Abstract

Rainbow techniques permit measurement of refractive indices, and hence the temperatures of liquid droplets through determination of the absolute angular position of a rainbow interference image in space. The Airy theory, which is commonly used to explain the rainbow effect, permits the determination of a unique refractive-index value, even in the presence of nonuniformities in the droplet. An extension of this theory to spheres that exhibit internal refractive-index gradients is proposed. The case of burning droplets is considered as an example of such spheres, and the results obtained are successfully compared with those presented in the literature.

© 2005 Optical Society of America

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References

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  1. N. Roth, K. Anders, and A. Frohn, and , J. Laser Appl. 2, 37 (1990).
    [CrossRef]
  2. G. Mie, Ann. Phys. (Paris) 25, 377 (1908).
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    [CrossRef] [PubMed]
  4. N. Damaschke, G. Gouesbet, G. Gréhan, and C. Tropea, Meas. Sci. Technol. 9, 137 (1998).
    [CrossRef]
  5. G. B. Airy, Trans. Cambridge Philos. Soc. 6, 397 (1838).
  6. R. Descartes, Discourseon Method, Optics, Geometry, and Meteorology, revised ed., translatedby P. J. Olscamp (Hackett,Indianapolis, Ind., 2001).
  7. M. Born and E. Wolf, Principles of Optics (Pergamon,London, 1959).
  8. H. C. vande Hulst, Light Scattering by Small Particles (Dover,New York, 1957).
  9. P. Massoli, Appl. Opt. 37, 3227 (1998).
    [CrossRef]
  10. C. K. Law, Combust. Flame 26, 17 (1976).
    [CrossRef]
  11. C. K. Law and W. A. Sirignano, Combust. Flame 28, 175 (1977).
    [CrossRef]
  12. L. A. Dombrovsky and S. S. Sazih, Int. Commun. Heat Mass Transfer 30, 787 (2003).
    [CrossRef]

2003 (1)

L. A. Dombrovsky and S. S. Sazih, Int. Commun. Heat Mass Transfer 30, 787 (2003).
[CrossRef]

1998 (2)

P. Massoli, Appl. Opt. 37, 3227 (1998).
[CrossRef]

N. Damaschke, G. Gouesbet, G. Gréhan, and C. Tropea, Meas. Sci. Technol. 9, 137 (1998).
[CrossRef]

1995 (1)

1990 (1)

N. Roth, K. Anders, and A. Frohn, and , J. Laser Appl. 2, 37 (1990).
[CrossRef]

1977 (1)

C. K. Law and W. A. Sirignano, Combust. Flame 28, 175 (1977).
[CrossRef]

1976 (1)

C. K. Law, Combust. Flame 26, 17 (1976).
[CrossRef]

1908 (1)

G. Mie, Ann. Phys. (Paris) 25, 377 (1908).

1838 (1)

G. B. Airy, Trans. Cambridge Philos. Soc. 6, 397 (1838).

Airy, G. B.

G. B. Airy, Trans. Cambridge Philos. Soc. 6, 397 (1838).

Anders, K.

N. Roth, K. Anders, and A. Frohn, and , J. Laser Appl. 2, 37 (1990).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon,London, 1959).

Damaschke, N.

N. Damaschke, G. Gouesbet, G. Gréhan, and C. Tropea, Meas. Sci. Technol. 9, 137 (1998).
[CrossRef]

Descartes, R.

R. Descartes, Discourseon Method, Optics, Geometry, and Meteorology, revised ed., translatedby P. J. Olscamp (Hackett,Indianapolis, Ind., 2001).

Dombrovsky, L. A.

L. A. Dombrovsky and S. S. Sazih, Int. Commun. Heat Mass Transfer 30, 787 (2003).
[CrossRef]

Frohn, A.

N. Roth, K. Anders, and A. Frohn, and , J. Laser Appl. 2, 37 (1990).
[CrossRef]

Gouesbet, G.

N. Damaschke, G. Gouesbet, G. Gréhan, and C. Tropea, Meas. Sci. Technol. 9, 137 (1998).
[CrossRef]

F. Onofri, G. Gréhan, and G. Gouesbet, Appl. Opt. 34, 7113 (1995).
[CrossRef] [PubMed]

Gréhan, G.

N. Damaschke, G. Gouesbet, G. Gréhan, and C. Tropea, Meas. Sci. Technol. 9, 137 (1998).
[CrossRef]

F. Onofri, G. Gréhan, and G. Gouesbet, Appl. Opt. 34, 7113 (1995).
[CrossRef] [PubMed]

Law, C. K.

C. K. Law and W. A. Sirignano, Combust. Flame 28, 175 (1977).
[CrossRef]

C. K. Law, Combust. Flame 26, 17 (1976).
[CrossRef]

Massoli, P.

Mie, G.

G. Mie, Ann. Phys. (Paris) 25, 377 (1908).

Onofri, F.

Roth, N.

N. Roth, K. Anders, and A. Frohn, and , J. Laser Appl. 2, 37 (1990).
[CrossRef]

Sazih, S. S.

L. A. Dombrovsky and S. S. Sazih, Int. Commun. Heat Mass Transfer 30, 787 (2003).
[CrossRef]

Sirignano, W. A.

C. K. Law and W. A. Sirignano, Combust. Flame 28, 175 (1977).
[CrossRef]

Tropea, C.

N. Damaschke, G. Gouesbet, G. Gréhan, and C. Tropea, Meas. Sci. Technol. 9, 137 (1998).
[CrossRef]

vande Hulst, H. C.

H. C. vande Hulst, Light Scattering by Small Particles (Dover,New York, 1957).

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon,London, 1959).

Ann. Phys. (Paris) (1)

G. Mie, Ann. Phys. (Paris) 25, 377 (1908).

Appl. Opt. (2)

Combust. Flame (2)

C. K. Law, Combust. Flame 26, 17 (1976).
[CrossRef]

C. K. Law and W. A. Sirignano, Combust. Flame 28, 175 (1977).
[CrossRef]

Int. Commun. Heat Mass Transfer (1)

L. A. Dombrovsky and S. S. Sazih, Int. Commun. Heat Mass Transfer 30, 787 (2003).
[CrossRef]

J. Laser Appl. (1)

N. Roth, K. Anders, and A. Frohn, and , J. Laser Appl. 2, 37 (1990).
[CrossRef]

Meas. Sci. Technol. (1)

N. Damaschke, G. Gouesbet, G. Gréhan, and C. Tropea, Meas. Sci. Technol. 9, 137 (1998).
[CrossRef]

Trans. Cambridge Philos. Soc. (1)

G. B. Airy, Trans. Cambridge Philos. Soc. 6, 397 (1838).

Other (3)

R. Descartes, Discourseon Method, Optics, Geometry, and Meteorology, revised ed., translatedby P. J. Olscamp (Hackett,Indianapolis, Ind., 2001).

M. Born and E. Wolf, Principles of Optics (Pergamon,London, 1959).

H. C. vande Hulst, Light Scattering by Small Particles (Dover,New York, 1957).

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Figures (4)

Fig. 1
Fig. 1

Comparison of the light path inside a sphere with a uniform refractive index (solid curves) and the light path inside a sphere with a spherically symmetric refractive index n [ r ̃ ] (dashed curves).

Fig. 2
Fig. 2

Decomposition of the light path in a sphere with a spherically symmetric refractive-index gradient.

Fig. 3
Fig. 3

Airy patterns for a sphere with a parabolic refractive-index gradient (solid curve) and with uniform refractive index (dashed–dotted and dotted curves).

Fig. 4
Fig. 4

Comparison of the rainbow temperatures obtained with the finely stratified sphere model and with the generalized Airy theory. The quantity t t 0 represents the heating time.

Equations (9)

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θ = 2 τ 2 ( N 1 ) τ ,
d d s ( n d r d s ) = n .
χ r ̃ = c r ( n [ r ̃ ] 2 r ̃ 2 c 2 ) 1 2 ,
χ 1 = c [ r ̃ m 2 n s n c ( r ̃ m 2 r ̃ 1 2 ) 1 2 ] 1 i Π { 1 r ̃ 2 2 r ̃ m 2 , arcsin [ ( r ̃ m 2 r ̃ 2 r ̃ m 2 r ̃ 2 2 ) 1 2 ] , r ̃ 2 2 r ̃ m 2 r ̃ 1 2 r ̃ m 2 } ,
θ v = 2 τ 2 ( N 1 ) ( χ * [ r ̃ = 1 ] χ * [ r ̃ = r ̃ m ] ) .
u = 2 3 2 θ τ 2 τ = τ r g 1 sin 2 τ r g v 3 D 2 = h v 3 D 2 ,
I Ω ( z ) 2 = 0 cos 1 2 π ( z η η 3 ) d η 2 ,
z = ( θ v θ v r g ) ( 16 D 2 h λ 2 ) 1 3 ,
h = 2 3 2 θ v τ 2 τ = τ v r g 1 sin 2 τ v r g ,

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