforms for various sensing applications, e.g., detection of DNA, bacteria, virus and binding kinetics [2

2. F. Vollmer and L. Yang, “Label-free detection with high-Q microcavities: a review of biosensing mechanisms for integrated devices,” Nanophotonics 267–291 (2012).

]. In these applications, it has been shown that a sensing event can be estimated by measuring a change in the resonant wavelength (Δλ) of the micro-cavity. However, the sensing event also induces a change in quality factor (ΔQ) 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

3. L. Maleki and V. S. Ilchenko, “Techniques and devices for sensing a sample by using a whispering gallery mode resonator,” US Patent 6490039 (2002).

5

5. A. M. Armani and K. J. Vahala, “Heavy water detection using ultra-high-Q microcavities,” Opt. Lett. 31, 1896–1898 (2006). [CrossRef] [PubMed]

]. 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

6. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(2003). [CrossRef] [PubMed]

], and one of its variants, Phase Shift-Cavity Ring Down Spectroscopy (PS-CRDS) [7

7. J. Barnes, B. Carver, J. M. Fraser, G. Gagliardi, H. P. Loock, Z. Tian, M. W. B. Wilson, S. Yam, and O. Yastrubshak, “Loss determination in microsphere resonators by phase-shift cavity ring-down measurements,” Opt. Express 16, 13158–13167 (2008). [CrossRef] [PubMed]

]. 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

8. M. I. Cheema, S. Mehrabani, A. A. Hayat, Y.-A. Peter, A. M. Armani, and A. G. Kirk, “Simultaneous measurement of quality factor and wavelength shift by phase shift microcavity ring down spectroscopy,” Opt. Express 20, 9090–9098 (2012). [CrossRef] [PubMed]

]. 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

8. M. I. Cheema, S. Mehrabani, A. A. Hayat, Y.-A. Peter, A. M. Armani, and A. G. Kirk, “Simultaneous measurement of quality factor and wavelength shift by phase shift microcavity ring down spectroscopy,” Opt. Express 20, 9090–9098 (2012). [CrossRef] [PubMed]

].

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, Δλ, or the change in the quality factor, ΔQ as a function of the sensing event) [2

2. F. Vollmer and L. Yang, “Label-free detection with high-Q microcavities: a review of biosensing mechanisms for integrated devices,” Nanophotonics 267–291 (2012).

]. In [8

8. M. I. Cheema, S. Mehrabani, A. A. Hayat, Y.-A. Peter, A. M. Armani, and A. G. Kirk, “Simultaneous measurement of quality factor and wavelength shift by phase shift microcavity ring down spectroscopy,” Opt. Express 20, 9090–9098 (2012). [CrossRef] [PubMed]

], 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.

The rest of the paper is organized as follows. In Section 2, we present the estimator model for both uncorrelated and correlated noise in Δλ and ΔQ, 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.

Fig. 1 The change in wavelength (Δλ) and quality factor (ΔQ) is induced by a sensing event. Each measurement has an experimental noise, i.e., Nλ, and NQ. The estimator estimates ΔX from the outputs (measurements) of the system.

Mathematically, we may describe the above system as:
Δλ=fλ(ΔX)+Nλ,
(1)
ΔQ=fQ(ΔX)+NQ,
(2)
i.e., the change in X (ΔX, the sensing event) induces changes in the wavelength (Δλ) and the quality factor (ΔQ) with corresponding noise, Nλ and NQ, in each of the measurements, respectively. Because the device physics is lumped into functions fλ and fQ, 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 ΔX), and ignoring any absorption due to the sensing event, we can simplify Eqs. (1)(2) as follows:
Δλ=mλΔX+Nλ,
(3)
ΔQ=mQΔX+NQ,
(4)
where mλ and mQ 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.

The objective is to estimate the sensing event, ΔX^, from the measured change in wavelength, Δλ, and change in quality factor, ΔQ, while taking into account the experimental noise. Therefore, rewriting Eqs. (3)(4) in the matrix form, we get:
y=ΔXm+N,
(5)
where
y=[ΔλΔQ],m=[mλmQ],N=[NλNQ].
(6)
We assume that the noise variables, Nλ, and NQ, follow the normal (Gaussian) distribution:
Nλ~𝒩[μλ,σλ2]=1σλ2πexp(12(Δλμλσλ)2),
(7)
NQ~𝒩[μQ,σQ2]=1σQ2πexp(12(ΔQμQσQ)2),
(8)
where σ and μ represent the standard deviation and mean for the corresponding modality as denoted by the subscript. Since the noise variables (Nλ, NQ) are assumed as Gaussian, the probability distributions of the change in wavelength, Δλ, and the change in quality factor, ΔQ, are also Gaussian, and are given by:
P(Δλ)=𝒩[ΔXmλ+μλ,σλ]2,
(9)
P(ΔQ)=𝒩[ΔXmQ+μQ,σQ2].
(10)
Depending upon the experimental setup, the two noise variables, Nλ and NQ, can be either un-correlated or correlated. For example, simultaneous measurements of Δλ and ΔQ and/or use of the same detector for each of them will result in correlated noise [8

8. M. I. Cheema, S. Mehrabani, A. A. Hayat, Y.-A. Peter, A. M. Armani, and A. G. Kirk, “Simultaneous measurement of quality factor and wavelength shift by phase shift microcavity ring down spectroscopy,” Opt. Express 20, 9090–9098 (2012). [CrossRef] [PubMed]

]. However, sequential measurements of Δλ and ΔQ 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, ΔX, for both uncorrelated and correlated noise sources.

2.1. Maximum likelihood estimation: Uncorrelated noise sources

In this section, we assume that the noise variables, Nλ and NQ, are uncorrelated. The probability density function of y (Eq. (5)), parameterized by the sensing event, ΔX, is a multivariate normal with mean my and covariance matrix, R:
fΔX(y)=12πdet(R)12exp(12(ymy)R1(ymy)),
(11)
where det represents determinant of a matrix,
my=[ΔXmλ+μλΔXmQ+μQ],R=[σλ200σQ2].
(12)
It is straightforward to show that Eq. (11) further reduces to
fΔX(y)=𝒩[ΔXmλ+μλ,σλ2]𝒩[ΔXmQ+μQ,σQ2],
(13)
i.e., the product of the two individual densities when the noises variables are uncorrelated. In order to derive the maximum likelihood estimate, we take the natural logarithm of both sides of Eq. (13),
lnfΔX(y)=ln1σλσQ(2π)(ΔλΔXmλμλ)22σλ2(ΔQΔXmQμQ)22σQ2.
(14)
According to the maximum likelihood principle [9

9. L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis (Addison-Wesley, 1991).

], the optimal estimate for the sensing event, ΔX, (denoted by ΔX^) can be found by
ΔXlnfΔX(y)=0.
(15)
It can be verified that the application of Eq. (15) to Eq. (14) results in:
ΔX^=(σQmQ)2(Δλμλmλ)+(σλmλ)2(ΔQμQmQ)(σλmλ)2+(σQmQ)2.
(16)

2.2. Maximum likelihood estimation: Correlated noise sources

2.3. Discussion of the estimator

The estimator in Eq. (16) and Eq. (18) provides the optimal estimate by considering both the underlying physics (mλ, mQ, Δλ, ΔQ) and the noise variables (Nλ, NQ) 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 σλ/mλ > σQ/mQ (i.e., change in wavelength, Δλ, is more noisy or less sensitive), then more weight is assigned to the ΔQ measurement for improved estimation, and vice versa. For the case of σλ/mλ = σQ/mQ, 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.

3. Estimation: Experimental results

The experimental setup is shown in Fig. 2. A 1530nm tunable laser (Tunics T100R, Yenista Optics) is used to couple light in a microtoroidal cavity [6

6. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(2003). [CrossRef] [PubMed]

] via tapered optical fiber. The laser is modulated in wavelength via FG1 (HP 8116A function generator, triangular waveform, 7Vpp, 100Hz for the Δλ measurements, 100mHz for the ΔQ measurements) and an external port (FSC) of the laser. The laser is amplitude modulated via FG2 (Agilent 33250A function generator, sinusoid waveform,, 200mVpp, 40MHz), and an electro-optic modulator (OC-192, JDSU). The change in wavelength, Δλ, is recorded by a combination of a photodetector Det1 (PDA10CS, Thorlabs), and an oscilloscope (Agilent DSO3034). The change in quality factor, ΔQ, is recorded by a combination of a photodetector Det2 (HP 11982), and a lock-in amplifier (Zurich Instruments HF2LI).

Fig. 2 (a) Experimental setup. FG-Function Generation, Det-Detector. Flow cell is same as the one used in [8] (b)Scanning electron microscope image of a typical microtoroidal cavity (c) Cross section of the microtoroidal cavity showing its dimensions: Major diameter, D, and minor diameter, d.

The microtoroidal cavity (major diameter 140μm, minor diameter 6μm) [6

6. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(2003). [CrossRef] [PubMed]

] is immersed in heavy water (D2O); because D2O has lower optical loss at 1530nm than H2O, the microcavity is able to maintain a high quality factor [11

11. A. Armani, D. Armani, B. Min, K. Vahala, and S. Spillane, “Ultra-high-Q microcavity operation in H2O and D2O,” Appl. Phys. Lett. 87(2005). [CrossRef]

]. 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 ΔX) 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 ΔX is then used to measure the response of the cavity in terms of a change in wavelength, Δλ, and the change in quality factor, ΔQ. 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: mλ = 53.6nm/RIU and mQ = 4.34 × 106RIU−1. The linearity of the Δλ measurements for similar range of refractive index change have also been previously reported in [12

12. N. Hanumegowda, C. Stica, B. Patel, I. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett. 87(2005). [CrossRef]

].

Fig. 3 Experimental results. Δλ is positive (i.e. λ shifts towards red) and ΔQ is negative (i.e. Q decreases) as ΔX increases. This behavior is consistent with the fundamental micro-cavity theory and is verified by simulations.

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, NQ, 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 (FG2). From the noise data, standard deviation, σQ, of the noise in ΔQ calculation is 30 for the quality factor of 5 × 104 [13

13. M. I. Cheema, “Towards optimal whispering gallery mode microcavity sensors: Novel techniques and analyses,” Ph.D. thesis, McGill University (2013). (To be published).

]. A similar distribution is obtained for the wavelength measurement, which gives standard deviation, σλ, of the noise in Δλ measurement of 46 fm. The noise, Nλ, 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, Nλ and NQ, is 1.163 × 10−3 i.e. the two noise sources have negligible correlation.

Fig. 4 A normal probability distribution fit (pdf) to the experimental noise data (σθ = 4.10 × 10−4 deg.) for phase shift (θ) of the waveguide signal w.r.t to the reference signal (FG2).The sum of the areas of the rectangles is equal to one. From the noise data, the noise (σQ) in ΔQ calculation is 30 for the quality factor of 5 × 104. The noise (σλ) in Δλ measurement is 46 fm. See Section 3 for further details and discussion.

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 (< 103) 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 (σλ = 6.76pm, σQ = 1572).

The application of the estimator is demonstrated by recording the response (Δλ = 91.91pm, ΔQ = 1.24 × 104) of the cavity for another refractive index change, ΔX, which is not included in Fig. 3. The results, obtained by using data (mλ, mQ, σλ, σQ) of Fig. 3, are shown in the Table 1. The value of ΔX is also measured from the commercial SPR system and is found to be 1.95 × 10−3.

Table 1. Estimation Results. From the commercial SPR system (GenOptics, France), ΔX = 1.95 × 10−3 RIU.

table-icon
View This Table

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 (Δλ, ΔQ) 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

14. M. I. Cheema and A. G. Kirk, “Accurate determination of the quality factor and tunneling distance of axisymmetric resonators for biosensing applications,” Opt. Express 21, 8724–8735 (2013). [CrossRef] [PubMed]

] to model the microtoriodal cavities [6

6. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(2003). [CrossRef] [PubMed]

] immersed in heavy water at 1530nm. The FEM model is used to calculate the change in the wavelength, Δλ, and the change in quality factor, ΔQ, as a function of the intrinsic quality factor of the fundamental TM mode of the cavity, in response to the refractive index change ΔX (10−4 − 10−3) in the outside medium (heavy water). The slopes (mλ, mQ) 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 (6μm).

The standard deviation, σλ of the wavelength noise, Nλ, is theoretically calculated by running Monte Carlo simulations at each quality factor used in the FEM model [13

13. M. I. Cheema, “Towards optimal whispering gallery mode microcavity sensors: Novel techniques and analyses,” Ph.D. thesis, McGill University (2013). (To be published).

]. On the other hand, the standard deviation, σQ, of the quality factor noise, NQ, is theoretically calculated by running simulations based upon the coupled mode theory of a resonator coupled with a waveguide [13

13. M. I. Cheema, “Towards optimal whispering gallery mode microcavity sensors: Novel techniques and analyses,” Ph.D. thesis, McGill University (2013). (To be published).

]. 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, σλ, and the slope, mλ, of the wavelength measurements decrease with the increase in quality factor [13

13. M. I. Cheema, “Towards optimal whispering gallery mode microcavity sensors: Novel techniques and analyses,” Ph.D. thesis, McGill University (2013). (To be published).

], whereas for the quality factor measurements, the standard deviation, σQ, increases with the increase in the quality factor [13

13. M. I. Cheema, “Towards optimal whispering gallery mode microcavity sensors: Novel techniques and analyses,” Ph.D. thesis, McGill University (2013). (To be published).

] and the slope, mλ, has a maximum value at a particular quality factor [14

14. M. I. Cheema and A. G. Kirk, “Accurate determination of the quality factor and tunneling distance of axisymmetric resonators for biosensing applications,” Opt. Express 21, 8724–8735 (2013). [CrossRef] [PubMed]

].

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 (Nλ, NQ) in the measurement system.

Fig. 5 Modeling Results. D−Δλ: Δλ is dominant contributor in the estimator. D−ΔQ: ΔQ is dominant contributor in the estimator. AEC: Approximately equal contribution from the two measurements. For details of the simulation and other symbols, see Section 2 and Section 4.

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 (σλ/mλ, σQ/mQ) 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).

5. Discussion

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

  1. We use microtoroidal cavities at 1530nm and employ PS-CRDS for change in quality factor, ΔQ, 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, σQ will be higher for FWHM measurements as compared to the PS-CRDS system [8

    8. M. I. Cheema, S. Mehrabani, A. A. Hayat, Y.-A. Peter, A. M. Armani, and A. G. Kirk, “Simultaneous measurement of quality factor and wavelength shift by phase shift microcavity ring down spectroscopy,” Opt. Express 20, 9090–9098 (2012). [CrossRef] [PubMed]

    , 13

    13. M. I. Cheema, “Towards optimal whispering gallery mode microcavity sensors: Novel techniques and analyses,” Ph.D. thesis, McGill University (2013). (To be published).

    ], 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

    15. T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” PNAS 108, 5976–5979 (2011). [CrossRef] [PubMed]

    ].
  2. The use of calibration curves to determine the slope of the linear response of the change in wavelength (mλ) and the quality factor (mQ) 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

    12. N. Hanumegowda, C. Stica, B. Patel, I. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett. 87(2005). [CrossRef]

    , 16

    16. A. Oraevsky, “Whispering-gallery waves,” Quantum Electron. 32, 377–400 (2002). [CrossRef]

    , 17

    17. S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in micro-spheres by protein adsorption,” Opt. Lett. 28, 272–274 (2003). [CrossRef] [PubMed]

    ].
  3. The proposed approach assumes linearity of the change in wavelength, Δλ, and the change in quality factor, ΔQ, for small change ΔX (i.e., sensing event). This assumption may not be valid for large range of ΔX [16

    16. A. Oraevsky, “Whispering-gallery waves,” Quantum Electron. 32, 377–400 (2002). [CrossRef]

    ]. The analysis presented in Section 2 can be extended to the case of general dependance of the change in wavelength, Δλ and the change in quality factor, ΔQ, on the sensing event, ΔX. 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, Nλ, NQ, and after setting the natural logarithm to 0):
    (Δλfλ(ΔX)μλ)fλ(ΔX)σλ2+(ΔQfQ(ΔX)μQ)fQ(ΔX)σQ2=0,
    (19)
    where f′X) represents the derivative of f w.r.t ΔX. Solution of Eq. (19) for ΔX gives the optimal estimate, ΔX^, for the sensing event.

    The simulations confirm that as ΔX (refractive index change in heavy water) increases, quadratic or cubic dependance of the change in wavelength, Δλ, and the change in quality factor, ΔQ, on ΔX lead to the fitting noise that is less than the experimental noise. For example, assuming a quadratic dependance of Δλ, and ΔQ in the following form:
    Δλ=mλNΔX2+mλLΔX+Nλ=fλ(ΔX)+Nλ,
    (20)
    ΔQ=mQNΔX2+mQLΔX+NQ=fQ(ΔX)+NQ,
    (21)
    where mλN, mλL, mQN, and mQL are constants then solution of Eq. (19) provides the best estimate for the sensing event.

  4. The development of a common metric to enable the comparison of different types of label-free sensors can involve various elements [18

    18. I. M. White and X. Fan, “On the performance quantification of resonant refractive index sensors,” Opt. Express 16, 1020–1028 (2008). [CrossRef] [PubMed]

    , 19

    19. J. Hu, X. Sun, A. Agarwal, and L. C. Kimerling, “Design guidelines for optical resonator biochemical sensors,” J. Opt. Soc. Am. B 26, 1032–1041 (2009). [CrossRef]

    ]. 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
    SM=m(ΔAμ)σ2,
    (22)
    where m is the slope of linear response of the sensor, ΔA is the measured quantity (e.g., change in quality factor in microcavity sensor or change in incidence angle/wavelength in surface plasmon sensors [20

    20. J. Homola, I. Koudela, and S. S. Yee, “Surface plasmon resonance sensors based on diffraction gratings and prism couplers: sensitivity comparison,” Sensors and Actuators B: Chemical 54, 16–24 (1999). [CrossRef]

    ]), 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

    16. A. Oraevsky, “Whispering-gallery waves,” Quantum Electron. 32, 377–400 (2002). [CrossRef]

    ]. 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.

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 (ΔA) in response to the same sensing event (ΔX). We estimate ΔX 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:
ΔA1=m1ΔX+N1,
(24)
ΔAn=mnΔX+Nn,
(25)
where Nn are the experimental noise (assuming normal distribution) measured in the response of a sensor corresponding to the sensing event (ΔX) and mn 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:
ΔX^=1Z(m1(ΔA1μ1)σ12Sensor1+m2(ΔA2μ2)σ22Sensor2++mn(ΔAnμn)σn2Sensorn),
(26)
where
Z=i=1i=nmi2σi2.
(27)
In Eq. (26), 1/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.

References and links

1.

H. K. Hunt and A. M. Armani, “Label-free biological and chemical sensors,” Nanoscale 2, 1544–1559 (2010). [CrossRef] [PubMed]

2.

F. Vollmer and L. Yang, “Label-free detection with high-Q microcavities: a review of biosensing mechanisms for integrated devices,” Nanophotonics 267–291 (2012).

3.

L. Maleki and V. S. Ilchenko, “Techniques and devices for sensing a sample by using a whispering gallery mode resonator,” US Patent 6490039 (2002).

4.

J. L. Nadeau, V. S. Ilchenko, D. Kossakovski, G. H. Bearman, and L. Maleki, “High-Q whispering-gallery mode sensor in liquids,” Proc. SPIE 4629, 172–180 (2002). [CrossRef]

5.

A. M. Armani and K. J. Vahala, “Heavy water detection using ultra-high-Q microcavities,” Opt. Lett. 31, 1896–1898 (2006). [CrossRef] [PubMed]

6.

D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421(2003). [CrossRef] [PubMed]

7.

J. Barnes, B. Carver, J. M. Fraser, G. Gagliardi, H. P. Loock, Z. Tian, M. W. B. Wilson, S. Yam, and O. Yastrubshak, “Loss determination in microsphere resonators by phase-shift cavity ring-down measurements,” Opt. Express 16, 13158–13167 (2008). [CrossRef] [PubMed]

8.

M. I. Cheema, S. Mehrabani, A. A. Hayat, Y.-A. Peter, A. M. Armani, and A. G. Kirk, “Simultaneous measurement of quality factor and wavelength shift by phase shift microcavity ring down spectroscopy,” Opt. Express 20, 9090–9098 (2012). [CrossRef] [PubMed]

9.

L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis (Addison-Wesley, 1991).

10.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 80, 4057–4059 (2002). [CrossRef]

11.

A. Armani, D. Armani, B. Min, K. Vahala, and S. Spillane, “Ultra-high-Q microcavity operation in H2O and D2O,” Appl. Phys. Lett. 87(2005). [CrossRef]

12.

N. Hanumegowda, C. Stica, B. Patel, I. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett. 87(2005). [CrossRef]

13.

M. I. Cheema, “Towards optimal whispering gallery mode microcavity sensors: Novel techniques and analyses,” Ph.D. thesis, McGill University (2013). (To be published).

14.

M. I. Cheema and A. G. Kirk, “Accurate determination of the quality factor and tunneling distance of axisymmetric resonators for biosensing applications,” Opt. Express 21, 8724–8735 (2013). [CrossRef] [PubMed]

15.

T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” PNAS 108, 5976–5979 (2011). [CrossRef] [PubMed]

16.

A. Oraevsky, “Whispering-gallery waves,” Quantum Electron. 32, 377–400 (2002). [CrossRef]

17.

S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in micro-spheres by protein adsorption,” Opt. Lett. 28, 272–274 (2003). [CrossRef] [PubMed]

18.

I. M. White and X. Fan, “On the performance quantification of resonant refractive index sensors,” Opt. Express 16, 1020–1028 (2008). [CrossRef] [PubMed]

19.

J. Hu, X. Sun, A. Agarwal, and L. C. Kimerling, “Design guidelines for optical resonator biochemical sensors,” J. Opt. Soc. Am. B 26, 1032–1041 (2009). [CrossRef]

20.

J. Homola, I. Koudela, and S. S. Yee, “Surface plasmon resonance sensors based on diffraction gratings and prism couplers: sensitivity comparison,” Sensors and Actuators B: Chemical 54, 16–24 (1999). [CrossRef]

21.

I. White, H. Oveys, and X. Fan, “Liquid-core optical ring-resonator sensors,” Opt. Lett. 31, 1319–1321 (2006). [CrossRef] [PubMed]

OCIS Codes
(280.1415) Remote sensing and sensors : Biological sensing and sensors
(280.1545) Remote sensing and sensors : Chemical analysis
(140.3945) Lasers and laser optics : Microcavities
(140.3948) Lasers and laser optics : Microcavity devices
(280.4788) Remote sensing and sensors : Optical sensing and sensors

ToC Category:
Sensors

History
Original Manuscript: June 26, 2013
Revised Manuscript: August 30, 2013
Manuscript Accepted: August 31, 2013
Published: September 20, 2013

Citation
M. Imran Cheema, Usman A. Khan, Andrea M. Armani, and Andrew G. Kirk, "Towards more accurate microcavity sensors: maximum likelihood estimation applied to a combination of quality factor and wavelength shifts," Opt. Express 21, 22817-22828 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-19-22817


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References

  1. H. K. Hunt and A. M. Armani, “Label-free biological and chemical sensors,” Nanoscale2, 1544–1559 (2010). [CrossRef] [PubMed]
  2. F. Vollmer and L. Yang, “Label-free detection with high-Q microcavities: a review of biosensing mechanisms for integrated devices,” Nanophotonics267–291 (2012).
  3. L. Maleki and V. S. Ilchenko, “Techniques and devices for sensing a sample by using a whispering gallery mode resonator,” US Patent6490039 (2002).
  4. J. L. Nadeau, V. S. Ilchenko, D. Kossakovski, G. H. Bearman, and L. Maleki, “High-Q whispering-gallery mode sensor in liquids,” Proc. SPIE4629, 172–180 (2002). [CrossRef]
  5. A. M. Armani and K. J. Vahala, “Heavy water detection using ultra-high-Q microcavities,” Opt. Lett.31, 1896–1898 (2006). [CrossRef] [PubMed]
  6. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature421(2003). [CrossRef] [PubMed]
  7. J. Barnes, B. Carver, J. M. Fraser, G. Gagliardi, H. P. Loock, Z. Tian, M. W. B. Wilson, S. Yam, and O. Yastrubshak, “Loss determination in microsphere resonators by phase-shift cavity ring-down measurements,” Opt. Express16, 13158–13167 (2008). [CrossRef] [PubMed]
  8. M. I. Cheema, S. Mehrabani, A. A. Hayat, Y.-A. Peter, A. M. Armani, and A. G. Kirk, “Simultaneous measurement of quality factor and wavelength shift by phase shift microcavity ring down spectroscopy,” Opt. Express20, 9090–9098 (2012). [CrossRef] [PubMed]
  9. L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis (Addison-Wesley, 1991).
  10. F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, “Protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett.80, 4057–4059 (2002). [CrossRef]
  11. A. Armani, D. Armani, B. Min, K. Vahala, and S. Spillane, “Ultra-high-Q microcavity operation in H2O and D2O,” Appl. Phys. Lett.87(2005). [CrossRef]
  12. N. Hanumegowda, C. Stica, B. Patel, I. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett.87(2005). [CrossRef]
  13. M. I. Cheema, “Towards optimal whispering gallery mode microcavity sensors: Novel techniques and analyses,” Ph.D. thesis, McGill University (2013). (To be published).
  14. M. I. Cheema and A. G. Kirk, “Accurate determination of the quality factor and tunneling distance of axisymmetric resonators for biosensing applications,” Opt. Express21, 8724–8735 (2013). [CrossRef] [PubMed]
  15. T. Lu, H. Lee, T. Chen, S. Herchak, J.-H. Kim, S. E. Fraser, R. C. Flagan, and K. Vahala, “High sensitivity nanoparticle detection using optical microcavities,” PNAS108, 5976–5979 (2011). [CrossRef] [PubMed]
  16. A. Oraevsky, “Whispering-gallery waves,” Quantum Electron.32, 377–400 (2002). [CrossRef]
  17. S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in micro-spheres by protein adsorption,” Opt. Lett.28, 272–274 (2003). [CrossRef] [PubMed]
  18. I. M. White and X. Fan, “On the performance quantification of resonant refractive index sensors,” Opt. Express16, 1020–1028 (2008). [CrossRef] [PubMed]
  19. J. Hu, X. Sun, A. Agarwal, and L. C. Kimerling, “Design guidelines for optical resonator biochemical sensors,” J. Opt. Soc. Am. B26, 1032–1041 (2009). [CrossRef]
  20. J. Homola, I. Koudela, and S. S. Yee, “Surface plasmon resonance sensors based on diffraction gratings and prism couplers: sensitivity comparison,” Sensors and Actuators B: Chemical54, 16–24 (1999). [CrossRef]
  21. I. White, H. Oveys, and X. Fan, “Liquid-core optical ring-resonator sensors,” Opt. Lett.31, 1319–1321 (2006). [CrossRef] [PubMed]

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