Abstract

We describe a method for quickly and easily measuring the size of small particles in suspensions. This method uses a self-mixing laser Doppler measurement with a laser-diode-pumped, thin-slice LiNdP4O12 laser with extremely high optical sensitivity. The average size of the particles in Brownian motion is determined by a Lorentz fitting of the measured power spectrum of the modulated self-mixing laser light resulting from the motion. The dependence of the measured power spectra on particle size and concentration was quantitatively identified from the results of a systematic investigation of small polystyrene latex particles with different diameters and concentrations. The sizes and ratios of particles with different diameters mixed in water were accurately measured. An application of this self-mixing laser method for estimation of the average size of plankton in seawater showed that it is a practical method for characterizing biological species.

©2006 Optical Society of America

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References

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  1. B. J. Berne and R. Pecora, Dynamic Light Scattering with Applications to Chemistry, Biology, and Physics (Dover, NY, 2000).
  2. H. Z. Cummins, N. Knable, and Y. Yeh, “Observation of diffusion broadening of Rayleight scattered light,” Phys. Rev. Lett. 12, 150–153 (1964).
    [Crossref]
  3. R. C. Youngquist, S. Carr, and D. E. N. Davies, “Optical coherence-domain reflectometry: a new optical evaluation technique,” Opt. Lett. 12, 158–160 (1987).
    [Crossref] [PubMed]
  4. D. A. Boas, K. K. Bizheva, and A. M. Siegel, “Using dynamic low-coherence interferometry to image Brownian motion with highly scattering media,” Opt. Lett. 23, 319–321 (1998).
    [Crossref]
  5. M. Harris, G. N. Pearson, C. A. Hill, and J. M. Vaughan, “The fractal character of Gaussian-Lorentzian light,” Opt. Commun. 116, 15–19 (1995).
    [Crossref]
  6. K. Otsuka, K. Abe, J.-Y. Ko, and T.-S. Lim, “Real-time nanometer-vibration measurement with a self-mixing microchip solid-state laser,” Opt. Lett. 27, 1339–1341 (2002).
    [Crossref]
  7. K. Abe, K. Otsuka, and J.-Y. Ko, “Self-mixing laser Doppler vibrometry with high optical sensitivity: application to real-time sound reproduction,” New J. Phys. 5, 8.1–8.9 (2003).
    [Crossref]
  8. K. Otsuka, K. Abe, N. Sano, S. Sudo, and J.-Y. Ko, “Two-channel self-mixing laser Doppler measurement with carrier-frequency-division multiplexing,” Appl. Opt. 44, 1709–1714 (2005).
    [Crossref] [PubMed]
  9. Z. Sun, C. D. Tomlin, and E. M. Sevick-Muraca, “Approach for particle sizing in dense polydisperse colloidal suspension using multiple scattered light,” Langmuir 17, 6142–6147 (2001).
    [Crossref]
  10. L. B. Aberle and W. Staude, “Three-dimensional cross correlation technique: influence of multiply scattered light in the Rayleigh-Gans regime,” Phys. Chem. Chem. Phys. 1, 3917–3921 (1999).
    [Crossref]
  11. R. Xu, Particle Characterization: Light Scattering Methods (Kluwer, London, 2000).
  12. L. B. Aberle, P. Hülstede, S. Wiegand, W. Schöer, and W. Staude, “Effective suppression of multiply scattered light in static and dynamic light scattering,” Appl. Opt. 37, 6511–6524 (1998).
    [Crossref]
  13. K. A. Stacey, Light-scattering in Physical Chemistry (Buttler Worth Scientific Publications, London, 1956).
  14. M. Bertero, P. Boccacci, and E.R. Pike, “On the recovery and resolution of exponential relaxation rates from experimental data: a singular-value analysis of the Laplace transform inversion in the presence of noise,” Proc. Royal Soc. London A 383, 15–29 (1982).
    [Crossref]

2005 (1)

2003 (1)

K. Abe, K. Otsuka, and J.-Y. Ko, “Self-mixing laser Doppler vibrometry with high optical sensitivity: application to real-time sound reproduction,” New J. Phys. 5, 8.1–8.9 (2003).
[Crossref]

2002 (1)

2001 (1)

Z. Sun, C. D. Tomlin, and E. M. Sevick-Muraca, “Approach for particle sizing in dense polydisperse colloidal suspension using multiple scattered light,” Langmuir 17, 6142–6147 (2001).
[Crossref]

1999 (1)

L. B. Aberle and W. Staude, “Three-dimensional cross correlation technique: influence of multiply scattered light in the Rayleigh-Gans regime,” Phys. Chem. Chem. Phys. 1, 3917–3921 (1999).
[Crossref]

1998 (2)

1995 (1)

M. Harris, G. N. Pearson, C. A. Hill, and J. M. Vaughan, “The fractal character of Gaussian-Lorentzian light,” Opt. Commun. 116, 15–19 (1995).
[Crossref]

1987 (1)

1982 (1)

M. Bertero, P. Boccacci, and E.R. Pike, “On the recovery and resolution of exponential relaxation rates from experimental data: a singular-value analysis of the Laplace transform inversion in the presence of noise,” Proc. Royal Soc. London A 383, 15–29 (1982).
[Crossref]

1964 (1)

H. Z. Cummins, N. Knable, and Y. Yeh, “Observation of diffusion broadening of Rayleight scattered light,” Phys. Rev. Lett. 12, 150–153 (1964).
[Crossref]

Abe, K.

Aberle, L. B.

L. B. Aberle and W. Staude, “Three-dimensional cross correlation technique: influence of multiply scattered light in the Rayleigh-Gans regime,” Phys. Chem. Chem. Phys. 1, 3917–3921 (1999).
[Crossref]

L. B. Aberle, P. Hülstede, S. Wiegand, W. Schöer, and W. Staude, “Effective suppression of multiply scattered light in static and dynamic light scattering,” Appl. Opt. 37, 6511–6524 (1998).
[Crossref]

Berne, B. J.

B. J. Berne and R. Pecora, Dynamic Light Scattering with Applications to Chemistry, Biology, and Physics (Dover, NY, 2000).

Bertero, M.

M. Bertero, P. Boccacci, and E.R. Pike, “On the recovery and resolution of exponential relaxation rates from experimental data: a singular-value analysis of the Laplace transform inversion in the presence of noise,” Proc. Royal Soc. London A 383, 15–29 (1982).
[Crossref]

Bizheva, K. K.

Boas, D. A.

Boccacci, P.

M. Bertero, P. Boccacci, and E.R. Pike, “On the recovery and resolution of exponential relaxation rates from experimental data: a singular-value analysis of the Laplace transform inversion in the presence of noise,” Proc. Royal Soc. London A 383, 15–29 (1982).
[Crossref]

Carr, S.

Cummins, H. Z.

H. Z. Cummins, N. Knable, and Y. Yeh, “Observation of diffusion broadening of Rayleight scattered light,” Phys. Rev. Lett. 12, 150–153 (1964).
[Crossref]

Davies, D. E. N.

Harris, M.

M. Harris, G. N. Pearson, C. A. Hill, and J. M. Vaughan, “The fractal character of Gaussian-Lorentzian light,” Opt. Commun. 116, 15–19 (1995).
[Crossref]

Hill, C. A.

M. Harris, G. N. Pearson, C. A. Hill, and J. M. Vaughan, “The fractal character of Gaussian-Lorentzian light,” Opt. Commun. 116, 15–19 (1995).
[Crossref]

Hülstede, P.

Knable, N.

H. Z. Cummins, N. Knable, and Y. Yeh, “Observation of diffusion broadening of Rayleight scattered light,” Phys. Rev. Lett. 12, 150–153 (1964).
[Crossref]

Ko, J.-Y.

Lim, T.-S.

Otsuka, K.

Pearson, G. N.

M. Harris, G. N. Pearson, C. A. Hill, and J. M. Vaughan, “The fractal character of Gaussian-Lorentzian light,” Opt. Commun. 116, 15–19 (1995).
[Crossref]

Pecora, R.

B. J. Berne and R. Pecora, Dynamic Light Scattering with Applications to Chemistry, Biology, and Physics (Dover, NY, 2000).

Pike, E.R.

M. Bertero, P. Boccacci, and E.R. Pike, “On the recovery and resolution of exponential relaxation rates from experimental data: a singular-value analysis of the Laplace transform inversion in the presence of noise,” Proc. Royal Soc. London A 383, 15–29 (1982).
[Crossref]

Sano, N.

Schöer, W.

Sevick-Muraca, E. M.

Z. Sun, C. D. Tomlin, and E. M. Sevick-Muraca, “Approach for particle sizing in dense polydisperse colloidal suspension using multiple scattered light,” Langmuir 17, 6142–6147 (2001).
[Crossref]

Siegel, A. M.

Stacey, K. A.

K. A. Stacey, Light-scattering in Physical Chemistry (Buttler Worth Scientific Publications, London, 1956).

Staude, W.

L. B. Aberle and W. Staude, “Three-dimensional cross correlation technique: influence of multiply scattered light in the Rayleigh-Gans regime,” Phys. Chem. Chem. Phys. 1, 3917–3921 (1999).
[Crossref]

L. B. Aberle, P. Hülstede, S. Wiegand, W. Schöer, and W. Staude, “Effective suppression of multiply scattered light in static and dynamic light scattering,” Appl. Opt. 37, 6511–6524 (1998).
[Crossref]

Sudo, S.

Sun, Z.

Z. Sun, C. D. Tomlin, and E. M. Sevick-Muraca, “Approach for particle sizing in dense polydisperse colloidal suspension using multiple scattered light,” Langmuir 17, 6142–6147 (2001).
[Crossref]

Tomlin, C. D.

Z. Sun, C. D. Tomlin, and E. M. Sevick-Muraca, “Approach for particle sizing in dense polydisperse colloidal suspension using multiple scattered light,” Langmuir 17, 6142–6147 (2001).
[Crossref]

Vaughan, J. M.

M. Harris, G. N. Pearson, C. A. Hill, and J. M. Vaughan, “The fractal character of Gaussian-Lorentzian light,” Opt. Commun. 116, 15–19 (1995).
[Crossref]

Wiegand, S.

Xu, R.

R. Xu, Particle Characterization: Light Scattering Methods (Kluwer, London, 2000).

Yeh, Y.

H. Z. Cummins, N. Knable, and Y. Yeh, “Observation of diffusion broadening of Rayleight scattered light,” Phys. Rev. Lett. 12, 150–153 (1964).
[Crossref]

Youngquist, R. C.

Appl. Opt. (2)

Langmuir (1)

Z. Sun, C. D. Tomlin, and E. M. Sevick-Muraca, “Approach for particle sizing in dense polydisperse colloidal suspension using multiple scattered light,” Langmuir 17, 6142–6147 (2001).
[Crossref]

New J. Phys. (1)

K. Abe, K. Otsuka, and J.-Y. Ko, “Self-mixing laser Doppler vibrometry with high optical sensitivity: application to real-time sound reproduction,” New J. Phys. 5, 8.1–8.9 (2003).
[Crossref]

Opt. Commun. (1)

M. Harris, G. N. Pearson, C. A. Hill, and J. M. Vaughan, “The fractal character of Gaussian-Lorentzian light,” Opt. Commun. 116, 15–19 (1995).
[Crossref]

Opt. Lett. (3)

Phys. Chem. Chem. Phys. (1)

L. B. Aberle and W. Staude, “Three-dimensional cross correlation technique: influence of multiply scattered light in the Rayleigh-Gans regime,” Phys. Chem. Chem. Phys. 1, 3917–3921 (1999).
[Crossref]

Phys. Rev. Lett. (1)

H. Z. Cummins, N. Knable, and Y. Yeh, “Observation of diffusion broadening of Rayleight scattered light,” Phys. Rev. Lett. 12, 150–153 (1964).
[Crossref]

Proc. Royal Soc. London A (1)

M. Bertero, P. Boccacci, and E.R. Pike, “On the recovery and resolution of exponential relaxation rates from experimental data: a singular-value analysis of the Laplace transform inversion in the presence of noise,” Proc. Royal Soc. London A 383, 15–29 (1982).
[Crossref]

Other (3)

B. J. Berne and R. Pecora, Dynamic Light Scattering with Applications to Chemistry, Biology, and Physics (Dover, NY, 2000).

R. Xu, Particle Characterization: Light Scattering Methods (Kluwer, London, 2000).

K. A. Stacey, Light-scattering in Physical Chemistry (Buttler Worth Scientific Publications, London, 1956).

Supplementary Material (3)

» Media 1: AIF (2239 KB)     
» Media 2: MOV (2374 KB)     
» Media 3: MOV (2476 KB)     

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Figures (9)

Fig. 1.
Fig. 1. Experimental setup for the self-mixing laser scattering spectroscopy of small particles in suspension.

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Fig. 2.
Fig. 2. Power spectrum of modulated output signal for 115-nm-diameter PS particles in water with 0.1 wt. % concentration. Black and red curves indicate power spectra obtained from the experiment and the calculation of a Lorentz-function curve fitting, respectively. [Media 1]

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Fig. 3.
Fig. 3. (a) Power spectra of modulated output signal for 115-nm-diameter PS particles in water with 0.001–10 wt. % concentrations. Black and red curves indicate power spectra obtained from the experiment and the calculation of a Lorentz-function curve fitting, respectively. (b) Plot of a proportionality constant of the Lorentz function A against PS concentration. Dashed line indicates a linear dependence of the proportionality constant against the PS concentration.

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Fig. 4.
Fig. 4. (a) Power spectra of modulated output signal for 115-, 262-, and 474-nm-diameter PS particles in water with 0.05 wt. % concentration. Black and red lines indicate power spectra obtained from the experiment and the calculation of a Lorentz-function curve-fitting, respectively. (b) Dependence of a proportionality constant of the Lorentz function on the particle diameter for concentrations of 0.05 and 0.1 wt. % and diameters of 20, 115, 204, 262, and 474 nm. Solid and dotted lines indicate theoretical curves derived from RD approximation and Rayleigh equation, respectively.

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Fig. 5.
Fig. 5. (a) Power spectra of modulated output signal for a 1:1 mixed sample of 115-and 474-nm-diameter PS particles in water with 0.05 wt. % concentration, together with those of single-component samples. Black and red lines indicate power spectra obtained from the experiment and the calculation of a Lorentz-function curve-fitting, respectively. Diameters estimated from the Lorentz-function curve-fitting were d115,mix = 121.9±4.7 nm and d474,mix = 497.2±30.5 nm, respectively. (b) Dependence of a proportionality constant of the Lorentz function on the ratio of number of PS particles.

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Fig. 6.
Fig. 6. Number ratio of PS particles obtained from experiment versus the theoretical ratio.

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Fig. 7.
Fig. 7. Calculated particle size distribution for a 1:1 mixed sample of 115-nm-diameter PS particles in water with 0.1 wt. % concentration and 474-nm-diameter PS particles in water with 0.2 wt. % concentration.

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Fig. 8.
Fig. 8. Power spectra of modulated outputs observed at different times for (a) Nannochloropsis oculata and (b) Tetraselmis tetrathele. Movie demonstrates time-dependent power spectra and sounds obtained from Nannochloropis (2.31 megabytes) [Media 2] and Tetraselmis (2.41 megabytes) [Media 3] in seawater.

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Fig. 9.
Fig. 9. Averaged power spectra of modulated output signals. (a) Nannochloropsis oculata; black line: experiment, red line: curve fitting. (b) Tetraselmis tetrathele; black line: experiment, red line: curve fitting by the summation of Lorenz (blue line) and Gauss (green line) functions.

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Equations (9)

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I ( k , ω ) = A Γ [ ( ω − 2 π f AOM ) 2 + Γ 2 ] ,
Γ = k 2 D ,
D = k B T 3 πηd ,
k ( θ ) = ( 4 πn λ ) sin ⁡ ( θ 2 ) ,
I ( θ ) ∝ V 2 NP ( θ ) ,
P ( θ ) = ∣ 3 ( sin ⁡ x − x cos ⁡ x ) x 2 ∣ 2 ,
x = d 2 ( 4 πn λ ) sin ⁡ ( θ 2 ) = dk ( θ ) 2 ,
I ( ω ) = A 1 , mix Γ 1 , mix [ ( ω − 2 π f AOM ) 2 + Γ 1 , mix 2 ] + A 2 , mix Γ 2 , mix [ ( ω − 2 π f AOM ) 2 + Γ 2 , mix 2 ] ,
N i , mix = N i , sin gle A i , mix A i , sin gle ,

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