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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 19 — Sep. 23, 2013
  • pp: 22012–22022
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Full-vectorial whispering-gallery-mode cavity analysis

Xuan Du, Serge Vincent, and Tao Lu  »View Author Affiliations


Optics Express, Vol. 21, Issue 19, pp. 22012-22022 (2013)
http://dx.doi.org/10.1364/OE.21.022012


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Abstract

We present a full-vectorial three-dimensional whispering-gallery-mode microcavity analysis technique. With this technique, optical properties such as resonance wavelength, quality factor, and electromagnetic field distribution of a microcavity in the presence of individual nanoparticle adsorption can be simulated with high accuracy, even in the presence of field distortion from plasmon effects at a wavelength close to plasmon resonance. This formulation is applicable to a wide variety of whispering-gallery related problems, such as waveguide to cavity coupling and full wave propagation analysis of a general whispering-gallery-mode microcavity where axisymmetry along the azimuthal direction is not required.

© 2013 OSA

1. Introduction

2. Theoretical formulations

2.1. Ideal whispering-gallery-mode microcavities

Fig. 1: (a) Light circulating along the azimuthal direction in a whispering-gallery-mode microcavity (e.g. a silica microtoroid). A cylindrical coordinate system is used for modelling purposes. (b) Light propagating from ϕ to ϕ + δϕ as it passes by a bound particle.

2.2. Non-ideal whispering-gallery-mode microcavities

At an azimuthal angle ϕ0, the electrical field distribution at the cross section E(ρ, z, ϕ0) can be expanded onto the normalized WGM mode ê(ρ, z, ϕ0) at ϕ0 according to
E(ρ,z,ϕ0)=A(ϕ0)e^(ρ,z,ϕ0)
(4)
Note that by analogy to the MMM in Cartesian coordinates, we may obtain a complete set of orthogonal modes at the azimuthal angle ϕ0 that satisfies the mode equation of a perfect WGM cavity
[2ρ2+1ρρ+2z2+(k02n2(ρ,z,ϕ0)m(ϕ0)2ρ2)]e^(ρ,z,ϕ0)=0
(5)
where both the mode profile and m have become ϕ-dependent. Additionally, given that (5) is identical to (2) for any fixed angle ϕ0, one may find the Mth order resonant wavelength λr(ϕ) at ϕ0 by replacing k0 with 2π/λr, m with M, and solving the mode equation above using a two-dimensional mode solver for ideal WGM’s (such as that in [20

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

]). Note λr(ϕ) is also a ϕ-dependent quantity. After propagating an infinitesimal azimuthal angle δϕ (Fig. 1(b)), we obtain the electrical field at ϕ0 + δϕ
E(ρ,z,ϕ0+δϕ)=A(ϕ0)e^(ρ,z,ϕ0)ejm(ϕ0)δϕ
(6)
Under the approximation that, on average, photons in the mode travel the same optical path length and experience the same loss as those at the resonance wavelength λr(ϕ0), we can obtain the real mr and imaginary mi part of m(ϕ0) after infinitesimal rotation δϕ via
mr(ϕ0)=Mλr(ϕ0)λ0mi(ϕ0)=mabs(ϕ0)+mrad(ϕ0)
(7)
Here, we ignore mcouple and msurf for a purpose explained in a later section. On the other hand, E(ϕ0 + δϕ) can be expanded onto the normalized eigen mode ê(ρ, z, ϕ0 + δϕ), defined at ϕ0 + δϕ,
E(ρ,z,ϕ0+δϕ)=A(ϕ0+δϕ)e^(ρ,z,ϕ0+δϕ)
(8)

By equating the right hand sides of (6) and (8), multiplying both sides by 12η0nr(ρ,z,ϕ0+δϕ)e^(ρ,z,ϕ0+δϕ), and integrating over the cross section at ϕ0 + δϕ, we obtain the evolution of A(ϕ0 + δϕ) according to
A(ϕ0+δϕ)=A(ϕ0)ej[M+δm(ϕ)]δϕ
(9)

We may then obtain an additional loss term mm(ϕ0) characterized by
mm(ϕ0)=limδϕ01δϕln[nr(ρ,z,ϕ0+δϕ)2η0e^*(ρ,z,ϕ0+δϕ)e^(ρ,z,ϕ)dσ]
(10)
arising from the mode mismatch between ê(ϕ0 + δϕ) and ê(ϕ0) in addition to the absorption and radiation loss derived from mi. For simplicity, we redefine mi(ϕ0) = mm(ϕ0) + mabs(ϕ0) + mrad (ϕ0) and define a mode order detuning term δm(ϕ) = [mr(ϕ)−M]+ jmi (ϕ). Consequently, the field E(ρ, z, 0) = A(0)ê(ρ, z, 0) propagating from ϕ = 0 to an azimuthal angle ϕ0 can be expressed as
E(ρ,z,ϕ0)=A(0)ej[Mϕ0+ϕ=0ϕ0δm(ϕ)dϕ]e^(ρ,z,ϕ0)
(11)

To satisfy the resonance condition, the overall phase change of the field after propagating through a 2π azimuthal angle should be 2:
ϕ=02πmr(ϕ)dϕ=2Mπ
(12)

As can be seen, in the absence of the axisymmetry, an additional loss term from the mode mismatch between neighbouring cross sections occurs. This yields the decrease of overall Qtot factor from the inclusion of the mode mismatch Qm.

3. Application and discussion

3.1. The ideal WGM cavity

To validate the formulated algorithm, we first examine the case of a WGM with perfect axisymmetry. It’s evident that the WGM mode at each cross section is identical and that the mode mismatch loss vanishes. In addition, λr is identical at any azimuthal angle ϕ and the cavity resonance wavelength λres = λr. Therefore, an ideal WGM cavity can be treated as a special case under the current MMM formulation.

In the presence of a tapered waveguide that couples the light in and out of the cavity, mode mismatch occurs within the cavity-taper interaction regime. The coupling loss can be viewed as a special case of mode mismatch loss triggered by the waveguide taper. Assuming the taper interacts with the cavity within an azimuthal angle between ϕ0 and ϕ0 + Δϕ, we obtain a new formulation of coupling Q as
1Qcoupling=2ϕ0ϕ0+Δϕmm(ϕ)dϕ2πM
(15)
By treating the tapered waveguide as part of an asymmetric WGM cavity, one may easily obtain a resonance wavelength shift Δλ = λresλ̃res due to the perturbation of the waveguide taper to the cavity according to
Δλ=ϕ0ϕ0+Δϕ[λr(ϕ)λ˜res]dϕ2π
(16)

Surface roughness of the cavity often contributes to the loss as an individual term [11

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

] in the context of MMM, wherein the loss also originates from the mode mismatch between cavity cross sections. Using the same approach, one can obtain the surface roughness associated Q using the equations obtained above, although an integral over a 2π azimuthal angle must be performed. Evidently, the surface roughness could perturb optical path length and yield an addition shift of cavity resonance wavelength. This shift, however, would usually be small in magnitude.

3.2. Nanoparticles on a WGM cavity

In this test case we examine the resonant wavelength shift and decrease of quality factor when N nanoparticles land on the surface of a WGM cavity. Here we assume each nanoparticle i occupies a space lying between ϕi and ϕi + Δϕi. From (13) and (14) we derive that the particle induced resonance wavelength shift Δλ and quality factor Q change follows the expression
Δλ=i=1Nϕiϕi+Δϕi[λr(ϕ)λ˜res]dϕ2π
(17)
1Qtot1Q˜tot=2i=1Nϕiϕi+Δϕi[mm(ϕ)+mabs(ϕ)+mrad(ϕ)m˜i]dϕ2πM
(18)

Figure 2 shows the intensity of the normalized toroid-bead hybrid modes at the azimuthal cross section, where the center of the beads are located. As shown in Fig. 2(a), where a polystyrene bead was adsorbed at the equator of the toroid, the intensity around the bead (c.f. Fig. 2(a) inset) displays little distortion from the unperturbed toroid mode. In this case, the resulting resonance shift of the cavity can be extracted with sufficient accuracy using first order perturbation theory [18

18. I. Teraoka, S. Arnold, and F. Vollmer, “Perturbation approach to resonance shifts of whispering-gallery modes in a dielectric microsphere as a probe of a surrounding medium,” J. Opt. Soc. Am. B 20, 1937–1946 (2003). [CrossRef]

]. Fig. 2(b) shows the WGM-SP hybrid mode excited at the gold-dielectric interface. As shown in Fig. 2(b)’s inset, there is strong field intensity stored in the surface plasmon resulting in two hot spots aligned along the same direction as that of the electric field (along the direction displayed for the TE mode). Due to the coupling of the WGM and the SP, the field intensity outside the cavity is stronger than that inside the cavity. Such significant distortion to the WGM precludes the treatment of such problems with first order perturbation theory. On the other hand, MMM intrinsically incorporates the change in mode field distribution and yields accurate evaluations of the perturbed system.

Fig. 2: The fundamental mode intensity distribution of a silica microtoroid with (a) a polystyrene bead and (b) a gold bead bound to the equator at a 633-nm wavelength. The modes are plotted at the azimuthal cross section where the center of the bead is located. The insets provide a zoomed-in view of the intensity distribution around the beads.

Figure 3 illustrates the calculated terms mr, mm, and mabs arising from the PS (Fig. 3(a)) and gold (Fig. 3(b)) nanoparticles. It is observed that the mode mismatch loss dominates the degradation of the Q factor over material absorption in both cases.

Fig. 3: The real and imaginary part of the mode order m along the propagation direction when (a) a 50-nm radius PS bead and (b) a 50-nm radius Au bead are placed at ϕ = 0.

In Fig. 4 we further investigate the convergence rate as a function of azimuthal angle discretization in the case of a 50-nm radius gold bead binding to the toroid. The percent error was calculated by comparing the results of each δϕ to the expected result obtained through the Richardson extrapolation procedure. One can observe that the simulated resonance wavelength shift converges on the order of O(δϕ2) (black line in Fig. 4(a)) while the convergence rate of Q is O(δϕ0.9)(black line in Fig. 4(b)). A higher convergence rate is possible by implementing higher order integration procedures to evaluate (17) and (18).

Fig. 4: Shift and Q factor vs. grid spacing δϕ along the ϕ̂ direction for a 50-nm radius gold bead. The last point is omitted for the creation of the line of best fit.

In Fig. 5, we display the resonance wavelength shifts and the degraded Q factors due to the binding of a polystyrene (Fig. 5(a)) and gold (Fig. 5(b)) bead of different radii, where an unperturbed toroid has a resonant wavelength of 632.747 nm and a theoretical Q factor of 1.65 × 109. The resonance shifts predicted by MMM (red solid square markers) are compared with those predicted by a first order perturbation method (black plus symbol markers). For PS beads, MMM predicts wavelength shifts which are in good agreement with those predicted by the first order perturbation method. The quality factor (blue solid circle markers) of the toroid is unaffected when a bead with radius smaller than 5 nm is attached to the surface, yet there is a drop to approximately 107 if a 50-nm radius bead is attached instead. The calculated wavelength shift and quality factor degradation is in line with the experimental observation reported in [13

13. 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. U. S. A. (2011). [CrossRef]

].

Fig. 5: Binding shift and Q factor degradation due to (a) a bound PS sphere and (b) a bound Au sphere for different bead radii.

For the case of a gold bead simulation, the excess polarizability is given by α = 4πR3εm(εpεm)/(εp + 2εm) [39

39. S. A. Maier, Plasmonics: Fundamentals and Applications(Berlin: Springer, 2007).

] where εp and εm are the permittivities of the particle and the medium, respectively. MMM predicts resonance shifts from 2×10−2 to 359 fm and Q factor degradation from 3 × 108 to 7.5 × 104 when the attached gold bead radius increases from 2.5 nm to 50 nm. As expected, at an off-plasmon resonance wavelength of 633 nm, the perturbation method matches the MMM results when the bead size is below 20 nm yet for large beads yields greater errors due to the non-negligible field distortion from the bead.

In Fig. 6 we also compared the shifts calculated by the two methods as a function of wavelengths. As illustrated, near the surface plasmon resonance of a gold nanobead (540 nm, as shown in the insert), the accuracy of the perturbation method drops below 50% due to the strong surface plasmon effect that was similarly observed in [19

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

].

Fig. 6: Cavity resonance shifts as a function of wavelengths for a 25-nm radius gold bead. The insert shows the excess polarizability of the bead at different wavelengths.

To verify our formulation we further compared our results with reported experiments. We simulated the binding of 27.5-nm gold beads to a 50-μm radius microsphere with a 4 nm inter-particle gap as reported in [16

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

, 17

17. M. A. Santiago-Cordoba, M. Cetinkaya, S. V. Boriskina, F. Vollmer, and M. C. Demirel, “Ultrasensitive detection of a protein by optical trapping in a photonic-plasmonic microcavity,” J. Biophotonics 5, 629–638 (2012). [CrossRef]

]. In this simulation, we obtained a resonance wavelength shift of 32.6 fm per bead adsorption. Compared to the total resonance shift in Fig. 1(d) of [16

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

], we predict that a total of 10 beads are adsorbed to the microsphere surface. Knowing that an unperturbed cavity quality factor of 3 × 106 was reported in the same paper, we estimated that the quality factor will drop to 5.8 × 105 by the adsorption of 10 beads. This value is in good agreement with the reported experimental value on the order of 105.

4. Conclusion

References and links

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2.

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

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J. Dominguez-Juarez, G. Kozyreff, and J. Martorell, “Whispering gallery microresonators for second harmonic light generation from a low number of small molecules,” Nat. Commun. 2, 1–8 (2010).

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H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics 6, 369–373 (2012). [CrossRef]

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Y. Sun, J. Liu, G. Frye-Mason, S.-j. Ja, A. K. Thompson, and X. Fan, “Optofluidic ring resonator sensors for rapid dnt vapor detection,” Analyst 134, 1386–1391 (2009). [CrossRef] [PubMed]

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G. Bahl, X. Fan, and T. Carmon, “Acoustic whispering-gallery modes in optomechanical shells,” New J. Phys. 14, 115026 (2012). [CrossRef]

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M. R. Lee and P. M. Fauchet, “Nanoscale microcavity sensor for single particle detection,” Opt. Lett. 32, 3284–3286 (2007). [CrossRef] [PubMed]

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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. 67, 033806 (2003). [CrossRef]

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S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold raman laser using a spherical dielectric microcavity,” Nature 415, 621–623 (2002). [CrossRef] [PubMed]

11.

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

12.

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

13.

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. U. S. A. (2011). [CrossRef]

14.

B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface plasmon polariton whispering gallery microcavity,” Nature 457, 455–458 (2009). [CrossRef] [PubMed]

15.

S. I. Shopova, R. Rajmangal, S. Holler, and S. Arnold, “Plasmonic enhancement of a whispering-gallery-mode biosensor for single nanoparticle detection,” Appl. Phys. Lett. 98, 243104 (2011). [CrossRef]

16.

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

17.

M. A. Santiago-Cordoba, M. Cetinkaya, S. V. Boriskina, F. Vollmer, and M. C. Demirel, “Ultrasensitive detection of a protein by optical trapping in a photonic-plasmonic microcavity,” J. Biophotonics 5, 629–638 (2012). [CrossRef]

18.

I. Teraoka, S. Arnold, and F. Vollmer, “Perturbation approach to resonance shifts of whispering-gallery modes in a dielectric microsphere as a probe of a surrounding medium,” J. Opt. Soc. Am. B 20, 1937–1946 (2003). [CrossRef]

19.

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

20.

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

21.

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]

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OCIS Codes
(040.1880) Detectors : Detection
(230.3990) Optical devices : Micro-optical devices
(230.5750) Optical devices : Resonators

ToC Category:
Optical Devices

History
Original Manuscript: July 23, 2013
Revised Manuscript: August 28, 2013
Manuscript Accepted: August 29, 2013
Published: September 11, 2013

Citation
Xuan Du, Serge Vincent, and Tao Lu, "Full-vectorial whispering-gallery-mode cavity analysis," Opt. Express 21, 22012-22022 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-19-22012


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References

  1. K. Vahala, “Optical microcavities,” Nature424, 839–846 (2003). [CrossRef] [PubMed]
  2. D. Armani, T. Kippenberg, S. Spillane, and K. Vahala, “Ultra-high-Q toroid microcavity on a chip,” Nature421, 925–928 (2003). [CrossRef] [PubMed]
  3. S. I. Shopova, R. Rajmangal, Y. Nishida, and S. Arnold, “Ultrasensitive nanoparticle detection using a portable whispering gallery mode biosensor driven by a periodically poled lithium-niobate frequency doubled distributed feedback laser,” Rev. Sci. Instrum.81, 103110 (2010). [CrossRef] [PubMed]
  4. J. Dominguez-Juarez, G. Kozyreff, and J. Martorell, “Whispering gallery microresonators for second harmonic light generation from a low number of small molecules,” Nat. Commun.2, 1–8 (2010).
  5. H. Lee, T. Chen, J. Li, K. Y. Yang, S. Jeon, O. Painter, and K. J. Vahala, “Chemically etched ultrahigh-Q wedge-resonator on a silicon chip,” Nat. Photonics6, 369–373 (2012). [CrossRef]
  6. Y. Sun, J. Liu, G. Frye-Mason, S.-j. Ja, A. K. Thompson, and X. Fan, “Optofluidic ring resonator sensors for rapid dnt vapor detection,” Analyst134, 1386–1391 (2009). [CrossRef] [PubMed]
  7. G. Bahl, X. Fan, and T. Carmon, “Acoustic whispering-gallery modes in optomechanical shells,” New J. Phys.14, 115026 (2012). [CrossRef]
  8. M. R. Lee and P. M. Fauchet, “Nanoscale microcavity sensor for single particle detection,” Opt. Lett.32, 3284–3286 (2007). [CrossRef] [PubMed]
  9. J. R. Buck and H. J. Kimble, “Optimal sizes of dielectric microspheres for cavity QED with strong coupling,” Phys. Rev. A.67, 033806 (2003). [CrossRef]
  10. S. M. Spillane, T. J. Kippenberg, and K. J. Vahala, “Ultralow-threshold raman laser using a spherical dielectric microcavity,” Nature415, 621–623 (2002). [CrossRef] [PubMed]
  11. M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, “Ultimate Q of optical microsphere resonators,” Opt. Lett.21, 453–455 (1996). [CrossRef] [PubMed]
  12. S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in microspheres by protein adsorption,” Opt. Lett.28, 272–274 (2003). [CrossRef] [PubMed]
  13. 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. U. S. A.(2011). [CrossRef]
  14. B. Min, E. Ostby, V. Sorger, E. Ulin-Avila, L. Yang, X. Zhang, and K. Vahala, “High-Q surface plasmon polariton whispering gallery microcavity,” Nature457, 455–458 (2009). [CrossRef] [PubMed]
  15. S. I. Shopova, R. Rajmangal, S. Holler, and S. Arnold, “Plasmonic enhancement of a whispering-gallery-mode biosensor for single nanoparticle detection,” Appl. Phys. Lett.98, 243104 (2011). [CrossRef]
  16. M. Santiago-Cordoba, S. Boriskina, F. Vollmer, and M. Demirel, “Nanoparticle-based protein detection by optical shift of a resonant microcavity,” Appl. Phys. Lett.99, 073701 (2011). [CrossRef]
  17. M. A. Santiago-Cordoba, M. Cetinkaya, S. V. Boriskina, F. Vollmer, and M. C. Demirel, “Ultrasensitive detection of a protein by optical trapping in a photonic-plasmonic microcavity,” J. Biophotonics5, 629–638 (2012). [CrossRef]
  18. I. Teraoka, S. Arnold, and F. Vollmer, “Perturbation approach to resonance shifts of whispering-gallery modes in a dielectric microsphere as a probe of a surrounding medium,” J. Opt. Soc. Am. B20, 1937–1946 (2003). [CrossRef]
  19. M. R. Foreman and F. Vollmer, “Theory of resonance shifts of whispering gallery modes by arbitrary plasmonic nanoparticles,” New J. Phys.15, 083006 (2013). [CrossRef]
  20. M. Oxborrow, “Traceable 2-D finite-element simulation of the whispering-gallery modes of axisymmetric electromagnetic resonators,” IEEE Trans. Microwave Theory Tech.55, 1209–1218 (2007). [CrossRef]
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  22. J. Y. Lee, X. Luo, and A. W. Poon, “Reciprocal transmissions and asymmetric modal distributions in waveguide-coupled spiral-shaped microdisk resonators,” Opt. Express15, 14650–14666 (2007). [CrossRef] [PubMed]
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