Abstract

Many proposed microfluidic biosensor designs are based on the measurement of the resonances of an optical microcavity. Fluorescence-based resonators tend to be simpler and more robust than setups that use evanescent coupling from tuneable laser to probe the cavity. In all sensor designs the detection limits depend on the wavelength resolution of the detection system, which is a limitation of fluorescence-based devices. In this work, we explore the ultimate resolution and detection limits of refractometric microcavity sensor structures. Because many periodic modes are collected simultaneously from fluorescent resonators, standard Fourier methods can be best suited for rapid and precise analysis of the resonance shifts. Simple numerical expressions to calculate the ultimate sensor resolution and detection limits were found, and the results compared to experiments in which the resonances of fluorescent-core microcapillaries responded to various sucrose concentrations in water.

© 2012 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. E. Herold and A. Rasooly, eds., Lab-on-a-Chip Technology: Fabrication and Microfluidics. Caister Academic Press, 2009.
  2. J. J. Amsden, P. Domachuk, A. Gopinath, and R. D. White, “L. D. Negro D. L Kaplan and F. G. Omenetto, “Biocompatible films: rapid nanoimprinting of silk fibroin films for biophotonic applications,” Adv. Mater. (Deerfield Beach Fla.)22, 1–4 (2010).
  3. A. H. Diercks, A. Ozinsky, C. L. Hansen, J. M. Spotts, D. J. Rodriguez, and A. Aderem, “A microfluidic device for multiplexed protein detection in nano-liter volumes,” Anal. Biochem.386(1), 30–35 (2009).
    [CrossRef] [PubMed]
  4. D. Erickson, S. Mandal, A. H. J. Yang, and B. Cordovez, “Nanobiosensors: optofluidic, electrical and mechanical approaches to biomolecular detection at the nanoscale,” Microfluid. Nanofluid.4(1-2), 33–52 (2008).
    [CrossRef] [PubMed]
  5. F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods5(7), 591–596 (2008).
    [CrossRef] [PubMed]
  6. A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science317(5839), 783–787 (2007).
    [CrossRef] [PubMed]
  7. N. Jokerst, M. Royal, S. Palit, L. Luan, S. Dhar, and T. Tyler, “Chip scale integrated microresonator sensing systems,” J. Biophotonics2(4), 212–226 (2009).
    [CrossRef] [PubMed]
  8. I. M. White, H. Oveys, and X. Fan, “Liquid-core optical ring-resonator sensors,” Opt. Lett.31(9), 1319–1321 (2006).
    [CrossRef] [PubMed]
  9. I. M. White, H. Zhu, J. D. Suter, N. M. Hanumegowda, H. Oveys, M. Zourob, and X. Fan, “Refractometric sensors for lab-on-a-chip based on optical ring resonators,” IEEE Sens. J.7(1), 28–35 (2007).
    [CrossRef]
  10. V. Zamora, A. Díez, M. V. Andrés, and B. Gimeno, “Refractometric sensor based on whispering-gallery modes of thin capillarie,” Opt. Express15(19), 12011–12016 (2007).
    [CrossRef] [PubMed]
  11. S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in microspheres by protein adsorption,” Opt. Lett.28(4), 272–274 (2003).
    [CrossRef] [PubMed]
  12. A. Francois and M. Himmelhaus, “Optical biosensor based on whispering gallery mode excitations in clusters of microparticles,” Appl. Phys. Lett.92(14), 141107 (2008).
    [CrossRef]
  13. F. Vollmer, S. Arnold, and D. Keng, “Single virus detection from the reactive shift of a whispering-gallery mode,” Proc. Natl. Acad. Sci. U.S.A.105(52), 20701–20704 (2008).
    [CrossRef] [PubMed]
  14. P. Bianucci, J. R. Rodríguez, C. M. Clements, J. G. C. Veinot, and A. Meldrum, “Silicon nanocrystal luminescence coupled to whispering gallery modes in optical fibers,” J. Appl. Phys.105(2), 023108 (2009).
    [CrossRef]
  15. I. M. White and X. Fan, “On the performance quantification of resonant refractive index sensors,” Opt. Express16(2), 1020–1028 (2008).
    [CrossRef] [PubMed]
  16. H. T. Beier, G. L. Coté, and K. E. Meissner, “Modeling whispering gallery modes in quantum dot embedded polystyrene microspheres,” J. Opt. Soc. Am. B27(3), 536–543 (2010).
    [CrossRef]
  17. S. Pang, R. E. Beckham, and K. E. Meissner, “Quantum dot-embedded microspheres for remote refractive index sensing,” Appl. Phys. Lett.92(22), 221108 (2008).
    [CrossRef] [PubMed]
  18. D. E. Gómez, I. Pastoriza-Santos, and P. Mulvaney, “Tunable whispering gallery mode emission from quantum-dot-doped microspheres,” Small1(2), 238–241 (2005).
    [CrossRef] [PubMed]
  19. A. Salinas-Castillo, M. Camprubí-Robles, and R. Mallavia, “Synthesis of a new fluorescent conjugated polymer microsphere for chemical sensing in aqueous media,” Chem. Commun. (Camb.)46(8), 1263–1265 (2010).
    [CrossRef] [PubMed]
  20. A. Beltaos and A. Meldrum, “Whispering gallery modes in silicon-nanocrystal-coated silica microspheres,” J. Lumin.126(2), 607–613 (2007).
    [CrossRef]
  21. E. Nuhiji and P. Mulvaney, “Detection of unlabeled oligonucleotide targets using whispering gallery modes in single, fluorescent microspheres,” Small3(8), 1408–1414 (2007).
    [CrossRef] [PubMed]
  22. A. Weller, F. C. Liu, R. Dahint, and M. Himmelhaus, “Whispering gallery mode biosensors in the low-Q limit,” Appl. Phys. B90(3-4), 561–567 (2008).
    [CrossRef]
  23. M. Charlebois, A. Paquet, L. S. Verret, K. Boissinot, M. Boissinot, M. G. Bergeron, and C. N. Allen, “Toward automatic label-free whispering gallery modes biodetection with a quantum dot-coated microsphere population,” Nanoscale Res. Lett.5(3), 524–532 (2010).
    [CrossRef] [PubMed]
  24. A. G. Piersol, “Time delay estimation using phase data,” IEEE Trans. Acoust. Speech Signal Process.29(3), 471–477 (1981).
    [CrossRef]
  25. W. C. Karl and H. H. Pien, “High-resolution biosensor spectral peak shift estimation,” IEEE Trans. Signal Process.53(12), 4631–4639 (2005).
    [CrossRef]
  26. M. Suematsu and M. Takeda, “Wavelength-shift interferometry for distance measurements using the Fourier transform technique for fringe analysis,” Appl. Opt.30(28), 4046–4055 (1991).
    [CrossRef] [PubMed]
  27. F. Bruyneel, H. De Smet, J. Vanfleteren, and A. Van Calster, “Method for measuring the cell gap in liquid-crystal displays,” Opt. Eng.40(2), 259–267 (2001).
    [CrossRef]
  28. C. M. Hessel, M. A. Summers, A. Meldrum, M. Malac, and J. G. C. Veinot, “Direct patterning, conformal coating, and erbium doping of luminescent nc-Si/SiO2 thin films from solution processable hydrogen silsesquioxane,” Adv. Mater. (Deerfield Beach Fla.)19(21), 3513–3516 (2007).
    [CrossRef]
  29. C. P. K. Manchee, V. Zamora, J. W. Silverstone, J. G. C. Veinot, and A. Meldrum, “Refractometric sensing with fluorescent-core microcapillaries,” Opt. Express19(22), 21540–21551 (2011).
    [CrossRef] [PubMed]
  30. A. W. Poon, R. K. Chang, and J. A. Lock, “Spiral morphology-dependent resonances in an optical fiber: effects of fiber tilt and focused Gaussian beam illumination,” Opt. Lett.23(14), 1105–1107 (1998).
    [CrossRef] [PubMed]
  31. A. L. Stancik and E. B. Brauns, “A simple asymmetric lineshape for fitting infrared absorption spectra,” Vib. Spectrosc.47(1), 66–69 (2008).
    [CrossRef]
  32. N. R. Lomb, “Least-squares frequency analysis of unequally spaced data,” Astrophys. Space Sci.39(2), 447–462 (1976).
    [CrossRef]
  33. Santa Barbara Instruments Group, http://www.sbig.com/site/ , Pers. Comm.
  34. V. Zamora, A. Diez, M. V. Andres, and B. V. Gimeno, “Cylindrical optical microcavities: basic properties and sensor applications,” Photonics and Nanostructures – Fundamentals and Applications 9, 149–158 (2011); also V. Zamora and A. Meldrum, unpublished results.

2011

2010

H. T. Beier, G. L. Coté, and K. E. Meissner, “Modeling whispering gallery modes in quantum dot embedded polystyrene microspheres,” J. Opt. Soc. Am. B27(3), 536–543 (2010).
[CrossRef]

J. J. Amsden, P. Domachuk, A. Gopinath, and R. D. White, “L. D. Negro D. L Kaplan and F. G. Omenetto, “Biocompatible films: rapid nanoimprinting of silk fibroin films for biophotonic applications,” Adv. Mater. (Deerfield Beach Fla.)22, 1–4 (2010).

A. Salinas-Castillo, M. Camprubí-Robles, and R. Mallavia, “Synthesis of a new fluorescent conjugated polymer microsphere for chemical sensing in aqueous media,” Chem. Commun. (Camb.)46(8), 1263–1265 (2010).
[CrossRef] [PubMed]

M. Charlebois, A. Paquet, L. S. Verret, K. Boissinot, M. Boissinot, M. G. Bergeron, and C. N. Allen, “Toward automatic label-free whispering gallery modes biodetection with a quantum dot-coated microsphere population,” Nanoscale Res. Lett.5(3), 524–532 (2010).
[CrossRef] [PubMed]

2009

A. H. Diercks, A. Ozinsky, C. L. Hansen, J. M. Spotts, D. J. Rodriguez, and A. Aderem, “A microfluidic device for multiplexed protein detection in nano-liter volumes,” Anal. Biochem.386(1), 30–35 (2009).
[CrossRef] [PubMed]

N. Jokerst, M. Royal, S. Palit, L. Luan, S. Dhar, and T. Tyler, “Chip scale integrated microresonator sensing systems,” J. Biophotonics2(4), 212–226 (2009).
[CrossRef] [PubMed]

P. Bianucci, J. R. Rodríguez, C. M. Clements, J. G. C. Veinot, and A. Meldrum, “Silicon nanocrystal luminescence coupled to whispering gallery modes in optical fibers,” J. Appl. Phys.105(2), 023108 (2009).
[CrossRef]

2008

S. Pang, R. E. Beckham, and K. E. Meissner, “Quantum dot-embedded microspheres for remote refractive index sensing,” Appl. Phys. Lett.92(22), 221108 (2008).
[CrossRef] [PubMed]

A. Francois and M. Himmelhaus, “Optical biosensor based on whispering gallery mode excitations in clusters of microparticles,” Appl. Phys. Lett.92(14), 141107 (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. U.S.A.105(52), 20701–20704 (2008).
[CrossRef] [PubMed]

D. Erickson, S. Mandal, A. H. J. Yang, and B. Cordovez, “Nanobiosensors: optofluidic, electrical and mechanical approaches to biomolecular detection at the nanoscale,” Microfluid. Nanofluid.4(1-2), 33–52 (2008).
[CrossRef] [PubMed]

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods5(7), 591–596 (2008).
[CrossRef] [PubMed]

A. Weller, F. C. Liu, R. Dahint, and M. Himmelhaus, “Whispering gallery mode biosensors in the low-Q limit,” Appl. Phys. B90(3-4), 561–567 (2008).
[CrossRef]

I. M. White and X. Fan, “On the performance quantification of resonant refractive index sensors,” Opt. Express16(2), 1020–1028 (2008).
[CrossRef] [PubMed]

A. L. Stancik and E. B. Brauns, “A simple asymmetric lineshape for fitting infrared absorption spectra,” Vib. Spectrosc.47(1), 66–69 (2008).
[CrossRef]

2007

C. M. Hessel, M. A. Summers, A. Meldrum, M. Malac, and J. G. C. Veinot, “Direct patterning, conformal coating, and erbium doping of luminescent nc-Si/SiO2 thin films from solution processable hydrogen silsesquioxane,” Adv. Mater. (Deerfield Beach Fla.)19(21), 3513–3516 (2007).
[CrossRef]

V. Zamora, A. Díez, M. V. Andrés, and B. Gimeno, “Refractometric sensor based on whispering-gallery modes of thin capillarie,” Opt. Express15(19), 12011–12016 (2007).
[CrossRef] [PubMed]

I. M. White, H. Zhu, J. D. Suter, N. M. Hanumegowda, H. Oveys, M. Zourob, and X. Fan, “Refractometric sensors for lab-on-a-chip based on optical ring resonators,” IEEE Sens. J.7(1), 28–35 (2007).
[CrossRef]

A. Beltaos and A. Meldrum, “Whispering gallery modes in silicon-nanocrystal-coated silica microspheres,” J. Lumin.126(2), 607–613 (2007).
[CrossRef]

E. Nuhiji and P. Mulvaney, “Detection of unlabeled oligonucleotide targets using whispering gallery modes in single, fluorescent microspheres,” Small3(8), 1408–1414 (2007).
[CrossRef] [PubMed]

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science317(5839), 783–787 (2007).
[CrossRef] [PubMed]

2006

2005

D. E. Gómez, I. Pastoriza-Santos, and P. Mulvaney, “Tunable whispering gallery mode emission from quantum-dot-doped microspheres,” Small1(2), 238–241 (2005).
[CrossRef] [PubMed]

W. C. Karl and H. H. Pien, “High-resolution biosensor spectral peak shift estimation,” IEEE Trans. Signal Process.53(12), 4631–4639 (2005).
[CrossRef]

2003

2001

F. Bruyneel, H. De Smet, J. Vanfleteren, and A. Van Calster, “Method for measuring the cell gap in liquid-crystal displays,” Opt. Eng.40(2), 259–267 (2001).
[CrossRef]

1998

1991

1981

A. G. Piersol, “Time delay estimation using phase data,” IEEE Trans. Acoust. Speech Signal Process.29(3), 471–477 (1981).
[CrossRef]

1976

N. R. Lomb, “Least-squares frequency analysis of unequally spaced data,” Astrophys. Space Sci.39(2), 447–462 (1976).
[CrossRef]

Aderem, A.

A. H. Diercks, A. Ozinsky, C. L. Hansen, J. M. Spotts, D. J. Rodriguez, and A. Aderem, “A microfluidic device for multiplexed protein detection in nano-liter volumes,” Anal. Biochem.386(1), 30–35 (2009).
[CrossRef] [PubMed]

Allen, C. N.

M. Charlebois, A. Paquet, L. S. Verret, K. Boissinot, M. Boissinot, M. G. Bergeron, and C. N. Allen, “Toward automatic label-free whispering gallery modes biodetection with a quantum dot-coated microsphere population,” Nanoscale Res. Lett.5(3), 524–532 (2010).
[CrossRef] [PubMed]

Amsden, J. J.

J. J. Amsden, P. Domachuk, A. Gopinath, and R. D. White, “L. D. Negro D. L Kaplan and F. G. Omenetto, “Biocompatible films: rapid nanoimprinting of silk fibroin films for biophotonic applications,” Adv. Mater. (Deerfield Beach Fla.)22, 1–4 (2010).

Andrés, M. V.

Armani, A. M.

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science317(5839), 783–787 (2007).
[CrossRef] [PubMed]

Arnold, S.

F. Vollmer, S. Arnold, and D. Keng, “Single virus detection from the reactive shift of a whispering-gallery mode,” Proc. Natl. Acad. Sci. U.S.A.105(52), 20701–20704 (2008).
[CrossRef] [PubMed]

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods5(7), 591–596 (2008).
[CrossRef] [PubMed]

S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in microspheres by protein adsorption,” Opt. Lett.28(4), 272–274 (2003).
[CrossRef] [PubMed]

Beckham, R. E.

S. Pang, R. E. Beckham, and K. E. Meissner, “Quantum dot-embedded microspheres for remote refractive index sensing,” Appl. Phys. Lett.92(22), 221108 (2008).
[CrossRef] [PubMed]

Beier, H. T.

Beltaos, A.

A. Beltaos and A. Meldrum, “Whispering gallery modes in silicon-nanocrystal-coated silica microspheres,” J. Lumin.126(2), 607–613 (2007).
[CrossRef]

Bergeron, M. G.

M. Charlebois, A. Paquet, L. S. Verret, K. Boissinot, M. Boissinot, M. G. Bergeron, and C. N. Allen, “Toward automatic label-free whispering gallery modes biodetection with a quantum dot-coated microsphere population,” Nanoscale Res. Lett.5(3), 524–532 (2010).
[CrossRef] [PubMed]

Bianucci, P.

P. Bianucci, J. R. Rodríguez, C. M. Clements, J. G. C. Veinot, and A. Meldrum, “Silicon nanocrystal luminescence coupled to whispering gallery modes in optical fibers,” J. Appl. Phys.105(2), 023108 (2009).
[CrossRef]

Boissinot, K.

M. Charlebois, A. Paquet, L. S. Verret, K. Boissinot, M. Boissinot, M. G. Bergeron, and C. N. Allen, “Toward automatic label-free whispering gallery modes biodetection with a quantum dot-coated microsphere population,” Nanoscale Res. Lett.5(3), 524–532 (2010).
[CrossRef] [PubMed]

Boissinot, M.

M. Charlebois, A. Paquet, L. S. Verret, K. Boissinot, M. Boissinot, M. G. Bergeron, and C. N. Allen, “Toward automatic label-free whispering gallery modes biodetection with a quantum dot-coated microsphere population,” Nanoscale Res. Lett.5(3), 524–532 (2010).
[CrossRef] [PubMed]

Brauns, E. B.

A. L. Stancik and E. B. Brauns, “A simple asymmetric lineshape for fitting infrared absorption spectra,” Vib. Spectrosc.47(1), 66–69 (2008).
[CrossRef]

Bruyneel, F.

F. Bruyneel, H. De Smet, J. Vanfleteren, and A. Van Calster, “Method for measuring the cell gap in liquid-crystal displays,” Opt. Eng.40(2), 259–267 (2001).
[CrossRef]

Camprubí-Robles, M.

A. Salinas-Castillo, M. Camprubí-Robles, and R. Mallavia, “Synthesis of a new fluorescent conjugated polymer microsphere for chemical sensing in aqueous media,” Chem. Commun. (Camb.)46(8), 1263–1265 (2010).
[CrossRef] [PubMed]

Chang, R. K.

Charlebois, M.

M. Charlebois, A. Paquet, L. S. Verret, K. Boissinot, M. Boissinot, M. G. Bergeron, and C. N. Allen, “Toward automatic label-free whispering gallery modes biodetection with a quantum dot-coated microsphere population,” Nanoscale Res. Lett.5(3), 524–532 (2010).
[CrossRef] [PubMed]

Clements, C. M.

P. Bianucci, J. R. Rodríguez, C. M. Clements, J. G. C. Veinot, and A. Meldrum, “Silicon nanocrystal luminescence coupled to whispering gallery modes in optical fibers,” J. Appl. Phys.105(2), 023108 (2009).
[CrossRef]

Cordovez, B.

D. Erickson, S. Mandal, A. H. J. Yang, and B. Cordovez, “Nanobiosensors: optofluidic, electrical and mechanical approaches to biomolecular detection at the nanoscale,” Microfluid. Nanofluid.4(1-2), 33–52 (2008).
[CrossRef] [PubMed]

Coté, G. L.

Dahint, R.

A. Weller, F. C. Liu, R. Dahint, and M. Himmelhaus, “Whispering gallery mode biosensors in the low-Q limit,” Appl. Phys. B90(3-4), 561–567 (2008).
[CrossRef]

De Smet, H.

F. Bruyneel, H. De Smet, J. Vanfleteren, and A. Van Calster, “Method for measuring the cell gap in liquid-crystal displays,” Opt. Eng.40(2), 259–267 (2001).
[CrossRef]

Dhar, S.

N. Jokerst, M. Royal, S. Palit, L. Luan, S. Dhar, and T. Tyler, “Chip scale integrated microresonator sensing systems,” J. Biophotonics2(4), 212–226 (2009).
[CrossRef] [PubMed]

Diercks, A. H.

A. H. Diercks, A. Ozinsky, C. L. Hansen, J. M. Spotts, D. J. Rodriguez, and A. Aderem, “A microfluidic device for multiplexed protein detection in nano-liter volumes,” Anal. Biochem.386(1), 30–35 (2009).
[CrossRef] [PubMed]

Díez, A.

Domachuk, P.

J. J. Amsden, P. Domachuk, A. Gopinath, and R. D. White, “L. D. Negro D. L Kaplan and F. G. Omenetto, “Biocompatible films: rapid nanoimprinting of silk fibroin films for biophotonic applications,” Adv. Mater. (Deerfield Beach Fla.)22, 1–4 (2010).

Erickson, D.

D. Erickson, S. Mandal, A. H. J. Yang, and B. Cordovez, “Nanobiosensors: optofluidic, electrical and mechanical approaches to biomolecular detection at the nanoscale,” Microfluid. Nanofluid.4(1-2), 33–52 (2008).
[CrossRef] [PubMed]

Fan, X.

Flagan, R. C.

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science317(5839), 783–787 (2007).
[CrossRef] [PubMed]

Francois, A.

A. Francois and M. Himmelhaus, “Optical biosensor based on whispering gallery mode excitations in clusters of microparticles,” Appl. Phys. Lett.92(14), 141107 (2008).
[CrossRef]

Fraser, S. E.

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science317(5839), 783–787 (2007).
[CrossRef] [PubMed]

Gimeno, B.

Gómez, D. E.

D. E. Gómez, I. Pastoriza-Santos, and P. Mulvaney, “Tunable whispering gallery mode emission from quantum-dot-doped microspheres,” Small1(2), 238–241 (2005).
[CrossRef] [PubMed]

Gopinath, A.

J. J. Amsden, P. Domachuk, A. Gopinath, and R. D. White, “L. D. Negro D. L Kaplan and F. G. Omenetto, “Biocompatible films: rapid nanoimprinting of silk fibroin films for biophotonic applications,” Adv. Mater. (Deerfield Beach Fla.)22, 1–4 (2010).

Hansen, C. L.

A. H. Diercks, A. Ozinsky, C. L. Hansen, J. M. Spotts, D. J. Rodriguez, and A. Aderem, “A microfluidic device for multiplexed protein detection in nano-liter volumes,” Anal. Biochem.386(1), 30–35 (2009).
[CrossRef] [PubMed]

Hanumegowda, N. M.

I. M. White, H. Zhu, J. D. Suter, N. M. Hanumegowda, H. Oveys, M. Zourob, and X. Fan, “Refractometric sensors for lab-on-a-chip based on optical ring resonators,” IEEE Sens. J.7(1), 28–35 (2007).
[CrossRef]

Hessel, C. M.

C. M. Hessel, M. A. Summers, A. Meldrum, M. Malac, and J. G. C. Veinot, “Direct patterning, conformal coating, and erbium doping of luminescent nc-Si/SiO2 thin films from solution processable hydrogen silsesquioxane,” Adv. Mater. (Deerfield Beach Fla.)19(21), 3513–3516 (2007).
[CrossRef]

Himmelhaus, M.

A. Francois and M. Himmelhaus, “Optical biosensor based on whispering gallery mode excitations in clusters of microparticles,” Appl. Phys. Lett.92(14), 141107 (2008).
[CrossRef]

A. Weller, F. C. Liu, R. Dahint, and M. Himmelhaus, “Whispering gallery mode biosensors in the low-Q limit,” Appl. Phys. B90(3-4), 561–567 (2008).
[CrossRef]

Holler, S.

Jokerst, N.

N. Jokerst, M. Royal, S. Palit, L. Luan, S. Dhar, and T. Tyler, “Chip scale integrated microresonator sensing systems,” J. Biophotonics2(4), 212–226 (2009).
[CrossRef] [PubMed]

Karl, W. C.

W. C. Karl and H. H. Pien, “High-resolution biosensor spectral peak shift estimation,” IEEE Trans. Signal Process.53(12), 4631–4639 (2005).
[CrossRef]

Keng, D.

F. Vollmer, S. Arnold, and D. Keng, “Single virus detection from the reactive shift of a whispering-gallery mode,” Proc. Natl. Acad. Sci. U.S.A.105(52), 20701–20704 (2008).
[CrossRef] [PubMed]

Khoshsima, M.

Kulkarni, R. P.

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science317(5839), 783–787 (2007).
[CrossRef] [PubMed]

Liu, F. C.

A. Weller, F. C. Liu, R. Dahint, and M. Himmelhaus, “Whispering gallery mode biosensors in the low-Q limit,” Appl. Phys. B90(3-4), 561–567 (2008).
[CrossRef]

Lock, J. A.

Lomb, N. R.

N. R. Lomb, “Least-squares frequency analysis of unequally spaced data,” Astrophys. Space Sci.39(2), 447–462 (1976).
[CrossRef]

Luan, L.

N. Jokerst, M. Royal, S. Palit, L. Luan, S. Dhar, and T. Tyler, “Chip scale integrated microresonator sensing systems,” J. Biophotonics2(4), 212–226 (2009).
[CrossRef] [PubMed]

Malac, M.

C. M. Hessel, M. A. Summers, A. Meldrum, M. Malac, and J. G. C. Veinot, “Direct patterning, conformal coating, and erbium doping of luminescent nc-Si/SiO2 thin films from solution processable hydrogen silsesquioxane,” Adv. Mater. (Deerfield Beach Fla.)19(21), 3513–3516 (2007).
[CrossRef]

Mallavia, R.

A. Salinas-Castillo, M. Camprubí-Robles, and R. Mallavia, “Synthesis of a new fluorescent conjugated polymer microsphere for chemical sensing in aqueous media,” Chem. Commun. (Camb.)46(8), 1263–1265 (2010).
[CrossRef] [PubMed]

Manchee, C. P. K.

Mandal, S.

D. Erickson, S. Mandal, A. H. J. Yang, and B. Cordovez, “Nanobiosensors: optofluidic, electrical and mechanical approaches to biomolecular detection at the nanoscale,” Microfluid. Nanofluid.4(1-2), 33–52 (2008).
[CrossRef] [PubMed]

Meissner, K. E.

H. T. Beier, G. L. Coté, and K. E. Meissner, “Modeling whispering gallery modes in quantum dot embedded polystyrene microspheres,” J. Opt. Soc. Am. B27(3), 536–543 (2010).
[CrossRef]

S. Pang, R. E. Beckham, and K. E. Meissner, “Quantum dot-embedded microspheres for remote refractive index sensing,” Appl. Phys. Lett.92(22), 221108 (2008).
[CrossRef] [PubMed]

Meldrum, A.

C. P. K. Manchee, V. Zamora, J. W. Silverstone, J. G. C. Veinot, and A. Meldrum, “Refractometric sensing with fluorescent-core microcapillaries,” Opt. Express19(22), 21540–21551 (2011).
[CrossRef] [PubMed]

P. Bianucci, J. R. Rodríguez, C. M. Clements, J. G. C. Veinot, and A. Meldrum, “Silicon nanocrystal luminescence coupled to whispering gallery modes in optical fibers,” J. Appl. Phys.105(2), 023108 (2009).
[CrossRef]

C. M. Hessel, M. A. Summers, A. Meldrum, M. Malac, and J. G. C. Veinot, “Direct patterning, conformal coating, and erbium doping of luminescent nc-Si/SiO2 thin films from solution processable hydrogen silsesquioxane,” Adv. Mater. (Deerfield Beach Fla.)19(21), 3513–3516 (2007).
[CrossRef]

A. Beltaos and A. Meldrum, “Whispering gallery modes in silicon-nanocrystal-coated silica microspheres,” J. Lumin.126(2), 607–613 (2007).
[CrossRef]

Mulvaney, P.

E. Nuhiji and P. Mulvaney, “Detection of unlabeled oligonucleotide targets using whispering gallery modes in single, fluorescent microspheres,” Small3(8), 1408–1414 (2007).
[CrossRef] [PubMed]

D. E. Gómez, I. Pastoriza-Santos, and P. Mulvaney, “Tunable whispering gallery mode emission from quantum-dot-doped microspheres,” Small1(2), 238–241 (2005).
[CrossRef] [PubMed]

Nuhiji, E.

E. Nuhiji and P. Mulvaney, “Detection of unlabeled oligonucleotide targets using whispering gallery modes in single, fluorescent microspheres,” Small3(8), 1408–1414 (2007).
[CrossRef] [PubMed]

Oveys, H.

I. M. White, H. Zhu, J. D. Suter, N. M. Hanumegowda, H. Oveys, M. Zourob, and X. Fan, “Refractometric sensors for lab-on-a-chip based on optical ring resonators,” IEEE Sens. J.7(1), 28–35 (2007).
[CrossRef]

I. M. White, H. Oveys, and X. Fan, “Liquid-core optical ring-resonator sensors,” Opt. Lett.31(9), 1319–1321 (2006).
[CrossRef] [PubMed]

Ozinsky, A.

A. H. Diercks, A. Ozinsky, C. L. Hansen, J. M. Spotts, D. J. Rodriguez, and A. Aderem, “A microfluidic device for multiplexed protein detection in nano-liter volumes,” Anal. Biochem.386(1), 30–35 (2009).
[CrossRef] [PubMed]

Palit, S.

N. Jokerst, M. Royal, S. Palit, L. Luan, S. Dhar, and T. Tyler, “Chip scale integrated microresonator sensing systems,” J. Biophotonics2(4), 212–226 (2009).
[CrossRef] [PubMed]

Pang, S.

S. Pang, R. E. Beckham, and K. E. Meissner, “Quantum dot-embedded microspheres for remote refractive index sensing,” Appl. Phys. Lett.92(22), 221108 (2008).
[CrossRef] [PubMed]

Paquet, A.

M. Charlebois, A. Paquet, L. S. Verret, K. Boissinot, M. Boissinot, M. G. Bergeron, and C. N. Allen, “Toward automatic label-free whispering gallery modes biodetection with a quantum dot-coated microsphere population,” Nanoscale Res. Lett.5(3), 524–532 (2010).
[CrossRef] [PubMed]

Pastoriza-Santos, I.

D. E. Gómez, I. Pastoriza-Santos, and P. Mulvaney, “Tunable whispering gallery mode emission from quantum-dot-doped microspheres,” Small1(2), 238–241 (2005).
[CrossRef] [PubMed]

Pien, H. H.

W. C. Karl and H. H. Pien, “High-resolution biosensor spectral peak shift estimation,” IEEE Trans. Signal Process.53(12), 4631–4639 (2005).
[CrossRef]

Piersol, A. G.

A. G. Piersol, “Time delay estimation using phase data,” IEEE Trans. Acoust. Speech Signal Process.29(3), 471–477 (1981).
[CrossRef]

Poon, A. W.

Rodriguez, D. J.

A. H. Diercks, A. Ozinsky, C. L. Hansen, J. M. Spotts, D. J. Rodriguez, and A. Aderem, “A microfluidic device for multiplexed protein detection in nano-liter volumes,” Anal. Biochem.386(1), 30–35 (2009).
[CrossRef] [PubMed]

Rodríguez, J. R.

P. Bianucci, J. R. Rodríguez, C. M. Clements, J. G. C. Veinot, and A. Meldrum, “Silicon nanocrystal luminescence coupled to whispering gallery modes in optical fibers,” J. Appl. Phys.105(2), 023108 (2009).
[CrossRef]

Royal, M.

N. Jokerst, M. Royal, S. Palit, L. Luan, S. Dhar, and T. Tyler, “Chip scale integrated microresonator sensing systems,” J. Biophotonics2(4), 212–226 (2009).
[CrossRef] [PubMed]

Salinas-Castillo, A.

A. Salinas-Castillo, M. Camprubí-Robles, and R. Mallavia, “Synthesis of a new fluorescent conjugated polymer microsphere for chemical sensing in aqueous media,” Chem. Commun. (Camb.)46(8), 1263–1265 (2010).
[CrossRef] [PubMed]

Silverstone, J. W.

Spotts, J. M.

A. H. Diercks, A. Ozinsky, C. L. Hansen, J. M. Spotts, D. J. Rodriguez, and A. Aderem, “A microfluidic device for multiplexed protein detection in nano-liter volumes,” Anal. Biochem.386(1), 30–35 (2009).
[CrossRef] [PubMed]

Stancik, A. L.

A. L. Stancik and E. B. Brauns, “A simple asymmetric lineshape for fitting infrared absorption spectra,” Vib. Spectrosc.47(1), 66–69 (2008).
[CrossRef]

Suematsu, M.

Summers, M. A.

C. M. Hessel, M. A. Summers, A. Meldrum, M. Malac, and J. G. C. Veinot, “Direct patterning, conformal coating, and erbium doping of luminescent nc-Si/SiO2 thin films from solution processable hydrogen silsesquioxane,” Adv. Mater. (Deerfield Beach Fla.)19(21), 3513–3516 (2007).
[CrossRef]

Suter, J. D.

I. M. White, H. Zhu, J. D. Suter, N. M. Hanumegowda, H. Oveys, M. Zourob, and X. Fan, “Refractometric sensors for lab-on-a-chip based on optical ring resonators,” IEEE Sens. J.7(1), 28–35 (2007).
[CrossRef]

Takeda, M.

Teraoka, I.

Tyler, T.

N. Jokerst, M. Royal, S. Palit, L. Luan, S. Dhar, and T. Tyler, “Chip scale integrated microresonator sensing systems,” J. Biophotonics2(4), 212–226 (2009).
[CrossRef] [PubMed]

Vahala, K. J.

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science317(5839), 783–787 (2007).
[CrossRef] [PubMed]

Van Calster, A.

F. Bruyneel, H. De Smet, J. Vanfleteren, and A. Van Calster, “Method for measuring the cell gap in liquid-crystal displays,” Opt. Eng.40(2), 259–267 (2001).
[CrossRef]

Vanfleteren, J.

F. Bruyneel, H. De Smet, J. Vanfleteren, and A. Van Calster, “Method for measuring the cell gap in liquid-crystal displays,” Opt. Eng.40(2), 259–267 (2001).
[CrossRef]

Veinot, J. G. C.

C. P. K. Manchee, V. Zamora, J. W. Silverstone, J. G. C. Veinot, and A. Meldrum, “Refractometric sensing with fluorescent-core microcapillaries,” Opt. Express19(22), 21540–21551 (2011).
[CrossRef] [PubMed]

P. Bianucci, J. R. Rodríguez, C. M. Clements, J. G. C. Veinot, and A. Meldrum, “Silicon nanocrystal luminescence coupled to whispering gallery modes in optical fibers,” J. Appl. Phys.105(2), 023108 (2009).
[CrossRef]

C. M. Hessel, M. A. Summers, A. Meldrum, M. Malac, and J. G. C. Veinot, “Direct patterning, conformal coating, and erbium doping of luminescent nc-Si/SiO2 thin films from solution processable hydrogen silsesquioxane,” Adv. Mater. (Deerfield Beach Fla.)19(21), 3513–3516 (2007).
[CrossRef]

Verret, L. S.

M. Charlebois, A. Paquet, L. S. Verret, K. Boissinot, M. Boissinot, M. G. Bergeron, and C. N. Allen, “Toward automatic label-free whispering gallery modes biodetection with a quantum dot-coated microsphere population,” Nanoscale Res. Lett.5(3), 524–532 (2010).
[CrossRef] [PubMed]

Vollmer, F.

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods5(7), 591–596 (2008).
[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. U.S.A.105(52), 20701–20704 (2008).
[CrossRef] [PubMed]

S. Arnold, M. Khoshsima, I. Teraoka, S. Holler, and F. Vollmer, “Shift of whispering-gallery modes in microspheres by protein adsorption,” Opt. Lett.28(4), 272–274 (2003).
[CrossRef] [PubMed]

Weller, A.

A. Weller, F. C. Liu, R. Dahint, and M. Himmelhaus, “Whispering gallery mode biosensors in the low-Q limit,” Appl. Phys. B90(3-4), 561–567 (2008).
[CrossRef]

White, I. M.

White, R. D.

J. J. Amsden, P. Domachuk, A. Gopinath, and R. D. White, “L. D. Negro D. L Kaplan and F. G. Omenetto, “Biocompatible films: rapid nanoimprinting of silk fibroin films for biophotonic applications,” Adv. Mater. (Deerfield Beach Fla.)22, 1–4 (2010).

Yang, A. H. J.

D. Erickson, S. Mandal, A. H. J. Yang, and B. Cordovez, “Nanobiosensors: optofluidic, electrical and mechanical approaches to biomolecular detection at the nanoscale,” Microfluid. Nanofluid.4(1-2), 33–52 (2008).
[CrossRef] [PubMed]

Zamora, V.

Zhu, H.

I. M. White, H. Zhu, J. D. Suter, N. M. Hanumegowda, H. Oveys, M. Zourob, and X. Fan, “Refractometric sensors for lab-on-a-chip based on optical ring resonators,” IEEE Sens. J.7(1), 28–35 (2007).
[CrossRef]

Zourob, M.

I. M. White, H. Zhu, J. D. Suter, N. M. Hanumegowda, H. Oveys, M. Zourob, and X. Fan, “Refractometric sensors for lab-on-a-chip based on optical ring resonators,” IEEE Sens. J.7(1), 28–35 (2007).
[CrossRef]

Adv. Mater. (Deerfield Beach Fla.)

J. J. Amsden, P. Domachuk, A. Gopinath, and R. D. White, “L. D. Negro D. L Kaplan and F. G. Omenetto, “Biocompatible films: rapid nanoimprinting of silk fibroin films for biophotonic applications,” Adv. Mater. (Deerfield Beach Fla.)22, 1–4 (2010).

C. M. Hessel, M. A. Summers, A. Meldrum, M. Malac, and J. G. C. Veinot, “Direct patterning, conformal coating, and erbium doping of luminescent nc-Si/SiO2 thin films from solution processable hydrogen silsesquioxane,” Adv. Mater. (Deerfield Beach Fla.)19(21), 3513–3516 (2007).
[CrossRef]

Anal. Biochem.

A. H. Diercks, A. Ozinsky, C. L. Hansen, J. M. Spotts, D. J. Rodriguez, and A. Aderem, “A microfluidic device for multiplexed protein detection in nano-liter volumes,” Anal. Biochem.386(1), 30–35 (2009).
[CrossRef] [PubMed]

Appl. Opt.

Appl. Phys. B

A. Weller, F. C. Liu, R. Dahint, and M. Himmelhaus, “Whispering gallery mode biosensors in the low-Q limit,” Appl. Phys. B90(3-4), 561–567 (2008).
[CrossRef]

Appl. Phys. Lett.

A. Francois and M. Himmelhaus, “Optical biosensor based on whispering gallery mode excitations in clusters of microparticles,” Appl. Phys. Lett.92(14), 141107 (2008).
[CrossRef]

S. Pang, R. E. Beckham, and K. E. Meissner, “Quantum dot-embedded microspheres for remote refractive index sensing,” Appl. Phys. Lett.92(22), 221108 (2008).
[CrossRef] [PubMed]

Astrophys. Space Sci.

N. R. Lomb, “Least-squares frequency analysis of unequally spaced data,” Astrophys. Space Sci.39(2), 447–462 (1976).
[CrossRef]

Chem. Commun. (Camb.)

A. Salinas-Castillo, M. Camprubí-Robles, and R. Mallavia, “Synthesis of a new fluorescent conjugated polymer microsphere for chemical sensing in aqueous media,” Chem. Commun. (Camb.)46(8), 1263–1265 (2010).
[CrossRef] [PubMed]

IEEE Sens. J.

I. M. White, H. Zhu, J. D. Suter, N. M. Hanumegowda, H. Oveys, M. Zourob, and X. Fan, “Refractometric sensors for lab-on-a-chip based on optical ring resonators,” IEEE Sens. J.7(1), 28–35 (2007).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process.

A. G. Piersol, “Time delay estimation using phase data,” IEEE Trans. Acoust. Speech Signal Process.29(3), 471–477 (1981).
[CrossRef]

IEEE Trans. Signal Process.

W. C. Karl and H. H. Pien, “High-resolution biosensor spectral peak shift estimation,” IEEE Trans. Signal Process.53(12), 4631–4639 (2005).
[CrossRef]

J. Appl. Phys.

P. Bianucci, J. R. Rodríguez, C. M. Clements, J. G. C. Veinot, and A. Meldrum, “Silicon nanocrystal luminescence coupled to whispering gallery modes in optical fibers,” J. Appl. Phys.105(2), 023108 (2009).
[CrossRef]

J. Biophotonics

N. Jokerst, M. Royal, S. Palit, L. Luan, S. Dhar, and T. Tyler, “Chip scale integrated microresonator sensing systems,” J. Biophotonics2(4), 212–226 (2009).
[CrossRef] [PubMed]

J. Lumin.

A. Beltaos and A. Meldrum, “Whispering gallery modes in silicon-nanocrystal-coated silica microspheres,” J. Lumin.126(2), 607–613 (2007).
[CrossRef]

J. Opt. Soc. Am. B

Microfluid. Nanofluid.

D. Erickson, S. Mandal, A. H. J. Yang, and B. Cordovez, “Nanobiosensors: optofluidic, electrical and mechanical approaches to biomolecular detection at the nanoscale,” Microfluid. Nanofluid.4(1-2), 33–52 (2008).
[CrossRef] [PubMed]

Nanoscale Res. Lett.

M. Charlebois, A. Paquet, L. S. Verret, K. Boissinot, M. Boissinot, M. G. Bergeron, and C. N. Allen, “Toward automatic label-free whispering gallery modes biodetection with a quantum dot-coated microsphere population,” Nanoscale Res. Lett.5(3), 524–532 (2010).
[CrossRef] [PubMed]

Nat. Methods

F. Vollmer and S. Arnold, “Whispering-gallery-mode biosensing: label-free detection down to single molecules,” Nat. Methods5(7), 591–596 (2008).
[CrossRef] [PubMed]

Opt. Eng.

F. Bruyneel, H. De Smet, J. Vanfleteren, and A. Van Calster, “Method for measuring the cell gap in liquid-crystal displays,” Opt. Eng.40(2), 259–267 (2001).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. Natl. Acad. Sci. U.S.A.

F. Vollmer, S. Arnold, and D. Keng, “Single virus detection from the reactive shift of a whispering-gallery mode,” Proc. Natl. Acad. Sci. U.S.A.105(52), 20701–20704 (2008).
[CrossRef] [PubMed]

Science

A. M. Armani, R. P. Kulkarni, S. E. Fraser, R. C. Flagan, and K. J. Vahala, “Label-free, single-molecule detection with optical microcavities,” Science317(5839), 783–787 (2007).
[CrossRef] [PubMed]

Small

D. E. Gómez, I. Pastoriza-Santos, and P. Mulvaney, “Tunable whispering gallery mode emission from quantum-dot-doped microspheres,” Small1(2), 238–241 (2005).
[CrossRef] [PubMed]

E. Nuhiji and P. Mulvaney, “Detection of unlabeled oligonucleotide targets using whispering gallery modes in single, fluorescent microspheres,” Small3(8), 1408–1414 (2007).
[CrossRef] [PubMed]

Vib. Spectrosc.

A. L. Stancik and E. B. Brauns, “A simple asymmetric lineshape for fitting infrared absorption spectra,” Vib. Spectrosc.47(1), 66–69 (2008).
[CrossRef]

Other

Santa Barbara Instruments Group, http://www.sbig.com/site/ , Pers. Comm.

V. Zamora, A. Diez, M. V. Andres, and B. V. Gimeno, “Cylindrical optical microcavities: basic properties and sensor applications,” Photonics and Nanostructures – Fundamentals and Applications 9, 149–158 (2011); also V. Zamora and A. Meldrum, unpublished results.

K. E. Herold and A. Rasooly, eds., Lab-on-a-Chip Technology: Fabrication and Microfluidics. Caister Academic Press, 2009.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

(a) a set of simulated spectra with different SNRs, offset for clarity. (b) the 3σ resolution (vertical axis) as a function of the SNR and the WGM peak width, Δfpeak, obtained by least squares Lorentzian curve fitting to one of the modes in each spectrum. The pitch was 47 GHz. The colored plane represents the best fit solution for the system (given by Eq. (3)), giving an r2 value of 0.97.

Fig. 2
Fig. 2

(a) the dominant sinusoidal component of the DFT spectrum (black), corresponding to the FSR of the resonances, along with the original Lorentzian WGMs (red). (b) the power spectrum (squared magnitude of the Fourier components) of the spectral waveform. The 15th component clearly dominates, reflecting the strongly periodic nature of the simulated WGM spectrum; (c) the 3σ resolution for the DFT method, as a function of the SNR and the WGM peak width. As in Fig. 2, the colored surface is the linear least squares best fit solution (given by Eq. (4)) to the numerical results for P = 47 GHz, showing in this case r2 = 0.99.

Fig. 3
Fig. 3

Diagram of an FCM consisting of a capillary with an high-index fluorescent QD layer (cyan) coated on the channel surface. The field directions for the TE polarization are shown in yellow and green.

Fig. 4
Fig. 4

The main panel shows the WGM spectrum for each 0.0012-RIU step in the solution refractive index, in wavelength units. The vertical dashed line is a guide to the eye, showing that the shift of a single mode is difficult to observe. The data are offset for clarity. The inset shows one of the peaks (converted to frequency units) with the skewed Lorentzian fit, showing the location of the peak and centre (f0) frequencies.

Fig. 5
Fig. 5

Wavelength shifts for the spectra in Fig. 4 (SNR = 3x105) obtained using different methods. Left: peak picking; Middle: curve fitting; Right: weighted fit of the Fourier phase components. The error bars are one standard deviation from 10 repetitions of a single-index measurement and shift analysis. The first two methods used a peak centered at 382 THz.

Fig. 6
Fig. 6

The same spectrum taken with four different collection times (240s, 10s, 1s, and 0.1s) with the corresponding SNR. All spectra were taken with water inside the capillary channel.

Fig. 7
Fig. 7

WGM peak shifts as a function of RI, for an SNR = 6. A reasonable correlation can still be extracted from the DFT method; whereas for peak picking the spectral shift information is lost. In this case, for technical reasons the data was collected only once. The errors in the CF and DFT method were drawn from their (assumed) 3σ normal distribution about the linear model, and for the peak picking method, the error was assumed the same as in Fig. 5a where the spectral data collection was repeated.

Tables (2)

Tables Icon

Table 1 Comparison of the ultimate resolution for fluorescence and evanescent WGMs.

Tables Icon

Table 2 Experimental and numerical results for a wide range of experimental SNRs. The subscripts n and e refer to numerical and experimental values, respectively.

Equations (1)

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

P(f)= 2A πγ(f) /[ 1+4 ( f f 0 γ(f) ) 2 ].

Metrics