Abstract

Up to now the application of rainbow thermometry has been limited to particle systems possessing a uniform refractive index. This is mostly due to the absence of an appropriate data inversion algorithm that takes into account the presence of a refractive index gradient. In this paper, for the first time to our knowledge, exploiting a generalization of the Airy theory, a data inversion algorithm for a single droplet, presenting a parabolic refractive index gradient, is proposed. This data inversion algorithm is used to compute the diameter and the refractive index at the core and at the surface of a simulated burning droplet. The results are compared to the analytical solutions showing a satisfactory agreement.

© 2005 Optical Society of America

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References

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  1. J. P. A. J. van Beeck, D. Giannoulis, L. Zimmer, M. L. Riethmuller, “Global rainbow thermometry for droplet-temperature measurement,” Opt. Lett. 24, 1696–1698 (1999).
    [CrossRef]
  2. N. Roth, K. Anders, A. Frohn, “Simultaneous measurement of temperature and size of droplets in the micro-metric range,” J. Laser Appl. 2, 37–42 (1990).
    [CrossRef]
  3. S. V. Sankar, D. H. Buermann, W. D. Bachalo, “An advanced Rainbow signal processor for improved accuracy in droplet measurement,” presented at the Eighth International Symposium on Application of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, 1996.
  4. J. P. A. J. van Beeck, M. L. Riethmuller, “Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity,” Appl. Opt. 35, 2259–2266 (1996).
    [CrossRef] [PubMed]
  5. C. L. Adler, J. A. Lock, I. P. Rafferty, W. Hickok, “Twin-rainbow metrology. I. Measurement of the thickness of a thin liquid film draining under gravity,” Appl. Opt. 42, 6584–6594 (2003).
    [CrossRef] [PubMed]
  6. C. L. Adler, J. A. Lock, J. K. Nash, K. W. Saunders, “Experimental observation of rainbow scattering by a coated cylinder: twin primary rainbows and thin-film interference,” Appl. Opt. 40, 1548– 1558 (2001).
    [CrossRef]
  7. J. A. Lock, J. M. Jamison, C.-Y. Lin, “Rainbow scattering by a coated sphere,” Appl. Opt. 33, 4677–4691 (1994).
    [CrossRef] [PubMed]
  8. F. Onofri, G. Grehan, G. Gousbet, “Electromagnetic scattering from a multilayered sphere located in an arbitrary beam,” Appl. Opt. 34, 7113–7124 (1995).
    [CrossRef] [PubMed]
  9. Y. P. Han, L. Méès, K. F. Ren, G. Gouesbet, S. Z. Wu, G. Gréhan, “Scattering light by spheroids: the far field case,” Opt. Commun. 210, 1–9 (2002).
    [CrossRef]
  10. Y. P. Han, G. Gréhan, G. Gouesbet, “Generalized Lorenz-Mie theory for a spheroidal particle with off-axis Gaussianbeam illumination,” Appl. Opt. 42, 6621–6629 (2003).
    [CrossRef] [PubMed]
  11. H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1957).
  12. M. R. Vetrano, J. van Beeck, M. Riethmuller, “Generalization of the rainbow Airy theory to nonuniform spheres,” Opt. Lett. 30, 658–660 (2005).
    [CrossRef] [PubMed]
  13. P. Massoli, “Rainbow refractometry to radially inhomogeneous spheres: the critical case of evaporating droplets,” Appl. Opt. 37, 3227–3235 (1998).
    [CrossRef]
  14. L. A. Dombrovsky, S. S. Sazih, “A simplified nonisothermal model for droplet heating and evaporation,” Int. Commun. Heat Mass Transfer 30, 787–796 (2003).
    [CrossRef]
  15. M. R. Vetrano, J. van Beeck, M. Riethmuller, “Global rainbow thermometry: improvements in the data inversion algorithm and validation technique in liquid–liquid suspension,” Appl. Opt. 43, 3600–3607 (2004).
    [CrossRef] [PubMed]
  16. C. K. Law, “Unsteady droplet combustion with droplet heating,” Combust. Flame 26, 17–22 (1976).
    [CrossRef]
  17. C. K. Law, W. A. Sirignano, “Unsteady droplet combustion with droplet heating-II: conduction limit,” Combust. Flame 28, 175–186 (1977).
    [CrossRef]

2005

2004

2003

2002

Y. P. Han, L. Méès, K. F. Ren, G. Gouesbet, S. Z. Wu, G. Gréhan, “Scattering light by spheroids: the far field case,” Opt. Commun. 210, 1–9 (2002).
[CrossRef]

2001

1999

1998

1996

1995

1994

1990

N. Roth, K. Anders, A. Frohn, “Simultaneous measurement of temperature and size of droplets in the micro-metric range,” J. Laser Appl. 2, 37–42 (1990).
[CrossRef]

1977

C. K. Law, W. A. Sirignano, “Unsteady droplet combustion with droplet heating-II: conduction limit,” Combust. Flame 28, 175–186 (1977).
[CrossRef]

1976

C. K. Law, “Unsteady droplet combustion with droplet heating,” Combust. Flame 26, 17–22 (1976).
[CrossRef]

Adler, C. L.

Anders, K.

N. Roth, K. Anders, A. Frohn, “Simultaneous measurement of temperature and size of droplets in the micro-metric range,” J. Laser Appl. 2, 37–42 (1990).
[CrossRef]

Bachalo, W. D.

S. V. Sankar, D. H. Buermann, W. D. Bachalo, “An advanced Rainbow signal processor for improved accuracy in droplet measurement,” presented at the Eighth International Symposium on Application of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, 1996.

Buermann, D. H.

S. V. Sankar, D. H. Buermann, W. D. Bachalo, “An advanced Rainbow signal processor for improved accuracy in droplet measurement,” presented at the Eighth International Symposium on Application of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, 1996.

Dombrovsky, L. A.

L. A. Dombrovsky, S. S. Sazih, “A simplified nonisothermal model for droplet heating and evaporation,” Int. Commun. Heat Mass Transfer 30, 787–796 (2003).
[CrossRef]

Frohn, A.

N. Roth, K. Anders, A. Frohn, “Simultaneous measurement of temperature and size of droplets in the micro-metric range,” J. Laser Appl. 2, 37–42 (1990).
[CrossRef]

Giannoulis, D.

Gouesbet, G.

Y. P. Han, G. Gréhan, G. Gouesbet, “Generalized Lorenz-Mie theory for a spheroidal particle with off-axis Gaussianbeam illumination,” Appl. Opt. 42, 6621–6629 (2003).
[CrossRef] [PubMed]

Y. P. Han, L. Méès, K. F. Ren, G. Gouesbet, S. Z. Wu, G. Gréhan, “Scattering light by spheroids: the far field case,” Opt. Commun. 210, 1–9 (2002).
[CrossRef]

Gousbet, G.

Grehan, G.

Gréhan, G.

Y. P. Han, G. Gréhan, G. Gouesbet, “Generalized Lorenz-Mie theory for a spheroidal particle with off-axis Gaussianbeam illumination,” Appl. Opt. 42, 6621–6629 (2003).
[CrossRef] [PubMed]

Y. P. Han, L. Méès, K. F. Ren, G. Gouesbet, S. Z. Wu, G. Gréhan, “Scattering light by spheroids: the far field case,” Opt. Commun. 210, 1–9 (2002).
[CrossRef]

Han, Y. P.

Y. P. Han, G. Gréhan, G. Gouesbet, “Generalized Lorenz-Mie theory for a spheroidal particle with off-axis Gaussianbeam illumination,” Appl. Opt. 42, 6621–6629 (2003).
[CrossRef] [PubMed]

Y. P. Han, L. Méès, K. F. Ren, G. Gouesbet, S. Z. Wu, G. Gréhan, “Scattering light by spheroids: the far field case,” Opt. Commun. 210, 1–9 (2002).
[CrossRef]

Hickok, W.

Jamison, J. M.

Law, C. K.

C. K. Law, W. A. Sirignano, “Unsteady droplet combustion with droplet heating-II: conduction limit,” Combust. Flame 28, 175–186 (1977).
[CrossRef]

C. K. Law, “Unsteady droplet combustion with droplet heating,” Combust. Flame 26, 17–22 (1976).
[CrossRef]

Lin, C.-Y.

Lock, J. A.

Massoli, P.

Méès, L.

Y. P. Han, L. Méès, K. F. Ren, G. Gouesbet, S. Z. Wu, G. Gréhan, “Scattering light by spheroids: the far field case,” Opt. Commun. 210, 1–9 (2002).
[CrossRef]

Nash, J. K.

Onofri, F.

Rafferty, I. P.

Ren, K. F.

Y. P. Han, L. Méès, K. F. Ren, G. Gouesbet, S. Z. Wu, G. Gréhan, “Scattering light by spheroids: the far field case,” Opt. Commun. 210, 1–9 (2002).
[CrossRef]

Riethmuller, M.

Riethmuller, M. L.

Roth, N.

N. Roth, K. Anders, A. Frohn, “Simultaneous measurement of temperature and size of droplets in the micro-metric range,” J. Laser Appl. 2, 37–42 (1990).
[CrossRef]

Sankar, S. V.

S. V. Sankar, D. H. Buermann, W. D. Bachalo, “An advanced Rainbow signal processor for improved accuracy in droplet measurement,” presented at the Eighth International Symposium on Application of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, 1996.

Saunders, K. W.

Sazih, S. S.

L. A. Dombrovsky, S. S. Sazih, “A simplified nonisothermal model for droplet heating and evaporation,” Int. Commun. Heat Mass Transfer 30, 787–796 (2003).
[CrossRef]

Sirignano, W. A.

C. K. Law, W. A. Sirignano, “Unsteady droplet combustion with droplet heating-II: conduction limit,” Combust. Flame 28, 175–186 (1977).
[CrossRef]

van Beeck, J.

van Beeck, J. P. A. J.

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1957).

Vetrano, M. R.

Wu, S. Z.

Y. P. Han, L. Méès, K. F. Ren, G. Gouesbet, S. Z. Wu, G. Gréhan, “Scattering light by spheroids: the far field case,” Opt. Commun. 210, 1–9 (2002).
[CrossRef]

Zimmer, L.

Appl. Opt.

J. A. Lock, J. M. Jamison, C.-Y. Lin, “Rainbow scattering by a coated sphere,” Appl. Opt. 33, 4677–4691 (1994).
[CrossRef] [PubMed]

P. Massoli, “Rainbow refractometry to radially inhomogeneous spheres: the critical case of evaporating droplets,” Appl. Opt. 37, 3227–3235 (1998).
[CrossRef]

F. Onofri, G. Grehan, G. Gousbet, “Electromagnetic scattering from a multilayered sphere located in an arbitrary beam,” Appl. Opt. 34, 7113–7124 (1995).
[CrossRef] [PubMed]

J. P. A. J. van Beeck, M. L. Riethmuller, “Rainbow phenomena applied to the measurement of droplet size and velocity and to the detection of nonsphericity,” Appl. Opt. 35, 2259–2266 (1996).
[CrossRef] [PubMed]

C. L. Adler, J. A. Lock, J. K. Nash, K. W. Saunders, “Experimental observation of rainbow scattering by a coated cylinder: twin primary rainbows and thin-film interference,” Appl. Opt. 40, 1548– 1558 (2001).
[CrossRef]

C. L. Adler, J. A. Lock, I. P. Rafferty, W. Hickok, “Twin-rainbow metrology. I. Measurement of the thickness of a thin liquid film draining under gravity,” Appl. Opt. 42, 6584–6594 (2003).
[CrossRef] [PubMed]

Y. P. Han, G. Gréhan, G. Gouesbet, “Generalized Lorenz-Mie theory for a spheroidal particle with off-axis Gaussianbeam illumination,” Appl. Opt. 42, 6621–6629 (2003).
[CrossRef] [PubMed]

M. R. Vetrano, J. van Beeck, M. Riethmuller, “Global rainbow thermometry: improvements in the data inversion algorithm and validation technique in liquid–liquid suspension,” Appl. Opt. 43, 3600–3607 (2004).
[CrossRef] [PubMed]

Combust. Flame

C. K. Law, “Unsteady droplet combustion with droplet heating,” Combust. Flame 26, 17–22 (1976).
[CrossRef]

C. K. Law, W. A. Sirignano, “Unsteady droplet combustion with droplet heating-II: conduction limit,” Combust. Flame 28, 175–186 (1977).
[CrossRef]

Int. Commun. Heat Mass Transfer

L. A. Dombrovsky, S. S. Sazih, “A simplified nonisothermal model for droplet heating and evaporation,” Int. Commun. Heat Mass Transfer 30, 787–796 (2003).
[CrossRef]

J. Laser Appl.

N. Roth, K. Anders, A. Frohn, “Simultaneous measurement of temperature and size of droplets in the micro-metric range,” J. Laser Appl. 2, 37–42 (1990).
[CrossRef]

Opt. Commun.

Y. P. Han, L. Méès, K. F. Ren, G. Gouesbet, S. Z. Wu, G. Gréhan, “Scattering light by spheroids: the far field case,” Opt. Commun. 210, 1–9 (2002).
[CrossRef]

Opt. Lett.

Other

H. C. van de Hulst, Light Scattering by Small Particles (Dover, 1957).

S. V. Sankar, D. H. Buermann, W. D. Bachalo, “An advanced Rainbow signal processor for improved accuracy in droplet measurement,” presented at the Eighth International Symposium on Application of Laser Techniques to Fluid Mechanics, Lisbon, Portugal, 1996.

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Figures (12)

Fig. 1
Fig. 1

Light path inside a particle presenting an internal spherical symmetric profile of refractive index. Due to this symmetry four equal branches constitute the light path (dashed curve, dotted curve, dashed–dotted curve, solid curve). θ and τ are, respectively, the scattering angle and the incident angle. r and χ radial are the polar coordinates of the spherical system of coordinates used.

Fig. 2
Fig. 2

Effect of the particle diameter on the standard rainbow pattern. The particle is supposed to possess a parabolic refractive index profile. The refractive indexes at the surface and at the center of the particle are, respectively, ns and nc, and λ is the light wavelength. The only point of the rainbow pattern independent from the particle diameter is the rainbow scattering angle θrg.

Fig. 3
Fig. 3

Effect of the particle internal profile of refractive index on the standard rainbow pattern.

Fig. 4
Fig. 4

Example of polynomial refractive index profiles inside a particle.

Fig. 5
Fig. 5

Representation of the rainbow scattering angle θrg as a function of the polynomial order m of the refractive index profile inside a particle. The dashed-dotted curve and the dotted curve represent, respectively, the value of the rainbow scattering angle in the case of a uniform refractive index equal to nc and ns.

Fig. 6
Fig. 6

Three-dimensional representation of the rainbow refractive index nrg as a function of the refractive indices nc and ns. This surface can be well approximated by Eq. (10).

Fig. 7
Fig. 7

Angular position of the first five maxima of a standard rainbow pattern as functions of the variable nc. The standard rainbow pattern is generated by a particle of 100 µm diameter with a constant surface refractive index ns.

Fig. 8
Fig. 8

Internal refractive index profiles of an n-octane droplet burning in a standard atmosphere. The quantity τ represents the nondimensional time t/t0, where t0 is the heating time.

Fig. 9
Fig. 9

Standard rainbow patterns obtained for the refractive index profiles of Fig. 8.

Fig. 10
Fig. 10

Rainbow temperature evolution, obtained with the former data inversion algorithm, as a function of the n-octane droplet heating time.

Fig. 11
Fig. 11

Comparison of the theoretical refractive index values in the core and at the surface of an n-octane droplet burning in a standard atmosphere and the values obtained with the old and with the new data inversion algorithm.

Fig. 12
Fig. 12

Comparison between the theoretical n-octane droplet diameter and the one obtained using the new data inversion algorithm.

Equations (12)

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

θ [ τ ] = 2 τ 2 ( N 1 ) τ [ τ , n ] ,
θ [ τ ] = 2 τ 2 ( N 1 ) χ 1 ,
χ 1 [ τ ] = 1 r m n e cos τ r n [ r ] 2 r 2 ( n e cos τ ) 2 r ,
I | Ω ( z ) | 2 = | 0 cos ( 1 2 π { [ ( θ θ r g ) × ( 16 D 2 h 1 λ 2 ) 1 / 3 ] η η 3 } ) d η | 2 ,
h = 2 3 2 θ 2 τ | τ = τ r s 1 sin 2 τ r g ,
θ r g = θ [ τ r g ] ,
χ 1 τ = [ N 1 ] 1 ,
n [ r ] = n c ( n c n s ) r m ,
θ r g = 2 ( n r g 2 1 3 ) 0.5 4 arccos [ 1 n r g cos ( n r g 2 1 3 ) 0.5 ] ,
n r g [ n c , n s ] = n s + δ n s 2 ( n c n s ) ,
( θ θ 1 r g ) h 1 1 / 3 ( θ θ 2 r g ) h 2 1 / 3 = k π k [ 1 , ] ,
θ 1 r g θ 2 r g = ( h 1 h 2 ) 1 / 3 ,

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