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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 7 — Apr. 8, 2013
  • pp: 8724–8735
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Accurate determination of the quality factor and tunneling distance of axisymmetric resonators for biosensing applications

M. Imran Cheema and Andrew G. Kirk  »View Author Affiliations


Optics Express, Vol. 21, Issue 7, pp. 8724-8735 (2013)
http://dx.doi.org/10.1364/OE.21.008724


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Abstract

Due to ultra high quality factor (106 − 109), axisymmetric optical microcavities are popular platforms for biosensing applications. It has been recently demonstrated that a microcavity biosensor can track a biodetection event as a function of its quality factor by using phase shift cavity ring down spectroscopy (PS-CRDS). However, to achieve maximum sensitivity, it is necessary to optimize the microcavity parameters for a given sensing application. Here, we introduce an improved finite element model which allows us to determine the optimized geometry for the PS-CRDS sensor. The improved model not only provides fast and accurate determination of quality factors but also determines the tunneling distance of axisymmetric resonators. The improved model is validated numerically, analytically, and experimentally.

© 2013 OSA

1. Introduction

Microcavities are widely used in many applications such as biosensing, nonlinear processes, lasers and optomechanics. Most previously published work with microcavity biosensors has employed measurement of the change in resonant frequency of the sensor as a function of a biodetection event [1

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

, 2

2. X. Fan, I. M. White, S. I. Shopoua, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Analytica Chimica Acta 620, 8–26 (2008) [CrossRef]

] with only a few instances of using the quality factor as a sensing metric [3

3. L. Maleki and V. S. Ilchenko, “Techniques and devices for sensing a sample by using a whispering gallery mode resonator,” California Institute of Technology, US Patent 6,490,039, (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] .

]. It has been shown that the quality factor of microcavities can be measured in the time domain using phase shift shift cavity ring down spectroscopy (PS-CRDS) [6

6. 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] .

]. This approach has recently been applied to biosensing applications [7

7. 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 order to extend this work to ultra sensitive biosensing applications (e.g. determination of binding kinetics, detection of pMfM concentrations, and proteins, viruses etc), it is important to design a cavity with the optimum parameters for high sensitivity. One of the goals of the current work, is to rapidly estimate the quality factors of the microcavities with high accuracy to address these applications.

Fig. 1 a) (False color) Logarithmic intensity of the fundamental TE mode of a silica microsphere in air. b) Normalized field intensity along the dashed line shown in (a). Dotted curve shows potential, normalized to k2. Tunneling distance is indicated by t.

In order to provide accurate determination of the WGM quality factor, we have improved Oxborrow’s model by modifying its master equation and implementing the PML along the boundaries of the computation domain. In the present work, geared towards sensing applications, we expand and refine the model presented in our earlier work [24

24. M. I. Cheema and A. G. Kirk, “Implementation of the perfectly matched layer to determine the quality factor of axisymmetric resonators in COMSOL,” in COMSOL conference,Boston, Oct 8 2010.

]. Other researchers [25

25. A. Arbabi, Y. M. Kang, and L. L. Goddard, “Analysis and Design of a Microring Inline Single Wavelength Reflector,” in FIO , October24, 2010.

] have subsequently reported the inclusion of PML in Oxborrow’s model for determining resonant frequencies and corresponding mode profiles of microring resonators but mathematical details have not been provided.

Our modified model does not have any of the drawbacks of Oxborrow’s model. Moreover, we have computed the quality factors of all the modes without using any fitting algorithms as opposed to the approach taken in [13

13. M. Karl, B. Kettner, S. Burger, F. Schmidt, H. Kalt, and M. Hetterich, “Dependencies of micropillar cavity quality factors calculated with finite element methods,” Opt. Express 2, (2009).

]. Furthermore, with the modified Oxborrow’s method, tunneling distances can also be accurately extracted for microcavities of various shapes. In the present work, a simple expression for computing tunneling distance of microtoroidal cavities is also provided.

In our model, we treat the PML as an anisotropic absorber and implement it in the cylindrical coordinate system. Our model is applicable to any axisymmetric resonator geometry but due to the availability of analytical expressions for spherical resonators, we have validated the model by determining the quality factors and tunneling distances of a silica microsphere in air. We have found that our simulation results are in excellent agreement with the analytical results. We also apply our model to microtoroidal cavities immersed in liquid and show that our results are consistent with those obtained by experiments. The model is then used to determine the optimum parameters of a microtoroidal cavity sensor based upon phase shift cavity ring down spectroscopy.

The paper is organized as follows. In section 2, an improved FEM model is introduced by incorporating the PML. The optimal parameters of the PML are discussed in section 2.1. The analytical expressions of the quality factor and the tunneling distance of microspheres are presented in section 2.3. These expressions are then used in section 3.1 for comparing the modeling results. In the same section, an empirical expression for tunneling distance of fundamental modes of a toroidal cavity is also provided. In section 3.2, the model is validated experimentally by measuring the quality factors of the microtoroidal cavities in liquid. In section 3.3, the model is validated numerically by running convergence test. In section 4, we show that the model can be applied for sensing applications. In this section, we show that for a sensor, whose sensing metric is change in the quality factor, an optimum geometry exists for achieving maximum sensitivity.

2. Mathematical description

Applying Galerkin’s method to the wave equation and after using the boundary conditions for open resonators, one can arrive at the FEM equation in the weak form [16

16. M. Oxborrow, “Traceable 2-D finite element simulation of the whispering gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. on Microwave Theory Tech. 55, 1209–1218 (2007) [CrossRef] .

]:
V(×H˜*)ε1(×H)α(H˜*)(H)+c2H˜*2Ht2dV)=0
(1)
where H⃗ represents the magnetic field of the resonator and H˜ represents the test magnetic field, an essential component of the weak form. The second term of Eq. (1) represents a penalty term to suppress false solutions. None of the field components will depend upon the azimuthal coordinate ϕ in the axisymmetric resonators, resulting in reduction of the 3D problem to a 2D problem.

2.1. Perfectly matched layer formulation

A PML can be treated as an anisotropic absorber in which the diagonal permittivity and permeability tensors of the absorber are modified according to Eq. (2) [26

26. F. L. Teixeira and W. C. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microwave Guided Wave Lett. 7, 371–373, (1997) [CrossRef] .

].
ε¯=εΛ¯,μ¯=μΛ¯
(2)
The radial and axial modification factors are represented by Λ̄, which is given by Eq. (3)
Λ¯=(r˜r)(szsr)r^+(rr˜)(szsr)ϕ^+(r˜r)(srsz)z^
(3)
where
sr={nmedium0rrpmlnmediumjG(rrpmltrpml)Nr>rpml
(4)
sz={nmediumjG(zlpmlztlpml)Nz<zlpmlnmediumzlpmlzzupmlnmediumjG(zzupmltupml)Nz>zupml
(5)
r˜={r0rrpmlrjG((rrpml)N+1(N+1)tpmlN)r>rpml
(6)

where trpml, tupml, tlpml are the PML thicknesses in the radial, +z and −z directions respectively and rpml, zupml, zlpml are the locations of the start of PML in the radial, +z and −z directions respectively. nmedium is refractive index of the medium, N is order of the PML, and G is a positive integer.

In the PML expressions (sr, sz, ), the imaginary component contributes to the attenuation of waves in the PML but at the same time, due to the discrete nature of the FEM mesh, a large imaginary component will introduce reflections at the interface between the PML and the medium. In order to determine the optimal value for the imaginary component, we have investigated linear, quadratic, and cubic PML of different thicknesses for various values of G by running many simulations for various sphere diameters. To deduce the optimum values of the parameters, we then compared the simulation results for QWGM of spherical cavities with the analytical ones. The simulation results show that a linear (i.e. N = 1), and λ/4 thick PML with a G value of 5 is optimum. We have also used these optimum values for the simulations of the microtoroidal cavities. The location of PML is also important to obtain accurate results. The radial PML should be greater than the tunneling distance (t) of the microcavity and the z PML should be greater than FWHM (wz) of WGM along the z axis. After running series of simulations we find the following as optimum values (w.r.t. (r,z) = (0,0), see Fig. 1):
rpml6t
(7)
z(u,l)pml|5.5wz|
(8)

2.2. Finite element method equation with perfectly matched layer

In order to incorporate the PML, we have reformulated Eq. (1) in the following way:
V((×H˜*)ε¯1(×H)α(×H˜*)(H)+c2H˜*μ¯2Ht2)dV=0
(9)
By casting Eq. (9) into the FEM software COMSOL, a full vectorial finite element model of a silica sphere in air can be obtained. By using the eigenvalue solver in COMSOL, resonant frequencies (fr) of all the modes can easily be determined. Quality factor due to the WGM radiation can be calculated as [16

16. M. Oxborrow, “Traceable 2-D finite element simulation of the whispering gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. on Microwave Theory Tech. 55, 1209–1218 (2007) [CrossRef] .

]:
Qwgm=(fr)2(fr)
(10)

2.3. Analytical expressions of the spherical resonator

2.3.1. Quality factor

The quality factor due to WGM radiation losses in a spherical microcavity can be written as [27

27. V. V. Datsyuk, “Some characteristics of resonant electromagnetic modes in a dielectric sphere,”Appl. Phys. B54, 184–187, (1992).

]:
Qwgm=12(m+12)p12M(p21)12e2Tm,m1
(11)
where
  • m = azimuthal mode number
p2=εsphereεmedium
(12)
Tm=(m+12)(mltanh(ml)
(13)
ml=cosh1(p(11m+12(tq0β+p12M(p21)))1)
(14)
β=(12(m+12))13
(15)
M={0ForTE1ForTM
(16)
tq0 is the qth root of equation Fairy(tq0)=0

m can be calculated using the characteristic equation for WGM frequencies [28

28. L. A. Weinstein, Open Resonators and Open Waveguides (The Golem Press, Boulder, Colorado, USA, 1969) pp. 298.

]:
p12Mjm'[pk0a]jm[pk0a]=hm'[k0a]hm[k0a]
(17)
where j, h are Bessel functions, k0 is the wave number (2π/λ0), and a is the radius of a microsphere.

It should be noted that Eq. (11) is an asymptotic solution for the Qwgm of the spherical resonator which requires m ≫ 1, however the error is less than 1% for m ≥ 19 [29

29. J. R. Buck and H. J. Kimble, “Optimal sizes of dielectric microspheres for cavity QED with strong coupling,” Phys. Rev. A 67033806 (2003) [CrossRef] .

]. In our comparison for the analytical results, we have applied Eq. (11) to silica spheres with mode numbers (m) ranging from 25 – 80.

2.3.2. Tunneling distance

The Schrodinger equation for a microsphere for a fundamental mode can be written in radial coordinates (r) as [30

30. B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,”J. Opt. Soc. Am. A 10, 2, (1993) [CrossRef] .

]:
d2Ψrdr2+VrΨr=EΨr
(18)
where E = k2 is the total energy and Ψr is a position probability function of a photon. The potential (Vr) is given by
Vr=k2(1pr2)+m(m+1)r2
(19)
where pr is ratio of relative permittivity of the sphere and the surrounding medium in radial coordinates. Analytically, tunneling distance for a fundamental mode of a microsphere can be written as [30

30. B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,”J. Opt. Soc. Am. A 10, 2, (1993) [CrossRef] .

]:
t=m(m+1)ka
(20)

3. Results

3.1. Comparison between simulations and analytical results

Figure 1 shows a fundamental TE mode of a silica spherical cavity in air.

Fig. 2 Modeling results at 1550nm. WGM Quality factors for various silica sphere diameters of fundamental TE mode. OUB and OLB represent upper and lower bounds that are calculated by using Oxborrow’s model [16].

Figure 3 shows the results for tunneling distance for both microsphere and microtoroidal cavities. An empirical relation for the tunneling distance for fundamental modes of the micro-toroidal cavities can be extracted from the simulation results and is given as:
t=m(m+1)kD+d2
(21)
where D and d are major and minor diameter of a microtoroid and are shown in Fig. 3(b). The tunneling distance results are also in agreement with experimental results presented in [11

11. M. Tomes, K. J. Vahala, and T. Carmon, “Direct imaging of tunneling from a potential well,” Opt. Express 17, 19160 (2009) [CrossRef] .

].

Fig. 3 Modeling results at 1550nm. Tunneling distance of a fundamental TE mode as a function of the microcavity geometries. Inset of Fig. 3(b) shows the cross section of a microtoroidal cavity.

3.2. Comparison between simulations and experimental results

Our model is also appropriate for other axisymmetric resonators where analytical solutions are not available. In order to test this, we have used our FEM model to determine the total quality factor of microtoroidal cavities [31

31. D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003) [CrossRef] [PubMed] .

] and have compared the simulation results with experimental measurement of the quality factor. Mathematically the total quality factor (Qtotal) can be represented by:
Qtotal=1Qwgm1+Qsurroundings1+Qmaterial1+Qcoupling1
(22)

In order to validate the model with reasonable range of quality factors, it is necessary to ensure suitable experimental conditions in which to observe these quality factors. We have performed the experiments with the microtoroids immersed in ethanol (refractive index: 1.3538 at 1550nm[32

32. S. Scheurich, S. Belle, R. Hellmann, S. So, I.J.G. Sparrow, and G. Emmerson, “Application of a silica-on-silicon planar optical waveguide Bragg grating sensor for organic liquid compound detection,” Proc. SPIE 7356 (2009) [CrossRef]

]) by using the fluidic cell described in [7

7. 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 preferred ethanol over water as ethanol not only provides low refractive index contrast between the silica cavity and the surroundings but also has low absorption as compared to water at 1550nm[33

33. M. Gallignani, S. Garrigues, and M. Guardia, “Direct determination of ethanol in all types of alcoholic beverages by near-infrared derivative spectrometry,”Analyst 118, (1993) [CrossRef] .

]. This will allow us to select a wide range of microtoroidal cavities whose Qtotal is mainly limited by the Qwgm.

A broadband source (peak wavelength:1530nm) is used to couple the light into microtoroids via a tapered optical fiber (taper waist: ≤ 1μm). The resonant peaks are observed on a high resolution optical spectrum analyzer (Apex AP 2443B, resolution: 0.16pm) and quality factor is determined by Lorentz curve fitting of the peaks (Q = λλ). The tapered fiber is positioned (using 10nm resolution nanostages) along the equator of the cavity to couple light into fundamental transverse modes. It is ensured that the tapered fiber does not touch the cavity during the measurements. In order to minimize the effect of Qcoupling on Qtotal, the power at the peak wavelength of the source is set around −48dbm (Optical spectrum analyzer sensitivity: −70dbm) and all the measurements are taken in highly undercoupled regime. The comparison between simulation and experimental results is shown in Fig. 4.

Fig. 4 Comparison between the modeling and the experimental results. WGM Quality factors of fundamental TM mode for various silica microtoroidal cavity diameters immersed in ethanol. Minor diameter d of each cavity is slightly different and is around 5μm ± 1μm. The experimental quality factors are determined by Lorentz curve fitting of the resonant peaks (Q = λλ) where λ ≈ 1530nm.

3.3. Computational speed and numerical accuracy

For the results presented in Figs.14, we have used the quadratic Lagrange elements of triangular shape with average element size (longest edge) of 0.3μm and the number of degrees of freedom of order 105. As an example, a microtoroidal cavity model (Computation domain size:25μm × 16μm, D = 30μm, d = 5μm) with the aforementioned statistics took only 35s to find the first 20 eigenvalues on a quad core 64 bit operating system PC.

Fig. 5 Convergence plot for the quality factor determination of a microsphere.

4. Discussion and application to PS-CRDS microcavity sensor

The results presented in section 3 show that our finite element model is both physically and numerically accurate. The quality factors for all the modes (fundamental and higher order modes) are also obtained by one single simulation rather than multiple simulations, which was the case for the original Oxborrow model. Moreover, no prior knowledge of any of the mode frequencies is required to obtain the quality factors. The model also gives fast results and without any fitting algorithms.

We have also applied our model to the PS-CRDS microtoroidal cavity sensor [7

7. 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 order to determine the optimum parameters for an application of refractometric sensing. In such a scheme, the microcavity is immersed into water and a small refractive index change (δn) is introduced. This change in refractive index will influence the Qtotal of the cavity and can easily be measured via PS-CRDS microtoroidal cavity sensor [7

7. 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] .

]. Figure 6 shows the modeling results for change in Qtotal of the cavity as a function of D. It can be clearly seen that to achieve maximum sensitivity there exists an optimum geometry for both λ = 633nm and λ = 1530nm. Due to high water absorption at 1530nm, ΔQ is lower than the one at 633nm. Since Qtotal varies with the wavelength, the optimum geometry will be different for each of the wavelength.

Fig. 6 Modeling results show the existence of an optimum geometry. Change in quality factor (ΔQ) of fundamental TM mode as a function of major diameter (D) of a microtoroidal cavity (minor diameter(d) = 6μm) immersed in water. ΔQ shows the difference between Qtotal for water and that measured when small refractive index change (δn = 10−3) is introduced into the water. (a)Modeling results for λ = 633nm (b)Modeling results for λ = 1530nm.

In order to understand that why an optimum geometry exists for a ΔQ measurement, we have plotted the individual terms of Eq. (22) as a function of the cavity geometry (Fig. 7). We have assumed Qcoupling = 0 which is a reasonable assumption as experimentally, contribution of Qcoupling can be minimized by taking measurements at multiple input power levels [36

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

]. Figure 7 shows that the optimum region exists close to the point of inflection of the Qtotal curve. Results in Fig. 7 show that as diameter increases, the WGM is better confined within the microcavity (see Fig. 1(a)) and so Qwgm is less influenced by the external environment, with the result that the total quality factor is limited by silica absorption.

Fig. 7 Modeling results. Various quality factors (Eq. (22)) for fundamental TM mode involved in a microtoroidal cavity immersed in water as a function of D (d = 6μm). In the optimum region the Qwgm is close to Qtotal.(a) Modeling results at λ = 633nm. (b) Modeling results at λ = 1530nm. Qsilica ≈ 1011 (Silica has very low absorption at 1530nm) and is omitted in the figure to reduce the range in y-axis.

We have formulated an empirical expression (Eq. (21)) for the tunneling distance of the fundamental modes of microtoroids. This expression is inspired by the analytical expression for the tunneling distances of microspheres by treating the microtoroid as a sphere of diameter (D + d). This formulation is not surprising as these equations are true only for fundamental modes which lie along the equatorial region of a microcavity. However, this treatment will not be true for higher order modes (non equatorial modes) as the curvature for a microsphere of diameter D + d will be quite different from a microtoroid with D and d as its dimensions (see Fig. 3(b)).

In summary, our finite element model, without any approximation in its master equation and coupled with a PML, not only gives accurate quality factors and tunneling distances, but can also determine the other important parameters (e.g. mode volumes) accurately for a wide range of applications based on axisymmetric microcavities such as biosensing, non-linear processes, and lasers.

Acknowledgments

We thank Prof. Jonathan Webb at McGill University for useful discussions which assisted in our understanding of PMLs. We are grateful to Prof. Andrea Armani at University of Southern California for her invaluable suggestions for improving the manuscript. We appreciate the efforts of Ashley Maker at University of Southern California, for fabricating the microtoroidal cavities used in this work. We are also grateful to Prof. Tal Carmon at University of Michigan for pointing us to tunneling distance calculations. This work is supported by the NSERC-CREATE training program in Integrated Sensor Systems, McGill Institute of Advanced Materials, Montreal, Canada.

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

2.

X. Fan, I. M. White, S. I. Shopoua, H. Zhu, J. D. Suter, and Y. Sun, “Sensitive optical biosensors for unlabeled targets: A review,” Analytica Chimica Acta 620, 8–26 (2008) [CrossRef]

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7.

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] .

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F. L. Teixeira and W. C. Chew, “Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates,” IEEE Microwave Guided Wave Lett. 7, 371–373, (1997) [CrossRef] .

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V. V. Datsyuk, “Some characteristics of resonant electromagnetic modes in a dielectric sphere,”Appl. Phys. B54, 184–187, (1992).

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J. R. Buck and H. J. Kimble, “Optimal sizes of dielectric microspheres for cavity QED with strong coupling,” Phys. Rev. A 67033806 (2003) [CrossRef] .

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B. R. Johnson, “Theory of morphology-dependent resonances: shape resonances and width formulas,”J. Opt. Soc. Am. A 10, 2, (1993) [CrossRef] .

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D. K. Armani, T. J. Kippenberg, S. M. Spillane, and K. J. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature 421, 925–928 (2003) [CrossRef] [PubMed] .

32.

S. Scheurich, S. Belle, R. Hellmann, S. So, I.J.G. Sparrow, and G. Emmerson, “Application of a silica-on-silicon planar optical waveguide Bragg grating sensor for organic liquid compound detection,” Proc. SPIE 7356 (2009) [CrossRef]

33.

M. Gallignani, S. Garrigues, and M. Guardia, “Direct determination of ethanol in all types of alcoholic beverages by near-infrared derivative spectrometry,”Analyst 118, (1993) [CrossRef] .

34.

A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations (Springer, 1997).

35.

V. S. Ilchenko, A. A. Savchenkov, A. B. Matsko, and L. Maleki, “Dispersion compensation in whispering-gallery modes,” J. Opt. Soc. Am. A 20, 157–162, (2003) [CrossRef] .

36.

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

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: January 11, 2013
Revised Manuscript: March 12, 2013
Manuscript Accepted: March 17, 2013
Published: April 2, 2013

Virtual Issues
Vol. 8, Iss. 5 Virtual Journal for Biomedical Optics

Citation
M. Imran Cheema and Andrew G. Kirk, "Accurate determination of the quality factor and tunneling distance of axisymmetric resonators for biosensing applications," Opt. Express 21, 8724-8735 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-7-8724


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