Abstract

We report a novel method of diffraction imaging flow cytometry to measure and analyze size distribution of microspheres. An automated and robust image processing software based on the short-time-Fourier-transform algorithm has been developed to analyze the characteristic and spatially varying oscillations of side scatters recorded as a diffraction image. Our results demonstrate that the new method allows accurate and rapid determination of single microspheres’ diameters ranging from 1 to 100μm. The capacity for analysis of light scattering by two-sphere aggregates has been demonstrated but analytical tools for characterization of aggregates by multiple microspheres remain to be developed.

© 2012 OSA

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References

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  1. K. K. Kim, D. W. Pack, M. Ferrari, A. P. Lee, and L. J. Lee, “Microspheres for drug delivery,” in Biological and Biomedical Nanotechnology, A. P. Lee, J. Lee, and M. Ferrari, eds. (Springer, 2006), pp. 19–50.
  2. D. Svoboda, M. Kozubek, and S. Stejskal, “Generation of digital phantoms of cell nuclei and simulation of image formation in 3D image cytometry,” Cytometry A 75A(6), 494–509 (2009).
    [CrossRef] [PubMed]
  3. B. P. Hanley, L. Xing, and R. H. Cheng, “Variance in multiplex suspension array assays: microsphere size variation impact,” Theor. Biol. Med. Model. 4(1), 31–38 (2007).
    [CrossRef] [PubMed]
  4. I. K. Ludlow and P. H. Kaye, “A scanning diffractometer for the rapid analysis of microparticles and biological cells,” J. Colloid Interface Sci. 69(3), 571–589 (1979).
    [CrossRef]
  5. S. L. Min and A. Gomez, “High-resolution size measurement of single spherical particles with a fast Fourier transform of the angular scattering intensity,” Appl. Opt. 35(24), 4919–4926 (1996).
    [CrossRef] [PubMed]
  6. J. T. Soini, A. V. Chernyshev, P. E. Hänninen, E. Soini, and V. P. Maltsev, “A new design of the flow cuvette and optical set-up for the scanning flow cytometer,” Cytometry 31(2), 78–84 (1998).
    [CrossRef] [PubMed]
  7. K. A. Semyanov, P. A. Tarasov, A. E. Zharinov, A. V. Chernyshev, A. G. Hoekstra, and V. P. Maltsev, “Single-particle sizing from light scattering by spectral decomposition,” Appl. Opt. 43(26), 5110–5115 (2004).
    [CrossRef] [PubMed]
  8. C. Godefroy and M. Adjouadi, “Particle sizing in a flow environment using light scattering patterns,” Part. Part. Syst. Charact. 17(2), 47–55 (2000).
    [CrossRef]
  9. B. Ovryn, “Three-dimensional forward scattering particle image velocimetry applied to a microscopic field-of-view,” Exp. Fluids 29(7), S175–S184 (2000).
    [CrossRef]
  10. Y. L. Xu and B. A. S. Gustafson, “Comparison between Multisphere Light-scattering Calculations: Rigorous Solution and Discrete-Dipole Approximation,” Astrophys. J. 513(2), 894–909 (1999).
    [CrossRef]
  11. K. M. Jacobs, J. Q. Lu, and X. H. Hu, “Development of a diffraction imaging flow cytometer,” Opt. Lett. 34(19), 2985–2987 (2009).
    [CrossRef] [PubMed]
  12. K. M. Jacobs, L. V. Yang, J. Ding, A. E. Ekpenyong, R. Castellone, J. Q. Lu, and X. H. Hu, “Diffraction imaging of spheres and melanoma cells with a microscope objective,” J Biophotonics 2(8-9), 521–527 (2009).
    [CrossRef] [PubMed]
  13. K. Dong, Y. Feng, K. M. Jacobs, J. Q. Lu, R. S. Brock, L. V. Yang, F. E. Bertrand, M. A. Farwell, and X. H. Hu, “Label-free classification of cultured cells through diffraction imaging,” Biomed. Opt. Express 2(6), 1717–1726 (2011).
    [CrossRef] [PubMed]
  14. M. Portnoff, “Time-frequency representation of digital signals and systems based on short-time Fourier analysis,” IEEE Trans. Acoust., Speech Signal Process. 28(1), 55–69 (1980).
    [CrossRef]
  15. J. Q. Lu, P. Yang, and X.-H. Hu, “Simulations of light scattering from a biconcave red blood cell using the finite-difference time-domain method,” J. Biomed. Opt. 10(2), 024022 (2005).
    [CrossRef] [PubMed]
  16. M. A. Yurkin, A. G. Hoekstra, R. S. Brock, and J. Q. Lu, “Systematic comparison of the discrete dipole approximation and the finite difference time domain method for large dielectric scatterers,” Opt. Express 15(26), 17902–17911 (2007).
    [CrossRef] [PubMed]
  17. X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X. H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48(24), 4165–4172 (2003).
    [CrossRef] [PubMed]
  18. J. G. Daugman, “Complete discrete 2-D Gabor transforms by neural networks for image analysis and compression,” IEEE Trans. Acoust., Speech, Signal Process., 36(7), 1169–1179 (1988).
    [CrossRef]

2011 (1)

2009 (3)

K. M. Jacobs, J. Q. Lu, and X. H. Hu, “Development of a diffraction imaging flow cytometer,” Opt. Lett. 34(19), 2985–2987 (2009).
[CrossRef] [PubMed]

D. Svoboda, M. Kozubek, and S. Stejskal, “Generation of digital phantoms of cell nuclei and simulation of image formation in 3D image cytometry,” Cytometry A 75A(6), 494–509 (2009).
[CrossRef] [PubMed]

K. M. Jacobs, L. V. Yang, J. Ding, A. E. Ekpenyong, R. Castellone, J. Q. Lu, and X. H. Hu, “Diffraction imaging of spheres and melanoma cells with a microscope objective,” J Biophotonics 2(8-9), 521–527 (2009).
[CrossRef] [PubMed]

2007 (2)

2005 (1)

J. Q. Lu, P. Yang, and X.-H. Hu, “Simulations of light scattering from a biconcave red blood cell using the finite-difference time-domain method,” J. Biomed. Opt. 10(2), 024022 (2005).
[CrossRef] [PubMed]

2004 (1)

2003 (1)

X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X. H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48(24), 4165–4172 (2003).
[CrossRef] [PubMed]

2000 (2)

C. Godefroy and M. Adjouadi, “Particle sizing in a flow environment using light scattering patterns,” Part. Part. Syst. Charact. 17(2), 47–55 (2000).
[CrossRef]

B. Ovryn, “Three-dimensional forward scattering particle image velocimetry applied to a microscopic field-of-view,” Exp. Fluids 29(7), S175–S184 (2000).
[CrossRef]

1999 (1)

Y. L. Xu and B. A. S. Gustafson, “Comparison between Multisphere Light-scattering Calculations: Rigorous Solution and Discrete-Dipole Approximation,” Astrophys. J. 513(2), 894–909 (1999).
[CrossRef]

1998 (1)

J. T. Soini, A. V. Chernyshev, P. E. Hänninen, E. Soini, and V. P. Maltsev, “A new design of the flow cuvette and optical set-up for the scanning flow cytometer,” Cytometry 31(2), 78–84 (1998).
[CrossRef] [PubMed]

1996 (1)

1988 (1)

J. G. Daugman, “Complete discrete 2-D Gabor transforms by neural networks for image analysis and compression,” IEEE Trans. Acoust., Speech, Signal Process., 36(7), 1169–1179 (1988).
[CrossRef]

1980 (1)

M. Portnoff, “Time-frequency representation of digital signals and systems based on short-time Fourier analysis,” IEEE Trans. Acoust., Speech Signal Process. 28(1), 55–69 (1980).
[CrossRef]

1979 (1)

I. K. Ludlow and P. H. Kaye, “A scanning diffractometer for the rapid analysis of microparticles and biological cells,” J. Colloid Interface Sci. 69(3), 571–589 (1979).
[CrossRef]

Adjouadi, M.

C. Godefroy and M. Adjouadi, “Particle sizing in a flow environment using light scattering patterns,” Part. Part. Syst. Charact. 17(2), 47–55 (2000).
[CrossRef]

Bertrand, F. E.

Brock, R. S.

Castellone, R.

K. M. Jacobs, L. V. Yang, J. Ding, A. E. Ekpenyong, R. Castellone, J. Q. Lu, and X. H. Hu, “Diffraction imaging of spheres and melanoma cells with a microscope objective,” J Biophotonics 2(8-9), 521–527 (2009).
[CrossRef] [PubMed]

Cheng, R. H.

B. P. Hanley, L. Xing, and R. H. Cheng, “Variance in multiplex suspension array assays: microsphere size variation impact,” Theor. Biol. Med. Model. 4(1), 31–38 (2007).
[CrossRef] [PubMed]

Chernyshev, A. V.

K. A. Semyanov, P. A. Tarasov, A. E. Zharinov, A. V. Chernyshev, A. G. Hoekstra, and V. P. Maltsev, “Single-particle sizing from light scattering by spectral decomposition,” Appl. Opt. 43(26), 5110–5115 (2004).
[CrossRef] [PubMed]

J. T. Soini, A. V. Chernyshev, P. E. Hänninen, E. Soini, and V. P. Maltsev, “A new design of the flow cuvette and optical set-up for the scanning flow cytometer,” Cytometry 31(2), 78–84 (1998).
[CrossRef] [PubMed]

Daugman, J. G.

J. G. Daugman, “Complete discrete 2-D Gabor transforms by neural networks for image analysis and compression,” IEEE Trans. Acoust., Speech, Signal Process., 36(7), 1169–1179 (1988).
[CrossRef]

Ding, J.

K. M. Jacobs, L. V. Yang, J. Ding, A. E. Ekpenyong, R. Castellone, J. Q. Lu, and X. H. Hu, “Diffraction imaging of spheres and melanoma cells with a microscope objective,” J Biophotonics 2(8-9), 521–527 (2009).
[CrossRef] [PubMed]

Dong, K.

Ekpenyong, A. E.

K. M. Jacobs, L. V. Yang, J. Ding, A. E. Ekpenyong, R. Castellone, J. Q. Lu, and X. H. Hu, “Diffraction imaging of spheres and melanoma cells with a microscope objective,” J Biophotonics 2(8-9), 521–527 (2009).
[CrossRef] [PubMed]

Farwell, M. A.

Feng, Y.

Godefroy, C.

C. Godefroy and M. Adjouadi, “Particle sizing in a flow environment using light scattering patterns,” Part. Part. Syst. Charact. 17(2), 47–55 (2000).
[CrossRef]

Gomez, A.

Gustafson, B. A. S.

Y. L. Xu and B. A. S. Gustafson, “Comparison between Multisphere Light-scattering Calculations: Rigorous Solution and Discrete-Dipole Approximation,” Astrophys. J. 513(2), 894–909 (1999).
[CrossRef]

Hanley, B. P.

B. P. Hanley, L. Xing, and R. H. Cheng, “Variance in multiplex suspension array assays: microsphere size variation impact,” Theor. Biol. Med. Model. 4(1), 31–38 (2007).
[CrossRef] [PubMed]

Hänninen, P. E.

J. T. Soini, A. V. Chernyshev, P. E. Hänninen, E. Soini, and V. P. Maltsev, “A new design of the flow cuvette and optical set-up for the scanning flow cytometer,” Cytometry 31(2), 78–84 (1998).
[CrossRef] [PubMed]

Hoekstra, A. G.

Hu, X. H.

K. Dong, Y. Feng, K. M. Jacobs, J. Q. Lu, R. S. Brock, L. V. Yang, F. E. Bertrand, M. A. Farwell, and X. H. Hu, “Label-free classification of cultured cells through diffraction imaging,” Biomed. Opt. Express 2(6), 1717–1726 (2011).
[CrossRef] [PubMed]

K. M. Jacobs, J. Q. Lu, and X. H. Hu, “Development of a diffraction imaging flow cytometer,” Opt. Lett. 34(19), 2985–2987 (2009).
[CrossRef] [PubMed]

K. M. Jacobs, L. V. Yang, J. Ding, A. E. Ekpenyong, R. Castellone, J. Q. Lu, and X. H. Hu, “Diffraction imaging of spheres and melanoma cells with a microscope objective,” J Biophotonics 2(8-9), 521–527 (2009).
[CrossRef] [PubMed]

X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X. H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48(24), 4165–4172 (2003).
[CrossRef] [PubMed]

Hu, X.-H.

J. Q. Lu, P. Yang, and X.-H. Hu, “Simulations of light scattering from a biconcave red blood cell using the finite-difference time-domain method,” J. Biomed. Opt. 10(2), 024022 (2005).
[CrossRef] [PubMed]

Jacobs, K. M.

K. Dong, Y. Feng, K. M. Jacobs, J. Q. Lu, R. S. Brock, L. V. Yang, F. E. Bertrand, M. A. Farwell, and X. H. Hu, “Label-free classification of cultured cells through diffraction imaging,” Biomed. Opt. Express 2(6), 1717–1726 (2011).
[CrossRef] [PubMed]

K. M. Jacobs, J. Q. Lu, and X. H. Hu, “Development of a diffraction imaging flow cytometer,” Opt. Lett. 34(19), 2985–2987 (2009).
[CrossRef] [PubMed]

K. M. Jacobs, L. V. Yang, J. Ding, A. E. Ekpenyong, R. Castellone, J. Q. Lu, and X. H. Hu, “Diffraction imaging of spheres and melanoma cells with a microscope objective,” J Biophotonics 2(8-9), 521–527 (2009).
[CrossRef] [PubMed]

X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X. H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48(24), 4165–4172 (2003).
[CrossRef] [PubMed]

Kaye, P. H.

I. K. Ludlow and P. H. Kaye, “A scanning diffractometer for the rapid analysis of microparticles and biological cells,” J. Colloid Interface Sci. 69(3), 571–589 (1979).
[CrossRef]

Kozubek, M.

D. Svoboda, M. Kozubek, and S. Stejskal, “Generation of digital phantoms of cell nuclei and simulation of image formation in 3D image cytometry,” Cytometry A 75A(6), 494–509 (2009).
[CrossRef] [PubMed]

Lu, J. Q.

K. Dong, Y. Feng, K. M. Jacobs, J. Q. Lu, R. S. Brock, L. V. Yang, F. E. Bertrand, M. A. Farwell, and X. H. Hu, “Label-free classification of cultured cells through diffraction imaging,” Biomed. Opt. Express 2(6), 1717–1726 (2011).
[CrossRef] [PubMed]

K. M. Jacobs, J. Q. Lu, and X. H. Hu, “Development of a diffraction imaging flow cytometer,” Opt. Lett. 34(19), 2985–2987 (2009).
[CrossRef] [PubMed]

K. M. Jacobs, L. V. Yang, J. Ding, A. E. Ekpenyong, R. Castellone, J. Q. Lu, and X. H. Hu, “Diffraction imaging of spheres and melanoma cells with a microscope objective,” J Biophotonics 2(8-9), 521–527 (2009).
[CrossRef] [PubMed]

M. A. Yurkin, A. G. Hoekstra, R. S. Brock, and J. Q. Lu, “Systematic comparison of the discrete dipole approximation and the finite difference time domain method for large dielectric scatterers,” Opt. Express 15(26), 17902–17911 (2007).
[CrossRef] [PubMed]

J. Q. Lu, P. Yang, and X.-H. Hu, “Simulations of light scattering from a biconcave red blood cell using the finite-difference time-domain method,” J. Biomed. Opt. 10(2), 024022 (2005).
[CrossRef] [PubMed]

X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X. H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48(24), 4165–4172 (2003).
[CrossRef] [PubMed]

Ludlow, I. K.

I. K. Ludlow and P. H. Kaye, “A scanning diffractometer for the rapid analysis of microparticles and biological cells,” J. Colloid Interface Sci. 69(3), 571–589 (1979).
[CrossRef]

Ma, X.

X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X. H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48(24), 4165–4172 (2003).
[CrossRef] [PubMed]

Maltsev, V. P.

K. A. Semyanov, P. A. Tarasov, A. E. Zharinov, A. V. Chernyshev, A. G. Hoekstra, and V. P. Maltsev, “Single-particle sizing from light scattering by spectral decomposition,” Appl. Opt. 43(26), 5110–5115 (2004).
[CrossRef] [PubMed]

J. T. Soini, A. V. Chernyshev, P. E. Hänninen, E. Soini, and V. P. Maltsev, “A new design of the flow cuvette and optical set-up for the scanning flow cytometer,” Cytometry 31(2), 78–84 (1998).
[CrossRef] [PubMed]

Min, S. L.

Ovryn, B.

B. Ovryn, “Three-dimensional forward scattering particle image velocimetry applied to a microscopic field-of-view,” Exp. Fluids 29(7), S175–S184 (2000).
[CrossRef]

Portnoff, M.

M. Portnoff, “Time-frequency representation of digital signals and systems based on short-time Fourier analysis,” IEEE Trans. Acoust., Speech Signal Process. 28(1), 55–69 (1980).
[CrossRef]

Semyanov, K. A.

Soini, E.

J. T. Soini, A. V. Chernyshev, P. E. Hänninen, E. Soini, and V. P. Maltsev, “A new design of the flow cuvette and optical set-up for the scanning flow cytometer,” Cytometry 31(2), 78–84 (1998).
[CrossRef] [PubMed]

Soini, J. T.

J. T. Soini, A. V. Chernyshev, P. E. Hänninen, E. Soini, and V. P. Maltsev, “A new design of the flow cuvette and optical set-up for the scanning flow cytometer,” Cytometry 31(2), 78–84 (1998).
[CrossRef] [PubMed]

Stejskal, S.

D. Svoboda, M. Kozubek, and S. Stejskal, “Generation of digital phantoms of cell nuclei and simulation of image formation in 3D image cytometry,” Cytometry A 75A(6), 494–509 (2009).
[CrossRef] [PubMed]

Svoboda, D.

D. Svoboda, M. Kozubek, and S. Stejskal, “Generation of digital phantoms of cell nuclei and simulation of image formation in 3D image cytometry,” Cytometry A 75A(6), 494–509 (2009).
[CrossRef] [PubMed]

Tarasov, P. A.

Xing, L.

B. P. Hanley, L. Xing, and R. H. Cheng, “Variance in multiplex suspension array assays: microsphere size variation impact,” Theor. Biol. Med. Model. 4(1), 31–38 (2007).
[CrossRef] [PubMed]

Xu, Y. L.

Y. L. Xu and B. A. S. Gustafson, “Comparison between Multisphere Light-scattering Calculations: Rigorous Solution and Discrete-Dipole Approximation,” Astrophys. J. 513(2), 894–909 (1999).
[CrossRef]

Yang, L. V.

K. Dong, Y. Feng, K. M. Jacobs, J. Q. Lu, R. S. Brock, L. V. Yang, F. E. Bertrand, M. A. Farwell, and X. H. Hu, “Label-free classification of cultured cells through diffraction imaging,” Biomed. Opt. Express 2(6), 1717–1726 (2011).
[CrossRef] [PubMed]

K. M. Jacobs, L. V. Yang, J. Ding, A. E. Ekpenyong, R. Castellone, J. Q. Lu, and X. H. Hu, “Diffraction imaging of spheres and melanoma cells with a microscope objective,” J Biophotonics 2(8-9), 521–527 (2009).
[CrossRef] [PubMed]

Yang, P.

J. Q. Lu, P. Yang, and X.-H. Hu, “Simulations of light scattering from a biconcave red blood cell using the finite-difference time-domain method,” J. Biomed. Opt. 10(2), 024022 (2005).
[CrossRef] [PubMed]

X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X. H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48(24), 4165–4172 (2003).
[CrossRef] [PubMed]

Yurkin, M. A.

Zharinov, A. E.

Appl. Opt. (2)

Astrophys. J. (1)

Y. L. Xu and B. A. S. Gustafson, “Comparison between Multisphere Light-scattering Calculations: Rigorous Solution and Discrete-Dipole Approximation,” Astrophys. J. 513(2), 894–909 (1999).
[CrossRef]

Biomed. Opt. Express (1)

Cytometry (1)

J. T. Soini, A. V. Chernyshev, P. E. Hänninen, E. Soini, and V. P. Maltsev, “A new design of the flow cuvette and optical set-up for the scanning flow cytometer,” Cytometry 31(2), 78–84 (1998).
[CrossRef] [PubMed]

Cytometry A (1)

D. Svoboda, M. Kozubek, and S. Stejskal, “Generation of digital phantoms of cell nuclei and simulation of image formation in 3D image cytometry,” Cytometry A 75A(6), 494–509 (2009).
[CrossRef] [PubMed]

Exp. Fluids (1)

B. Ovryn, “Three-dimensional forward scattering particle image velocimetry applied to a microscopic field-of-view,” Exp. Fluids 29(7), S175–S184 (2000).
[CrossRef]

IEEE Trans. Acoust., Speech Signal Process. (1)

M. Portnoff, “Time-frequency representation of digital signals and systems based on short-time Fourier analysis,” IEEE Trans. Acoust., Speech Signal Process. 28(1), 55–69 (1980).
[CrossRef]

IEEE Trans. Acoust., Speech, Signal Process., (1)

J. G. Daugman, “Complete discrete 2-D Gabor transforms by neural networks for image analysis and compression,” IEEE Trans. Acoust., Speech, Signal Process., 36(7), 1169–1179 (1988).
[CrossRef]

J Biophotonics (1)

K. M. Jacobs, L. V. Yang, J. Ding, A. E. Ekpenyong, R. Castellone, J. Q. Lu, and X. H. Hu, “Diffraction imaging of spheres and melanoma cells with a microscope objective,” J Biophotonics 2(8-9), 521–527 (2009).
[CrossRef] [PubMed]

J. Biomed. Opt. (1)

J. Q. Lu, P. Yang, and X.-H. Hu, “Simulations of light scattering from a biconcave red blood cell using the finite-difference time-domain method,” J. Biomed. Opt. 10(2), 024022 (2005).
[CrossRef] [PubMed]

J. Colloid Interface Sci. (1)

I. K. Ludlow and P. H. Kaye, “A scanning diffractometer for the rapid analysis of microparticles and biological cells,” J. Colloid Interface Sci. 69(3), 571–589 (1979).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Part. Part. Syst. Charact. (1)

C. Godefroy and M. Adjouadi, “Particle sizing in a flow environment using light scattering patterns,” Part. Part. Syst. Charact. 17(2), 47–55 (2000).
[CrossRef]

Phys. Med. Biol. (1)

X. Ma, J. Q. Lu, R. S. Brock, K. M. Jacobs, P. Yang, and X. H. Hu, “Determination of complex refractive index of polystyrene microspheres from 370 to 1610 nm,” Phys. Med. Biol. 48(24), 4165–4172 (2003).
[CrossRef] [PubMed]

Theor. Biol. Med. Model. (1)

B. P. Hanley, L. Xing, and R. H. Cheng, “Variance in multiplex suspension array assays: microsphere size variation impact,” Theor. Biol. Med. Model. 4(1), 31–38 (2007).
[CrossRef] [PubMed]

Other (1)

K. K. Kim, D. W. Pack, M. Ferrari, A. P. Lee, and L. J. Lee, “Microspheres for drug delivery,” in Biological and Biomedical Nanotechnology, A. P. Lee, J. Lee, and M. Ferrari, eds. (Springer, 2006), pp. 19–50.

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Figures (6)

Fig. 1
Fig. 1

(a) Schematic of the experimental setup, Obj = microscope objective, Tub = tube lens; (b) examples of diffraction image by single polystyrene microspheres with nominal diameter, d, labeled on the left lower corner and microsphere ID number N on the right lower corner.

Fig. 2
Fig. 2

The 2D STFT power spectra and comparison of the 1D STFT and FFT power spectra of two microspheres. All spectral data were obtained from single rows with Δ as the pixel separation: (a) d = 7.9μm, N = 118 and row number Y = 185; (b) d = 2.5μm, N = 400 and Y = 272. The red arrow for each PSTFT(f) curve indicates the frequency location of sideband.

Fig. 3
Fig. 3

The sideband frequency fs versus the row number Y on selected rows of (a) d = 7.9μm, and N = 118; (b) d = 2.5μm and N = 400. The solid lines represents the maximum sideband frequency fs,max used to determine the microsphere diameter.

Fig. 4
Fig. 4

Histograms of maximum sideband frequency fs,max for four groups of 1000 microspheres with nominal diameter d. Inset: the mean values (symbols) and standard deviations (error bars) of fs,max versus d, the solid line represents a linear regression with r2 as the coefficient of determination.

Fig. 5
Fig. 5

The diffraction images by two and three microspheres of d = 2.5μm measured at different off-focus positions or simulated with specified Δθ at wavelength λ = 532nm: (a) measured, two-sphere, Δx = 50μm; (b) measured, three-sphere, Δx = 50μm; (c) measured, two-sphere, Δx = 150μm; (d) simulated, two-sphere, Δθs = 70°, θ0 = 3π/4, ϕ0 = 7π/4, refractive indices of sphere nsp = 1.588 and host nh = 1.334 [17].

Fig. 6
Fig. 6

The Gabor power spectral images PG(u, v) with fixed window width w and different sampling vector angle γ: (a) PG obtained from the measured diffraction image in Fig. 5(c) with w = 38Δ; (b) PG obtained from the simulated diffraction image in Fig. 5(d) with w = 76Δ. Among each pair of PG images, the left one shows the sidebands of maximum peaks and the right one shows the sidebands of minimal peaks for comparison.

Equations (5)

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

S(f;Z,Y)= I(Z',Y)g(ZZ')exp(2πifZ')dZ ',
g(ZZ')=exp{ π (ZZ') 2 w 2 }.
P STFT (f)=max{P(f,Z)}
d s =248.24 f s,max 0.71410.
G(u,v;f,γ,w)= exp{2πi(uZ+vY)}dZdY I(Z',Y') exp{ π w 2 [ (Z'Z) 2 + (Y'Y) 2 ]}cos{2π( u 0 (Z'Z)+ v 0 (Y'Y))}dZ'dY,

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