Abstract

We theoretically and experimentally analyze the biomolecule detection capability of the liquid core optical ring resonator (LCORR) as a label-free bio/chemical sensor. We first establish a simple and general linear relationship between the LCORR’s bulk refractive index sensitivity (BRIS) and its response to molecule deposition onto the surface, which enables us to easily characterize the LCORR sensing performance. Then, biosensing experiments are performed with bovine serum albumin (BSA) and LCORRs of various BRISs. The experimental results are in good agreement with the theoretical prediction. Further analysis shows that the LCORR is capable of detecting BSA below 10 pM with sub-picogram/mm2 mass detection limit.

© 2007 Optical Society of America

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  1. A. Serpenguzel, S. Arnold, and G. Griffel, "Excitation of resonances of microspheres on an optical fiber," Opt. Lett. 20, 654-656 (1995).
    [CrossRef] [PubMed]
  2. F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," App. Phys. Lett. 80, 4057-4059 (2002).
    [CrossRef]
  3. 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]
  4. 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, 1937-1947 (2003).
    [CrossRef]
  5. N. M. Hanumegowda, C. J. Stica, B. C. Patel, I. M. White, and X. Fan, "Refractometric sensors based on microsphere resonators," Appl. Phys. Lett. 87, 201107 (2005).
    [CrossRef]
  6. I. M. White, N. M. Hanumegowda, and X. Fan, "Subfemtomole detection of small molecules with microsphere sensors," Opt. Lett. 30, 3189-3191 (2005).
    [CrossRef] [PubMed]
  7. N. M. Hanumegowda, I. M. White, H. Oveys, and X. Fan, "Label-Free Protease Sensors based on Optical Microsphere Resonators," Sensors Lett. 3, 315-319 (2005).
    [CrossRef]
  8. I. M. White, H. Oveys, and X. Fan, "Liquid Core Optical Ring Resonator Sensors," Opt. Lett. 31, 1319-1321 (2006).
    [CrossRef] [PubMed]
  9. X. Fan, I. M. White, H. Zhu, J. D. Suter, and H. Oveys, "Overview of novel integrated optical ring resonator bio/chemical sensors," Proc. SPIE 6452, 6452M (2007).
  10. A. Yalcin, K. C. Popat, O. C. Aldridge, T. A. Desai, J. Hryniewicz, N. Chbouki, B. E. Little, O. King, V. Van, S. Chu, D. Gill, M. Anthes-Washburn, M. S. Unlu, and B. B. Goldberg, "Optical Sensing of Biomolecules Using Microring Resonators," IEEE J. Sel. Top. Quantum Electron. 12, 148-155 (2006).
    [CrossRef]
  11. A. M. Armani and K. J. Vahala, "Heavy water detection using ultra-high-Q microcavities," Opt. Lett. 31, 1896-1898, 2006.
    [CrossRef] [PubMed]
  12. C.-Y. Chao, W. Fung, and L. J. Guo, "Polymer microring resonators for biochemical sensing applications," IEEE J. Sel. Top. Quantum Electron. 12, 134-142, 2006.
    [CrossRef]
  13. A. Ksendzov and Y. Lin, "Integrated optics ring-resonator sensors for protein detection," Opt. Lett. 30, 3344-3346 (2005).
    [CrossRef]
  14. R. K. Chang and A. J. Campillo, Optical Processes in Microcavities (World Scientific, Singapore, 1996).
    [CrossRef]
  15. M. L. Gorodetsky, A. A. Savchenkov, and V. S. Ilchenko, "Ultimate Q of optical microsphere resonators," Opt. Lett. 21, 453-455 (1996).
    [CrossRef] [PubMed]
  16. H. Zhu, I. M. White, J. D. Suter, M. Zourob, and X. Fan, "An Integrated Refractive Index Optical Ring Resonator Detector for Capillary Electrophoresis," Anal. Chem. 79, 930-937 (2007).
    [CrossRef] [PubMed]

2007 (2)

X. Fan, I. M. White, H. Zhu, J. D. Suter, and H. Oveys, "Overview of novel integrated optical ring resonator bio/chemical sensors," Proc. SPIE 6452, 6452M (2007).

H. Zhu, I. M. White, J. D. Suter, M. Zourob, and X. Fan, "An Integrated Refractive Index Optical Ring Resonator Detector for Capillary Electrophoresis," Anal. Chem. 79, 930-937 (2007).
[CrossRef] [PubMed]

2006 (3)

I. M. White, H. Oveys, and X. Fan, "Liquid Core Optical Ring Resonator Sensors," Opt. Lett. 31, 1319-1321 (2006).
[CrossRef] [PubMed]

A. Yalcin, K. C. Popat, O. C. Aldridge, T. A. Desai, J. Hryniewicz, N. Chbouki, B. E. Little, O. King, V. Van, S. Chu, D. Gill, M. Anthes-Washburn, M. S. Unlu, and B. B. Goldberg, "Optical Sensing of Biomolecules Using Microring Resonators," IEEE J. Sel. Top. Quantum Electron. 12, 148-155 (2006).
[CrossRef]

C.-Y. Chao, W. Fung, and L. J. Guo, "Polymer microring resonators for biochemical sensing applications," IEEE J. Sel. Top. Quantum Electron. 12, 134-142, 2006.
[CrossRef]

2005 (4)

A. Ksendzov and Y. Lin, "Integrated optics ring-resonator sensors for protein detection," Opt. Lett. 30, 3344-3346 (2005).
[CrossRef]

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

I. M. White, N. M. Hanumegowda, and X. Fan, "Subfemtomole detection of small molecules with microsphere sensors," Opt. Lett. 30, 3189-3191 (2005).
[CrossRef] [PubMed]

N. M. Hanumegowda, I. M. White, H. Oveys, and X. Fan, "Label-Free Protease Sensors based on Optical Microsphere Resonators," Sensors Lett. 3, 315-319 (2005).
[CrossRef]

2003 (2)

2002 (1)

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," App. Phys. Lett. 80, 4057-4059 (2002).
[CrossRef]

1996 (1)

1995 (1)

Aldridge, O. C.

A. Yalcin, K. C. Popat, O. C. Aldridge, T. A. Desai, J. Hryniewicz, N. Chbouki, B. E. Little, O. King, V. Van, S. Chu, D. Gill, M. Anthes-Washburn, M. S. Unlu, and B. B. Goldberg, "Optical Sensing of Biomolecules Using Microring Resonators," IEEE J. Sel. Top. Quantum Electron. 12, 148-155 (2006).
[CrossRef]

Anthes-Washburn, M.

A. Yalcin, K. C. Popat, O. C. Aldridge, T. A. Desai, J. Hryniewicz, N. Chbouki, B. E. Little, O. King, V. Van, S. Chu, D. Gill, M. Anthes-Washburn, M. S. Unlu, and B. B. Goldberg, "Optical Sensing of Biomolecules Using Microring Resonators," IEEE J. Sel. Top. Quantum Electron. 12, 148-155 (2006).
[CrossRef]

Arnold, S.

Braun, D.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," App. Phys. Lett. 80, 4057-4059 (2002).
[CrossRef]

Chao, C.-Y.

C.-Y. Chao, W. Fung, and L. J. Guo, "Polymer microring resonators for biochemical sensing applications," IEEE J. Sel. Top. Quantum Electron. 12, 134-142, 2006.
[CrossRef]

Chbouki, N.

A. Yalcin, K. C. Popat, O. C. Aldridge, T. A. Desai, J. Hryniewicz, N. Chbouki, B. E. Little, O. King, V. Van, S. Chu, D. Gill, M. Anthes-Washburn, M. S. Unlu, and B. B. Goldberg, "Optical Sensing of Biomolecules Using Microring Resonators," IEEE J. Sel. Top. Quantum Electron. 12, 148-155 (2006).
[CrossRef]

Chu, S.

A. Yalcin, K. C. Popat, O. C. Aldridge, T. A. Desai, J. Hryniewicz, N. Chbouki, B. E. Little, O. King, V. Van, S. Chu, D. Gill, M. Anthes-Washburn, M. S. Unlu, and B. B. Goldberg, "Optical Sensing of Biomolecules Using Microring Resonators," IEEE J. Sel. Top. Quantum Electron. 12, 148-155 (2006).
[CrossRef]

Desai, T. A.

A. Yalcin, K. C. Popat, O. C. Aldridge, T. A. Desai, J. Hryniewicz, N. Chbouki, B. E. Little, O. King, V. Van, S. Chu, D. Gill, M. Anthes-Washburn, M. S. Unlu, and B. B. Goldberg, "Optical Sensing of Biomolecules Using Microring Resonators," IEEE J. Sel. Top. Quantum Electron. 12, 148-155 (2006).
[CrossRef]

Fan, X.

X. Fan, I. M. White, H. Zhu, J. D. Suter, and H. Oveys, "Overview of novel integrated optical ring resonator bio/chemical sensors," Proc. SPIE 6452, 6452M (2007).

H. Zhu, I. M. White, J. D. Suter, M. Zourob, and X. Fan, "An Integrated Refractive Index Optical Ring Resonator Detector for Capillary Electrophoresis," Anal. Chem. 79, 930-937 (2007).
[CrossRef] [PubMed]

I. M. White, H. Oveys, and X. Fan, "Liquid Core Optical Ring Resonator Sensors," Opt. Lett. 31, 1319-1321 (2006).
[CrossRef] [PubMed]

I. M. White, N. M. Hanumegowda, and X. Fan, "Subfemtomole detection of small molecules with microsphere sensors," Opt. Lett. 30, 3189-3191 (2005).
[CrossRef] [PubMed]

N. M. Hanumegowda, I. M. White, H. Oveys, and X. Fan, "Label-Free Protease Sensors based on Optical Microsphere Resonators," Sensors Lett. 3, 315-319 (2005).
[CrossRef]

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

Fung, W.

C.-Y. Chao, W. Fung, and L. J. Guo, "Polymer microring resonators for biochemical sensing applications," IEEE J. Sel. Top. Quantum Electron. 12, 134-142, 2006.
[CrossRef]

Gill, D.

A. Yalcin, K. C. Popat, O. C. Aldridge, T. A. Desai, J. Hryniewicz, N. Chbouki, B. E. Little, O. King, V. Van, S. Chu, D. Gill, M. Anthes-Washburn, M. S. Unlu, and B. B. Goldberg, "Optical Sensing of Biomolecules Using Microring Resonators," IEEE J. Sel. Top. Quantum Electron. 12, 148-155 (2006).
[CrossRef]

Goldberg, B. B.

A. Yalcin, K. C. Popat, O. C. Aldridge, T. A. Desai, J. Hryniewicz, N. Chbouki, B. E. Little, O. King, V. Van, S. Chu, D. Gill, M. Anthes-Washburn, M. S. Unlu, and B. B. Goldberg, "Optical Sensing of Biomolecules Using Microring Resonators," IEEE J. Sel. Top. Quantum Electron. 12, 148-155 (2006).
[CrossRef]

Gorodetsky, M. L.

Griffel, G.

Guo, L. J.

C.-Y. Chao, W. Fung, and L. J. Guo, "Polymer microring resonators for biochemical sensing applications," IEEE J. Sel. Top. Quantum Electron. 12, 134-142, 2006.
[CrossRef]

Hanumegowda, N. M.

N. M. Hanumegowda, I. M. White, H. Oveys, and X. Fan, "Label-Free Protease Sensors based on Optical Microsphere Resonators," Sensors Lett. 3, 315-319 (2005).
[CrossRef]

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

I. M. White, N. M. Hanumegowda, and X. Fan, "Subfemtomole detection of small molecules with microsphere sensors," Opt. Lett. 30, 3189-3191 (2005).
[CrossRef] [PubMed]

Holler, S.

Hryniewicz, J.

A. Yalcin, K. C. Popat, O. C. Aldridge, T. A. Desai, J. Hryniewicz, N. Chbouki, B. E. Little, O. King, V. Van, S. Chu, D. Gill, M. Anthes-Washburn, M. S. Unlu, and B. B. Goldberg, "Optical Sensing of Biomolecules Using Microring Resonators," IEEE J. Sel. Top. Quantum Electron. 12, 148-155 (2006).
[CrossRef]

Ilchenko, V. S.

Khoshsima, M.

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]

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," App. Phys. Lett. 80, 4057-4059 (2002).
[CrossRef]

King, O.

A. Yalcin, K. C. Popat, O. C. Aldridge, T. A. Desai, J. Hryniewicz, N. Chbouki, B. E. Little, O. King, V. Van, S. Chu, D. Gill, M. Anthes-Washburn, M. S. Unlu, and B. B. Goldberg, "Optical Sensing of Biomolecules Using Microring Resonators," IEEE J. Sel. Top. Quantum Electron. 12, 148-155 (2006).
[CrossRef]

Ksendzov, A.

Libchaber, A.

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," App. Phys. Lett. 80, 4057-4059 (2002).
[CrossRef]

Lin, Y.

Little, B. E.

A. Yalcin, K. C. Popat, O. C. Aldridge, T. A. Desai, J. Hryniewicz, N. Chbouki, B. E. Little, O. King, V. Van, S. Chu, D. Gill, M. Anthes-Washburn, M. S. Unlu, and B. B. Goldberg, "Optical Sensing of Biomolecules Using Microring Resonators," IEEE J. Sel. Top. Quantum Electron. 12, 148-155 (2006).
[CrossRef]

Oveys, H.

X. Fan, I. M. White, H. Zhu, J. D. Suter, and H. Oveys, "Overview of novel integrated optical ring resonator bio/chemical sensors," Proc. SPIE 6452, 6452M (2007).

I. M. White, H. Oveys, and X. Fan, "Liquid Core Optical Ring Resonator Sensors," Opt. Lett. 31, 1319-1321 (2006).
[CrossRef] [PubMed]

N. M. Hanumegowda, I. M. White, H. Oveys, and X. Fan, "Label-Free Protease Sensors based on Optical Microsphere Resonators," Sensors Lett. 3, 315-319 (2005).
[CrossRef]

Patel, B. C.

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

Popat, K. C.

A. Yalcin, K. C. Popat, O. C. Aldridge, T. A. Desai, J. Hryniewicz, N. Chbouki, B. E. Little, O. King, V. Van, S. Chu, D. Gill, M. Anthes-Washburn, M. S. Unlu, and B. B. Goldberg, "Optical Sensing of Biomolecules Using Microring Resonators," IEEE J. Sel. Top. Quantum Electron. 12, 148-155 (2006).
[CrossRef]

Savchenkov, A. A.

Serpenguzel, A.

Stica, C. J.

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

Suter, J. D.

X. Fan, I. M. White, H. Zhu, J. D. Suter, and H. Oveys, "Overview of novel integrated optical ring resonator bio/chemical sensors," Proc. SPIE 6452, 6452M (2007).

H. Zhu, I. M. White, J. D. Suter, M. Zourob, and X. Fan, "An Integrated Refractive Index Optical Ring Resonator Detector for Capillary Electrophoresis," Anal. Chem. 79, 930-937 (2007).
[CrossRef] [PubMed]

Teraoka, I.

Unlu, M. S.

A. Yalcin, K. C. Popat, O. C. Aldridge, T. A. Desai, J. Hryniewicz, N. Chbouki, B. E. Little, O. King, V. Van, S. Chu, D. Gill, M. Anthes-Washburn, M. S. Unlu, and B. B. Goldberg, "Optical Sensing of Biomolecules Using Microring Resonators," IEEE J. Sel. Top. Quantum Electron. 12, 148-155 (2006).
[CrossRef]

Van, V.

A. Yalcin, K. C. Popat, O. C. Aldridge, T. A. Desai, J. Hryniewicz, N. Chbouki, B. E. Little, O. King, V. Van, S. Chu, D. Gill, M. Anthes-Washburn, M. S. Unlu, and B. B. Goldberg, "Optical Sensing of Biomolecules Using Microring Resonators," IEEE J. Sel. Top. Quantum Electron. 12, 148-155 (2006).
[CrossRef]

Vollmer, F.

White, I. M.

X. Fan, I. M. White, H. Zhu, J. D. Suter, and H. Oveys, "Overview of novel integrated optical ring resonator bio/chemical sensors," Proc. SPIE 6452, 6452M (2007).

H. Zhu, I. M. White, J. D. Suter, M. Zourob, and X. Fan, "An Integrated Refractive Index Optical Ring Resonator Detector for Capillary Electrophoresis," Anal. Chem. 79, 930-937 (2007).
[CrossRef] [PubMed]

I. M. White, H. Oveys, and X. Fan, "Liquid Core Optical Ring Resonator Sensors," Opt. Lett. 31, 1319-1321 (2006).
[CrossRef] [PubMed]

I. M. White, N. M. Hanumegowda, and X. Fan, "Subfemtomole detection of small molecules with microsphere sensors," Opt. Lett. 30, 3189-3191 (2005).
[CrossRef] [PubMed]

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

N. M. Hanumegowda, I. M. White, H. Oveys, and X. Fan, "Label-Free Protease Sensors based on Optical Microsphere Resonators," Sensors Lett. 3, 315-319 (2005).
[CrossRef]

Yalcin, A.

A. Yalcin, K. C. Popat, O. C. Aldridge, T. A. Desai, J. Hryniewicz, N. Chbouki, B. E. Little, O. King, V. Van, S. Chu, D. Gill, M. Anthes-Washburn, M. S. Unlu, and B. B. Goldberg, "Optical Sensing of Biomolecules Using Microring Resonators," IEEE J. Sel. Top. Quantum Electron. 12, 148-155 (2006).
[CrossRef]

Zhu, H.

X. Fan, I. M. White, H. Zhu, J. D. Suter, and H. Oveys, "Overview of novel integrated optical ring resonator bio/chemical sensors," Proc. SPIE 6452, 6452M (2007).

H. Zhu, I. M. White, J. D. Suter, M. Zourob, and X. Fan, "An Integrated Refractive Index Optical Ring Resonator Detector for Capillary Electrophoresis," Anal. Chem. 79, 930-937 (2007).
[CrossRef] [PubMed]

Zourob, M.

H. Zhu, I. M. White, J. D. Suter, M. Zourob, and X. Fan, "An Integrated Refractive Index Optical Ring Resonator Detector for Capillary Electrophoresis," Anal. Chem. 79, 930-937 (2007).
[CrossRef] [PubMed]

Anal. Chem. (1)

H. Zhu, I. M. White, J. D. Suter, M. Zourob, and X. Fan, "An Integrated Refractive Index Optical Ring Resonator Detector for Capillary Electrophoresis," Anal. Chem. 79, 930-937 (2007).
[CrossRef] [PubMed]

App. Phys. Lett. (1)

F. Vollmer, D. Braun, A. Libchaber, M. Khoshsima, I. Teraoka, and S. Arnold, "Protein detection by optical shift of a resonant microcavity," App. Phys. Lett. 80, 4057-4059 (2002).
[CrossRef]

Appl. Phys. Lett. (1)

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

IEEE J. Sel. Top. Quantum Electron. (2)

A. Yalcin, K. C. Popat, O. C. Aldridge, T. A. Desai, J. Hryniewicz, N. Chbouki, B. E. Little, O. King, V. Van, S. Chu, D. Gill, M. Anthes-Washburn, M. S. Unlu, and B. B. Goldberg, "Optical Sensing of Biomolecules Using Microring Resonators," IEEE J. Sel. Top. Quantum Electron. 12, 148-155 (2006).
[CrossRef]

C.-Y. Chao, W. Fung, and L. J. Guo, "Polymer microring resonators for biochemical sensing applications," IEEE J. Sel. Top. Quantum Electron. 12, 134-142, 2006.
[CrossRef]

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

Opt. Lett. (6)

Proc. SPIE (1)

X. Fan, I. M. White, H. Zhu, J. D. Suter, and H. Oveys, "Overview of novel integrated optical ring resonator bio/chemical sensors," Proc. SPIE 6452, 6452M (2007).

Sensors Lett. (1)

N. M. Hanumegowda, I. M. White, H. Oveys, and X. Fan, "Label-Free Protease Sensors based on Optical Microsphere Resonators," Sensors Lett. 3, 315-319 (2005).
[CrossRef]

Other (2)

R. K. Chang and A. J. Campillo, Optical Processes in Microcavities (World Scientific, Singapore, 1996).
[CrossRef]

A. M. Armani and K. J. Vahala, "Heavy water detection using ultra-high-Q microcavities," Opt. Lett. 31, 1896-1898, 2006.
[CrossRef] [PubMed]

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Figures (5)

Fig. 1.
Fig. 1.

Structure of the LCORR. ε1=n2 1, ε2=n2 2, ε3=n2 3 dielectric constant for surrounding medium, wall, and core. R: LCORR inner radius.

Fig. 2.
Fig. 2.

Two representative BRIS curves of the LCORRs used in experiment. Solid curves are linear fits. Inset (A): Response of the LCORR to various concentrations of water-ethanol mixtures. (B): Response of the LCORR to BSA binding to the surface.

Fig. 3.
Fig. 3.

WGM spectral shift vs.BSA concentration for the LCORR with BRIS of 31 nm/RIU. The LCORR starts to saturate at 200 nM and the saturation WGM shift is 129 pm. Half saturation concentration: 8 nM. Inset: log-log scale. Solid line is the linear fit in log-log scale. Red dot: detection limit for BSA concentration.

Fig. 4.
Fig. 4.

Saturation WGM spectral shift vs. LCORR BRIS. The solid curve is the theoretical calculation based on Eq. (5). n2=1.45, n3=1.333. σp=2.9×1012 cm-2 and αex=(4πε0)×(3.85×10-21)cm3 are obtained from Ref. [3].

Fig. 5.
Fig. 5.

BSA molecules bind to the LCORR surface, followed by the cleavage by trypsin. BSA concentration: 0.1 mg/mL. Trypsin concentration: 0.1 mg/mL. Arrow indicates the time when trypsin is added.

Tables (1)

Tables Icon

Table 1. Comparison between Ssim and Sest. n2=1.45.

Equations (8)

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

2 π R · [ n 1 η 1 + n 3 η 3 + n 2 ( 1 η 1 η 3 ) ] l · λ ,
S = ( δλ δ n 3 ) 2 π R l η 3 λ n 2 η 3 ,
η 3 = 2 π h × 0 R ε 0 ε 3 E 0 2 · e ( R r ) L rdr ε 0 ε ( r ) · E ( r ) 2 dV 2 π Rh ε 0 n 3 2 E 0 2 L ε 0 ε ( r ) · E ( r ) 2 dV ,
L = λ 4 π 1 n 2 2 n 3 2 .
δλ λ = σ p α ex · 2 π Rh · E 0 2 2 ε 0 ε ( r ) · E ( r ) 2 dV = σ p α ex 2 π n 2 2 n 3 2 ε 0 λ 2 n 2 n 3 2 S ,
σ min = ( n 3 2 n 2 n 2 2 n 3 2 ε 0 λ 2 π α ex ) Δ min ,
S λ 2 n 1 2 π R ( n 2 2 n 1 2 ) 3 .
δλ λ = ( σ p α ex ε 0 R 1 n 2 2 n 1 2 ) n 2 n 1 .

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