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

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  1. M. Baaske and F. Vollmer, “Optical Resonator Biosensors: Molecular Diagnostic and Nanoparticle Detection on an Integrated Platform,” ChemPhysChem 13, 427–436 (2012).
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  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).
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  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).
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    [Crossref] [PubMed]
  42. J. Topolancik and F. Vollmer, “Photoinduced transformations in bacteriorhodopsin membrane monitored with optical microcavities,” Biophys. J. 92, 2223–2229 (2007).
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    [Crossref]
  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]
  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]
  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).
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    [Crossref]

2014 (1)

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

2013 (6)

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]

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]

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]

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]

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

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]

2012 (6)

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

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]

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]

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]

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]

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]

2011 (3)

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]

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]

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]

2010 (4)

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,  7682, 76820C (2010).
[Crossref]

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]

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

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]

2008 (3)

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]

S. Arnold, R. Ramjit, D. Keng, V. Kolchenko, and I. Teraoka, “Microparticle photophysics illuminates viral biosensing,” Faraday Disc. 137, 65–83, (2008).
[Crossref]

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]

2007 (4)

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

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]

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]

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]

2006 (2)

2005 (2)

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]

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]

2004 (2)

2003 (5)

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]

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]

K. J. Vahala, “Optical microcavities” Nature 424, 839–846 (2003).
[Crossref] [PubMed]

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]

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

2002 (2)

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]

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

2000 (1)

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]

1999 (1)

1996 (1)

1995 (1)

1992 (1)

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]

1991 (1)

1990 (1)

A. Zeira and A. Nehorai, “Frequency domain Cramer-Rao bound for Gaussian Processes,” IEEE Trans. Acoust. Speech. Sig. Proc. 381063–1066 (1990).
[Crossref]

1973 (1)

Abramowitz, M.

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

Ahn, W.

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]

Arnold, S.

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]

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]

S. Arnold, R. Ramjit, D. Keng, V. Kolchenko, and I. Teraoka, “Microparticle photophysics illuminates viral biosensing,” Faraday Disc. 137, 65–83, (2008).
[Crossref]

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]

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]

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]

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]

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]

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]

Baaske, M.

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

Barber, P.W.

Barbre, C.

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]

Bashkatov, A. N.

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]

Bass, M.

M. Bass, C. M. DeCusatis, and J. M. Enoch, Handbook of Optics, 3rd ed., Vol. 4. (McGraw-Hill, USA2009).

Bearman, G. H.

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

Benson, O.

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]

Bercher, J.-F.

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

Boriskina, S. V.

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]

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]

Bowen, W. P.

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[Crossref]

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]

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]

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]

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]

Topolancik, J.

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

Török, P.

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

Vahala, K.

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]

Vahala, K. J.

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]

K. J. Vahala, “Optical microcavities” Nature 424, 839–846 (2003).
[Crossref] [PubMed]

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]

Vignet, C.

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

Vollmer, F.

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

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]

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]

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

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]

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]

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]

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

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]

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]

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]

Vollmer., F.

Wang, W.

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]

Weiss, D. S.

White, I.

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]

Wu, X.

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,  7682, 76820C (2010).
[Crossref]

Xiao, Y.

Xiao, Y.-F.

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]

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]

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]

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]

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

Xu, L.

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,  7682, 76820C (2010).
[Crossref]

Yang, L.

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]

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]

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]

Yi, X.

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]

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]

Young, K.

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]

Yu, N.

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]

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]

Yu, X.-C.

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]

Zeira, A.

A. Zeira and A. Nehorai, “Frequency domain Cramer-Rao bound for Gaussian Processes,” IEEE Trans. Acoust. Speech. Sig. Proc. 381063–1066 (1990).
[Crossref]

Zhu, J.

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]

Zumofen, G.

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]

ACS Nano. (1)

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]

Adv. Mat. (1)

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]

Appl. Opt. (2)

Appl. Phys. Lett. (4)

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]

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]

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]

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]

Biophys. J. (2)

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]

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

ChemPhysChem (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]

Faraday Disc. (1)

S. Arnold, R. Ramjit, D. Keng, V. Kolchenko, and I. Teraoka, “Microparticle photophysics illuminates viral biosensing,” Faraday Disc. 137, 65–83, (2008).
[Crossref]

IEEE Trans. Acoust. Speech. Sig. Proc. (1)

A. Zeira and A. Nehorai, “Frequency domain Cramer-Rao bound for Gaussian Processes,” IEEE Trans. Acoust. Speech. Sig. Proc. 381063–1066 (1990).
[Crossref]

J. Appl. Phys. (2)

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]

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]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (4)

J. Opt. Soc. Am. B, (1)

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]

J. Opt. Soc. Am. B. (1)

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]

Nano. Lett. (1)

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]

Nanophot. (1)

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]

Nat. Methods (1)

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]

Nat. Phot. (1)

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]

Nature (1)

K. J. Vahala, “Optical microcavities” Nature 424, 839–846 (2003).
[Crossref] [PubMed]

New. J. Phys. (1)

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]

Opt. Commun. (1)

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

Opt. Express (3)

Opt. Lett (1)

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]

Opt. Lett. (3)

Phys. Lett. A (1)

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

Phys. Rev. A (3)

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

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

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]

Phys. Rev. Lett. (2)

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]

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]

Proc. Natl. Acad. Sci. USA (2)

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]

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]

Proc. SPIE (3)

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,  7682, 76820C (2010).
[Crossref]

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

Sci. Rep. (1)

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]

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Figures (6)

Fig. 1
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.

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Fig. 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 ΔΓ d,t (normalised to resonance linewidth) with WGM Q factor.

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Fig. 3
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.

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Fig. 4
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

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

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Fig. 6
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.

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Tables (1)

Tables Icon

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.

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Equations (46)

Equations on this page are rendered with MathJax. Learn more.

I ( ω ) = I 0 [ 1 − A Γ 2 / 4 ( ω − ω 0 ) 2 + Γ 2 / 4 ] ,
p I d ( I d , j ; ω 0 , Γ , A ) = 1 2 π σ d 2 exp [ − 1 2 ( I d , j − I j σ d ) 2 ] ,
p ω t ( ω t ) = 1 2 π σ t 2 exp [ − 1 2 ω t 2 σ t 2 ] ,
p I d ( I d , j ; ω 0 , Γ , A ) = ∫ − ∞ ∞ δ ( I d , j − I j ) p ω t ( ω t ) d ω t
= 1 2 π σ t 2 I 0 A Γ j 2 4 Λ j ( I 0 − I d , j ) 2 exp [ − 1 2 Δ j 2 σ t 2 ] exp [ − 1 2 Λ j 2 σ t 2 ] cosh [ Λ j Δ j σ t 2 ] ,
𝕂 w ≥ 𝕁 w − 1 ,
[ 𝕁 I ] i j = δ i , j ∫ p I d , j ( I d , j ) ( ∂ ln p I d , j ( I d , j ; ω 0 , Γ , A ) ∂ I j ) 2 d I d , j ,
J ω 0 , ω 0 = ∑ j = 1 N Ω 1 σ d 2 ( ∂ I j ∂ ω 0 ) 2 ≈ A 2 I 0 2 4 σ d 2 Δ Ω Γ 4 ∫ − ∞ ∞ ( ω 0 − ω ) 2 [ ( ω 0 − ω ) 2 + Γ 2 / 4 ] 4 d ω = A 2 I 0 2 2 σ d 2 Δ Ω π Γ .
𝕁 w = π I 0 2 8 σ d 2 Δ Ω ( 4 A 2 / Γ 0 0 0 A 2 / Γ A 0 A 2 Γ ) .
𝕁 w ′ = ( J ω 0 , ω 0 − J ω 0 , A 2 J A , A J ω 0 , Γ − J Γ , A J ω 0 , A J A , A J ω 0 , Γ − J Γ , A J ω 0 , A J A , A J Γ , Γ − J Γ , A 2 J A , A ) = A 2 I 0 2 16 σ d 2 Δ Ω π Γ ( 8 0 0 1 ) .
Δ ω d = Δ Γ d 2 = 2 β π σ d I 0 A Γ .
J ω 0 , ω 0 = ∑ j = 1 N Ω ∫ 0 ∞ [ 1 σ t 2 − Λ j 2 σ t 4 sech 2 ( Λ j Δ j σ t 2 ) ]   p I d , j d I d , j .
J ω 0 , ω 0 ≈ N Ω σ t 2 − 1 Δ Ω σ t 5 2 π ∫ 0 ∞ Λ 2 exp ( − Λ 2 2 σ t 2 ) ∫ − ∞ ∞ sech ( Λ Δ σ t 2 ) exp ( − Δ 2 2 σ t 2 ) d Δ d Λ ,
J ω 0 , ω 0 ≈ 1 σ t [ N Ω σ t − π Δ Ω U ( 1 2 , 0 , 1 2 ) ] = 1 σ t Δ Ω [ Ω σ t − 1.416 ] ,
Δ ω t = β Γ σ t W Γ / σ t − 1.416 ≈ σ t β W ,
J Γ , Γ ≈ 2 σ t 3 Γ 2 Δ Ω Ω 2 σ t [ 3 + ( Ω 2 σ t ) 2 ] ,
Δ Γ t = 2 σ t 3 β W Γ 2 24 σ t 2 + W 2 Γ 2 ≈ 2 σ t 3 β W 3 ,
[ 𝕁 I ] j k ≈ 1 2 π ∫ − π π 1 Φ ( s ) exp [ i s ( j − k ) ] d s ,
[ 𝕁 I ] j k ≈ π A p 2 ( cos p + p sin p − 1 ) ,
δ ω ω 0 = − ε 0 ε s Re [ α ] | E ( r p ) | 2 4 U ,
N = Δ ω | δ ω | = ( n c 2 − n s 2 ) Re [ α ] R 3 | Y l l ( π / 2 ) | 2 F Q ,
| Y l l ( θ ) | 2 ≈ 1 4 π 3 / 2 2 l + 1 l 1 / 2 sin 2 l θ
Q c = 2 π 5 n c n p 2 − n c 2 ( n c 2 − n s 2 ) ( R λ ) 3 / 2 exp [ 2 γ d ] ,
N = ( n c 2 − n s 2 ) Re [ α ] R 3 | Y l l ( π / 2 ) | 2 F 0 Q 0 ( 1 + Q c / Q 0 ) 3 4 Q c 2 / Q 0 2 .
N = σ t ( n c 2 − n s 2 ) Re [ α ] ω 0 R 3 | Y l l ( π / 2 ) | 2 β W .
δ ω ω 0 = − ε s ( ε c − ε s ) R Re [ α ] σ s .
δ Γ abs ω 0 = ε s | Y l l ( π / 2 ) | 2 ( ε c − ε s ) R 3 Im [ α ] ,
δ Γ sca ω 0 = ω 0 3 n s 5 | Y l l ( π / 2 ) | 2 3 π c 3 ( ε c − ε s ) R 3 | α | 2 ,
N = Δ Γ δ Γ sca = 6 π c 3 ( n c 2 − n s 2 ) | α | 2 n s 5 ω 0 3 R 3 | Y l l ( π / 2 ) | 2 F 0 Q 0 ( 1 + Q c / Q 0 ) 3 4 Q c 2 / Q 0 2
N = σ t 6 π c 3 ( n c 2 − n s 2 ) | α | 2 n s 5 ω 0 4 R 3 | Y l l ( π / 2 ) | 2 3 β W 3
Q c Q 0 = 1 2 [ 1 + 1 + 9 Q m 1 + Q m ] ≈ 2 − 2 3 Q m ,
δ R ≈ 2 C ( R opt ) Q 0 Q m ∂ Q m ∂ R [ ( 3 Q m + 2 ) ∂ 2 [ C ( R ) / Q 0 ] ∂ R 2 | R = R opt ] − 1 .
[ n s z h l ( n s z ) ] ′ h l ( n s z ) = N [ n c z j l ( n c z ) ] ′ j l ( n c z ) ,
z 0 ≈ l + 1 / 2 n c + ⋯
Γ ≈ 2 c a ( n c 2 − n s 2 ) 1 n s z 0 2 y l 2 ( n s z 0 )
ρ = d Γ / d T d ω 0 / d T = d Γ / d n c d ω 0 / d n c = 2 Q ( n s 2 n c 2 − n s 2 − n s z 0 l y l − 1 ( n s z 0 ) − ( l + 1 ) y l + 1 ( n s z 0 ) ( 2 l + 1 ) y l ( n s z 0 ) ) .
J Γ , Γ = ∑ j = 1 N Ω ∫ 0 ∞ [ 1 + Λ j 2 σ t 2 − Λ j 2 Δ j σ t 4 sech 2 ( Λ j Δ j σ t 2 ) ] 1 Γ 2 p I d , j d I d , j .
J Γ , Γ ≈ N Ω Γ 2 + 2 / π Γ 2 Δ Ω σ t { ∫ 0 ∞ exp ( − Λ 2 2 σ t 2 ) ∫ − Ω / 2 Ω / 2 [ Λ 2 σ t 2 cosh ( Λ Δ σ t 2 ) ] exp ( − Δ 2 2 σ t 2 ) d Δ d Λ − ∫ 0 ∞ exp ( − Λ 2 2 σ t 2 ) ∫ − Ω / 2 Ω / 2 [ Λ 2 Δ 2 σ t 4 sech ( Λ Δ σ t 2 ) ] exp ( − Δ 2 2 σ t 2 ) d Δ d Λ } .
J Γ , Γ ≈ N Ω Γ 2 + 2 / π Γ 2 Δ Ω σ t ∫ 0 ∞ exp ( − Λ 2 2 σ t 2 ) ∫ − X Ω X Ω cosh x exp ( − x 2 2 σ t 2 Λ 2 ) d x d Λ − 0.563 σ t Γ 2 Δ Ω
1 Γ 2 Δ Ω σ t 2 ∫ 0 ∞ Λ 2 [ erf ( Z σ t + Λ 2 σ t ) + erf ( Z σ t − Λ 2 σ t ) ]   d Λ
J Γ , Γ ≈ N Ω Γ 2 + 2 σ t 3 Γ 2 Δ Ω Z ( 3 + Z 2 ) − 0.563 σ t Γ 2 Δ Ω = Ω Γ 3 6 Z + Z 3 − 0.845 3 β Z .
C ( R ) = ( n c 2 − n s 2 ) Re [ α ] R 3 | Y l l ( π / 2 ) | 2 .
N = C ( R ) F 0 Q 0 ( 1 + Q c / Q 0 ) 2 4 Q c / Q 0 ( 1 + Q c / Q 0 Q c / Q 0 + 1 Q m ) .
d N d R = ∂ f 1 ∂ R f 2 + f 1 ∂ f 2 ∂ Q m ∂ Q m ∂ R + f 1 ∂ f 2 ∂ x ∂ x ∂ R ,
∂ f 1 ∂ R ( 27 16 + 9 8 Q m ) − f 1 ∂ Q m ∂ R 9 8 Q m 2 = 0 ,
∂ 2 f 1 ∂ R 2 | R = R opt δ R ( 27 16 + 9 8 Q m ) − f 1 ( R opt ) ∂ Q m ∂ R = 0 ,

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