Abstract

Theory is developed for frequency shift and linewidth-broadening induced by rodlike bacteria bound to micro-optical resonators. Optical shift of whispering gallery modes (WGMs) is modeled by introducing a form factor that accounts for random horizontal orientation of cylindrical bacteria bound by their high refractive index cell walls. Linewidth-broadening is estimated from scattering losses. Analytic results are confirmed by measurement using E.Coli as model system (~102 bacteria/mm2 sensitivity), establishing the WGM biosensor as sensitive technique for detection and analysis of micro-organisms.

© 2007 Optical Society of America

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References

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  1. M. L. Gorodetsky, A. Savechnkov, and V. S. Ilchenko, "Ultimate Q of optical microsphere resonators," Opt. Lett. 21,453-455 (1996).
    [CrossRef] [PubMed]
  2. A.M. Armani, K.J. Vahala, "Heavy water detection using ultra-high-Q microcavities," Opt. Lett. 31, 1896-1898 (2006).
    [CrossRef] [PubMed]
  3. 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, 201107-201107-3 (2005).
    [CrossRef]
  4. 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]
  5. 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]
  6. I. Teraoka, S. Arnold, and F. Vollmer, "Perturbation Approach to Shift of Whispering-Gallery-Modes in Microspheres by Protein Adsorption," J. Opt. Soc. Am. B 20, 1937-1946 (2003).
    [CrossRef]
  7. I. Teraoka, 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]
  8. M. Noto, F. Vollmer, I. Teraoka, and S. Arnold, "Nanolayer characterization through wavelength multiplexing of a microsphere resonator," Opt. Lett. 30, 510-512 (2005).
    [CrossRef] [PubMed]
  9. 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]
  10. J. Topolancik, F. Vollmer, "Photoinduced Transformations in Bacteriorhodopsin Membrane Monitored with Optical Microcavities," Biophys. J. 92, 2223-2229 (2007).
    [CrossRef] [PubMed]
  11. C.-Y. Chao, W. Fung, L.J. Guo, "Polymer Microring Resonators for Biochemical Sensing Applications," IEEE J. Selected Topics Quantum Electron. 12, 134-142 (2006).
    [CrossRef]
  12. A.M. Armani, R.P. Kulkarni, S.E. Fraser, R.C. Flagan, K.J. Vahala, "Label-Free, Single-Molecule Detection with Optical Microcavities," Science 10, 783-787 (2007).
    [CrossRef]
  13. W. Knoll, "Interfaces and thin films as seen by bound electromagnetic waves," Annu. Rev. Phys. Chem. 49, 569-638 (1998).
    [CrossRef]
  14. R. Karlsson, R. Stahlberg, "Surface-plasmon resonance detection and multispot sensing for direct monitoring of interactions involving low-molecular weight analytes for determination of low affinities," Anal. Biochem. 228, 274-280 (1995).
    [CrossRef] [PubMed]
  15. J. W. Costerton, Z. Lewandowski, D. E. Caldwell, D. R. Korber, and H. M. Lappin-Scott, "Microbial Biofilms," Annu. Rev. Microbiol. 49, 711-45 (1995).
    [CrossRef] [PubMed]
  16. P. S. Mead, P. M. Griffin, "Escherichia coli O157:H7," Lancet 352, 1207-1212 (1998).
    [CrossRef] [PubMed]
  17. J. W. Costerton, P. S. Stewart, E. P. Greenberg, "Bacterial Biofilms: A Common Cause of Persistent Infections," Science 284, 1318-1322 (1999).
    [CrossRef] [PubMed]
  18. K. H. Seo, J. F. Frank, "Attachement of E.Coli O157:H7 to lettuce leaf surface and bacterial viability in response to chlorine treatment as demonstrated by using confocal scanning laser microscopy," J. Food Protect. 62, 3-9 (1999).
  19. C. Lam, P. T. Leung and K. Young, "Explicit asymptotic formulas for the position, width and strength of resonances in Mie scattering," J. Opt. Soc. Am. B 9, 1585-1590 (1992).
    [CrossRef]
  20. J. D. Jackson, Classical electrodynamics (John Wiley & Son Inc, 1975), Chap. 9 and Chap. 16.
  21. N. Nanninga, Molecular Cytology of Escherichia coli (Academic Press, 1985), Chap. 1ff.
  22. D. S. Goodsell, "Inside a Living Cell," Trends. Biochem. Sci. 16, 203-206 (1991).
    [CrossRef] [PubMed]
  23. M. Ardhammer, P. Lincoln, B. Norden, "In visible liposomes: Refractive index matching with sucrose enables flow dichroism assessment of peptide orientation in lipid vesicle membrane," Proc. Natl. Acad. Sci. USA 99, 15313-15317 (2002).
    [CrossRef]
  24. P. J. Wyatt, "Cell Wall Thickness, Size Distribution, Refractive Index Ratio and Dry Weight Content of Living Bacteria," Nature 226, 277-279 (1970).
    [CrossRef] [PubMed]
  25. H. C. Berg, in E.Coli in Motion (Springer, 2003) Chap. 1ff.
  26. F. Vollmer, "Taking Detection to the Limit," B.I.F. Futura 20, 239-244 (2005). http://www.bifonds.de/public/inhaltf4.htm
  27. A. M. Armani, D. K. Armani, B. Min, K. J. Vahala, S. M. Spillane, "Ultra-high-Q microcavity operation in H20 and D20," Appl. Phys. Lett . 87, 151118-151118-3 (2005).
    [CrossRef]
  28. G. N. Watson, A Treatise on the Theory of Bessel functions (Cambridge University Press, 1966), Chapter VIII.

2007 (2)

A.M. Armani, R.P. Kulkarni, S.E. Fraser, R.C. Flagan, K.J. Vahala, "Label-Free, Single-Molecule Detection with Optical Microcavities," Science 10, 783-787 (2007).
[CrossRef]

J. Topolancik, F. Vollmer, "Photoinduced Transformations in Bacteriorhodopsin Membrane Monitored with Optical Microcavities," Biophys. J. 92, 2223-2229 (2007).
[CrossRef] [PubMed]

2006 (3)

2005 (1)

2003 (3)

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]

M. Ardhammer, P. Lincoln, B. Norden, "In visible liposomes: Refractive index matching with sucrose enables flow dichroism assessment of peptide orientation in lipid vesicle membrane," Proc. Natl. Acad. Sci. USA 99, 15313-15317 (2002).
[CrossRef]

1999 (2)

J. W. Costerton, P. S. Stewart, E. P. Greenberg, "Bacterial Biofilms: A Common Cause of Persistent Infections," Science 284, 1318-1322 (1999).
[CrossRef] [PubMed]

K. H. Seo, J. F. Frank, "Attachement of E.Coli O157:H7 to lettuce leaf surface and bacterial viability in response to chlorine treatment as demonstrated by using confocal scanning laser microscopy," J. Food Protect. 62, 3-9 (1999).

1998 (2)

P. S. Mead, P. M. Griffin, "Escherichia coli O157:H7," Lancet 352, 1207-1212 (1998).
[CrossRef] [PubMed]

W. Knoll, "Interfaces and thin films as seen by bound electromagnetic waves," Annu. Rev. Phys. Chem. 49, 569-638 (1998).
[CrossRef]

1996 (1)

1995 (2)

R. Karlsson, R. Stahlberg, "Surface-plasmon resonance detection and multispot sensing for direct monitoring of interactions involving low-molecular weight analytes for determination of low affinities," Anal. Biochem. 228, 274-280 (1995).
[CrossRef] [PubMed]

J. W. Costerton, Z. Lewandowski, D. E. Caldwell, D. R. Korber, and H. M. Lappin-Scott, "Microbial Biofilms," Annu. Rev. Microbiol. 49, 711-45 (1995).
[CrossRef] [PubMed]

1992 (1)

1991 (1)

D. S. Goodsell, "Inside a Living Cell," Trends. Biochem. Sci. 16, 203-206 (1991).
[CrossRef] [PubMed]

1970 (1)

P. J. Wyatt, "Cell Wall Thickness, Size Distribution, Refractive Index Ratio and Dry Weight Content of Living Bacteria," Nature 226, 277-279 (1970).
[CrossRef] [PubMed]

Ardhammer, M.

M. Ardhammer, P. Lincoln, B. Norden, "In visible liposomes: Refractive index matching with sucrose enables flow dichroism assessment of peptide orientation in lipid vesicle membrane," Proc. Natl. Acad. Sci. USA 99, 15313-15317 (2002).
[CrossRef]

Armani, A.M.

A.M. Armani, R.P. Kulkarni, S.E. Fraser, R.C. Flagan, K.J. Vahala, "Label-Free, Single-Molecule Detection with Optical Microcavities," Science 10, 783-787 (2007).
[CrossRef]

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

Arnold, S.

Braun, D.

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]

Caldwell, D. E.

J. W. Costerton, Z. Lewandowski, D. E. Caldwell, D. R. Korber, and H. M. Lappin-Scott, "Microbial Biofilms," Annu. Rev. Microbiol. 49, 711-45 (1995).
[CrossRef] [PubMed]

Chao, C.-Y.

C.-Y. Chao, W. Fung, L.J. Guo, "Polymer Microring Resonators for Biochemical Sensing Applications," IEEE J. Selected Topics Quantum Electron. 12, 134-142 (2006).
[CrossRef]

Costerton, J. W.

J. W. Costerton, P. S. Stewart, E. P. Greenberg, "Bacterial Biofilms: A Common Cause of Persistent Infections," Science 284, 1318-1322 (1999).
[CrossRef] [PubMed]

J. W. Costerton, Z. Lewandowski, D. E. Caldwell, D. R. Korber, and H. M. Lappin-Scott, "Microbial Biofilms," Annu. Rev. Microbiol. 49, 711-45 (1995).
[CrossRef] [PubMed]

Flagan, R.C.

A.M. Armani, R.P. Kulkarni, S.E. Fraser, R.C. Flagan, K.J. Vahala, "Label-Free, Single-Molecule Detection with Optical Microcavities," Science 10, 783-787 (2007).
[CrossRef]

Frank, J. F.

K. H. Seo, J. F. Frank, "Attachement of E.Coli O157:H7 to lettuce leaf surface and bacterial viability in response to chlorine treatment as demonstrated by using confocal scanning laser microscopy," J. Food Protect. 62, 3-9 (1999).

Fraser, S.E.

A.M. Armani, R.P. Kulkarni, S.E. Fraser, R.C. Flagan, K.J. Vahala, "Label-Free, Single-Molecule Detection with Optical Microcavities," Science 10, 783-787 (2007).
[CrossRef]

Fung, W.

C.-Y. Chao, W. Fung, L.J. Guo, "Polymer Microring Resonators for Biochemical Sensing Applications," IEEE J. Selected Topics Quantum Electron. 12, 134-142 (2006).
[CrossRef]

Goodsell, D. S.

D. S. Goodsell, "Inside a Living Cell," Trends. Biochem. Sci. 16, 203-206 (1991).
[CrossRef] [PubMed]

Gorodetsky, M. L.

Greenberg, E. P.

J. W. Costerton, P. S. Stewart, E. P. Greenberg, "Bacterial Biofilms: A Common Cause of Persistent Infections," Science 284, 1318-1322 (1999).
[CrossRef] [PubMed]

Griffin, P. M.

P. S. Mead, P. M. Griffin, "Escherichia coli O157:H7," Lancet 352, 1207-1212 (1998).
[CrossRef] [PubMed]

Guo, L.J.

C.-Y. Chao, W. Fung, L.J. Guo, "Polymer Microring Resonators for Biochemical Sensing Applications," IEEE J. Selected Topics Quantum Electron. 12, 134-142 (2006).
[CrossRef]

Holler, S.

Ilchenko, V. S.

Karlsson, R.

R. Karlsson, R. Stahlberg, "Surface-plasmon resonance detection and multispot sensing for direct monitoring of interactions involving low-molecular weight analytes for determination of low affinities," Anal. Biochem. 228, 274-280 (1995).
[CrossRef] [PubMed]

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," Appl. Phys. Lett. 80, 4057-4059 (2002).
[CrossRef]

Knoll, W.

W. Knoll, "Interfaces and thin films as seen by bound electromagnetic waves," Annu. Rev. Phys. Chem. 49, 569-638 (1998).
[CrossRef]

Korber, D. R.

J. W. Costerton, Z. Lewandowski, D. E. Caldwell, D. R. Korber, and H. M. Lappin-Scott, "Microbial Biofilms," Annu. Rev. Microbiol. 49, 711-45 (1995).
[CrossRef] [PubMed]

Kulkarni, R.P.

A.M. Armani, R.P. Kulkarni, S.E. Fraser, R.C. Flagan, K.J. Vahala, "Label-Free, Single-Molecule Detection with Optical Microcavities," Science 10, 783-787 (2007).
[CrossRef]

Lam, C.

Lappin-Scott, H. M.

J. W. Costerton, Z. Lewandowski, D. E. Caldwell, D. R. Korber, and H. M. Lappin-Scott, "Microbial Biofilms," Annu. Rev. Microbiol. 49, 711-45 (1995).
[CrossRef] [PubMed]

Leung, P. T.

Lewandowski, Z.

J. W. Costerton, Z. Lewandowski, D. E. Caldwell, D. R. Korber, and H. M. Lappin-Scott, "Microbial Biofilms," Annu. Rev. Microbiol. 49, 711-45 (1995).
[CrossRef] [PubMed]

Libchaber, A.

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]

Lincoln, P.

M. Ardhammer, P. Lincoln, B. Norden, "In visible liposomes: Refractive index matching with sucrose enables flow dichroism assessment of peptide orientation in lipid vesicle membrane," Proc. Natl. Acad. Sci. USA 99, 15313-15317 (2002).
[CrossRef]

Mead, P. S.

P. S. Mead, P. M. Griffin, "Escherichia coli O157:H7," Lancet 352, 1207-1212 (1998).
[CrossRef] [PubMed]

Norden, B.

M. Ardhammer, P. Lincoln, B. Norden, "In visible liposomes: Refractive index matching with sucrose enables flow dichroism assessment of peptide orientation in lipid vesicle membrane," Proc. Natl. Acad. Sci. USA 99, 15313-15317 (2002).
[CrossRef]

Noto, M.

Savechnkov, A.

Seo, K. H.

K. H. Seo, J. F. Frank, "Attachement of E.Coli O157:H7 to lettuce leaf surface and bacterial viability in response to chlorine treatment as demonstrated by using confocal scanning laser microscopy," J. Food Protect. 62, 3-9 (1999).

Stahlberg, R.

R. Karlsson, R. Stahlberg, "Surface-plasmon resonance detection and multispot sensing for direct monitoring of interactions involving low-molecular weight analytes for determination of low affinities," Anal. Biochem. 228, 274-280 (1995).
[CrossRef] [PubMed]

Stewart, P. S.

J. W. Costerton, P. S. Stewart, E. P. Greenberg, "Bacterial Biofilms: A Common Cause of Persistent Infections," Science 284, 1318-1322 (1999).
[CrossRef] [PubMed]

Teraoka, I.

Topolancik, J.

J. Topolancik, F. Vollmer, "Photoinduced Transformations in Bacteriorhodopsin Membrane Monitored with Optical Microcavities," Biophys. J. 92, 2223-2229 (2007).
[CrossRef] [PubMed]

Vahala, K.J.

A.M. Armani, R.P. Kulkarni, S.E. Fraser, R.C. Flagan, K.J. Vahala, "Label-Free, Single-Molecule Detection with Optical Microcavities," Science 10, 783-787 (2007).
[CrossRef]

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

Vollmer, F.

J. Topolancik, F. Vollmer, "Photoinduced Transformations in Bacteriorhodopsin Membrane Monitored with Optical Microcavities," Biophys. J. 92, 2223-2229 (2007).
[CrossRef] [PubMed]

M. Noto, F. Vollmer, I. Teraoka, and S. Arnold, "Nanolayer characterization through wavelength multiplexing of a microsphere resonator," Opt. Lett. 30, 510-512 (2005).
[CrossRef] [PubMed]

I. Teraoka, S. Arnold, and F. Vollmer, "Perturbation Approach to Shift of Whispering-Gallery-Modes in Microspheres by Protein Adsorption," J. Opt. Soc. Am. B 20, 1937-1946 (2003).
[CrossRef]

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, 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]

Wyatt, P. J.

P. J. Wyatt, "Cell Wall Thickness, Size Distribution, Refractive Index Ratio and Dry Weight Content of Living Bacteria," Nature 226, 277-279 (1970).
[CrossRef] [PubMed]

Young, K.

Anal. Biochem. (1)

R. Karlsson, R. Stahlberg, "Surface-plasmon resonance detection and multispot sensing for direct monitoring of interactions involving low-molecular weight analytes for determination of low affinities," Anal. Biochem. 228, 274-280 (1995).
[CrossRef] [PubMed]

Annu. Rev. Microbiol. (1)

J. W. Costerton, Z. Lewandowski, D. E. Caldwell, D. R. Korber, and H. M. Lappin-Scott, "Microbial Biofilms," Annu. Rev. Microbiol. 49, 711-45 (1995).
[CrossRef] [PubMed]

Annu. Rev. Phys. Chem. (1)

W. Knoll, "Interfaces and thin films as seen by bound electromagnetic waves," Annu. Rev. Phys. Chem. 49, 569-638 (1998).
[CrossRef]

Appl. 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," Appl. Phys. Lett. 80, 4057-4059 (2002).
[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, F. Vollmer, "Photoinduced Transformations in Bacteriorhodopsin Membrane Monitored with Optical Microcavities," Biophys. J. 92, 2223-2229 (2007).
[CrossRef] [PubMed]

IEEE J. Selected Topics Quantum Electron. (1)

C.-Y. Chao, W. Fung, L.J. Guo, "Polymer Microring Resonators for Biochemical Sensing Applications," IEEE J. Selected Topics Quantum Electron. 12, 134-142 (2006).
[CrossRef]

J. Food Protect. (1)

K. H. Seo, J. F. Frank, "Attachement of E.Coli O157:H7 to lettuce leaf surface and bacterial viability in response to chlorine treatment as demonstrated by using confocal scanning laser microscopy," J. Food Protect. 62, 3-9 (1999).

J. Opt. Soc. Am. B (3)

Lancet (1)

P. S. Mead, P. M. Griffin, "Escherichia coli O157:H7," Lancet 352, 1207-1212 (1998).
[CrossRef] [PubMed]

Nature (1)

P. J. Wyatt, "Cell Wall Thickness, Size Distribution, Refractive Index Ratio and Dry Weight Content of Living Bacteria," Nature 226, 277-279 (1970).
[CrossRef] [PubMed]

Opt. Lett. (4)

Proc. Natl. Acad. Sci. USA (1)

M. Ardhammer, P. Lincoln, B. Norden, "In visible liposomes: Refractive index matching with sucrose enables flow dichroism assessment of peptide orientation in lipid vesicle membrane," Proc. Natl. Acad. Sci. USA 99, 15313-15317 (2002).
[CrossRef]

Science (2)

J. W. Costerton, P. S. Stewart, E. P. Greenberg, "Bacterial Biofilms: A Common Cause of Persistent Infections," Science 284, 1318-1322 (1999).
[CrossRef] [PubMed]

A.M. Armani, R.P. Kulkarni, S.E. Fraser, R.C. Flagan, K.J. Vahala, "Label-Free, Single-Molecule Detection with Optical Microcavities," Science 10, 783-787 (2007).
[CrossRef]

Trends. Biochem. Sci. (1)

D. S. Goodsell, "Inside a Living Cell," Trends. Biochem. Sci. 16, 203-206 (1991).
[CrossRef] [PubMed]

Other (7)

J. D. Jackson, Classical electrodynamics (John Wiley & Son Inc, 1975), Chap. 9 and Chap. 16.

N. Nanninga, Molecular Cytology of Escherichia coli (Academic Press, 1985), Chap. 1ff.

H. C. Berg, in E.Coli in Motion (Springer, 2003) Chap. 1ff.

F. Vollmer, "Taking Detection to the Limit," B.I.F. Futura 20, 239-244 (2005). http://www.bifonds.de/public/inhaltf4.htm

A. M. Armani, D. K. Armani, B. Min, K. J. Vahala, S. M. Spillane, "Ultra-high-Q microcavity operation in H20 and D20," Appl. Phys. Lett . 87, 151118-151118-3 (2005).
[CrossRef]

G. N. Watson, A Treatise on the Theory of Bessel functions (Cambridge University Press, 1966), Chapter VIII.

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, 201107-201107-3 (2005).
[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

(a) A whispering gallery mode (WGM) is excited at ~1.3 µm wavelength in a silica microsphere resonator. The mode is confined by total internal reflection and produces an exponentially decaying evanescent field. Adsorption of rodlike E.coli bacteria perturb the field and cause observable shift of resonance frequency as well as broadening of linewidth. Theory is developed for cylindrical bacteria bound with their axis aligned parallel (horizontal) to the surface but otherwise at arbitrary orientation. Ncav, nsol, nw, nint are the refractive indices of the cavity, the surrounding aqueous solution, the bacteria cell wall and interior (protoplasm) of the bacteria; respectively. (b) Fluorescent image of green fluorescent protein (GFP)-labeled bacteria. The micrograph confirms the orientation of the bound bacteria with cylindrical axis aligned parallel to the microsphere surface. Only few bacteria (arrows) are bound at their tip.

Fig. 2.
Fig. 2.

Shift of resonance wavelength and broadening of linewidth of a WGM due to adsorption of E.coli bacteria. The spectra were recorded in 100 s intervals. In parallel, bacteria surface densities where determined from optical micrographs.

Fig. 3.
Fig. 3.

The plot shows the linear relationship between measured fractional wavelength shift and surface density of adsorbed E.coli bacteria (data points). Analytic theory is plotted for TE and TM modes (lines). Also plotted are analytic results in the long wavelength limit (κ=1, TE mode).

Fig. 4.
Fig. 4.

Measured fractional linewidth shift versus E.coli surface density (data points). The lines are plots of the analytic results for TE and TM modes. Also plotted is the calculated increase of fractional linewidth in the long wavelength limit (κ=1, TE mode).

Equations (55)

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

· D = 0 × E = i ω B · B = 0 × B = i ω D
ε ( r ) = ε background ( r ) + δ ε ( r )
ε background = { n cav 2 for r < R ; n sol 2 for r > R .
δ ε ¯ = σ [ Δ ε A + ( Δ ε ' Δ ε ) A ' ] ,
E lm ( r ) = g l ( ω r ) X lm ( r ̂ )
B lm ( r ) = i ω × E lm ( r ) ,
1 r 2 d d r ( r 2 d g l d r ) + l ( l + 1 ) r 2 g l ε ω 2 g l = 0
1 r 2 d d r ( r 2 d g l ( 0 ) d r ) + l ( l + 1 ) r 2 g l ( 0 ) ε background ω 2 g l ( 0 ) = 0
g l ( 0 ) = { A j l ( n cav ω r ) for r < R e i δ l ( 0 ) [ j l ( n sol ω r ) cos δ l ( 0 ) + n l ( n sol ω r ) sin δ l ( 0 ) ] for r R
g l ( 0 ) 1 n sol . ω r sin ( n sol . ω r 1 2 l π + δ l ( 0 ) ) .
e 2 i δ l ( 0 ) = D l * ( ω ) D l ( ω )
D l ( ω ) = j l ( ς ) h l ( 1 ) ( z ) ς j l ' ( ς ) z h l ( 1 ) ' ( z ) .
n cav ω R = v + O ( v 1 3 ) ,
j l ( n cav ω r ) j l ( z ) e n cav 2 n sol 2 ω ( r R )
n l ( n cav ω r ) n l ( z ) e n cav 2 n sol 2 ω ( r R )
Δ ω = Δ δ l d δ l ( 0 ) d ω
0 d r r 2 [ g l ( 0 ) × Eq . 6 g l × Eq . 7 ]
sin Δ δ l = n sol . ω 3 R d r r 2 δ ε ( r ) ¯ g l ( 0 ) ( r ) g l ( r ) ,
Δ λ λ = Δ ω ω = n sol . ω 3 R d r r 2 δ ε ( r ) ¯ n l 2 ( n sol . ω r ) z ( ς 2 z 2 ) n l 2 ( z )
( Δ λ λ ) = σ [ Δ ε ν + ( Δ ε ' Δ ε ) ν ' ] ( n cav 2 n sol 2 ) R ρ κ
κ = Δ ε ν f ( z ) + ( Δ ε Δ ε ' ) f ( z ' ) Δ ε ν + ( Δ ε ' Δ ε ) ν ' e z
d ( r ) = δ ε ( r ) E 0 ( r )
E rad . ( r ) = ω 2 4 π r e ikr n × [ n × p ( n ) ]
B rad . ( r ) = n sol . n × E rad . ( r )
p ( n ) = v d 3 r ' e i n sol . ω n · r ' d ( r ' )
P ( R ̂ ) = 1 2 lim r r 2 d 2 n E ( r ) × B * ( r ) = n sol . * ω 4 32 π 2 d 2 n ( p · p * n · p 2 ) .
P = σ R 2 d 2 R ̂ P ( R ̂ ) ¯
Γ = 16 π 3 σ [ Δ ε ν + ( Δ ε ' Δ ε ) ν ' ] 2 3 n sol . ( n cav 2 n sol 2 ) λ 4 R ρ η
δ λ λ A fp σ b L ( n cav 2 n sol 2 ) R [ ( 1 e t L ) ( n w 2 n sol 2 ) + e t L ( n int 2 n sol 2 ) ]
j l ( ν sech α ) 1 2 ν ( sech α tanh α ) 1 2 e ν ( tanh α α )
n l ( ν sech α ) 1 2 ν ( sech α tanh α ) 1 2 e ν ( α tanh α ) .
sech α = n sol . ω r ν = n sol . n cav . ( 1 + r R R )
α α 0 r R R coth α 0
j l ( ς ) n l ( z ) ς j l ' ( ς ) z n l ' ( z ) = 0
D l ( ω ) = D l * ( ω )
d δ l ( 0 ) d ω = d d ω j l ( ς ) n l ( z ) ς j l ' ( ς ) z n l ' ( z ) j l ( ς ) j l ( z ) ς j l ' ( ς ) z j l ' ( z )
j l ( ς ) = ς n l ( z ) z n l ' ( z ) j l ' ( ς ) .
j l ( ς ) j l ( z ) ς j l ' ( ς ) z j l ' ( z ) = z j l ( ς ) j l ' ( z ) ς j l ' ( ς ) j l ( z ) = ς j l ' ( ς ) n l ' ( z ) j l ( ς ) n l ( z ) j l ' ( ς ) n l ' ( z )
= ς z 2 j l ' ( ς ) n l ' ( z )
j l ( ς ) n l ( z ) j l ' ( ς ) n l ' ( z ) = 1 z 2
1 z [ z j l ( z ) ] + [ 1 l ( l + 1 ) z 2 ] j l ( z ) = 0
[ ς j l ( ς ) ] = j l ( ς ) + ς [ l ( l + 1 ) ς 2 1 ] j l ( ς )
[ z n l ( z ) ] = n l ( z ) + z [ l ( l + 1 ) z 2 1 ] n l ( z ) .
d d ω j l ( ς ) n l ( z ) ς j l ( ς ) z n l ( z ) = 1 ω { ς j l ( ς ) [ z n l ( z ) ] [ ς j l ( ς ) ] z n l ( z ) }
= ( ς 2 z 2 ) j 1 ( ς ) n l ( z ) = ( ς 2 z 2 ) ς j l ' ( ς ) z n l ' ( z ) n l 2 ( z )
d δ l ( 0 ) d ω = 1 ω z ( ς 2 z 2 ) n l 2 ( z )
B lm ( r ) = f l ( ω r ) X lm ( r ̂ )
E lm ( r ) = i ε ω × B lm ( r )
1 r 2 d d r ( r 2 d f l d r ) + l ( l + 1 ) r 2 f l ε ω 2 f l + 1 ε d δ ε ¯ d r 1 r d d r ( r f l ) = 0
f l r = R = f l r = R +
1 ε cav . d d r ( r f l ) r = R = 1 ε sol . d d r ( r f l ) r = R +
sin Δ δ l = n sol . ω R d r r 2 f l ( 0 ) [ ω 2 δ ε ¯ f l 1 ε r d δ ε ¯ d r d d r ( r f l ) ]
e 2 i δ l ( 0 ) = D l ' * ( ω ) D l ' ( ω )
D l ' ( ω ) = j l ( ς ) h l ( 1 ) ( z ) 1 ς 2 [ ς j l ( ς ) ] ' 1 z 2 [ z h l ( 1 ) ( z ) ] ' .
Δ λ λ = Δ ω ω = R d r r 2 δ ε ¯ [ l ( l + 1 ) f l ( 0 ) 2 + ( d d r ( r f l ( 0 ) ) ) 2 ] n sol . d δ l ( 0 ) d ω .

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