Abstract

When a dielectric circular cylinder is obliquely illuminated, the scattering angle associated with the Airy caustics of the cylinder’s primary rainbow depends on the tilt of the cylinder. We display records of the scattering pattern for a transparent poly(methyl methacrylate) fiber ranging from small values of tilt through values of tilt that are sufficiently large for the Airy caustics from both sides of the fiber to merge in a meridional plane containing the incident wave vector and the fiber’s axis. The records are compared directly with the evolution of the caustic projected onto the observation plane, and certain qualitative features of the global evolution of the caustics are confirmed. Although the observations used laser illumination, they are relevant to anticipating the scattering by sunlit transparent tilted cylinders.

© 1998 Optical Society of America

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References

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  1. P. L. Marston, “Descartes glare points in scattering by icicles: color photographs and a tilted dielectric cylinder model of caustic and glare point evolution,” Appl. Opt. 37, 1551–1556 (1998).
    [CrossRef]
  2. J. Brandrup, E. H. Immergut, Polymer Handbook, 2nd ed. (Wiley Interscience, New York, 1975), pp. v/77–v/80.
  3. C. L. Adler, J. A. Lock, B. R. Stone, C. J. Garcia, “Higher-order interior caustics produced in scattering of diagonally incident plane wave by a circular cylinder,” J. Opt. Soc. Am. A 14, 1305–1315 (1997).
    [CrossRef]
  4. J. A. Lock, C. L. Adler, “Debye series analysis of the first-order rainbow produced in scattering of a diagonally incident plane wave by a circular cylinder,” J. Opt. Soc. Am. A 14, 1316–1328 (1997).
    [CrossRef]
  5. W. J. Humphreys, Physics of the Air (Dover, New York, 1964), pp. 476–506.
  6. R. A. R. Tricker, Introduction to Meteorological Optics (American Elsevier, New York, 1970).
  7. H. M. Presby, “Refractive index and diameter measurements of unclad optical fibers,” J. Opt. Soc. Am. 64, 280–284 (1974).
    [CrossRef]
  8. D. Marcuse, Principles of Optical Fiber Measurement (Academic, New York, 1981).
  9. C. M. Mount, P. L. Marston, “Glare points in the refracted-wave scattering by icicles and other tilted dielectric cylinders and the caustic-merging transition,” in Light and Color in the Open Air, Vol. 4 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 14–16.
  10. J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955).
    [CrossRef]
  11. P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” in Physical Acoustics, A. D. Pierce, R. N. Thurston, eds. (Academic, Boston, 1992), Vol. 21, pp. 1–234.
  12. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 199–201.
  13. D. W. Garvey, K. Zimmerman, P. Young, J. Tostenrude, J. S. Townsend, Z. Zhou, M. Lobel, M. Dayton, R. Wittorf, M. G. Kuzyk, J. Sounick, C. W. Dirk, “Single-mode nonlinear-optical polymer fibers,” J. Opt. Soc. Am. B 13, 2017–2023 (1996).
    [CrossRef]

1998 (1)

1997 (2)

1996 (1)

1974 (1)

1955 (1)

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955).
[CrossRef]

Adler, C. L.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 199–201.

Brandrup, J.

J. Brandrup, E. H. Immergut, Polymer Handbook, 2nd ed. (Wiley Interscience, New York, 1975), pp. v/77–v/80.

Dayton, M.

Dirk, C. W.

Garcia, C. J.

Garvey, D. W.

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 199–201.

Humphreys, W. J.

W. J. Humphreys, Physics of the Air (Dover, New York, 1964), pp. 476–506.

Immergut, E. H.

J. Brandrup, E. H. Immergut, Polymer Handbook, 2nd ed. (Wiley Interscience, New York, 1975), pp. v/77–v/80.

Kuzyk, M. G.

Lobel, M.

Lock, J. A.

Marcuse, D.

D. Marcuse, Principles of Optical Fiber Measurement (Academic, New York, 1981).

Marston, P. L.

P. L. Marston, “Descartes glare points in scattering by icicles: color photographs and a tilted dielectric cylinder model of caustic and glare point evolution,” Appl. Opt. 37, 1551–1556 (1998).
[CrossRef]

C. M. Mount, P. L. Marston, “Glare points in the refracted-wave scattering by icicles and other tilted dielectric cylinders and the caustic-merging transition,” in Light and Color in the Open Air, Vol. 4 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 14–16.

P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” in Physical Acoustics, A. D. Pierce, R. N. Thurston, eds. (Academic, Boston, 1992), Vol. 21, pp. 1–234.

Mount, C. M.

C. M. Mount, P. L. Marston, “Glare points in the refracted-wave scattering by icicles and other tilted dielectric cylinders and the caustic-merging transition,” in Light and Color in the Open Air, Vol. 4 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 14–16.

Presby, H. M.

Sounick, J.

Stone, B. R.

Tostenrude, J.

Townsend, J. S.

Tricker, R. A. R.

R. A. R. Tricker, Introduction to Meteorological Optics (American Elsevier, New York, 1970).

Wait, J. R.

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955).
[CrossRef]

Wittorf, R.

Young, P.

Zhou, Z.

Zimmerman, K.

Appl. Opt. (1)

Can. J. Phys. (1)

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195 (1955).
[CrossRef]

J. Opt. Soc. Am. (1)

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

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

Other (7)

P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” in Physical Acoustics, A. D. Pierce, R. N. Thurston, eds. (Academic, Boston, 1992), Vol. 21, pp. 1–234.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 199–201.

J. Brandrup, E. H. Immergut, Polymer Handbook, 2nd ed. (Wiley Interscience, New York, 1975), pp. v/77–v/80.

W. J. Humphreys, Physics of the Air (Dover, New York, 1964), pp. 476–506.

R. A. R. Tricker, Introduction to Meteorological Optics (American Elsevier, New York, 1970).

D. Marcuse, Principles of Optical Fiber Measurement (Academic, New York, 1981).

C. M. Mount, P. L. Marston, “Glare points in the refracted-wave scattering by icicles and other tilted dielectric cylinders and the caustic-merging transition,” in Light and Color in the Open Air, Vol. 4 of 1997 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1997), pp. 14–16.

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Figures (8)

Fig. 1
Fig. 1

Cylinder geometry considered for tilted illumination. In the meridional plane that contains the axis of the cylinder and the incident wave vector k i , the angle between the surface normal and k i is the tilt γ. For a ray incident upon an arbitrary point U, the projected angle of incidence ϕ (used in Appendix A) is less than the actual value i unless γ = 0.

Fig. 2
Fig. 2

Diagram of the apparatus used for projecting and recording the scattering patterns. The screen is viewed by a video camera located above and to the left of the goniometer.

Fig. 3
Fig. 3

Image of the apparatus obtained with a video camera offset from the one used to record the scattering. The goniometer is on the lower left, and the projection screen covers most of the right-hand side of the image.

Fig. 4
Fig. 4

Sequence of scattering patterns recorded at 1° intervals for tilt angles γ of 28.5°–51.5°. The pattern is partially blocked on the lower left (see text).

Fig. 5
Fig. 5

Sequence as in Fig. 4 but for γ from 9.6 to 33.6°.

Fig. 6
Fig. 6

Sequence of superposed images from Fig. 4 but with the contrast reversed so that bright regions appear dark. The cusp-shaped curve is the projected location of the Airy caustic for the corresponding range of tilts beginning at 28.5° up through the merging of the caustics in the meridional plane at γ c = 50.4°.

Fig. 7
Fig. 7

Sequence of records from Fig. 5 but with the contrast reversed and the computed caustic location superposed, as explained for Fig. 6 (see text).

Fig. 8
Fig. 8

(a) Laboratory-fixed coordinates used in the calculation of the intersection of the caustic with the projection screen. The point (x, y) is in the plane of the screen. (b) Fiber-fixed coordinates also show the incident and the scattered wave vectors in the meridional plane. The meridional plane is the y′ z′ plane in (b) and in the yz plane in (a).

Equations (8)

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γ c = arccos n 2 - 1 / 3 1 / 2 ,
n γ ,   n = n 2 - sin 2   γ 1 / 2 / cos   γ n .
x 2 = D p y c - y 3
θ = 180 ° + 2 ϕ - 2 ϕ ,
cos   δ = cos 180 ° - θ cos   γ .
cos   β = cos   β   cos   ψ + cos   δ   sin   ψ ,
cos   δ = - cos   β   sin   ψ + cos   δ   cos   ψ .
cos 2   ϕ D = n 2 - 1 / 3 .

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