Abstract

Generalized Lorenz–Mie theory (GLMT) for a multilayered sphere is used to simulate holograms produced by evaporating spherical droplets with refractive index gradient in the surrounding air/vapor mixture. Simulated holograms provide a physical interpretation of experimental holograms produced by evaporating Diethyl Ether droplets with diameter in the order of 50 μm and recorded in a digital in-line holography configuration with a divergent beam. Refractive index gradients in the surrounding medium lead to a modification of the center part of the droplet holograms, where the first fringe is unusually bright. GLMT simulations reproduce this modification well, assuming an exponential decay of the refractive index from the droplet surface to infinity. The diverging beam effect is also considered. In both evaporating and nonevaporating cases, an equivalence is found between Gaussian beam and plane wave illuminations, simply based on a magnification ratio to be applied to the droplets’ parameters.

© 2013 Optical Society of America

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  1. S. Guella, S. Alexandrova, and A. Saboni, “Evaporation of a free falling droplet,” Int. J. Therm. Sci. 47, 886–898 (2008).
    [CrossRef]
  2. J. S. Wu, Y. J. Liu, and H. J. Sheen, “Effect of ambient turbulence and fuel properties on the evaporation rate of single droplets,” Int. J. Heat Mass Transfer 44, 4593–4603 (2001).
    [CrossRef]
  3. C. Maqua, G. Castanet, F. Grisch, F. Lemoine, T. Kristyadi, and S. Sazhin, “Monodisperse droplet heating and evaporation: experimental study and modelling,” Int. J. Heat Mass Transfer 51, 3932–3945 (2008).
    [CrossRef]
  4. F. Soulez, L. Denis, C. Fournier, E. Thiébaut, and C. Goepfert, “Inverse-problem approach for particle digital holography: accurate location based on local optimization,” J. Opt. Soc. Am. 24, 1164–1171 (2007).
    [CrossRef]
  5. F. Soulez, L. Denis, C. Fournier, E. Thiébaut, and C. Goepfert, “Inverse problem approach in particle digital holography: Out-of-field particle detection made possible,” J. Opt. Soc. Am. 24, 3708–3716 (2007).
    [CrossRef]
  6. S. H. Lee, Y. Roichman, G. R. Yi, S. H. Kim, S. M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. Express 15, 18275–18282 (2007).
    [CrossRef]
  7. D. Chareyron, J. L. Marié, C. Fournier, G. Gire, N. Grosjean, L. Denis, M. Lance, and L. Méès, “Testing an in-line digital holography “inverse method” for the Lagrangian tracking of evaporating droplets in homogeneous nearly isotropic turbulence,” New J. Phys. 14, 043039 (2012).
    [CrossRef]
  8. L. Lorenz, “Lysbevaegelsen i og uden for en af plane Lysbølger belyst Kulge,” Vidensk. Selk. Skr. 6, 1–62 (1890).
  9. G. Mie, “Beiträge zur Optik Trüber Medien speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908).
    [CrossRef]
  10. P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 335, 57–136 (1909).
    [CrossRef]
  11. G. Gouesbet and G. Gréhan, “Generalized Lorenz–Mie theories, from past to future,” Atomiz. Spr. 10, 277–333 (2000).
  12. G. Gouesbet and G. Gréhan, Generalized Lorenz–Mie Theories (Springer, 2011).
  13. F. Onofri, G. Gréhan, and G. Gouesbet, “Electromagnetic scattering from a multilayered sphere located in an arbitrary beam,” Appl. Opt. 34, 7113–7124 (1995).
    [CrossRef]
  14. G. Gouesbet, B. Maheu, and G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,” J. Opt. 16, 239–247 (1985).
    [CrossRef]
  15. B. Maheu, G. Gouesbet, and G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. 19, 59–67 (1988).
    [CrossRef]
  16. G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz–Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516–2525 (1994).
    [CrossRef]
  17. C. S. Vikram and M. L. Billet, “Some salient features of in-line Fraunhofer holography with divergent beams,” Optik 78, 80–86 (1988).
  18. D. Lebrun, D. Allano, L. Méès, F. Walle, R. Boucheron, F. Corbin, and D. Fréchou, “Size measurement of bubbles in a cavitation tunnel by digital in-line holography,” Appl. Opt. 50, H1–H9 (2011).
    [CrossRef]
  19. S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakon, Y. Biscos, N. Garcia, G. Lavergne, L. Méès, G. Gouesbet, and G. Gréhan, “Rainbow refractometry on particles with radial refractive index gradients,” Exp. Fluids 43, 595–601 (2007).
    [CrossRef]
  20. G. R. Toker and J. Stricker, “Holographic study of suspended vaporizing volatile liquid droplets in still air,” Int. J. Heat Mass Transfer 39, 3475–3482 (1996).
    [CrossRef]
  21. G. R. Toker and J. Stricker, “Study of suspended vaporizing volatile liquid droplets by an enhanced sensitivity holographic technique: additional results,” Int. J. Heat Mass Transfer. 41, 2553–2555 (1998).
    [CrossRef]

2012 (1)

D. Chareyron, J. L. Marié, C. Fournier, G. Gire, N. Grosjean, L. Denis, M. Lance, and L. Méès, “Testing an in-line digital holography “inverse method” for the Lagrangian tracking of evaporating droplets in homogeneous nearly isotropic turbulence,” New J. Phys. 14, 043039 (2012).
[CrossRef]

2011 (1)

2008 (2)

S. Guella, S. Alexandrova, and A. Saboni, “Evaporation of a free falling droplet,” Int. J. Therm. Sci. 47, 886–898 (2008).
[CrossRef]

C. Maqua, G. Castanet, F. Grisch, F. Lemoine, T. Kristyadi, and S. Sazhin, “Monodisperse droplet heating and evaporation: experimental study and modelling,” Int. J. Heat Mass Transfer 51, 3932–3945 (2008).
[CrossRef]

2007 (4)

F. Soulez, L. Denis, C. Fournier, E. Thiébaut, and C. Goepfert, “Inverse-problem approach for particle digital holography: accurate location based on local optimization,” J. Opt. Soc. Am. 24, 1164–1171 (2007).
[CrossRef]

F. Soulez, L. Denis, C. Fournier, E. Thiébaut, and C. Goepfert, “Inverse problem approach in particle digital holography: Out-of-field particle detection made possible,” J. Opt. Soc. Am. 24, 3708–3716 (2007).
[CrossRef]

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakon, Y. Biscos, N. Garcia, G. Lavergne, L. Méès, G. Gouesbet, and G. Gréhan, “Rainbow refractometry on particles with radial refractive index gradients,” Exp. Fluids 43, 595–601 (2007).
[CrossRef]

S. H. Lee, Y. Roichman, G. R. Yi, S. H. Kim, S. M. Yang, A. van Blaaderen, P. van Oostrum, and D. G. Grier, “Characterizing and tracking single colloidal particles with video holographic microscopy,” Opt. Express 15, 18275–18282 (2007).
[CrossRef]

2001 (1)

J. S. Wu, Y. J. Liu, and H. J. Sheen, “Effect of ambient turbulence and fuel properties on the evaporation rate of single droplets,” Int. J. Heat Mass Transfer 44, 4593–4603 (2001).
[CrossRef]

2000 (1)

G. Gouesbet and G. Gréhan, “Generalized Lorenz–Mie theories, from past to future,” Atomiz. Spr. 10, 277–333 (2000).

1998 (1)

G. R. Toker and J. Stricker, “Study of suspended vaporizing volatile liquid droplets by an enhanced sensitivity holographic technique: additional results,” Int. J. Heat Mass Transfer. 41, 2553–2555 (1998).
[CrossRef]

1996 (1)

G. R. Toker and J. Stricker, “Holographic study of suspended vaporizing volatile liquid droplets in still air,” Int. J. Heat Mass Transfer 39, 3475–3482 (1996).
[CrossRef]

1995 (1)

1994 (1)

1988 (2)

B. Maheu, G. Gouesbet, and G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. 19, 59–67 (1988).
[CrossRef]

C. S. Vikram and M. L. Billet, “Some salient features of in-line Fraunhofer holography with divergent beams,” Optik 78, 80–86 (1988).

1985 (1)

G. Gouesbet, B. Maheu, and G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,” J. Opt. 16, 239–247 (1985).
[CrossRef]

1909 (1)

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 335, 57–136 (1909).
[CrossRef]

1908 (1)

G. Mie, “Beiträge zur Optik Trüber Medien speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908).
[CrossRef]

1890 (1)

L. Lorenz, “Lysbevaegelsen i og uden for en af plane Lysbølger belyst Kulge,” Vidensk. Selk. Skr. 6, 1–62 (1890).

Alexandrova, S.

S. Guella, S. Alexandrova, and A. Saboni, “Evaporation of a free falling droplet,” Int. J. Therm. Sci. 47, 886–898 (2008).
[CrossRef]

Allano, D.

Billet, M. L.

C. S. Vikram and M. L. Billet, “Some salient features of in-line Fraunhofer holography with divergent beams,” Optik 78, 80–86 (1988).

Biscos, Y.

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakon, Y. Biscos, N. Garcia, G. Lavergne, L. Méès, G. Gouesbet, and G. Gréhan, “Rainbow refractometry on particles with radial refractive index gradients,” Exp. Fluids 43, 595–601 (2007).
[CrossRef]

Boucheron, R.

Castanet, G.

C. Maqua, G. Castanet, F. Grisch, F. Lemoine, T. Kristyadi, and S. Sazhin, “Monodisperse droplet heating and evaporation: experimental study and modelling,” Int. J. Heat Mass Transfer 51, 3932–3945 (2008).
[CrossRef]

Chareyron, D.

D. Chareyron, J. L. Marié, C. Fournier, G. Gire, N. Grosjean, L. Denis, M. Lance, and L. Méès, “Testing an in-line digital holography “inverse method” for the Lagrangian tracking of evaporating droplets in homogeneous nearly isotropic turbulence,” New J. Phys. 14, 043039 (2012).
[CrossRef]

Charinpanitkul, T.

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakon, Y. Biscos, N. Garcia, G. Lavergne, L. Méès, G. Gouesbet, and G. Gréhan, “Rainbow refractometry on particles with radial refractive index gradients,” Exp. Fluids 43, 595–601 (2007).
[CrossRef]

Corbin, F.

Debye, P.

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 335, 57–136 (1909).
[CrossRef]

Denis, L.

D. Chareyron, J. L. Marié, C. Fournier, G. Gire, N. Grosjean, L. Denis, M. Lance, and L. Méès, “Testing an in-line digital holography “inverse method” for the Lagrangian tracking of evaporating droplets in homogeneous nearly isotropic turbulence,” New J. Phys. 14, 043039 (2012).
[CrossRef]

F. Soulez, L. Denis, C. Fournier, E. Thiébaut, and C. Goepfert, “Inverse-problem approach for particle digital holography: accurate location based on local optimization,” J. Opt. Soc. Am. 24, 1164–1171 (2007).
[CrossRef]

F. Soulez, L. Denis, C. Fournier, E. Thiébaut, and C. Goepfert, “Inverse problem approach in particle digital holography: Out-of-field particle detection made possible,” J. Opt. Soc. Am. 24, 3708–3716 (2007).
[CrossRef]

Fournier, C.

D. Chareyron, J. L. Marié, C. Fournier, G. Gire, N. Grosjean, L. Denis, M. Lance, and L. Méès, “Testing an in-line digital holography “inverse method” for the Lagrangian tracking of evaporating droplets in homogeneous nearly isotropic turbulence,” New J. Phys. 14, 043039 (2012).
[CrossRef]

F. Soulez, L. Denis, C. Fournier, E. Thiébaut, and C. Goepfert, “Inverse problem approach in particle digital holography: Out-of-field particle detection made possible,” J. Opt. Soc. Am. 24, 3708–3716 (2007).
[CrossRef]

F. Soulez, L. Denis, C. Fournier, E. Thiébaut, and C. Goepfert, “Inverse-problem approach for particle digital holography: accurate location based on local optimization,” J. Opt. Soc. Am. 24, 1164–1171 (2007).
[CrossRef]

Fréchou, D.

Garcia, N.

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakon, Y. Biscos, N. Garcia, G. Lavergne, L. Méès, G. Gouesbet, and G. Gréhan, “Rainbow refractometry on particles with radial refractive index gradients,” Exp. Fluids 43, 595–601 (2007).
[CrossRef]

Gire, G.

D. Chareyron, J. L. Marié, C. Fournier, G. Gire, N. Grosjean, L. Denis, M. Lance, and L. Méès, “Testing an in-line digital holography “inverse method” for the Lagrangian tracking of evaporating droplets in homogeneous nearly isotropic turbulence,” New J. Phys. 14, 043039 (2012).
[CrossRef]

Goepfert, C.

F. Soulez, L. Denis, C. Fournier, E. Thiébaut, and C. Goepfert, “Inverse-problem approach for particle digital holography: accurate location based on local optimization,” J. Opt. Soc. Am. 24, 1164–1171 (2007).
[CrossRef]

F. Soulez, L. Denis, C. Fournier, E. Thiébaut, and C. Goepfert, “Inverse problem approach in particle digital holography: Out-of-field particle detection made possible,” J. Opt. Soc. Am. 24, 3708–3716 (2007).
[CrossRef]

Gouesbet, G.

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakon, Y. Biscos, N. Garcia, G. Lavergne, L. Méès, G. Gouesbet, and G. Gréhan, “Rainbow refractometry on particles with radial refractive index gradients,” Exp. Fluids 43, 595–601 (2007).
[CrossRef]

G. Gouesbet and G. Gréhan, “Generalized Lorenz–Mie theories, from past to future,” Atomiz. Spr. 10, 277–333 (2000).

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

G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz–Mie theory. II. Off-axis beams,” J. Opt. Soc. Am. A 11, 2516–2525 (1994).
[CrossRef]

B. Maheu, G. Gouesbet, and G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. 19, 59–67 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, and G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,” J. Opt. 16, 239–247 (1985).
[CrossRef]

G. Gouesbet and G. Gréhan, Generalized Lorenz–Mie Theories (Springer, 2011).

Gréhan, G.

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakon, Y. Biscos, N. Garcia, G. Lavergne, L. Méès, G. Gouesbet, and G. Gréhan, “Rainbow refractometry on particles with radial refractive index gradients,” Exp. Fluids 43, 595–601 (2007).
[CrossRef]

G. Gouesbet and G. Gréhan, “Generalized Lorenz–Mie theories, from past to future,” Atomiz. Spr. 10, 277–333 (2000).

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

B. Maheu, G. Gouesbet, and G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. 19, 59–67 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, and G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,” J. Opt. 16, 239–247 (1985).
[CrossRef]

G. Gouesbet and G. Gréhan, Generalized Lorenz–Mie Theories (Springer, 2011).

Grier, D. G.

Grisch, F.

C. Maqua, G. Castanet, F. Grisch, F. Lemoine, T. Kristyadi, and S. Sazhin, “Monodisperse droplet heating and evaporation: experimental study and modelling,” Int. J. Heat Mass Transfer 51, 3932–3945 (2008).
[CrossRef]

Grosjean, N.

D. Chareyron, J. L. Marié, C. Fournier, G. Gire, N. Grosjean, L. Denis, M. Lance, and L. Méès, “Testing an in-line digital holography “inverse method” for the Lagrangian tracking of evaporating droplets in homogeneous nearly isotropic turbulence,” New J. Phys. 14, 043039 (2012).
[CrossRef]

Guella, S.

S. Guella, S. Alexandrova, and A. Saboni, “Evaporation of a free falling droplet,” Int. J. Therm. Sci. 47, 886–898 (2008).
[CrossRef]

Kim, S. H.

Kristyadi, T.

C. Maqua, G. Castanet, F. Grisch, F. Lemoine, T. Kristyadi, and S. Sazhin, “Monodisperse droplet heating and evaporation: experimental study and modelling,” Int. J. Heat Mass Transfer 51, 3932–3945 (2008).
[CrossRef]

Lance, M.

D. Chareyron, J. L. Marié, C. Fournier, G. Gire, N. Grosjean, L. Denis, M. Lance, and L. Méès, “Testing an in-line digital holography “inverse method” for the Lagrangian tracking of evaporating droplets in homogeneous nearly isotropic turbulence,” New J. Phys. 14, 043039 (2012).
[CrossRef]

Lavergne, G.

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakon, Y. Biscos, N. Garcia, G. Lavergne, L. Méès, G. Gouesbet, and G. Gréhan, “Rainbow refractometry on particles with radial refractive index gradients,” Exp. Fluids 43, 595–601 (2007).
[CrossRef]

Lebrun, D.

Lee, S. H.

Lemoine, F.

C. Maqua, G. Castanet, F. Grisch, F. Lemoine, T. Kristyadi, and S. Sazhin, “Monodisperse droplet heating and evaporation: experimental study and modelling,” Int. J. Heat Mass Transfer 51, 3932–3945 (2008).
[CrossRef]

Liu, Y. J.

J. S. Wu, Y. J. Liu, and H. J. Sheen, “Effect of ambient turbulence and fuel properties on the evaporation rate of single droplets,” Int. J. Heat Mass Transfer 44, 4593–4603 (2001).
[CrossRef]

Lock, J. A.

Lorenz, L.

L. Lorenz, “Lysbevaegelsen i og uden for en af plane Lysbølger belyst Kulge,” Vidensk. Selk. Skr. 6, 1–62 (1890).

Maheu, B.

B. Maheu, G. Gouesbet, and G. Gréhan, “A concise presentation of the generalized Lorenz–Mie theory for arbitrary location of the scatterer in an arbitrary incident profile,” J. Opt. 19, 59–67 (1988).
[CrossRef]

G. Gouesbet, B. Maheu, and G. Gréhan, “The order of approximation in a theory of the scattering of a Gaussian beam by a Mie scatter center,” J. Opt. 16, 239–247 (1985).
[CrossRef]

Maqua, C.

C. Maqua, G. Castanet, F. Grisch, F. Lemoine, T. Kristyadi, and S. Sazhin, “Monodisperse droplet heating and evaporation: experimental study and modelling,” Int. J. Heat Mass Transfer 51, 3932–3945 (2008).
[CrossRef]

Marié, J. L.

D. Chareyron, J. L. Marié, C. Fournier, G. Gire, N. Grosjean, L. Denis, M. Lance, and L. Méès, “Testing an in-line digital holography “inverse method” for the Lagrangian tracking of evaporating droplets in homogeneous nearly isotropic turbulence,” New J. Phys. 14, 043039 (2012).
[CrossRef]

Méès, L.

D. Chareyron, J. L. Marié, C. Fournier, G. Gire, N. Grosjean, L. Denis, M. Lance, and L. Méès, “Testing an in-line digital holography “inverse method” for the Lagrangian tracking of evaporating droplets in homogeneous nearly isotropic turbulence,” New J. Phys. 14, 043039 (2012).
[CrossRef]

D. Lebrun, D. Allano, L. Méès, F. Walle, R. Boucheron, F. Corbin, and D. Fréchou, “Size measurement of bubbles in a cavitation tunnel by digital in-line holography,” Appl. Opt. 50, H1–H9 (2011).
[CrossRef]

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakon, Y. Biscos, N. Garcia, G. Lavergne, L. Méès, G. Gouesbet, and G. Gréhan, “Rainbow refractometry on particles with radial refractive index gradients,” Exp. Fluids 43, 595–601 (2007).
[CrossRef]

Mie, G.

G. Mie, “Beiträge zur Optik Trüber Medien speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908).
[CrossRef]

Onofri, F.

Roichman, Y.

Saboni, A.

S. Guella, S. Alexandrova, and A. Saboni, “Evaporation of a free falling droplet,” Int. J. Therm. Sci. 47, 886–898 (2008).
[CrossRef]

Saengkaew, S.

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakon, Y. Biscos, N. Garcia, G. Lavergne, L. Méès, G. Gouesbet, and G. Gréhan, “Rainbow refractometry on particles with radial refractive index gradients,” Exp. Fluids 43, 595–601 (2007).
[CrossRef]

Sazhin, S.

C. Maqua, G. Castanet, F. Grisch, F. Lemoine, T. Kristyadi, and S. Sazhin, “Monodisperse droplet heating and evaporation: experimental study and modelling,” Int. J. Heat Mass Transfer 51, 3932–3945 (2008).
[CrossRef]

Sheen, H. J.

J. S. Wu, Y. J. Liu, and H. J. Sheen, “Effect of ambient turbulence and fuel properties on the evaporation rate of single droplets,” Int. J. Heat Mass Transfer 44, 4593–4603 (2001).
[CrossRef]

Soulez, F.

F. Soulez, L. Denis, C. Fournier, E. Thiébaut, and C. Goepfert, “Inverse problem approach in particle digital holography: Out-of-field particle detection made possible,” J. Opt. Soc. Am. 24, 3708–3716 (2007).
[CrossRef]

F. Soulez, L. Denis, C. Fournier, E. Thiébaut, and C. Goepfert, “Inverse-problem approach for particle digital holography: accurate location based on local optimization,” J. Opt. Soc. Am. 24, 1164–1171 (2007).
[CrossRef]

Stricker, J.

G. R. Toker and J. Stricker, “Study of suspended vaporizing volatile liquid droplets by an enhanced sensitivity holographic technique: additional results,” Int. J. Heat Mass Transfer. 41, 2553–2555 (1998).
[CrossRef]

G. R. Toker and J. Stricker, “Holographic study of suspended vaporizing volatile liquid droplets in still air,” Int. J. Heat Mass Transfer 39, 3475–3482 (1996).
[CrossRef]

Tanthapanichakon, W.

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakon, Y. Biscos, N. Garcia, G. Lavergne, L. Méès, G. Gouesbet, and G. Gréhan, “Rainbow refractometry on particles with radial refractive index gradients,” Exp. Fluids 43, 595–601 (2007).
[CrossRef]

Thiébaut, E.

F. Soulez, L. Denis, C. Fournier, E. Thiébaut, and C. Goepfert, “Inverse-problem approach for particle digital holography: accurate location based on local optimization,” J. Opt. Soc. Am. 24, 1164–1171 (2007).
[CrossRef]

F. Soulez, L. Denis, C. Fournier, E. Thiébaut, and C. Goepfert, “Inverse problem approach in particle digital holography: Out-of-field particle detection made possible,” J. Opt. Soc. Am. 24, 3708–3716 (2007).
[CrossRef]

Toker, G. R.

G. R. Toker and J. Stricker, “Study of suspended vaporizing volatile liquid droplets by an enhanced sensitivity holographic technique: additional results,” Int. J. Heat Mass Transfer. 41, 2553–2555 (1998).
[CrossRef]

G. R. Toker and J. Stricker, “Holographic study of suspended vaporizing volatile liquid droplets in still air,” Int. J. Heat Mass Transfer 39, 3475–3482 (1996).
[CrossRef]

van Blaaderen, A.

van Oostrum, P.

Vanisri, H.

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakon, Y. Biscos, N. Garcia, G. Lavergne, L. Méès, G. Gouesbet, and G. Gréhan, “Rainbow refractometry on particles with radial refractive index gradients,” Exp. Fluids 43, 595–601 (2007).
[CrossRef]

Vikram, C. S.

C. S. Vikram and M. L. Billet, “Some salient features of in-line Fraunhofer holography with divergent beams,” Optik 78, 80–86 (1988).

Walle, F.

Wu, J. S.

J. S. Wu, Y. J. Liu, and H. J. Sheen, “Effect of ambient turbulence and fuel properties on the evaporation rate of single droplets,” Int. J. Heat Mass Transfer 44, 4593–4603 (2001).
[CrossRef]

Yang, S. M.

Yi, G. R.

Ann. Phys. (2)

G. Mie, “Beiträge zur Optik Trüber Medien speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908).
[CrossRef]

P. Debye, “Der Lichtdruck auf Kugeln von beliebigem Material,” Ann. Phys. 335, 57–136 (1909).
[CrossRef]

Appl. Opt. (2)

Atomiz. Spr. (1)

G. Gouesbet and G. Gréhan, “Generalized Lorenz–Mie theories, from past to future,” Atomiz. Spr. 10, 277–333 (2000).

Exp. Fluids (1)

S. Saengkaew, T. Charinpanitkul, H. Vanisri, W. Tanthapanichakon, Y. Biscos, N. Garcia, G. Lavergne, L. Méès, G. Gouesbet, and G. Gréhan, “Rainbow refractometry on particles with radial refractive index gradients,” Exp. Fluids 43, 595–601 (2007).
[CrossRef]

Int. J. Heat Mass Transfer (3)

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S. Guella, S. Alexandrova, and A. Saboni, “Evaporation of a free falling droplet,” Int. J. Therm. Sci. 47, 886–898 (2008).
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Figures (10)

Fig. 1.
Fig. 1.

Holographic image of freon droplets evaporating in a turbulent flow. The experiment setup is detailed in [7].

Fig. 2.
Fig. 2.

Multilayered particle illuminated by a Gaussian beam.

Fig. 3.
Fig. 3.

Experimental setup. A divergent beam is used to record evaporating Diethyl Ether droplet holograms.

Fig. 4.
Fig. 4.

Comparison between holograms produced under focused beam and plane wave illumination. The plane wave equivalence consists in applying the magnification ratio G defined in Eq. (29) on both droplet radius and recording distance.

Fig. 5.
Fig. 5.

Evaporating droplet holograms for varying refractive index deviation at droplet surface. The droplet is illuminated by a Gaussian beam (ω0=1μm, λ=532nm and z0=(zSzE)=193.5mm). The droplet radius is rD=20μm, refractive index decreases exponentially in the surrounding medium following Eq. (30) with variable ns and a constant decay parameter σ=80μm.

Fig. 6.
Fig. 6.

Evaporating droplet holograms for varying decay parameter σ. The droplet is illuminated by a Gaussian beam (ω0=1μm, λ=532nm and z0=(zSzE)=193.5mm). The droplet radius is rD=20μm and the refractive index decreases exponentially in the surrounding medium following Eq. (30) with variable σ from 0 to 320 μm and a constant refractive index deviation at droplet surface ns=5×105.

Fig. 7.
Fig. 7.

Effect of the refractive index gradient on holograms for different droplet sizes. The holograms are produced by different droplets of radius rD=10μm, rD=20μm and rD=30μm, with and without surrounding refractive index gradients, which are defined by a constant deviation at droplet surface ns=5×105 and a width parameter proportional to the droplet radius σ=4rD.

Fig. 8.
Fig. 8.

Effect of the surrounding refractive index gradient on holograms for different droplet sizes. The holograms are produced by droplets of radius rD=50μm and rD=100μm, with and without surrounding refractive index gradients, which are defined by a constant deviation at droplet surface ns=5×105 and a width parameter proportional to the droplet radius σ=4rD.

Fig. 9.
Fig. 9.

Experimental hologram compared to computed holograms, with and without surrounding refractive index gradient. For both evaporating and nonevaporating case, the droplet diameter is rD=21μm and the recording distance is z=360.426mm. For the evaporating droplet, the surrounding refractive index gradient is defined by σ=80μm and ns=8.5×105.

Fig. 10.
Fig. 10.

Comparison between holograms computed for a divergent beam illumination and equivalent holograms computed for a plane wave illumination and magnified droplet parameters.

Equations (41)

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Eri=E0Ψ0[cosφsinθ(12Qlrcosθ)+2Qlx0cosθ]exp(K),
Eθi=E0Ψ0[cosφ(cosθ+2Qlrsin2θ)2Qlx0sinθ]exp(K),
Eφi=E0Ψ0sinφexp(K),
Hri=H0Ψ0[sinφsinθ(12Qlrcosθ)+2Qly0cosθ]exp(K),
Hθi=H0Ψ0[sinφ(cosθ+2Qlrsin2θ)2Qly0sinθ]exp(K),
Hφi=H0Ψ0cosφexp(K),
K=ik0(rcosθz0),Ψ0=iQexp(iQr2sin2θω02)exp(iQx02+y02ω02)×exp[2iQω02rsinθ(x0cosφ+y0sinφ)],Q=(i+2zz0k0ω02)1.
Ers=E0k02r2n=1n=m=nm=+ncnpwangn,TMmn(n+1)ξn(k0r)×Pn|m|(cosθ)exp(imφ),
Eθs=E0k0rn=1n=m=nm=+ncnpw[angn,TMmξn(k0r)τn|m|(cosθ)+mbngn,TEmξn(k0r)πn|m|(cosθ)]exp(imφ),
Eφs=iE0k0rn=1n=m=nm=+ncnpw[mangn,TMmξn(k0r)πn|m|(cosθ)+bngn,TEmξn(k0r)τn|m|(cosθ)]exp(imφ),
Hrs=H0k02r2n=1n=m=nm=+ncnpwbngn,TEmn(n+1)ξn(k0r)×Pn|m|(cosθ)exp(imφ),
Hθs=H0k0rn=1n=m=nm=+ncnpw[mangn,TMmξn(k0r)πn|m|(cosθ)bngn,TEmξn(k0r)τn|m|(cosθ)]exp(imφ),
Hφs=iH0k0rn=1n=m=nm=+ncnpw[angn,TMmξn(k0r)τn|m|(cosθ)mbngn,TEmξn(k0r)πn|m|(cosθ)]exp(imφ),
cnpw=(i)n2n+1n(n+1).
ξn(kr)=krΨn(4)(kr)=πkr2Hn+12(2)(kr).
τnm(cosθ)=ddθPnm(cosθ),πnm(cosθ)=Pnm(cosθ)sinθ.
[gn,TMmgn,TEm]=12(1+2iz0+)exp(iz0+s2x0+2+y0+21+2iz0+)×exp((n+12)2s21+2iz0+)×Rnm(i)|m|[iFn,TMmFn,TEm],
Rnm=(22n+1)|m|1,|m|1Rn0=2n(2n+1)2n+1,[Fn,TM0Fn,TE0]=[2x0+2iy0+]j=0a2j+1XjX+jj!(j+1)!,
[Fn,TMmFn,TEm]=am1Xm1(m1)!+j=ma2jm+1XjX+jmj!(jm)!×[X+jm+1+Xj+1X+jm+1Xj+1],m>0
[Fn,TMmFn,TEm]=a1mX1m(1m)!+j=ma2j+m+1Xj+mX+jj!(j+m)!×[Xj+m+1+X+j+1Xj+m+1X+j+1],m<0
a=(n+1/2)s1+2iz0+,
X=x0+iy0+,X+=x0++iy0+,
x0+=x0ω0,y0+=y0ω0,z0+=z0k0ω02.
an=ψn(xL)Hna(xL)mLψn(xL)ξn(xL)Hna(xL)mLξn(xL),
bn=mLψn(xL)Hnb(xL)ψn(xL)mLξn(xL)Hnb(xL)ξn(xL),
ψn(kr)=krΨn(1)(kr)=πkr2Jn+12(kr)
Hna(xj)=Hnb(xj)=ψn(x1)ψn(x1),
Hna(xj)=ψn(xj)Rjnaχn(xj)ψn(xj)Rjnaχn(xj),
Hnb(xj)=ψn(xj)Rjnbχn(xj)ψn(xj)Rjnbχn(xj)
R1na=R1nb=0,
R2na=m2ψn(x1)ψn(x1)m1ψn(x1)ψn(x1)m2χn(x1)ψn(x1)m1χn(x1)ψn(x1),
R2nb=m1ψn(x1)ψn(x1)m2ψn(x1)ψn(x1)m1χn(x1)ψn(x1)m2χn(x1)ψn(x1),
Rjna=mjψn(xj1)Hna(xj1)mj1ψn(xj1)mjχn(xj1)Hna(xj1)mj1χn(xj1),
Rjnb=mj1ψn(xj1)Hnb(xj1)mjψn(xj1)mj1χn(xj1)Hnb(xj1)mjχn(xj1),
χn(kr)=krΨn(2)(kr)=πkr2Yn+12(kr)
St=12Re[Et×Ht*].
St=12Re[cosθ(EθtHφt*EφtHθt*)sinθ(EφtHrt*ErtHφt*)].
IN=SSiSi,
Si=12Re[cosθ(EθiHφi*EφiHθi*)sinθ(EφiHri*EriHφi*)].
G=zSzSzE.
n(r)=n0+nsexp(rrDσ),

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