Abstract

This paper presents the possibility of measuring the three-dimensional (3D) relative locations and diameters of a set of spherical particles and discusses the behavior of the light recorded around the rainbow angle, an essential step toward refractive index measurements. When a set of particles is illuminated by a pulsed incident wave, the particles act as spherical light wave sources. When the pulse duration is short enough to fix the particle location (typically about 10 ns), interference fringes between these different spherical waves can be recorded. The Fourier transform of the fringes divides the complex fringe systems into a series of spots, with each spot characterizing the interference between a pair of particles. The analyses of these spots (in position and shape) potentially allow the measurement of particle characteristics (3D relative position, particle diameter, and particle refractive index value).

© 2012 Optical Society of America

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References

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  1. V. Devarakonda and A. K. Ray, “Effect of inter-particle interactions on evaporation of droplets in a linear array,” J. Aerosol Sci. 34, 837–857 (2003).
    [CrossRef]
  2. W. D. Bachalo and M. J. Houser, “Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions,” Opt. Eng. 23, 583–590 (1984).
    [CrossRef]
  3. S. Saengkaew, T. Charinpanikul, C. Laurent, Y. Biscos, G. Lavergne, G. Gouesbet, and G. Gréhan, “Processing of individual rainbow signals,” Exp. Fluids 48, 111–119 (2010).
    [CrossRef]
  4. J. P. A. J. Van Beeck, L. Zimmer, and M. L. Riethmuller, “Global rainbow thermometry for mean temperature and size measurement of spray droplets,” Part. Part. Syst. Charact. 18, 196–204 (2001).
    [CrossRef]
  5. P. Briard, S. Saengkaew, X. C. Wu, S. Meunier-Guttin-Cluzel, L. H. Chen, K. F. Cen, and G. Gréhan, “Measurements of 3D relative locations of particles by Fourier interferometry imaging (FII),” Opt. Express 19, 12700–12718 (2011).
    [CrossRef]
  6. E. Darakis, T. Khanam, and A. Rajendran, “Microparticle characterization using digital holography,” Chem. Eng. Sci. 65, 1037–1044 (2010).
    [CrossRef]
  7. X. Wu, S. Meunier-Guttin-Cluzel, Y. Wu, S. Saengkaew, D. Lebrun, M. Brunel, L. Chen, S. Coetmellec, K. Cen, and G. Gréhan, “Holography and micro-holography of particle fields: a numerical standard,” Opt. Commun. 285, 3013–3020 (2012).
    [CrossRef]
  8. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: the Art of Scientific Computing (Cambridge University, 1986).
  9. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).
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    [CrossRef]

2012

X. Wu, S. Meunier-Guttin-Cluzel, Y. Wu, S. Saengkaew, D. Lebrun, M. Brunel, L. Chen, S. Coetmellec, K. Cen, and G. Gréhan, “Holography and micro-holography of particle fields: a numerical standard,” Opt. Commun. 285, 3013–3020 (2012).
[CrossRef]

2011

2010

E. Darakis, T. Khanam, and A. Rajendran, “Microparticle characterization using digital holography,” Chem. Eng. Sci. 65, 1037–1044 (2010).
[CrossRef]

S. Saengkaew, T. Charinpanikul, C. Laurent, Y. Biscos, G. Lavergne, G. Gouesbet, and G. Gréhan, “Processing of individual rainbow signals,” Exp. Fluids 48, 111–119 (2010).
[CrossRef]

2003

V. Devarakonda and A. K. Ray, “Effect of inter-particle interactions on evaporation of droplets in a linear array,” J. Aerosol Sci. 34, 837–857 (2003).
[CrossRef]

2001

J. P. A. J. Van Beeck, L. Zimmer, and M. L. Riethmuller, “Global rainbow thermometry for mean temperature and size measurement of spray droplets,” Part. Part. Syst. Charact. 18, 196–204 (2001).
[CrossRef]

1984

W. D. Bachalo and M. J. Houser, “Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions,” Opt. Eng. 23, 583–590 (1984).
[CrossRef]

1981

Bachalo, W. D.

W. D. Bachalo and M. J. Houser, “Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions,” Opt. Eng. 23, 583–590 (1984).
[CrossRef]

Biscos, Y.

S. Saengkaew, T. Charinpanikul, C. Laurent, Y. Biscos, G. Lavergne, G. Gouesbet, and G. Gréhan, “Processing of individual rainbow signals,” Exp. Fluids 48, 111–119 (2010).
[CrossRef]

Briard, P.

Brunel, M.

X. Wu, S. Meunier-Guttin-Cluzel, Y. Wu, S. Saengkaew, D. Lebrun, M. Brunel, L. Chen, S. Coetmellec, K. Cen, and G. Gréhan, “Holography and micro-holography of particle fields: a numerical standard,” Opt. Commun. 285, 3013–3020 (2012).
[CrossRef]

Cen, K.

X. Wu, S. Meunier-Guttin-Cluzel, Y. Wu, S. Saengkaew, D. Lebrun, M. Brunel, L. Chen, S. Coetmellec, K. Cen, and G. Gréhan, “Holography and micro-holography of particle fields: a numerical standard,” Opt. Commun. 285, 3013–3020 (2012).
[CrossRef]

Cen, K. F.

Charinpanikul, T.

S. Saengkaew, T. Charinpanikul, C. Laurent, Y. Biscos, G. Lavergne, G. Gouesbet, and G. Gréhan, “Processing of individual rainbow signals,” Exp. Fluids 48, 111–119 (2010).
[CrossRef]

Chen, L.

X. Wu, S. Meunier-Guttin-Cluzel, Y. Wu, S. Saengkaew, D. Lebrun, M. Brunel, L. Chen, S. Coetmellec, K. Cen, and G. Gréhan, “Holography and micro-holography of particle fields: a numerical standard,” Opt. Commun. 285, 3013–3020 (2012).
[CrossRef]

Chen, L. H.

Chen, S.-H.

Coetmellec, S.

X. Wu, S. Meunier-Guttin-Cluzel, Y. Wu, S. Saengkaew, D. Lebrun, M. Brunel, L. Chen, S. Coetmellec, K. Cen, and G. Gréhan, “Holography and micro-holography of particle fields: a numerical standard,” Opt. Commun. 285, 3013–3020 (2012).
[CrossRef]

Darakis, E.

E. Darakis, T. Khanam, and A. Rajendran, “Microparticle characterization using digital holography,” Chem. Eng. Sci. 65, 1037–1044 (2010).
[CrossRef]

Devarakonda, V.

V. Devarakonda and A. K. Ray, “Effect of inter-particle interactions on evaporation of droplets in a linear array,” J. Aerosol Sci. 34, 837–857 (2003).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: the Art of Scientific Computing (Cambridge University, 1986).

Glantschnig, W. J.

Gouesbet, G.

S. Saengkaew, T. Charinpanikul, C. Laurent, Y. Biscos, G. Lavergne, G. Gouesbet, and G. Gréhan, “Processing of individual rainbow signals,” Exp. Fluids 48, 111–119 (2010).
[CrossRef]

Gréhan, G.

X. Wu, S. Meunier-Guttin-Cluzel, Y. Wu, S. Saengkaew, D. Lebrun, M. Brunel, L. Chen, S. Coetmellec, K. Cen, and G. Gréhan, “Holography and micro-holography of particle fields: a numerical standard,” Opt. Commun. 285, 3013–3020 (2012).
[CrossRef]

P. Briard, S. Saengkaew, X. C. Wu, S. Meunier-Guttin-Cluzel, L. H. Chen, K. F. Cen, and G. Gréhan, “Measurements of 3D relative locations of particles by Fourier interferometry imaging (FII),” Opt. Express 19, 12700–12718 (2011).
[CrossRef]

S. Saengkaew, T. Charinpanikul, C. Laurent, Y. Biscos, G. Lavergne, G. Gouesbet, and G. Gréhan, “Processing of individual rainbow signals,” Exp. Fluids 48, 111–119 (2010).
[CrossRef]

Houser, M. J.

W. D. Bachalo and M. J. Houser, “Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions,” Opt. Eng. 23, 583–590 (1984).
[CrossRef]

Khanam, T.

E. Darakis, T. Khanam, and A. Rajendran, “Microparticle characterization using digital holography,” Chem. Eng. Sci. 65, 1037–1044 (2010).
[CrossRef]

Laurent, C.

S. Saengkaew, T. Charinpanikul, C. Laurent, Y. Biscos, G. Lavergne, G. Gouesbet, and G. Gréhan, “Processing of individual rainbow signals,” Exp. Fluids 48, 111–119 (2010).
[CrossRef]

Lavergne, G.

S. Saengkaew, T. Charinpanikul, C. Laurent, Y. Biscos, G. Lavergne, G. Gouesbet, and G. Gréhan, “Processing of individual rainbow signals,” Exp. Fluids 48, 111–119 (2010).
[CrossRef]

Lebrun, D.

X. Wu, S. Meunier-Guttin-Cluzel, Y. Wu, S. Saengkaew, D. Lebrun, M. Brunel, L. Chen, S. Coetmellec, K. Cen, and G. Gréhan, “Holography and micro-holography of particle fields: a numerical standard,” Opt. Commun. 285, 3013–3020 (2012).
[CrossRef]

Meunier-Guttin-Cluzel, S.

X. Wu, S. Meunier-Guttin-Cluzel, Y. Wu, S. Saengkaew, D. Lebrun, M. Brunel, L. Chen, S. Coetmellec, K. Cen, and G. Gréhan, “Holography and micro-holography of particle fields: a numerical standard,” Opt. Commun. 285, 3013–3020 (2012).
[CrossRef]

P. Briard, S. Saengkaew, X. C. Wu, S. Meunier-Guttin-Cluzel, L. H. Chen, K. F. Cen, and G. Gréhan, “Measurements of 3D relative locations of particles by Fourier interferometry imaging (FII),” Opt. Express 19, 12700–12718 (2011).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: the Art of Scientific Computing (Cambridge University, 1986).

Rajendran, A.

E. Darakis, T. Khanam, and A. Rajendran, “Microparticle characterization using digital holography,” Chem. Eng. Sci. 65, 1037–1044 (2010).
[CrossRef]

Ray, A. K.

V. Devarakonda and A. K. Ray, “Effect of inter-particle interactions on evaporation of droplets in a linear array,” J. Aerosol Sci. 34, 837–857 (2003).
[CrossRef]

Riethmuller, M. L.

J. P. A. J. Van Beeck, L. Zimmer, and M. L. Riethmuller, “Global rainbow thermometry for mean temperature and size measurement of spray droplets,” Part. Part. Syst. Charact. 18, 196–204 (2001).
[CrossRef]

Saengkaew, S.

X. Wu, S. Meunier-Guttin-Cluzel, Y. Wu, S. Saengkaew, D. Lebrun, M. Brunel, L. Chen, S. Coetmellec, K. Cen, and G. Gréhan, “Holography and micro-holography of particle fields: a numerical standard,” Opt. Commun. 285, 3013–3020 (2012).
[CrossRef]

P. Briard, S. Saengkaew, X. C. Wu, S. Meunier-Guttin-Cluzel, L. H. Chen, K. F. Cen, and G. Gréhan, “Measurements of 3D relative locations of particles by Fourier interferometry imaging (FII),” Opt. Express 19, 12700–12718 (2011).
[CrossRef]

S. Saengkaew, T. Charinpanikul, C. Laurent, Y. Biscos, G. Lavergne, G. Gouesbet, and G. Gréhan, “Processing of individual rainbow signals,” Exp. Fluids 48, 111–119 (2010).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: the Art of Scientific Computing (Cambridge University, 1986).

Van Beeck, J. P. A. J.

J. P. A. J. Van Beeck, L. Zimmer, and M. L. Riethmuller, “Global rainbow thermometry for mean temperature and size measurement of spray droplets,” Part. Part. Syst. Charact. 18, 196–204 (2001).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: the Art of Scientific Computing (Cambridge University, 1986).

Wu, X.

X. Wu, S. Meunier-Guttin-Cluzel, Y. Wu, S. Saengkaew, D. Lebrun, M. Brunel, L. Chen, S. Coetmellec, K. Cen, and G. Gréhan, “Holography and micro-holography of particle fields: a numerical standard,” Opt. Commun. 285, 3013–3020 (2012).
[CrossRef]

Wu, X. C.

Wu, Y.

X. Wu, S. Meunier-Guttin-Cluzel, Y. Wu, S. Saengkaew, D. Lebrun, M. Brunel, L. Chen, S. Coetmellec, K. Cen, and G. Gréhan, “Holography and micro-holography of particle fields: a numerical standard,” Opt. Commun. 285, 3013–3020 (2012).
[CrossRef]

Zimmer, L.

J. P. A. J. Van Beeck, L. Zimmer, and M. L. Riethmuller, “Global rainbow thermometry for mean temperature and size measurement of spray droplets,” Part. Part. Syst. Charact. 18, 196–204 (2001).
[CrossRef]

Appl. Opt.

Chem. Eng. Sci.

E. Darakis, T. Khanam, and A. Rajendran, “Microparticle characterization using digital holography,” Chem. Eng. Sci. 65, 1037–1044 (2010).
[CrossRef]

Exp. Fluids

S. Saengkaew, T. Charinpanikul, C. Laurent, Y. Biscos, G. Lavergne, G. Gouesbet, and G. Gréhan, “Processing of individual rainbow signals,” Exp. Fluids 48, 111–119 (2010).
[CrossRef]

J. Aerosol Sci.

V. Devarakonda and A. K. Ray, “Effect of inter-particle interactions on evaporation of droplets in a linear array,” J. Aerosol Sci. 34, 837–857 (2003).
[CrossRef]

Opt. Commun.

X. Wu, S. Meunier-Guttin-Cluzel, Y. Wu, S. Saengkaew, D. Lebrun, M. Brunel, L. Chen, S. Coetmellec, K. Cen, and G. Gréhan, “Holography and micro-holography of particle fields: a numerical standard,” Opt. Commun. 285, 3013–3020 (2012).
[CrossRef]

Opt. Eng.

W. D. Bachalo and M. J. Houser, “Phase/Doppler spray analyzer for simultaneous measurements of drop size and velocity distributions,” Opt. Eng. 23, 583–590 (1984).
[CrossRef]

Opt. Express

Part. Part. Syst. Charact.

J. P. A. J. Van Beeck, L. Zimmer, and M. L. Riethmuller, “Global rainbow thermometry for mean temperature and size measurement of spray droplets,” Part. Part. Syst. Charact. 18, 196–204 (2001).
[CrossRef]

Other

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: the Art of Scientific Computing (Cambridge University, 1986).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1981).

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Figures (14)

Fig. 1.
Fig. 1.

Fourier interferometry imaging setup and the two coordinate systems that are used.

Fig. 2.
Fig. 2.

Numerical simulation of interferences fringes created by three water droplets (forward scattering off-axis, θ=20°±5°).

Fig. 3.
Fig. 3.

(a), (b) Magnitude Fourier spectrum and (c), (d) phase Fourier spectrum of interference fringes created by three particles (Fig. 2) without using [(a) and (c)] and after using [(b) and (d)] the Blackman–Harris apodization function.

Fig. 4.
Fig. 4.

Profile of a spot [red line in Fig. 3(b)] for off-axis forward scattering.

Fig. 5.
Fig. 5.

Reflection and refraction of light rays by a pair of particles and associated reference rays.

Fig. 6.
Fig. 6.

Δfη1 versus particle radius. The studied parameter is the scattering angle θ0. The points correspond to the Lorenz–Mie computation, and dashed lines correspond to geometrical optics computations [Eqs. (4) and (5)].

Fig. 7.
Fig. 7.

(a) and (b) Interference fringes created by six water droplets with two forward scattering angles and (c) and (d) their magnitude Fourier spectrums. The scattering angles θ0 are equal to 20° [(a) and (c)] and 90° [(b) and (d)].

Fig. 8.
Fig. 8.

Initial (red circle) and reconstructed (blue triangle) particle field.

Fig. 9.
Fig. 9.

(a) Simulation using the Lorenz–Mie theory of interferences fringes created by three water droplets close to their rainbow angle, (b) the associated magnitude, and (c) phase Fourier spectrums.

Fig. 10.
Fig. 10.

Filtered Fourier space (a) magnitude spectrum and (b) phase spectrum using a rectangular filter. The phases of the number complexes equal to zero are not defined. They are represented in gray in Fig. 9(b).

Fig. 11.
Fig. 11.

Composite rainbow associated with a pair of particles and calculated from the inversion of filtered Fourier space in Fig. 10.

Fig. 12.
Fig. 12.

Composite rainbow (green dots) and standard rainbow (red curve) created by two pairs of identical particles (refractive index value equal to 1.33, diameter equal to 130 μm).

Fig. 13.
Fig. 13.

Composite rainbow (green dots), associated standard rainbows (black curve I1 and blue curve I2), and [I1I2]1/2 function created by two particles with the same refractive index (refractive index equal to 1.33) and different diameters (130 and 100 μm). The composite rainbow has been normalized by the maximum of [I1I2]1/2.

Fig. 14.
Fig. 14.

Composite rainbows (red curves) and associated standard rainbows (blue and black curves) created by two particles with different refractive indices: 1.32 and 1.33; 1.33 and 1.36. The pair of particles has the same diameter (equal to 130 μm) and diameters equal to 130 and 100 μm. The scattering angle is equal to 140°±5°.

Tables (1)

Tables Icon

Table 1. Diameters and 3D Relative Locations of Initial and Reconstructed Particle Fields

Equations (6)

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

I=i=1NtotIi+k=1Ntotl=k+1Ntot2IkIlcos(ϕlϕk),
δp1δp2=(δp1δref,1)(δp2δref,2)+(δref,1δref,2),
[fηp1=1,p2=1fηp1=0,p2=0]±[π|θmaxθmin|180λ[R1N1sin(θ02)1+N122N1cos(θ02)R2N2sin(θ02)1+N222N2cos(θ02)(R2R1)cos(θ02)]+fη].
Δfη1π|θmaxθmin|180λ|N1sin(θ02)1+N122N1cosθ02+cos(θ02)|R1,
Δfη2π|θmaxθmin|180λ|N2sin(θ02)1+N222N2cos(θ02)+cos(θ02)|R2.
[fηfξ]=±π|θmaxθmin|180λ[(xlxk)cosθ0+(zlzk)sinθ0ylyk].

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