## Optimizing detection limits in whispering gallery mode biosensing |

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

1. M. Baaske and F. Vollmer, “Optical Resonator Biosensors: Molecular Diagnostic and Nanoparticle Detection on an Integrated Platform,” ChemPhysChem **13**, 427–436 (2012). [CrossRef] [PubMed]

3. F. Vollmer, S. Arnold, and D. Keng, “Single virus detection from the reactive shift of a whispering-gallery mode,” Proc. Natl. Acad. Sci. USA **105**, 20701–20704 (2008). [CrossRef] [PubMed]

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]

5. F. Vollmer, S. Arnold, D. Braun, I. Teraoka, and A. Libchaber, “Multiplexed DNA quantification by spectroscopic shift of two microsphere cavities,” Biophys. J. **85**, 1974–1979 (2003). [CrossRef] [PubMed]

6. 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]

7. 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]

8. W. Ahn, S. V. Boriskina, Y. Hong, and B. M. Reinhard, “Photonic-plasmonic mode coupling in on-chip integrated optoplasmonic molecules,” ACS Nano. **6**, 951–960 (2012). [CrossRef]

9. F. Vollmer and L. Yang, “Label-free detection with high-Q microcavities: a review of biosensing mechanisms for integrated devices,” Nanophot. **1**, 267–291 (2012). [CrossRef]

*Q*) resonances (fundamentally limited to ∼ 10

^{10}[10

10. M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. **21**, 453–455 (1996). [CrossRef] [PubMed]

11. M. R. Foreman and F. Vollmer, “Theory of resonance shifts of whispering gallery mode sensors by arbitrary plasmonic nanoparticles,” New. J. Phys. **15**, 083006 (2013). [CrossRef]

13. V. R. Dantham, S. Holler, C. Barbre, D. Keng, V. Kolchenko, and S. Arnold, “Label-free detection of single protein using a nanoplasmonic-photonic hybrid microcavity,” Nano. Lett. **13**, 3347–3351 (2013). [CrossRef] [PubMed]

## 2. Measurement and noise in WGM sensing

1. M. Baaske and F. Vollmer, “Optical Resonator Biosensors: Molecular Diagnostic and Nanoparticle Detection on an Integrated Platform,” ChemPhysChem **13**, 427–436 (2012). [CrossRef] [PubMed]

*ω*is tuned close to a resonance of the microcavity (with frequency

*ω*

_{0}and full width at half maximum (FWHM) Γ), a Lorentzian dip is observed in the transmission profile of the input beam (see Fig. 1) as described by where

*I*

_{0}is the incident beam power and

*A*describes the coupling efficiency to the microcavity (or transmission depth). An important point to note, however, is that whilst Eq. (1) describes a continuous lineshape, in practice discrete samples are taken at a fixed interval ΔΩ ≜

*β*Γ over a finite bandwidth Ω ≜

*W*Γ. In what follows we shall denote the discrete sampling frequencies and associated power level as

*ω*and

_{j}*I*=

_{j}*I*(

*ω*) respectively. For a single laser sweep a total of

_{j}*N*

_{Ω}data points,

*I*, are collected (

_{d,j}*j*= 1, 2,...,

*N*

_{Ω}), which for convenience we stack into a data vector

**I**

*= (*

_{d}*I*

_{d}_{,1},

*I*

_{d}_{,2},...,

*I*

_{d,NΩ}). For each laser scan the parameters

*ω*

_{0}, Γ and

*A*can then be estimated, for example, by numerical fitting. Determination of resonance shifts and broadening from, say, the presence of a biomolecule [6

6. 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]

7. 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]

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]

*I*can be described by the continuous PDF where

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

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]

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]

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]

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]

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. I. Teraoka, “Analysis of thermal stabilization of whispering gallery mode resonance,” Opt. Commun. **310**212–216 (2014). [CrossRef]

*ω*

_{0}→

*ω*

_{0}+

*ω*in Eq. (1), where

_{t}*ω*represents the thermorefractive resonance shift from the unperturbed resonance frequency, allows us to describe the effect of thermorefractive noise via the PDF where a Gaussian PDF is again assumed. Equivalently, the resultant random fluctuations in the measured power are described by where Eq. (4) derives from the law of total probability, Δ

_{t}*=*

_{j}*ω*−

_{j}*ω*

_{0}and

*ω*∈ [

*ω*

_{0}− Ω/2,

*ω*

_{0}+ Ω/2]) achieves greatest measurement accuracy. Only this case is therefore considered in what follows.

## 3. Fisher information and detection limits

**w**= (

*ω*

_{0}, Γ,

*A*), but instead infer them from noisy power readings. As such the Fisher information matrix can be written in the form 𝕁

**= 𝔾**

_{w}*𝕁*

^{T}**𝔾, where 𝔾 =**

_{I}*∂*

**I**/

*∂*

**w**is a matrix of derivatives,

*denotes the matrix transpose and 𝕁*

^{T}**is the Fisher information matrix associated with estimation of the transmitted power**

_{I}**I**. Initially, we assume that the noise on each data point is statistically independent such that we can write

*δ*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

_{i,j}**]**

_{I}*=*

_{jj}*J*

_{Ij}and

**G**

*=*

_{j}*∂I*

_{j}/∂**w**has been used.

*p*

_{Id,j}

*dI*= 1, however, to evaluate the second term we again approximate the summation over

_{d,j}*j*as an integration over frequency, yielding where a change in the integration variable has also been performed. Further approximating sech

*x*≈ exp[−

*x*

^{2}/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 where

*U*(

*a*,

*b*,

*z*) is the confluent hypergeometric function [34]. 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

*σ*. 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

_{t}*ω*

_{0}(in the presence of thermorefractive noise/laser jitter) in most cases. Our detection limit is then given by where the second approximation holds for

*σ*≪

_{t}*W*Γ. Eq. (15) shows that the detection limit scales with the number of sampling points as

*J*

_{Γ,Γ}follows in a similar manner to that shown here for

*J*

_{ω0,ω0}, however greater care must be taken with the size of the integration window, so as to avoid divergent integrals. For completeness the derivation is presented in Appendix B and only the final result is given here; chiefly such that where the second approximation again holds when

*σ*≪

_{t}*W*Γ. By virtue of the dependence of Δ

*ω*and ΔΓ

_{t,d}*on the resonance linewidth, the detection limits imposed by noise inherit a dependence on the size of the microresonator, as depicted in Fig. 2. Specifically Fig. 2(a) shows the dependence of the intrinsic (i.e. limited by radiation and absorption losses) quality factor*

_{t,d}*Q*

_{0}=

*ω*

_{0}/Γ for a fused silica microsphere resonator of radius

*R*immersed in water, supporting a fundamental TE WGM (first radial order) resonance at 780 nm. Curves in Fig. 2 were found by numerically solving the exact Mie resonance conditions of a microsphere [11

11. M. R. Foreman and F. Vollmer, “Theory of resonance shifts of whispering gallery mode sensors by arbitrary plasmonic nanoparticles,” New. J. Phys. **15**, 083006 (2013). [CrossRef]

*μ*m it can be seen that radiation losses dictate the resonance lifetime, whilst for larger resonators water absorption becomes dominant, producing an approximate linear relationship between intrinsic quality factor and resonator size. This linear dependence arises since for larger

*l*WGMs a greater proportion of the mode volume lies within the cavity and not in the surrounding absorbing host. It should be noted that for very large resonators of radius ≳ 1 mm, absorption within the cavity volume dominates, however for common biosensing applications resonators of this size are not desirable and hence we neglect cavity absorption in what follows. Figure 2(b), meanwhile, plots the size dependence of the various detection limits Δ

*ω*and ΔΓ

_{t,d}*(normalised to linewidth). Simulation parameters were*

_{t,d}*β*= 10

^{−3},

*W*= 20,

*σ*=

_{d}*I*

_{0}/5. To model the thermorefractive noise/laser jitter an equivalent temperature fluctuation of Δ

*T*= 5 mK was taken such that

*σ*= (

_{t}*ω*

_{0}Δ

*T /n*)

_{c}*dn*, where

_{c}/dT*ñ*=

_{c}*n*+

_{c}*iκ*is the complex refractive index of the microcavity and

_{c}*dn*= 1.45×10

_{c}/dT^{−5}[25

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

*T*gives comparable noise levels to detector noise. Specifically, for these simulation parameters we find Δ

*ω*≈ 2.5 fm, ΔΓ

_{d}*≈ 5 fm, Δ*

_{d}*ω*≈ 2 fm and ΔΓ

_{t}*≈ 0.3 fm for*

_{t}*Q*= 10

^{7}at 780 nm. Clear differences between detector and thermorefractive noise can be seen in Fig. 2. In particular, the detection limits for detector noise scale linearly with linewidth (normalised detection limits are hence a constant function of

*Q*), whilst those for thermorefractive noise exhibit a more complicated dependence, albeit monotonically increasing, i.e. smaller spheres are less susceptible to thermorefractive noise due to a small modal volume (a point further discussed later). We note, however, that suppression of laser jitter can be achieved experimentally [35

35. 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,” Proc. Natl. Acad. Sci. USA **108**, 5976–5979 (2011). [CrossRef] [PubMed]

*T*∼ 30

*μ*K is more appropriate [25

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

*f*) noise, whilst thermorefractive noise inherently has a non-uniform power spectrum [25

**21**, 697–705 (2004). [CrossRef]

**(and hence 𝕁**

_{I}**) is no longer diagonal. Knowledge of the spectral power density of the noise sources, e.g. [15**

_{I}15. J. Knittel, J. D. Swaim, D. L. McAuslan, G. A. Brawley, and W. P. Bowen, “Back-scatter based whispering gallery mode sensing,” Sci. Rep. **3**, 2974 (2013). [CrossRef] [PubMed]

**21**, 697–705 (2004). [CrossRef]

## 4. Optimal whispering gallery mode biosensors

### 4.1. Resonance shifts

*N*is minimised when

*Q*

_{c}/Q_{0}= 2 (or equivalently

*A*= 0.89) as depicted in Fig. 4(a). In contradiction to common wisdom,

*Q*

_{c}/Q_{0}= 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]

*Q*factors for under coupled modes.

*R*= 46.8

*μ*m with a coupling distance of 1.17

*μ*m 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,

*σ*, merely scales the detection limit

_{d}*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

*μ*m whereby

*N*≈ 10

^{−1.7}. Evidently, these figures are below the single InfA viron limit given the noise levels chosen.

*μ*m, 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.

*F*= (

*σ*/Γ)(

_{t}*β/W*)

^{1/2}and detection limit given by 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

*R*

^{3}dependence. Smaller resonators are thus less susceptible to thermorefractive noise for fixed

*σ*. Furthermore, smaller cavities imply smaller mode volumes such that the variance of temperature fluctuations,

_{t}*σ*are also smaller [25

_{t}**21**, 697–705 (2004). [CrossRef]

*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

*R*

^{3}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.

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

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. 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]

*σ*is 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

_{s}*R*(as opposed to

*R*

^{3}) and differing prefactors. The optimal coupling distances is hence once again given by

*Q*

_{c}/Q_{0}= 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

*μ*m. 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)).

### 4.2. Linewidth changes

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]

*δ*Γ

_{abs}and

*δ*Γ

_{sca}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. 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]

*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. 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]

*F*

_{0}is as defined above and 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.

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

*δ*Γ =

*δ*Γ

_{sca}+

*δ*Γ

_{abs}+

*δ*Γ

_{cc}. The additional

*δ*Γ

_{cc}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]

*δ*Γ

_{cc}= 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. 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]

*μ*m, 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

**25**, 5616–5620 (2013). [CrossRef]

## 5. Thermal stability

*N*(

*T*) =

*N*(

*T*

_{0}) + (

*T*−

*T*

_{0})

*dN/dT*+ ···, such that we can use the temperature gradient of the minimum detectable number of particles

*dN/dT*to quantitatively assess thermal stability of our previously derived detection limits. Restricting to detector noise only, the temperature derivatives of Eqs. (24) and (25) can be determined (for simplicity we consider only the temperature variation of the real part of the refractive indices of the cavity and surrounding water, where we take

*dn*= −8.33 × 10

_{s}/dT^{−5}[52

52. A. N. Bashkatov and E. A. Genina, “Water refractive index in dependence on temperature and wavelength: a simple approximation,” Proc. SPIE **5068**, 393–395 (2003). [CrossRef]

*N*. Use of larger resonators is thus seen to be experimentally beneficial, however the choice of cavity size, will depend on the relative importance assigned to detection sensitivity and temperature stability. Naturally, other considerations, such as space restrictions for on-chip integration and fabrication capabilities may also play a determining role.

## 6. Additional losses

*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*)

*F*

_{0}/

*AQ*, where now

*Q*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 where the latter approximation holds when

_{m}*∂Q*, all quantities in Eq. (32) are positive. Consequently,

_{m}/∂R*∂Q*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,

_{m}/∂R*∂Q*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

_{m}/∂R*∂Q*is positive, hence motivating the use of larger resonators. Equivalent results follow when considering line broadening as a sensing mechanism.

_{m}/∂R## 7. Conclusions

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

*N*= 1 or (

*n*)

_{s}/n_{c}^{2}for TE or TM modes respectively,

*j*(

_{l}*x*) and

*h*(

_{l}*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

*n*(

_{s}*n*) 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

_{c}*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]

*y*(

_{l}*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]

*z*

_{0}∼

*l*∼ 10

^{2}, 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 *J*_{Γ,Γ} in the presence of thermorefractive noise

*x*≈ exp[−

*x*

^{2}/2] to yield where the change of variables

*x*the second term becomes where 2

*Z*= Ω/

*σ*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

_{t}*Z*≫ 0.845 for realistic scenarios Eq. (16) quickly follows.

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

*N*=

*C*(

*R*)

*F*

_{0}/

*AQ*where Substituting in

*A*= 4

*Q*

_{0}

*Q*/(

_{c}*Q*

_{0}+

*Q*)

_{c}^{2}and

*f*

_{1}(

*R*) =

*C*(

*R*)

*F*

_{0}/

*Q*

_{0}and

*f*

_{2}=

*N/f*

_{1}, we can write where

*x*=

*Q*

_{c}/Q_{0}. We wish to find the conditions under which

*dN/dR*= 0. Variation of the coupling losses

*Q*by means of adjusting the coupling distance allows us to first zero the third term of Eq. (44) by setting

_{c}*∂f*

_{2}/

*∂x*= 0. Noting the requirement that

*x*> 0 it quickly follows that

*∂f*

_{2}/

*∂x*= 0 when Eq. (31) holds. In the limit that 1/

*Q*→ 0 the result of Section 4 (i.e.

_{m}*Q*

_{c}/Q_{0}= 2) is restored. Upon optimisation of the coupling distance we thus determine the optimal microcavity radius by solution of which follows by substitution of Eq. (31) into Eq. (44), expanding in terms of

*f*

_{1}(

*R*) about the optimal radius size for

*R*

_{opt}) and noting

*∂f*

_{1}/

*∂R*|

_{R=Ropt}= 0 yields where

*δR*is the change in the optimal microcavity radius from introduction of

*Q*. Rearrangement of Eq. (46) yields Eq. (32).

_{m}## Acknowledgments

## References and links

1. | M. Baaske and F. Vollmer, “Optical Resonator Biosensors: Molecular Diagnostic and Nanoparticle Detection on an Integrated Platform,” ChemPhysChem |

2. | K. J. Vahala, “Optical microcavities” Nature |

3. | F. Vollmer, S. Arnold, and D. Keng, “Single virus detection from the reactive shift of a whispering-gallery mode,” Proc. Natl. Acad. Sci. USA |

4. | M. Noto, F. Vollmer, D. Keng, I. Teraoka, and S. Arnold, “Nanolayer characterization through wavelength multiplexing of a microsphere resonator,” Opt. Lett |

5. | F. Vollmer, S. Arnold, D. Braun, I. Teraoka, and A. Libchaber, “Multiplexed DNA quantification by spectroscopic shift of two microsphere cavities,” Biophys. J. |

6. | 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. |

7. | S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer., “Shift of whispering-gallery modes in micro-spheres by protein adsorption,” Opt. Lett. |

8. | W. Ahn, S. V. Boriskina, Y. Hong, and B. M. Reinhard, “Photonic-plasmonic mode coupling in on-chip integrated optoplasmonic molecules,” ACS Nano. |

9. | F. Vollmer and L. Yang, “Label-free detection with high-Q microcavities: a review of biosensing mechanisms for integrated devices,” Nanophot. |

10. | M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett. |

11. | M. R. Foreman and F. Vollmer, “Theory of resonance shifts of whispering gallery mode sensors by arbitrary plasmonic nanoparticles,” New. J. Phys. |

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. |

13. | V. R. Dantham, S. Holler, C. Barbre, D. Keng, V. Kolchenko, and S. Arnold, “Label-free detection of single protein using a nanoplasmonic-photonic hybrid microcavity,” Nano. Lett. |

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. |

15. | J. Knittel, J. D. Swaim, D. L. McAuslan, G. A. Brawley, and W. P. Bowen, “Back-scatter based whispering gallery mode sensing,” Sci. Rep. |

16. | L. Xu, H. Li, X. Wu, L. Shang, and L. Liu, “Ultra-sensitive label-free biosensing by using single-mode coupled microcavity laser,” Proc. SPIE , |

17. | T. McGarvey, A. Conjusteau, and H. Mabuchi, “Finesse and sensitivity gain in cavity-enhanced absorption spectroscopy of biomolecules in solution,” Opt. Express , |

18. | J. D. Swaim, J. Knittel, and W. P. Bowen, “Detection limits in whispering gallery biosensors with plasmonic enhancement,” Appl. Phys. Lett. |

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. |

20. | S. Arnold, R. Ramjit, D. Keng, V. Kolchenko, and I. Teraoka, “Microparticle photophysics illuminates viral biosensing,” Faraday Disc. |

21. | J. L. Nadeau, V. S. Ilchenko, D. Kossakovski, G. H. Bearman, and L. Maleki, “High-Q whispering-gallery mode sensor in liquids,” Proc. SPIE |

22. | N. M. Hanumegowda, C. J. Stica, B. C. Patel, I. White, and X. Fan, “Refractometric sensors based on microsphere resonators,” Appl. Phys. Lett. |

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. |

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 |

25. | M. L. Gorodetsky and I. S. Grudinin, “Fundamental thermal fluctuations in microspheres,” J. Opt. Soc. Am. B |

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. |

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. |

28. | T. Carmon, L. Yang, and K. J. Vahala, “Dynamical thermal behavior and thermal self-stability of microcavities,” Opt. Express |

29. | I. Teraoka, “Analysis of thermal stabilization of whispering gallery mode resonance,” Opt. Commun. |

30. | L. L. Scharf, |

31. | C. Vignet and J.-F. Bercher, “Analysis of signals in the Fisher-Shannon information plane,” Phys. Lett. A |

32. | H. Cramér, |

33. | M. R. Foreman and P. Török, “Information and resolution in electromagnetic optical systems,” Phys. Rev. A |

34. | M. Abramowitz and I. Stegun, |

35. | 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,” Proc. Natl. Acad. Sci. USA |

36. | A. Zeira and A. Nehorai, “Frequency domain Cramer-Rao bound for Gaussian Processes,” IEEE Trans. Acoust. Speech. Sig. Proc. |

37. | Y.-F. Xiao, “Optical cavity QED in Solid-State Systems - Theory to Realization,” PhD thesis (University of Science and Technology of China, 2007). |

38. | M. L. Gorodetsky and V. S. Ilchenko, “Optical microsphere resonators: optimal coupling to high-Q whispering-gallery modes,” J. Opt. Soc. Am. B |

39. | M. Bass, C. M. DeCusatis, and J. M. Enoch, |

40. | G. M. Hale and M. R. Querry, “Optical Constants of Water in the 200-nm to 200-μm Wavelength Region,” Appl. Opt. |

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. |

42. | J. Topolancik and F. Vollmer, “Photoinduced transformations in bacteriorhodopsin membrane monitored with optical microcavities,” Biophys. J. |

43. | D. Q. Chowdhury, S. C. Hill, and P.W. Barber, “Morphology-dependent resonances in radially inhomogeneous spheres,” J. Opt. Soc. Am. A |

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 |

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 |

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. |

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. |

48. | U. Resch-Genger, M. Grabolle, S. Cavaliere-Jaricot, R. Nitschke, and T. Nann, “Quantum dots versus organic dyes as fluorescent labels,” Nat. Methods |

49. | M. R. Foreman and F. Vollmer, “Level repulsion in hybrid photonic-plasmonic microresonators for enhanced biodetection,” Phys. Rev. A |

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. |

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 |

52. | A. N. Bashkatov and E. A. Genina, “Water refractive index in dependence on temperature and wavelength: a simple approximation,” Proc. SPIE |

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, |

**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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### References

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- A. Mazzei A, S. Götzinger, L. de S. Menezes, G. Zumofen, O. Benson, 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]
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