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Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 9, Iss. 5 — Apr. 29, 2014
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Optimizing detection limits in whispering gallery mode biosensing

Matthew R. Foreman, Wei-Liang Jin, and Frank Vollmer  »View Author Affiliations


Optics Express, Vol. 22, Issue 5, pp. 5491-5511 (2014)
http://dx.doi.org/10.1364/OE.22.005491


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Abstract

A theoretical analysis of detection limits in swept-frequency whispering gallery mode biosensing modalities is presented based on application of the Cramér-Rao lower bound. Measurement acuity factors are derived assuming the presence of uncoloured and 1/ f Gaussian technical noise. Frequency fluctuations, for example arising from laser jitter or thermorefractive noise, are also considered. Determination of acuity factors for arbitrary coloured noise by means of the asymptotic Fisher information matrix is highlighted. Quantification and comparison of detection sensitivity for both resonance shift and broadening sensing modalities are subsequently given. Optimal cavity and coupling geometries are furthermore identified, whereby it is found that slightly under-coupled cavities outperform critically and over coupled ones.

© 2014 Optical Society of America

1. Introduction

WGM based sensors can employ a number of different operational principles, for example, frequency locking or swept-frequency modalities, transmission vs. back-scattering detection, passive or active cavities and also ring-down measurements. Perhaps the more commonly used and easiest to implement of these strategies is the swept-frequency modality, in which the transmission of a source coupled to a resonator is monitored as the frequency of the source is varied. Whilst detection limits have been extensively studied for a number of these configurations [14

14. J. Zhu, S. J Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Phot. 4, 46–49 (2010). [CrossRef]

17

17. T. McGarvey, A. Conjusteau, and H. Mabuchi, “Finesse and sensitivity gain in cavity-enhanced absorption spectroscopy of biomolecules in solution,” Opt. Express , 14, 10441–10451 (2006). [CrossRef] [PubMed]

], analysis of swept-frequency based biosensors is inherently more complex and has thus not yet been comprehensively covered. To date, attempts to quantify the detection limits of swept-frequency WGM sensors have predominantly been based on either qualitative arguments or empirical estimates [18

18. J. D. Swaim, J. Knittel, and W. P. Bowen, “Detection limits in whispering gallery biosensors with plasmonic enhancement,” Appl. Phys. Lett. 99, 243109 (2011). [CrossRef]

22

22. N. M. Hanumegowda, C. J. Stica, B. C. Patel, I. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett. 87, 201107 (2005). [CrossRef]

] and hence lack the satisfaction associated with more rigorous treatments. A number of papers have attempted to adopt a more theoretical approach with Lopez-Yglesias et. al. [23

23. X. Lopez-Yglesias, J. M. Gamba, and R. C. Flagan, “The physics of extreme sensitivity in whispering gallery mode optical biosensors,” J. Appl. Phys. 111, 084701 (2012). [CrossRef]

], for example, considering different physical interactions between a WGM sensor and an adsorbed molecule which can affect induced sensing signals. Noise sources were, however, given only minimal consideration. Various fundamental noise sources, such as thermorefractive noise, were considered in the works of Matsko et. al. [24

24. A. B. Matsko, A. A. Savchenkov, N. Yu, and L. Maleki, “Whispering gallery mode resonators as frequency references. I. Fundamental limitations,” J. Opt. Soc. Am. B 24, 1324–1335 (2007). [CrossRef]

] and Gorodetsky et. al. [25

25. M. L. Gorodetsky and I. S. Grudinin, “Fundamental thermal fluctuations in microspheres,” J. Opt. Soc. Am. B 21, 697–705 (2004). [CrossRef]

], however these treatments are not immediately transferable to a biosensing context thus hampering their utility. This paper thus attempts to bridge this gap and formulate biodetection limits when using swept-frequency WGM microsensors based upon a rigorous consideration of the noise sources deriving from information theoretic tools. Attention is limited to two dominant noise sources, namely additive Gaussian detector noise and thermorefractive noise/laser fluctuations, since these are frequently the limiting factors within biosensing experiments. Noise is assumed to be uncoloured, however calculation for arbitrary power spectra is discussed and illustrated for pink (1/ f) noise. Moreover, to the authors knowledge, all preceding treatments, perhaps with the exception of [19

19. L. Shao, X.-F. Jiang, X.-C. Yu, B.-B. Li, W. R. Clements, F. Vollmer, W. Wang, Y.-F. Xiao, and Q. Gong, “Detection of single nanoparticles and lentiviruses using microcavity resonance broadening,” Adv. Mat. 25, 5616–5620 (2013). [CrossRef]

], have considered detection limits based upon monitoring reactive resonance shifts only. Measurements of linewidth change are however currently emerging as a complementary sensing mechanism. We therefore also formulate detection limits in this regard. Beyond the insight into system performance that quantification of detection limits affords, such knowledge allows for optimisation of sensor geometry and other operational parameters. Optimisation of this nature therefore constitutes a further aim of this work.

The structure of this article is as follows. In Section 2 we first formulate the measurement problem and describe the origin and nature of relevant noise sources, before proceeding to describe fundamental detection limits imposed on measuring reactive shifts and linewidth changes by means of an information theoretic analysis in Section 3. Detection limits derived in Section 3, although sufficient to describe the problem, are rather formal in nature, therefore in Section 4 we apply our results to biosensing experiments. Consequently, optimal WGM sensing configurations are identified for different sensing scenarios, and the performance of reactive shift and line broadening sensing modalities compared. Section 5 continues by considering robustness of optimal configurations to environmental temperature changes, before Section 6 finally considers how optimisation results are altered in the presence of additional loss mechanisms, such as surface roughness or nanoantenna induced heating and scattering losses. Final conclusions are drawn in Section 7. A number of appendices are also given providing further elaboration on our mathematical derivations where necessary.

2. Measurement and noise in WGM sensing

Fig. 1 Schematic of observed transmission lineshape, induced red shift and line broadening upon binding of biomolecules to the microcavity surface, illustrating definitions of quantities used in this work.

Technical noise arises from poor experimental setup and can include factors such as stray light impinging upon photodetectors and vibrations. Generally, such noise sources are statistically independent and are well described by a Gaussian probability distribution function (PDF) by virtue of the Central Limit Theorem. Noting that typical power levels in WGM biosensing experiments are ≲ 1 mW, it is furthermore reasonable to adopt a classical description of light, such that the net effect of technical noise on the measured power Id,j can be described by the continuous PDF
pId(Id,j;ω0,Γ,A)=12πσd2exp[12(Id,jIjσd)2],
(2)
where σd2 is the variance of the detected intensity. If multiple independent noise sources contribute to the total technical noise, σd2 is found by adding the associated variances of individual noise sources. Detector noise is perhaps the most intuitive and common noise such that we shall hereafter refer to technical noise simply as detector noise.

Fundamental noise limitations, on the other hand, can arise from a number of different physical phenomena. Thermodynamic fluctuations within the cavity and surrounding medium, for example, give rise to both thermorefractive and thermoelastic noise, from variations in the refractive index and cavity size respectively [25

25. M. L. Gorodetsky and I. S. Grudinin, “Fundamental thermal fluctuations in microspheres,” J. Opt. Soc. Am. B 21, 697–705 (2004). [CrossRef]

, 26

26. W.-L. Jin, X. Yi, Y. Hu, B. Li, and Y. Xiao, “Temperature-insensitive detection of low-concentration nanoparticles using a functionalized high-Q microcavity,” Appl. Opt. 52, 155–161 (2013). [CrossRef] [PubMed]

]. Additionally, temperature fluctuations can cause changes in the coupling distance, albeit these can be negated by means of free space coupling [19

19. L. Shao, X.-F. Jiang, X.-C. Yu, B.-B. Li, W. R. Clements, F. Vollmer, W. Wang, Y.-F. Xiao, and Q. Gong, “Detection of single nanoparticles and lentiviruses using microcavity resonance broadening,” Adv. Mat. 25, 5616–5620 (2013). [CrossRef]

]. Microresonators are furthermore interrogated by means of laser light which, due to its quantised nature, can result in photothermal, optoelastic and phase noise. A good discussion of the effects of these noise sources can be found in [24

24. A. B. Matsko, A. A. Savchenkov, N. Yu, and L. Maleki, “Whispering gallery mode resonators as frequency references. I. Fundamental limitations,” J. Opt. Soc. Am. B 24, 1324–1335 (2007). [CrossRef]

], however, for our purposes it is sufficient to consider only thermorefractive noise since this was found to be dominant (although methods of suppressing its influence have been proposed [27

27. A. A. Savchenkov, A. B. Matsko, V. S. Ilchenko, N. Yu, and L. Maleki, “Whispering-gallery-mode resonators as frequency references. II. Stabilization,” J. Opt. Soc. Am. B. 24, 2099–2997 (2007). [CrossRef]

]). Fluctuations in the refractive index within the mode volume have a dual effect of modifying both the frequency and linewidth of the microcavity resonance. Frequently in the literature, fluctuations in the resonance linewidth are assumed to be negligible; an assumption based on empirical observation [28

28. T. Carmon, L. Yang, and K. J. Vahala, “Dynamical thermal behavior and thermal self-stability of microcavities,” Opt. Express 12, 4742–4750 (2004). [CrossRef] [PubMed]

, 29

29. I. Teraoka, “Analysis of thermal stabilization of whispering gallery mode resonance,” Opt. Commun. 310212–216 (2014). [CrossRef]

]. Theoretical justification of this approximation is, however, given in Appendix A, such that we too only consider fluctuations in the resonance frequency. Making the replacement ω0ω0 + ωt in Eq. (1), where ωt represents the thermorefractive resonance shift from the unperturbed resonance frequency, allows us to describe the effect of thermorefractive noise via the PDF
pωt(ωt)=12πσt2exp[12ωt2σt2],
(3)
where a Gaussian PDF is again assumed. Equivalently, the resultant random fluctuations in the measured power are described by
pId(Id,j;ω0,Γ,A)=δ(Id,jIj)pωt(ωt)dωt
(4)
=12πσt2I0AΓj24Λj(I0Id,j)2exp[12Δj2σt2]exp[12Λj2σt2]cosh[ΛjΔjσt2],
(5)
where Eq. (4) derives from the law of total probability, Δj = ωjω0 and Λj=Γ2[I0A/(I0Id,j)1]1/2. Whilst we have considered fluctuations in the central resonance frequency of the WGM arising from thermorefractive noise, it is worthwhile noting that frequency fluctuations can also arise from other sources, such as instabilities in the laser (typically ∼100 kHz – 1 MHz for tunable laser sources). Results given for thermorefractive noise are hence equally applicable to laser fluctuations (jitter). In the presence of both thermorefractive noise and laser jitter, σt2 is again found by adding the associated variances of each noise source. Whilst not explicitly discussed here, our results have shown that symmetric sweeping of the resonance peak (i.e. the laser frequency sweeps the interval ω ∈ [ω0 − Ω/2, ω0 + Ω/2]) achieves greatest measurement accuracy. Only this case is therefore considered in what follows.

3. Fisher information and detection limits

Fisher information is a natural metric emerging from the field of statistical estimation, which quantifies the performance of an ideal observer in estimating an original signal given a noise corrupted version [30

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

]. Although Fisher information shares many properties with the better known Shannon information, such as superadditivity and positivity [31

31. C. Vignet and J.-F. Bercher, “Analysis of signals in the Fisher-Shannon information plane,” Phys. Lett. A 312, 27–33 (2003). [CrossRef]

], the former is more suitable for describing the measurement problem (as opposed to the information transmission problem). In particular, the covariance matrix, 𝕂w, for estimation of a parameter vector w, which provides a convenient parameterisation of measurement precision, is lower bounded by the inverse of the Fisher information matrix 𝕁w viz.
𝕂w𝕁w1,
(6)
where the inequality implies the difference of the two matrices is positive definite and does not necessarily hold element-wise. In our case w = (ω0, Γ, A). Eq. (6) is known as the Cramér-Rao lower bound (CRLB) [32

32. H. Cramér, Mathematical Methods of Statistics, (Princeton University Press, USA, 1946).

] and also implies the weaker set of inequalities σwi21/[𝕁w]ii where σwi2 is the variance for estimation of wi and wi ([𝕁w]ii) denotes the ith (diagonal) element of w (𝕁w). It is important to note that the CRLB as expressed by Eq. (6) explicitly quantifies the uncertainty achievable by any unbiased estimator and hence represents a fundamental limit to measurement precision. Furthermore, by use of the maximum likelihood estimator the CRLB can be asymptotically achieved [30

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

]. A fuller discussion of Fisher information and properties of the maximum likelihood estimator can be found in [30

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

, 33

33. M. R. Foreman and P. Török, “Information and resolution in electromagnetic optical systems,” Phys. Rev. A 82, 043835 (2010). [CrossRef]

] and references therein.

Within the context of biosensing, we do not directly measure the parameters w = (ω0, Γ, A), but instead infer them from noisy power readings. As such the Fisher information matrix can be written in the form 𝕁w = 𝔾T 𝕁I𝔾, where 𝔾 = I/w is a matrix of derivatives, T denotes the matrix transpose and 𝕁I is the Fisher information matrix associated with estimation of the transmitted power I. Initially, we assume that the noise on each data point is statistically independent such that we can write pId(Id)=Πj=1NΩpId(Id,j;ω0,Γ,A), yielding
[𝕁I]ij=δi,jpId,j(Id,j)(lnpId,j(Id,j;ω0,Γ,A)Ij)2dId,j,
(7)
where δi,j is the Kronecker delta. The case of dependent data samples will be discussed below. Since each data sample is statistically independent we can invoke additivity of Fisher information to give 𝕁w=j=1NΩJIjGjGj where ⊗ denotes the outer product and the shorthand notation [𝕁I]jj = JIj and Gj = ∂Ij/∂w has been used.

In principle, the PDFs given by Eqs. (2) and (5) can be substituted directly into Eq. (7) and calculated numerically, however, to gain further insight into the noise limitations we make a number of approximations, such that analytic results follow. First considering the simpler case of detector noise, as described by Eq. (2), we assume that the laser sweeping window has a large bandwidth relative to the FWHM of the WGM resonance i.e. Ω > 2Γ, and that the sampling interval is small i.e. ΔΩ ≪ Γ. Under these conditions the summation in Eq. (7) can be approximated by a continuous integral over frequency. Taking evaluation of [𝕁w]11 = Jω0,ω0 as an illustrative example we find
Jω0,ω0=j=1NΩ1σd2(Ijω0)2A2I024σd2ΔΩΓ4(ω0ω)2[(ω0ω)2+Γ2/4]4dω=A2I022σd2ΔΩπΓ.
(8)
Similarly, evaluation of other elements of the Fisher information matrix can be performed, ultimately yielding
𝕁w=πI028σd2ΔΩ(4A2/Γ000A2/ΓA0A2Γ).
(9)
Off diagonal elements of the Fisher information matrix represent correlations that exist between estimates of different parameters, such that here we see that estimates of the transmission depth and FWHM are correlated. Such a correlation is to be expected, since determination of the FWHM requires an estimate of the dip depth. In practical biosensing, the transmission depth is, however, of marginal or no interest and can hence be treated as a nuisance parameter [30

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

]. Nuisance parameters, by virtue of the correlations mentioned above, reduce the amount of information any measurement yields about the parameters of interest, and thus reduce the detection limits. Within the framework of statistical estimation theory it can be shown that a reduced Fisher information matrix can be defined, which in our case, and for the sake of completeness, takes the form
𝕁w=(Jω0,ω0Jω0,A2JA,AJω0,ΓJΓ,AJω0,AJA,AJω0,ΓJΓ,AJω0,AJA,AJΓ,ΓJΓ,A2JA,A)=A2I0216σd2ΔΩπΓ(8001).
(10)
In the remainder of this work, we will, however, not consider calculation of the reduced Fisher information matrix, since it adds a level of mathematical complexity, which provides little additional insight. Instead, attention will be given to calculation of [𝕁w]11 = Jω0,ω0 and [𝕁w]22 = JΓ,Γ. Applying the inequality σwi21/[𝕁w]ii thus yields an expression for the minimum detectable resonance shift, Δωd, and linewidth change, ΔΓd,
Δωd=ΔΓd2=2βπσdI0AΓ.
(11)

We can approach the problem for derivation of the detection limits in the presence of thermorefractive noise/laser jitter in a similar manner, however additional approximations and care must be taken. We start by considering
Jω0,ω0=j=1NΩ0[1σt2Λj2σt4sech2(ΛjΔjσt2)]pId,jdId,j.
(12)
The summation for the first term can be easily performed, since by definition ∫ pId,j dId,j = 1, however, to evaluate the second term we again approximate the summation over j as an integration over frequency, yielding
Jω0,ω0NΩσt21ΔΩσt52π0Λ2exp(Λ22σt2)sech(ΛΔσt2)exp(Δ22σt2)dΔdΛ,
(13)
where a change in the integration variable has also been performed. Further approximating sechx ≈ exp[−x2/2], where the FWHM of the Gaussian has been chosen such that the Taylor expansion of the functions match up to quadratic order, allows evaluation of Eq. (13) such that
Jω0,ω01σt[NΩσtπΔΩU(12,0,12)]=1σtΔΩ[Ωσt1.416],
(14)
where U(a, b, z) is the confluent hypergeometric function [34

34. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, (Dover Publications, New York, 1970).

]. Finite integration limits for integration over Δ can also be taken, leading to an additional term in the kernel of the integration over Λ of the form erf[Ω22σt1+Λ2σt2]. Noting, however, that the argument of the error function is large, we can approximate this term as unity, hence also yielding Eq. (14). We note that due to the approximations taken the Fisher information expressed by Eq. (14) can adopt negative (i.e. unphysical) values when Ω ≤ 1.416σt. Fortunately, this condition is only satisfied in pathological noise scenarios, such that Eq. (14) gives a good estimate of the Fisher information for estimation of ω0 (in the presence of thermorefractive noise/laser jitter) in most cases. Our detection limit is then given by
Δωt=βΓσtWΓ/σt1.416σtβW,
(15)
where the second approximation holds for σtWΓ. Eq. (15) shows that the detection limit scales with the number of sampling points as NΩ1/2=(β/W)1/2.

Fig. 2 (a) Variation of intrinsic quality factor Q0 of a fused silica WGM resonance in water with microcavity radius R. (b) Variation of detection limits Δωd,t and ΔΓd,t (normalised to resonance linewidth) with WGM Q factor.

4. Optimal whispering gallery mode biosensors

4.1. Resonance shifts

Fig. 3 (a) Minimum detectable number of influenza A (InfA) virons, N as a function of microcavity radius R as set by detector noise for different coupling distances, d. Solid blue curve corresponds to the optimal coupling distance. (b) As (a) for the optimal coupling distance albeit with the addition of thermorefractive noise of varying magnitude as set by the temperature fluctuations ΔT. Solid blue curve corresponds to detector noise only. Dashed curves show detection limits associated with the presence of thermorefractive noise alone. Solid blue lines in (a) and (b) are equivalent.

Coupling distance can, however, generally be adjusted and provides a further degree of freedom for optimisation in any experimental setup. It can easily be shown analytically that N is minimised when Qc/Q0 = 2 (or equivalently A = 0.89) as depicted in Fig. 4(a). In contradiction to common wisdom, Qc/Q0 = 2 implies that optimal detection can be achieved when the microresonator is slightly under coupled rather than critically coupled. This result arises since we have shown that optimal detection simultaneously requires a large transmission depth A and a narrow linewidth. Although critical coupling offers the largest transmission depth [41

41. M. Cai, O. Painter, and K. J. Vahala, “Observation of Critical Coupling in a Fiber Taper to a Silica-Microsphere Whispering-Gallery Mode System,” Phys. Rev. Lett. 85, 74–77 (2000). [CrossRef] [PubMed]

], it also has a slightly larger linewidth than the under coupled regime. Furthermore, with regards to Fig. 4(b) it is evident that the under coupled regime always outperforms the over coupled case by virtue of the higher Q factors for under coupled modes.

Fig. 4 (a) Variation of (1 + Qc/Q0)3/(Qc/Q0)2 factor, describing coupling loss dependence of minimum number of detectable particles, with Qc/Q0. A clear minimum is exhibited at Qc = 2Q0. (b) Variation of (1 + Qc/Q0)3/(Qc/Q0)2 with transmission depth A in the over- and under-coupled regime. Dashed black line corresponds to A = 0.89, i.e. Qc = 2Q0

For the parameters given above, the optimal sphere radius is found to be R = 46.8 μm with a coupling distance of 1.17 μm implying that, at best, ∼ 10−2.2 InfA virons can be detected at 780 nm, as shown by the solid blue curve in Fig. 3. These calculations were performed assuming only detector noise to be present. Variation of the magnitude of the detector noise, σd, merely scales the detection limit N in a linear fashion and does not alter the optimal cavity size. Accordingly, it is also noted that the same optimal sphere radius follows in the case of pink noise. Allowing for the presence of thermorefractive noise (arising from 2.5 mK temperature fluctuations), however, causes the optimal cavity size to drop to 41.6 μm whereby N ≈ 10−1.7. Evidently, these figures are below the single InfA viron limit given the noise levels chosen.

Whilst Fig. 4 shows an optimal microcavity radius of 46.8 μm, it is important to mention that this value is strongly dependent on the operating wavelength, principally due to wavelength dependent absorption and dispersion of water. To highlight this point, Table 1 shows the calculated globally optimal microcavity size and coupling distance for a set of common wavelengths ranging from the blue to the infrared end of the optical spectrum. Decreased water absorption in the blue region of the spectrum gives significantly better detection limits than in the red region as would be expected. For example, over two orders of magnitude reduction in N between operating wavelengths of 780 nm and 410 nm can be seen such that when operating at 410 nm detection of a single bovine serum albumin (BSA) is possible. For wavelengths smaller than 410 nm detection limits fall due to increased water absorption. Furthermore, our calculations show that smaller microcavities allow more sensitive measurements in the blue region of the spectrum.

Table 1. Calculated optimal parameters for differing wavelengths. Optimal parameters for detection of BSA monolayer for λ = 1550 nm and 1300 nm were beyond computational bounds.

table-icon
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Turning attention to the case of thermorefractive noise, the above analysis can be applied in a similar manner, yielding a measurement acuity of F = (σt/Γ)(β/W)1/2 and detection limit given by
N=σt(nc2ns2)Re[α]ω0R3|Yll(π/2)|2βW.
(25)
Interestingly, we here see stark differences when compared to the behaviour of detector noise. For example, we note that the detection limit set by thermorefractive noise is independent of coupling and cavity losses and hence scales with an approximate R3 dependence. Smaller resonators are thus less susceptible to thermorefractive noise for fixed σt. Furthermore, smaller cavities imply smaller mode volumes such that the variance of temperature fluctuations, σt are also smaller [25

25. M. L. Gorodetsky and I. S. Grudinin, “Fundamental thermal fluctuations in microspheres,” J. Opt. Soc. Am. B 21, 697–705 (2004). [CrossRef]

]. Illustration of the dependence of N in the presence of pure thermorefractive noise is shown in Fig. 3(b) by the green and red dashed curves for temperature fluctuations of 2.5 and 5 mK respectively. A true R3 relation is not seen in Fig. 3(b) due to the weak size dependence of the WGM index l. Given the monotonic dependence on R it is immediately apparent, that in contrast to the case of detector noise, no optimal microcavity size exists when thermorefractive noise is dominant.

Realistically, any experimental setup will be subject to both technical and fundamental noise sources. Accordingly the experimental detection limit and optimal microcavity size is set by competing requirements of both noise sources. Formally, noting that the individual noise sources are independent and additive we can employ Stam’s inequality and the properties of Fisher information to show Δω2Δωd2+Δωt2. Since we are concerned with the best case scenario we shall hereafter assume equality holds. Intuitively, given the earlier results we would anticipate that the optimal cavity size, in the presence of both detector noise and thermorefractive noise, would be smaller than that for detector noise alone. This expectation is indeed borne out in numerical calculations as shown by the solid curves in Fig. 3(b) for differing magnitude of thermal fluctuations, whereby it can be seen that for large cavities thermorefractive noise is dominant over detector noise, whilst for small cavities the converse holds. Simulation parameters are as above. Given, however, that the optimal cavity size is now set by a balance of competing noise effects it is important to note that the optimal cavity size is dependent on both the sweeping window and sampling rate, since these determine the limit imposed by thermal noise/jitter.

Hitherto we have considered perturbation of a WGM by a single particle, however, studies of monolayers may also be of interest, for example in monitoring self-assembled biological membranes and monolayers [4

4. M. Noto, F. Vollmer, D. Keng, I. Teraoka, and S. Arnold, “Nanolayer characterization through wavelength multiplexing of a microsphere resonator,” Opt. Lett 30, 510–512 (2005). [CrossRef] [PubMed]

,42

42. J. Topolancik and F. Vollmer, “Photoinduced transformations in bacteriorhodopsin membrane monitored with optical microcavities,” Biophys. J. 92, 2223–2229 (2007). [CrossRef] [PubMed]

]. In this case it can easily be shown [43

43. D. Q. Chowdhury, S. C. Hill, and P.W. Barber, “Morphology-dependent resonances in radially inhomogeneous spheres,” J. Opt. Soc. Am. A 8, 1702–1705 (1991). [CrossRef]

, 44

44. I. Teraoka and S. Arnold, “Theory of resonance shifts in TE and TM whispering gallery modes by nonradial perturbations for sensing applications,” J. Opt. Soc. Am. B 23, 1381–1389 (2006). [CrossRef]

] that the resonance shift of the WGM for a uniform monolayer of surface density σs is
δωω0=εs(εcεs)RRe[α]σs.
(26)
Letting this shift equal the smallest detectable shift as before, allows the minimum detectable surface density to be determined. We note that the final result is of the same form as Eqs. (24) and (25), albeit for a linear dependence on R (as opposed to R3) and differing prefactors. The optimal coupling distances is hence once again given by Qc/Q0 = 2. Figure 5 shows an example calculation of the minimum surface density for a monolayer of BSA. Optimality, for detection of monolayers, is again determined by a balance of competing factors, however for monolayer detection a greater resonator surface area is also desirable because it yields larger absolute resonance shifts. Optimal resonator size is, therefore, significantly larger than for detection of a single particle and, for the same simulations as above, is R = 202.5 μm. Optimal cavity size is, however, seen to be much more sensitive to variations in the coupling distance than for single particle detection (Fig. 5(a)). Addition of thermorefractive noise is seen to reduce the optimal resonator radius as before (Fig. 5(b)).

Fig. 5 As Fig. 3, but for detection of a monolayer of BSA molecules.

4.2. Linewidth changes

Upon binding to a microresonator, a particle introduces additional loss mechanisms, namely absorption losses in the particle itself and increased scattering losses. In many scenarios, the associated linewidth change can be detected and provides an alternative and complimentary sensing mechanism to reactive shifts [19

19. L. Shao, X.-F. Jiang, X.-C. Yu, B.-B. Li, W. R. Clements, F. Vollmer, W. Wang, Y.-F. Xiao, and Q. Gong, “Detection of single nanoparticles and lentiviruses using microcavity resonance broadening,” Adv. Mat. 25, 5616–5620 (2013). [CrossRef]

]. Whilst Eq. (21) (and subsequent) describe the minimum detectable number of particles when sensing via reactive wavelength shifts, equivalent expressions for monitoring of resonance linewidth can be found. Denoting the linewidth change associated with particle absorption and scattering by δΓabs and δΓsca respectively, we have [45

45. S. Arnold, S. I. Shopova, and S. Holler, “Whispering gallery mode bio-sensor for label-free detection of single molecules: thermo-optic vs. reactive mechanism,” Opt. Express 18, 281–287 (2010). [CrossRef] [PubMed]

, 46

46. A. Mazzei A, S. Götzinger, L. de S. Menezes, G. Zumofen, O. Benson, and V. Sandoghdar, “Controlled Coupling of Counterpropagating Whispering-Gallery Modes by a Single Rayleigh Scatterer: A Classical Problem in a Quantum Optical Light,” Phys. Rev. Lett. 99, 173603 (2007). [CrossRef] [PubMed]

]
δΓabsω0=εs|Yll(π/2)|2(εcεs)R3Im[α],
(27)
δΓscaω0=ω03ns5|Yll(π/2)|23πc3(εcεs)R3|α|2,
(28)
where c is the speed of light. Emphasis must be made, however, that these expressions are not valid in the case of mode splitting, which has also been proposed as a further sensing mechanism [14

14. J. Zhu, S. J Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Phot. 4, 46–49 (2010). [CrossRef]

, 47

47. X. Yi, Y.-F. Xiao, Y. Feng, D.-Y. Qiu, J.-Y. Fan, Y. Li, and Q. Gong, “Mode-splitting-based optical label-free biosensing with a biorecognition-covered microcavity,” J. Appl. Phys. 111, 114702 (2012). [CrossRef]

]. For dielectric particles at wavelengths far from molecular resonances, the latter is dominant and yields a minimum detectable number of particles of
N=ΔΓδΓsca=6πc3(nc2ns2)|α|2ns5ω03R3|Yll(π/2)|2F0Q0(1+Qc/Q0)34Qc2/Q02
(29)
for detector noise, where F0 is as defined above and
N=σt6πc3(nc2ns2)|α|2ns5ω04R3|Yll(π/2)|23βW3
(30)
for thermorefractive noise. Blue curves in Fig. 6(a) depict the limits imposed on detection of single InfA virons using linewidth broadening when scattering losses are the dominant broadening process (dashed curve corresponds to detector noise only, dotted curve corresponds to thermorefractive noise only and solid curve corresponds to the presence of both noise sources). Of particular note is that thermorefractive noise plays a less significant role than detector noise, such that an optimal cavity radius can again be identified.

Fig. 6 (a) Comparison of minimum detectable number of influenza virons, N as a function of microcavity radius R when monitoring linewidth changes (blue and green) or resonance frequency shift (red). Blue curves depict detection limits when broadening is dominated by particle induced scattering losses for different noise sources. Green curves show detection limits for broadening considering all particle induced broadening mechanisms. (b) As (a) albeit for a 60 nm radius gold-silica nanoshells with resonance tuned to match the probing WGM frequency. (inset) Dependence of dN/dT with respect to microcavity radius.

Absorption in the particle can also significantly affect the linewidth broadening of a WGM, especially if the analyte comprises metallic nanoparticles or fluorescent emitters (or if these are used as labels) [12

12. M. A. Santiago-Cordoba, S. V. Boriskina, F. Vollmer, and M. C. Demirel, “Nanoparticle-based protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett. 99, 073701 (2011). [CrossRef]

, 48

48. U. Resch-Genger, M. Grabolle, S. Cavaliere-Jaricot, R. Nitschke, and T. Nann, “Quantum dots versus organic dyes as fluorescent labels,” Nat. Methods 5, 763–775 (2008). [CrossRef] [PubMed]

, 49

49. M. R. Foreman and F. Vollmer, “Level repulsion in hybrid photonic-plasmonic microresonators for enhanced biodetection,” Phys. Rev. A 88, 023831 (2013). [CrossRef]

]. In this case the linewidth change is given by δΓ = δΓsca + δΓabs + δΓcc. The additional δΓcc term has also been included since the presence of a scattering particle on a microresonator couples light into a (initially degenerate) counter-propagating WGM [50

50. D. S. Weiss, V. Sandoghdar, J. Hare, V. Lefèvre-Seguin, J. M. Raimond, and S. Haroche, “Splitting of high-Q Mie modes induced by light backscattering in silica microspheres,” Opt. Lett. 20, 1835–1837 (1995). [CrossRef] [PubMed]

]. When coupling is not strong enough to induced mode splitting, the unresolved splitting of the two counter-propagating WGMs yields an additional broadening of magnitude δΓcc = 2|δω| [14

14. J. Zhu, S. J Ozdemir, Y.-F. Xiao, L. Li, L. He, D.-R. Chen, and L. Yang, “On-chip single nanoparticle detection and sizing by mode splitting in an ultrahigh-Q microresonator,” Nat. Phot. 4, 46–49 (2010). [CrossRef]

, 51

51. Y.-F. Xiao, Y.-C. Liu, B.-B. Li, Y.-L. Chen, Y. Li, and Q. Gong, “Strongly enhanced light-matter interaction in a hybrid photonic-plasmonic resonator,” Phys. Rev. A 85, 031805 (2012). [CrossRef]

]. Since these additional induced loss mechanisms cause greater line broadening, the particle induced changes in lineshape are larger and thus the associated detection limit is better than when only scattering losses are present. Figure 6(a) highlights this point by considering the detection limits, for influenza virons, associated with scattering losses only (blue curves), as compared to inclusion of all broadening mechanisms, i.e. scattering, absorption and coupling to counter-propagating modes (solid green curve). Additionally, and for comparison purposes, the detection limit for sensing using reactive shifts is also shown in Fig. 6(a) (red curve). Similar detection sensitivity is exhibited by both detection modalities for cavity sizes ≲ 39 μm, however, as the microresonator size increases thermorefractive noise begins to dominant the reactive shift detection limit, whilst sensing via line broadening is only weakly affected (as discussed above). Ultimately biosensing via mode broadening is predicted to have the better detection limit, with appropriate optimisation of cavity size and coupling distance. Similar results were found in [19

19. L. Shao, X.-F. Jiang, X.-C. Yu, B.-B. Li, W. R. Clements, F. Vollmer, W. Wang, Y.-F. Xiao, and Q. Gong, “Detection of single nanoparticles and lentiviruses using microcavity resonance broadening,” Adv. Mat. 25, 5616–5620 (2013). [CrossRef]

]. When particle absorption is large, e.g. for resonant plasmonic nanoparticles, sensing via monitoring linewidth changes is superior for all cavity sizes. To illustrate this point Fig. 6(b) shows the limits for detection of 60 nm radius gold-silica nanoshells with plasmon resonance tuned to 780 nm i.e. to match the WGM frequency, whereby over an order of magnitude is gained in detection sensitivity compared to the shift based modality.

5. Thermal stability

6. Additional losses

Commonly, in any experimental realisation of a WGM biosensing experiment, the observed Q factor is lower than theoretical expectations when considering only the intrinsic, absorption and coupling losses. Such additional losses, can for example arise from surface roughness, scattering defects in the resonator structure (e.g. air bubbles) and non-sphericity of the resonator. Furthermore, in plasmon enhanced sensing the presence of the nanoantenna gives rise to additional scattering and heating losses (see Section 4.2). The question then arises as to how additional losses affect the preceding results. To address this problem it is necessary to return to Eq. (21), which we write in the form N = C(R)F0/AQ, where now Q1=Q01+Qc1+Qm1 and Qm denotes the additional miscellaneous losses. As before we can consider the optimisation of the coupling distance and the resonator size independently. Fuller mathematical details are given in Appendix C, however, we find that the new optimal coupling distance satisfies the relation
QcQ0=12[1+1+9Qm1+Qm]223Qm,
(31)
where the latter approximation holds when Qm11. Subsequently optimisation of the resonator size yields a shift in the optimal resonator size of
δR2C(Ropt)Q0QmQmR[(3Qm+2)2[C(R)/Q0]R2|R=Ropt]1.
(32)
With the possible exception of ∂Qm/∂R, all quantities in Eq. (32) are positive. Consequently, ∂Qm/∂R dictates whether the optimal resonator size increases or decreases. For example, for mechanisms whereby losses increase with the resonator size, such as surface roughness and defect scattering, ∂Qm/∂R is negative, such the optimal resonator size decreases. In contrast, however, given that larger microcavities have a larger proportion of the mode lying within the resonator, scattering losses from nanoantenna decrease with larger resonator size. Consequently ∂Qm/∂R is positive, hence motivating the use of larger resonators. Equivalent results follow when considering line broadening as a sensing mechanism.

7. Conclusions

To facilitate utility within a biosensing context, we further applied the derived detection limits to the question of the minimum number of detectable bioparticles (and surface density of a monolayer). Numerical calculations were also presented based on detection of InfA virons and BSA monolayers. In the presence of detector noise a clear optimal microcavity size can be identified arising from balancing the requirements of narrow linewidth, small mode volume and large surface intensity. Attention was limited to first radial order WGMs, however, it is important to mention that the optimal cavity size increases as the radial number increases. Coupling to the correct mode is therefore important in realising potential gains offered by optimising microcavity size. Sensor surface area was also found to play an important role for detection of monolayers, such that optimal cavity sizes were found to be significantly larger than for single particle detection. Interestingly, our results show that from an experimental point of view it can be better to air on the side of caution when fabricating microspheres so as to produce larger than desired spheres. This paradigm can be understood by observing that loss of detection sensitivity is more severe for smaller than optimal cavity sizes than the reverse case (see e.g. Fig. 3). This point is further seen when considering the stability of detection limits to long term temperature drifts. No optimal resonator size exists in the case of thermorefractive noise alone, however when both noise sources are present, the optimum size is reduced, dependent on the relative magnitude of each noise source. In addition to optimising the microresonator radius, use of Fisher information underlined the role of coupling losses in setting detection limits, hence allowing a further degree of freedom in system design. Specifically, and in opposition to common opinion, a critically coupled cavity was found to be less sensitive than a moderately under coupled one.

The analysis and discussion given, was found to hold true when either induced changes in resonance frequency or linewidth were monitored. Relative performance of the sensing modalities however is dependent on the properties of the particles of interest. Non-absorbing particles imply comparable performance between both modalities for smaller resonators, albeit broadening based sensing surpasses the more common reactive shift based sensing for larger microcavities and when absorption in the particle is strong. Careful assessment of the specific detection task at hand is therefore necessary in system design.

A. Appendix A - Justification for neglecting temperature dependent line broadening

To study the dependence of the resonance frequency and linewidth of a WGM in a microsphere, we must recall the resonance condition for Mie resonances i.e.
[nszhl(nsz)]hl(nsz)=N[nczjl(ncz)]jl(ncz),
(33)
where N = 1 or (ns/nc)2 for TE or TM modes respectively, jl(x) and hl(x) are the spherical Bessel and Hankel functions of the first kind, z = ka, k is the (complex) vacuum wavenumber, a is the resonator radius, prime denotes differentiation with respect to the argument of the respective Hankel or Bessel function and ns (nc) is the refractive index of the surrounding medium (resonator). From Eq. (33) it has previously been shown that the spectral position and linewidth (neglecting absorption) of high Q WGMs can be expressed by asymptotic expressions [53

53. C. Lam, P. T. Leung, and K. Young, “Explicit asymptotic formulas for the positions, widths, and strengths of resonances in mie scattering,” J. Opt. Soc. Am. B, 9, 1585–1592 (1992). [CrossRef]

], which when written in our notation take the form
z0l+1/2nc+
(34)
Γ2ca(nc2ns2)1nsz02yl2(nsz0)
(35)
where yl(x) are the spherical Neumann functions. We note we make no restriction to resonators in air as done in [53

53. C. Lam, P. T. Leung, and K. Young, “Explicit asymptotic formulas for the positions, widths, and strengths of resonances in mie scattering,” J. Opt. Soc. Am. B, 9, 1585–1592 (1992). [CrossRef]

]. Considering then TE modes for definiteness we can show by differentiating Eqs. (34) and (35) and using standard properties of the spherical Neumann functions [34

34. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, (Dover Publications, New York, 1970).

], that the ratio
ρ=dΓ/dTdω0/dT=dΓ/dncdω0/dnc=2Q(ns2nc2ns2nsz0lyl1(nsz0)(l+1)yl+1(nsz0)(2l+1)yl(nsz0)).
(36)
Terms within the parentheses are of order z0l ∼ 102, such that the 1/Q factor dominates. Eq. (36) therefore demonstrates that the variation in the linewidth from temperature fluctuations is many orders of magnitude smaller than the variation in the resonance frequency since WGMs possess high Q factors. Whilst algebraically more involved, the derivation for TM modes similar conclusions.

B. Appendix B - Derivation of JΓ,Γ in the presence of thermorefractive noise

Our derivation starts by noting
JΓ,Γ=j=1NΩ0[1+Λj2σt2Λj2Δjσt4sech2(ΛjΔjσt2)]1Γ2pId,jdId,j.
(37)
Evaluating the summation for the first term exactly and approximating the summation as an integral in the second and third terms as before yields
JΓ,ΓNΩΓ2+2/πΓ2ΔΩσt{0exp(Λ22σt2)Ω/2Ω/2[Λ2σt2cosh(ΛΔσt2)]exp(Δ22σt2)dΔdΛ0exp(Λ22σt2)Ω/2Ω/2[Λ2Δ2σt4sech(ΛΔσt2)]exp(Δ22σt2)dΔdΛ}.
(38)
The third term can be treated in an analogous manner to the integral in Eq. (13) by again making the approximation sechx ≈ exp[−x2/2] to yield
JΓ,ΓNΩΓ2+2/πΓ2ΔΩσt0exp(Λ22σt2)XΩXΩcoshxexp(x22σt2Λ2)dxdΛ0.563σtΓ2ΔΩ
(39)
where the change of variables x=ΛΔ/σt2 has been made. Upon performing the integration over x the second term becomes
1Γ2ΔΩσt20Λ2[erf(Zσt+Λ2σt)+erf(ZσtΛ2σt)]dΛ
(40)
where 2Z = Ω/σt and erf denotes the error function. Noting that the error function is an odd function and that erf(2) = 0.995 ≈ 1 the kernel can be taken as zero for Λ>(Z+22)σt. Further neglecting exp(−4) terms after integration of the resulting finite definite integral (40) yields
JΓ,ΓNΩΓ2+2σt3Γ2ΔΩZ(3+Z2)0.563σtΓ2ΔΩ=ΩΓ36Z+Z30.8453βZ.
(41)
Finally noting that Z ≫ 0.845 for realistic scenarios Eq. (16) quickly follows.

C. Appendix C - Derivation of Eqs. (31) and (32)

Consider Eq. (21) which we write in the form N = C(R)F0/AQ where
C(R)=(nc2ns2)Re[α]R3|Yll(π/2)|2.
(42)
Substituting in A = 4Q0Qc/(Q0 + Qc)2 and Q1=Q01+Qc1+Qm1 yields
N=C(R)F0Q0(1+Qc/Q0)24Qc/Q0(1+Qc/Q0Qc/Q0+1Qm).
(43)
Letting f1(R) = C(R)F0/Q0 and f2 = N/f1, we can write
dNdR=f1Rf2+f1f2QmQmR+f1f2xxR,
(44)
where x = Qc/Q0. We wish to find the conditions under which dN/dR = 0. Variation of the coupling losses Qc by means of adjusting the coupling distance allows us to first zero the third term of Eq. (44) by setting ∂f2/∂x = 0. Noting the requirement that x > 0 it quickly follows that ∂f2/∂x = 0 when Eq. (31) holds. In the limit that 1/Qm → 0 the result of Section 4 (i.e. Qc/Q0 = 2) is restored. Upon optimisation of the coupling distance we thus determine the optimal microcavity radius by solution of
f1R(2716+98Qm)f1QmR98Qm2=0,
(45)
which follows by substitution of Eq. (31) into Eq. (44), expanding in terms of Qm1 and equating to zero. Further performing a Taylor expansion of f1(R) about the optimal radius size for Qm1=0 (i.e. Ropt) and noting ∂f1/∂R|R=Ropt = 0 yields
2f1R2|R=RoptδR(2716+98Qm)f1(Ropt)QmR=0,
(46)
where δR is the change in the optimal microcavity radius from introduction of Qm. Rearrangement of Eq. (46) yields Eq. (32).

Acknowledgments

We gratefully acknowledge financial support for this work from the Max Planck Society (FV), the Alexander von Humboldt Foundation (MRF), the National Undergraduate Innovational Experimentation Program and the National Fund for Fostering Talents of Basic Science (WLJ -Grant No. J1030310 and No. J1103205).

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OCIS Codes
(000.5490) General : Probability theory, stochastic processes, and statistics
(260.5740) Physical optics : Resonance
(280.1415) Remote sensing and sensors : Biological sensing and sensors
(110.3055) Imaging systems : Information theoretical analysis
(240.3990) Optics at surfaces : Micro-optical devices

ToC Category:
Sensors

History
Original Manuscript: November 6, 2013
Revised Manuscript: January 6, 2014
Manuscript Accepted: February 3, 2014
Published: March 3, 2014

Virtual Issues
Vol. 9, Iss. 5 Virtual Journal for Biomedical Optics

Citation
Matthew R. Foreman, Wei-Liang Jin, and Frank Vollmer, "Optimizing detection limits in whispering gallery mode biosensing," Opt. Express 22, 5491-5511 (2014)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-22-5-5491


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