Abstract

An experimental method is presented that detects whether a droplet is spherical. The method is based on a comparison between two droplet diameters deduced from two different optical interference patterns observed in a rainbow that is created by a droplet scattering laser light. Experimental validation has been carried out with a CCD camera. Once a rainbow pattern has been identified as coming from a spherical droplet, we can derive a reliable droplet velocity and diameter from the same interference patterns, using theories for the rainbow that are valid only for spherical droplets. Preliminary experiments have been carried out with a laser beam and a photomultiplier.

© 1996 Optical Society of America

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References

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  1. N. Roth, K. Anders, A. Frohn, “Simultaneous measurement of temperature and size of droplets in the micrometer range,” J. Laser Appl. 2, 37–42 (1990).
  2. S. V. Sankar, D. H. Buermann, W. D. Bachalo, “Simultaneous measurements of droplet size, velocity, and temperature in a swirl-stabilized spray flame,” in Proceedings of the Seventh International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon (1994), Vol. 1, pp. 12.31–12.39.
  3. J. P. A. J. van Beeck, M. L. Riethmuller, “Nonintrusive measurements of temperature and size of single falling raindrops,” Appl. Opt. 34, 1633–1639 (1995).
  4. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 4, pp. 82–101.
  5. E. Hecht, Optics (Addison-Wesley, Reading, Mass., 1987), Chap. 11, pp. 498–505.
  6. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 13, pp. 240–246.
  7. J. P. A. J. van Beeck, M. L. Riethmuller, “Détermination non-intrusive de la dimension et de la témperature des gouttes dans une pulvérisation,” in Recueil des actes du 4e Congrès Francophone de Vélocimétrie Laser (Laboratoire de Chimie Physique de la Combustion, Université de Poitiers-CNRS, 1994).
  8. J. R. Probert-Jones, “Surface waves in backscattering and the localization principle,”J. Opt. Soc. Am. 73, 503 (1983).
  9. J. A. Lock, “Cooperative effects among partial waves in Mie scattering,” J. Opt. Soc. Am. A 5, 2032–2044 (1988).
  10. H. C. van de Hulst, R. T. Wang, “Glare points,” Appl. Opt. 30, 4755–4763 (1991).
  11. J. A. Lock, “Theory of the observation made of high-order rainbows,” Appl. Opt. 26, 5291–5297 (1987).
  12. G. Gouesbet, B. Maheu, G. Gréhan, “Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation,” J. Opt. Soc. Am. A 5, 1427–1443 (1988).
  13. N. Roth, K. Anders, A. Frohn, “Simultaneous determination of refractive index and droplet size using Mie theory,” in Proceedings of the Sixth International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon (1992), pp. 15.5.1–15.5.5.
  14. J. P. A. J. van Beeck, M. L. Riethmuller, “Simultaneous determination of temperature and size of droplets from the rainbow using Airy theory,” in Proceedings of the Seventh International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon (1994), pp. 21.5.1–21.5.6.
  15. P. L. Marston, “Rainbow phenomena and the detection of nonsphericity in drops,” Appl. Opt. 19, 680–685 (1980).
  16. W. Möbius, “Zur Theorie des Regenbogens und ihrer experimentellen Prüfung,” Ann. Phys. (Leipzig) 33, 1493–1558 (1910).
  17. J. A. Lock, J. H. Andrews, “Optical caustics in natural phenomena,” Am. J. Phys. 60, 397–407 (1992).

1995 (1)

1992 (1)

J. A. Lock, J. H. Andrews, “Optical caustics in natural phenomena,” Am. J. Phys. 60, 397–407 (1992).

1991 (1)

1990 (1)

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

1988 (2)

1987 (1)

1983 (1)

1980 (1)

1910 (1)

W. Möbius, “Zur Theorie des Regenbogens und ihrer experimentellen Prüfung,” Ann. Phys. (Leipzig) 33, 1493–1558 (1910).

Anders, K.

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

N. Roth, K. Anders, A. Frohn, “Simultaneous determination of refractive index and droplet size using Mie theory,” in Proceedings of the Sixth International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon (1992), pp. 15.5.1–15.5.5.

Andrews, J. H.

J. A. Lock, J. H. Andrews, “Optical caustics in natural phenomena,” Am. J. Phys. 60, 397–407 (1992).

Bachalo, W. D.

S. V. Sankar, D. H. Buermann, W. D. Bachalo, “Simultaneous measurements of droplet size, velocity, and temperature in a swirl-stabilized spray flame,” in Proceedings of the Seventh International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon (1994), Vol. 1, pp. 12.31–12.39.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 4, pp. 82–101.

Buermann, D. H.

S. V. Sankar, D. H. Buermann, W. D. Bachalo, “Simultaneous measurements of droplet size, velocity, and temperature in a swirl-stabilized spray flame,” in Proceedings of the Seventh International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon (1994), Vol. 1, pp. 12.31–12.39.

Frohn, A.

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

N. Roth, K. Anders, A. Frohn, “Simultaneous determination of refractive index and droplet size using Mie theory,” in Proceedings of the Sixth International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon (1992), pp. 15.5.1–15.5.5.

Gouesbet, G.

Gréhan, G.

Hecht, E.

E. Hecht, Optics (Addison-Wesley, Reading, Mass., 1987), Chap. 11, pp. 498–505.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 4, pp. 82–101.

Lock, J. A.

Maheu, B.

Marston, P. L.

Möbius, W.

W. Möbius, “Zur Theorie des Regenbogens und ihrer experimentellen Prüfung,” Ann. Phys. (Leipzig) 33, 1493–1558 (1910).

Probert-Jones, J. R.

Riethmuller, M. L.

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

J. P. A. J. van Beeck, M. L. Riethmuller, “Détermination non-intrusive de la dimension et de la témperature des gouttes dans une pulvérisation,” in Recueil des actes du 4e Congrès Francophone de Vélocimétrie Laser (Laboratoire de Chimie Physique de la Combustion, Université de Poitiers-CNRS, 1994).

J. P. A. J. van Beeck, M. L. Riethmuller, “Simultaneous determination of temperature and size of droplets from the rainbow using Airy theory,” in Proceedings of the Seventh International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon (1994), pp. 21.5.1–21.5.6.

Roth, N.

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

N. Roth, K. Anders, A. Frohn, “Simultaneous determination of refractive index and droplet size using Mie theory,” in Proceedings of the Sixth International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon (1992), pp. 15.5.1–15.5.5.

Sankar, S. V.

S. V. Sankar, D. H. Buermann, W. D. Bachalo, “Simultaneous measurements of droplet size, velocity, and temperature in a swirl-stabilized spray flame,” in Proceedings of the Seventh International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon (1994), Vol. 1, pp. 12.31–12.39.

van Beeck, J. P. A. J.

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

J. P. A. J. van Beeck, M. L. Riethmuller, “Détermination non-intrusive de la dimension et de la témperature des gouttes dans une pulvérisation,” in Recueil des actes du 4e Congrès Francophone de Vélocimétrie Laser (Laboratoire de Chimie Physique de la Combustion, Université de Poitiers-CNRS, 1994).

J. P. A. J. van Beeck, M. L. Riethmuller, “Simultaneous determination of temperature and size of droplets from the rainbow using Airy theory,” in Proceedings of the Seventh International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon (1994), pp. 21.5.1–21.5.6.

van de Hulst, H. C.

H. C. van de Hulst, R. T. Wang, “Glare points,” Appl. Opt. 30, 4755–4763 (1991).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 13, pp. 240–246.

Wang, R. T.

Am. J. Phys. (1)

J. A. Lock, J. H. Andrews, “Optical caustics in natural phenomena,” Am. J. Phys. 60, 397–407 (1992).

Ann. Phys. (Leipzig) (1)

W. Möbius, “Zur Theorie des Regenbogens und ihrer experimentellen Prüfung,” Ann. Phys. (Leipzig) 33, 1493–1558 (1910).

Appl. Opt. (4)

J. Laser Appl. (1)

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

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Other (7)

S. V. Sankar, D. H. Buermann, W. D. Bachalo, “Simultaneous measurements of droplet size, velocity, and temperature in a swirl-stabilized spray flame,” in Proceedings of the Seventh International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon (1994), Vol. 1, pp. 12.31–12.39.

N. Roth, K. Anders, A. Frohn, “Simultaneous determination of refractive index and droplet size using Mie theory,” in Proceedings of the Sixth International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon (1992), pp. 15.5.1–15.5.5.

J. P. A. J. van Beeck, M. L. Riethmuller, “Simultaneous determination of temperature and size of droplets from the rainbow using Airy theory,” in Proceedings of the Seventh International Symposium on Applications of Laser Techniques to Fluid Mechanics, Lisbon (1994), pp. 21.5.1–21.5.6.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Chap. 4, pp. 82–101.

E. Hecht, Optics (Addison-Wesley, Reading, Mass., 1987), Chap. 11, pp. 498–505.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957), Chap. 13, pp. 240–246.

J. P. A. J. van Beeck, M. L. Riethmuller, “Détermination non-intrusive de la dimension et de la témperature des gouttes dans une pulvérisation,” in Recueil des actes du 4e Congrès Francophone de Vélocimétrie Laser (Laboratoire de Chimie Physique de la Combustion, Université de Poitiers-CNRS, 1994).

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Figures (18)

Fig. 1
Fig. 1

Supernumerary bows in the monochromatic first-order rainbow created by a nearly spherical droplet suspended from a small tube with its point ground flat. The camera is focused at infinity so that the pattern corresponds to that in the far field.

Fig. 2
Fig. 2

Far-field Lorenz-Mie-scattered light-intensity distribution resembling the first-order monochromatic rainbow. D = 1 mm, the wavelength of the incident plane wave is λ = 632.8 nm, and the refractive index is m = 1.33534 + 0i.

Fig. 3
Fig. 3

Magnitude squared of the Fourier transform of Fig. 2.

Fig. 4
Fig. 4

Main geometric rays contributing to the first-order rainbow.

Fig. 5
Fig. 5

Scattering problem with Gaussian illumination for which the first-order rainbow has been calculated by generalized Lorenz–Mie theory (see Fig. 6).

Fig. 6
Fig. 6

Rainbow pattern corresponding to the situation outlined in Fig. 5; D = 150 μm, λ = 632.8 nm, m = 1.33, the beamwaist radius is ω0 = 20 μm, and the center of the laser beam is 64 μm off-axis.

Fig. 7
Fig. 7

Magnitude squared of the Fourier transform in Fig. 6.

Fig. 8
Fig. 8

Severely distorted supernumerary bows; an intrusion of a cusp-shaped second-order rainbow into the first-order rainbow can be seen. The picture was taken by a camera, focused at infinity, pointed at a hanging nonspherical tear-shaped drop.

Fig. 9
Fig. 9

Angular frequencies related to the supernumerary bows [F 1, Eq. (2)] and to the ripple structure [F 3, Eq. (4)] as a function of the droplet diameter.

Fig. 10
Fig. 10

Outline of the experimental setup for the measurement of droplet temperature and diameter.

Fig. 11
Fig. 11

Rainbow signal coming from a droplet that is spherical, i.e., the droplet diameter D 1, derived from the distance between the supernumerary bows, equal to diameter D 3 that is derived from the ripple structure.

Fig. 12
Fig. 12

Magnitude squared of the Fourier transform of the rainbow in Fig. 11.

Fig. 13
Fig. 13

Rainbow signal coming from a nonspherical droplet, i.e., the droplet diameter D 1 [Eq. (2)] unequal to D 3 [Eq. (4)].

Fig. 14
Fig. 14

Magnitude squared of the Fourier transform of the rainbow in Fig. 13.

Fig. 15
Fig. 15

Ratio between the temporal frequencies dN 3/dt and dN 1/dt, which equals F 3/F 1 [Eqs. (2) and (4)]; this ratio depends only on the droplet diameter and not on the droplet velocity.

Fig. 16
Fig. 16

Outline of the experimental setup for the simultaneous measurement of droplet velocity and diameter.

Fig. 17
Fig. 17

Monochromatic rainbow in time, recorded with the aid of the setup depicted in Fig. 16.

Fig. 18
Fig. 18

Spectrum of Fig. 17.

Equations (10)

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F i = δ i λ π 180 .
F 1 = sin τ r g 2 . 3794 ( 16 D 2 λ 2 cos τ r g ) 1 / 3 π 180 with sin τ r g = ( m 2 1 3 ) 1 / 2 ,
F 2 = D 2 λ ( cos τ r g + cos θ r g 2 ) π 180 F 1 / 2 , with θ r g = 2 τ r g + 4 arccos ( cos τ r g m ) ,
F 3 = F 2 + F 1 .
F 4 = D λ π 180 .
g ( x ) = F { S 1 ( θ ) } .
i 1 ( θ ) = | S 1 ( θ ) | 2 = | G 1 ( θ ) | 2 ,
g ( x ) g * ( x ) = F 1 { | G ( θ ) | 2 } ,
| F { i 1 ( θ ) } | 2 = | F { | G 1 ( θ ) | 2 } | 2 = | F 1 { | G ( θ ) | 2 } | 2 = | g ( x ) g * ( x ) | 2 .
υ = d t l cos θ 0 π 180 = d N i d t 1 F i l cos θ 0 π 180 ,

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