Abstract

A ray-theoretic account of the passage of light through a radially inhomogeneous transparent sphere has been used to establish the existence of multiple primary rainbows for some refractive index profiles. The existence of such additional bows is a consequence of a sufficiently attractive potential in the interior of the drop, i.e., the refractive index gradient should be sufficiently negative there. The profiles for which this gradient is monotonically increasing do not result in this phenomenon, but nonmonotone profiles can do so, depending on the form of n. Sufficiently oscillatory profiles can lead to apparently singular behavior in the deviation angle (within the geometrical optics approximation) as well as multiple rainbows. These results also apply to systems with circular cylindrical cross sections, and may be of value in the field of rainbow refractometry.

© 2007 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  23. M. R. Vetrano, J. P. A. J. van Beeck, and M. L. Riethmuller, "Assessment of refractive index gradients by standard rainbow thermometry," Appl. Opt. 44, 7275-7281 (2005).
    [CrossRef]
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2005 (2)

2003 (1)

2002 (1)

2001 (1)

1999 (1)

1998 (3)

1997 (1)

1996 (3)

1995 (1)

1994 (2)

L. Kai, P. Massoli, and A. D'Alessio, "Some far-field scattering characteristics of radially inhomogeneous particles," Part. Part. Syst. Charact. 11, 385-390 (1994).
[CrossRef]

M. Schneider and E. D. Hirleman, "Influence of internal refractive index gradients on size measurements of spherically symmetric particles by phase Doppler anemometry," Appl. Opt. 33, 2379-2388 (1994).

1980 (1)

1977 (1)

1975 (2)

1974 (1)

Adler, C. L.

Alexopoulos, N. G.

Anders, K.

K. Anders, N. Roth, and A. Frohn, "Influence of refractive index gradients within droplets on rainbow position and implications for rainbow refractometry," Part. Part. Syst. Charact. 13, 125-129 (1996).
[CrossRef]

N. Roth, K. Anders, and A. Frohn, "Size insensitive rainbow refractometry: theoretical aspects," presented at the Eighth International Symposium on Applications of Laser Techniques to Fluid Mechanics (Lisbon, 1996).

Born, M.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 2002).

Brockman, C. L.

Chan, C. W.

Chigier, N.

Choudury, D. Q.

M. Schneider, E. D. Hirleman, H. Salaheen, D. Q. Choudury, and S. C. Hill, "Rainbows and radially inhomogeneous droplets," in Proceedings of the Third International Congress on Optical Particle Sizing, M. Maeda, ed. (Yokohama, 1993), pp. 323-326.

D'Alessio, A.

L. Kai, P. Massoli, and A. D'Alessio, "Some far-field scattering characteristics of radially inhomogeneous particles," Part. Part. Syst. Charact. 11, 385-390 (1994).
[CrossRef]

Frohn, A.

K. Anders, N. Roth, and A. Frohn, "Influence of refractive index gradients within droplets on rainbow position and implications for rainbow refractometry," Part. Part. Syst. Charact. 13, 125-129 (1996).
[CrossRef]

N. Roth, K. Anders, and A. Frohn, "Size insensitive rainbow refractometry: theoretical aspects," presented at the Eighth International Symposium on Applications of Laser Techniques to Fluid Mechanics (Lisbon, 1996).

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980).

Hattori, H.

Hickok, W.

Hill, S. C.

M. Schneider, E. D. Hirleman, H. Salaheen, D. Q. Choudury, and S. C. Hill, "Rainbows and radially inhomogeneous droplets," in Proceedings of the Third International Congress on Optical Particle Sizing, M. Maeda, ed. (Yokohama, 1993), pp. 323-326.

Hirleman, E. D.

M. Schneider and E. D. Hirleman, "Influence of internal refractive index gradients on size measurements of spherically symmetric particles by phase Doppler anemometry," Appl. Opt. 33, 2379-2388 (1994).

M. Schneider, E. D. Hirleman, H. Salaheen, D. Q. Choudury, and S. C. Hill, "Rainbows and radially inhomogeneous droplets," in Proceedings of the Third International Congress on Optical Particle Sizing, M. Maeda, ed. (Yokohama, 1993), pp. 323-326.

Hom, J.

Kagawa, K.

Kai, L.

L. Kai, P. Massoli, and A. D'Alessio, "Some far-field scattering characteristics of radially inhomogeneous particles," Part. Part. Syst. Charact. 11, 385-390 (1994).
[CrossRef]

Kakui, K.

Kurniawan, H.

Lee, W. K.

Lit, J. W. Y.

Lock, J. A.

Luneberg, R. K.

R. K. Luneberg, Mathematical Theory of Optics (U. California Press, 1966).

Marcuse, D.

Marston, P. L.

Massoli, P.

P. Massoli, "Rainbow refractometry applied to radially inhomogeneous spheres: the critical case of evaporating droplets," Appl. Opt. 37, 3227-3235 (1998).

L. Kai, P. Massoli, and A. D'Alessio, "Some far-field scattering characteristics of radially inhomogeneous particles," Part. Part. Syst. Charact. 11, 385-390 (1994).
[CrossRef]

P. Massoli, "Temperature and size of droplets inferred by light scattering methods: a theoretical analysis of the influence of internal inhomogeneities," presented at the 13th Annual Conference on Liquid Atomization and Spray Systems (ILASS- Europe, Florence, 1997).

Nash, J.

Phipps, D.

Presby, H. M.

Rafferty, I. P.

Riethmuller, M. L.

Roth, N.

K. Anders, N. Roth, and A. Frohn, "Influence of refractive index gradients within droplets on rainbow position and implications for rainbow refractometry," Part. Part. Syst. Charact. 13, 125-129 (1996).
[CrossRef]

N. Roth, K. Anders, and A. Frohn, "Size insensitive rainbow refractometry: theoretical aspects," presented at the Eighth International Symposium on Applications of Laser Techniques to Fluid Mechanics (Lisbon, 1996).

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980).

Salaheen, H.

M. Schneider, E. D. Hirleman, H. Salaheen, D. Q. Choudury, and S. C. Hill, "Rainbows and radially inhomogeneous droplets," in Proceedings of the Third International Congress on Optical Particle Sizing, M. Maeda, ed. (Yokohama, 1993), pp. 323-326.

Saunders, K.

Schneider, M.

M. Schneider and E. D. Hirleman, "Influence of internal refractive index gradients on size measurements of spherically symmetric particles by phase Doppler anemometry," Appl. Opt. 33, 2379-2388 (1994).

M. Schneider, E. D. Hirleman, H. Salaheen, D. Q. Choudury, and S. C. Hill, "Rainbows and radially inhomogeneous droplets," in Proceedings of the Third International Congress on Optical Particle Sizing, M. Maeda, ed. (Yokohama, 1993), pp. 323-326.

Stone, B. R.

van Beeck, J. P. A. J.

Vetrano, M. R.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 2002).

Yamanaka, H.

Yokoi, S.

Appl. Opt. (15)

H. Hattori, H. Yamanaka, H. Kurniawan, S. Yokoi, and K. Kagawa, "Using minimum deviation of a secondary rainbow and its application to water analysis in a high-precision refractive-index comparator for liquids," Appl. Opt. 36, 5552-5556 (1997).

H. Hattori, K. Kakui, H. Kurniawan, and K. Kagawa, "Liquid refractometry by the rainbow method," Appl. Opt. 37, 4123-4129 (1998).

H. Hattori, "Simulation study on refractometry by the rainbow method," Appl. Opt. 38, 4037-4046 (1999).

J. Hom and N. Chigier, "Rainbow refractometry: simultaneous measurement of temperature, refractive index, and size of droplets," Appl. Opt. 41, 1899-1907 (2002).

C. L. Adler, J. A. Lock, and B. R. Stone, "Rainbow scattering by a cylinder with a nearly elliptical cross section," Appl. Opt. 37, 1540-1550 (1998).

C. L. Adler, J. A. Lock, D. Phipps, K. Saunders, and J. Nash, "Supernumerary spacings of rainbows produced by an elliptical cross-section cylinder. II: Experiment," Appl. Opt. 40, 2535-2545 (2001).

P. Massoli, "Rainbow refractometry applied to radially inhomogeneous spheres: the critical case of evaporating droplets," Appl. Opt. 37, 3227-3235 (1998).

M. Schneider and E. D. Hirleman, "Influence of internal refractive index gradients on size measurements of spherically symmetric particles by phase Doppler anemometry," Appl. Opt. 33, 2379-2388 (1994).

P. L. Marston, "Rainbow phenomena and the detection of nonsphericity in drops," Appl. Opt. 19, 680-685 (1980).

J. P. A. J. van Beeck and 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).

J. P. A. J. van Beeck and M. L. Riethmuller, "Nonintrusive measurements of temperature and size of single falling raindrops," Appl. Opt. 34, 1633-1639 (1995).

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

M. R. Vetrano, J. P. A. J. van Beeck, and M. L. Riethmuller, "Assessment of refractive index gradients by standard rainbow thermometry," Appl. Opt. 44, 7275-7281 (2005).
[CrossRef]

D. Marcuse, "Light scattered from unclad fibers: ray theory," Appl. Opt. 14, 1528-1532 (1975).

C. L. Brockman and N. G. Alexopoulos, "Geometrical optics of inhomogeneous particles: glory ray and the rainbow revisited," Appl. Opt. 16, 166-174 (1977).

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. B (1)

Opt. Lett. (1)

Part. Part. Syst. Charact. (2)

L. Kai, P. Massoli, and A. D'Alessio, "Some far-field scattering characteristics of radially inhomogeneous particles," Part. Part. Syst. Charact. 11, 385-390 (1994).
[CrossRef]

K. Anders, N. Roth, and A. Frohn, "Influence of refractive index gradients within droplets on rainbow position and implications for rainbow refractometry," Part. Part. Syst. Charact. 13, 125-129 (1996).
[CrossRef]

Other (6)

N. Roth, K. Anders, and A. Frohn, "Size insensitive rainbow refractometry: theoretical aspects," presented at the Eighth International Symposium on Applications of Laser Techniques to Fluid Mechanics (Lisbon, 1996).

M. Schneider, E. D. Hirleman, H. Salaheen, D. Q. Choudury, and S. C. Hill, "Rainbows and radially inhomogeneous droplets," in Proceedings of the Third International Congress on Optical Particle Sizing, M. Maeda, ed. (Yokohama, 1993), pp. 323-326.

P. Massoli, "Temperature and size of droplets inferred by light scattering methods: a theoretical analysis of the influence of internal inhomogeneities," presented at the 13th Annual Conference on Liquid Atomization and Spray Systems (ILASS- Europe, Florence, 1997).

M. Born and E. Wolf, Principles of Optics (Cambridge U. Press, 2002).

R. K. Luneberg, Mathematical Theory of Optics (U. California Press, 1966).

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products (Academic, 1980).

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