Abstract

Glare points associated with the Airy caustics of once and twice internally reflected rays are visible in the scattering by sunlit icicles. Supporting color photographs include an image of the far-field scattering. Relevant rays are analogous to the Descartes rays of primary and secondary rainbows of drops; however, the caustic conditions for the icicle are predicted to be affected by tilt of the illumination relative to the axis of the icicle. A model for the caustic evolution, given for a circular dielectric cylinder, manifests a transition in which the Airy caustic (and associated glare points) merge in the meridional plane at a critical tilt. At this critical tilt the merged glare point is predicted to be very bright. The calculations use the Bravais effective refractive index and generalized ray tracing.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. B. Airy, “On the intensity of light in the neighborhood of a caustic,” Trans. Cambridge Philos. Soc. 6, 397–403 and plate 7 (1838) (reprinted in Ref. 2 below).
  2. P. L. Marston, ed. Geometrical Aspects of Scattering, Vol. MS89 of SPIE Milestone Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1994).
  3. H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, Cambridge, 1992).
    [CrossRef]
  4. 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.
  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. D. K. Lynch, W. Livingston, Color and Light in Nature (Cambridge U. Press, Cambridge, 1995).
  8. Y. Takano, M. Tanaka, “Phase matrix and cross sections for single scattering by circular cylinders: a comparison of ray optics and wave theory,” Appl. Opt. 19, 2781–2793 (1980).
    [CrossRef] [PubMed]
  9. H. C. van de Hulst, R. T. Wang, “Glare points,” Appl. Opt. 30, 4755–4763 (1991).
    [CrossRef] [PubMed]
  10. C. B. Boyer, The Rainbow from Myth to Mathematics (Princeton U. Press, Princeton, N.J., 1987).
  11. P. O. Pedersen, “On the surface-tension of liquids investigated by the method of jet vibration,” Philos. Trans. R. Soc. London Ser. A 207B, 341–392 (1908).
  12. S. F. Morse, D. B. Thiessen, P. L. Marston, “Capillary bridge modes driven with modulated ultrasonic radiation pressure,” Phys. Fluids 8, 3–5 (1996).
    [CrossRef]
  13. 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]
  14. 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]
  15. C. M. Mount, D. B. Thiessen, P. L. Marston, “Scattering observations for tilted transparent fibers: evolution of Airy caustics with cylinder tilt and the caustic-merging transaction,” Appl. Opt. 37, 1534–1539 (1998).
    [CrossRef]
  16. O. N. Stravroudis, “Simpler derivation of the formulas for generalized ray tracing,” J. Opt. Soc. Am. 66, 1330–1333 (1976).
    [CrossRef]
  17. C. E. Dean, P. L. Marston, “Opening rate of the transverse cusp diffraction catastrophe in light scattered by oblate spheroidal drops,” Appl. Opt. 30, 3443–3451 (1991).
    [CrossRef] [PubMed]

1998

1997

1996

S. F. Morse, D. B. Thiessen, P. L. Marston, “Capillary bridge modes driven with modulated ultrasonic radiation pressure,” Phys. Fluids 8, 3–5 (1996).
[CrossRef]

1991

1980

1976

1908

P. O. Pedersen, “On the surface-tension of liquids investigated by the method of jet vibration,” Philos. Trans. R. Soc. London Ser. A 207B, 341–392 (1908).

1838

G. B. Airy, “On the intensity of light in the neighborhood of a caustic,” Trans. Cambridge Philos. Soc. 6, 397–403 and plate 7 (1838) (reprinted in Ref. 2 below).

Adler, C. L.

Airy, G. B.

G. B. Airy, “On the intensity of light in the neighborhood of a caustic,” Trans. Cambridge Philos. Soc. 6, 397–403 and plate 7 (1838) (reprinted in Ref. 2 below).

Boyer, C. B.

C. B. Boyer, The Rainbow from Myth to Mathematics (Princeton U. Press, Princeton, N.J., 1987).

Dean, C. E.

Garcia, C. J.

Humphreys, W. J.

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

Livingston, W.

D. K. Lynch, W. Livingston, Color and Light in Nature (Cambridge U. Press, Cambridge, 1995).

Lock, J. A.

Lynch, D. K.

D. K. Lynch, W. Livingston, Color and Light in Nature (Cambridge U. Press, Cambridge, 1995).

Marston, P. L.

C. M. Mount, D. B. Thiessen, P. L. Marston, “Scattering observations for tilted transparent fibers: evolution of Airy caustics with cylinder tilt and the caustic-merging transaction,” Appl. Opt. 37, 1534–1539 (1998).
[CrossRef]

S. F. Morse, D. B. Thiessen, P. L. Marston, “Capillary bridge modes driven with modulated ultrasonic radiation pressure,” Phys. Fluids 8, 3–5 (1996).
[CrossRef]

C. E. Dean, P. L. Marston, “Opening rate of the transverse cusp diffraction catastrophe in light scattered by oblate spheroidal drops,” Appl. Opt. 30, 3443–3451 (1991).
[CrossRef] [PubMed]

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.

Morse, S. F.

S. F. Morse, D. B. Thiessen, P. L. Marston, “Capillary bridge modes driven with modulated ultrasonic radiation pressure,” Phys. Fluids 8, 3–5 (1996).
[CrossRef]

Mount, C. M.

Nussenzveig, H. M.

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, Cambridge, 1992).
[CrossRef]

Pedersen, P. O.

P. O. Pedersen, “On the surface-tension of liquids investigated by the method of jet vibration,” Philos. Trans. R. Soc. London Ser. A 207B, 341–392 (1908).

Stone, B. R.

Stravroudis, O. N.

Takano, Y.

Tanaka, M.

Thiessen, D. B.

Tricker, R. A. R.

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

van de Hulst, H. C.

Wang, R. T.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Philos. Trans. R. Soc. London Ser. A

P. O. Pedersen, “On the surface-tension of liquids investigated by the method of jet vibration,” Philos. Trans. R. Soc. London Ser. A 207B, 341–392 (1908).

Phys. Fluids

S. F. Morse, D. B. Thiessen, P. L. Marston, “Capillary bridge modes driven with modulated ultrasonic radiation pressure,” Phys. Fluids 8, 3–5 (1996).
[CrossRef]

Trans. Cambridge Philos. Soc.

G. B. Airy, “On the intensity of light in the neighborhood of a caustic,” Trans. Cambridge Philos. Soc. 6, 397–403 and plate 7 (1838) (reprinted in Ref. 2 below).

Other

P. L. Marston, ed. Geometrical Aspects of Scattering, Vol. MS89 of SPIE Milestone Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1994).

H. M. Nussenzveig, Diffraction Effects in Semiclassical Scattering (Cambridge U. Press, Cambridge, 1992).
[CrossRef]

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.

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. K. Lynch, W. Livingston, Color and Light in Nature (Cambridge U. Press, Cambridge, 1995).

C. B. Boyer, The Rainbow from Myth to Mathematics (Princeton U. Press, Princeton, N.J., 1987).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Geometry considered in the circular cylinder approximation for the refraction and the reflection of light by an icicle. The z axis of the cylinder is taken to be vertical, as is the axis of the icicle. The illumination is incident with a wave vector that is tilted by an angle γ relative to a horizontal base plane. Although the actual angle of incidence is i for an illuminated surface point U (where the local cylinder normal is N), the angle of incidence projected on the base plane is ϕ.

Fig. 2
Fig. 2

Form of the projected refracted, internally reflected, and scattered rays in the base plane for the usual situation in which the tilt γ is less than the critical value given by Eq. (A3). The projected angles of incidence and refraction are ϕ and ϕ′, and the axis of the cylinder is at C. A virtual wave front is shown that is defined to be orthogonal to the extrapolated projection of the outgoing rays. The wave front is locally flat when the projected scattering angle θ has the Descartes value θ D , which is the solid line.

Fig. 3
Fig. 3

Calculated evolution of the Descartes ray projections for a circular cylinder as a function of tilt γ for ice (solid curves) and for poly(methyl methacrylate) (PMMA) plastic (dashed curve). See Section 2 and Appendix A, in which the application to plastic fibers is discussed. The circle is taken from a calculation for ice given in Ref. 8.

Fig. 4
Fig. 4

Primary and secondary Descartes rays projected onto the base plane for a circular ice cylinder with n = 1.31 and illuminated at γ = 20°. As shown for this case in Fig. 3, the Descartes ray scattering angle θ D for p = 2 exceeds θ D for p = 3. The projected specular reflection and bisector of the cylinder are shown from the perspective of a distant observer at the θ D of the primary rainbow. The meridional plane that contains the cylinder’s axis and the incident wave vector is shown. Glare points associated with specular reflection were not as bright as those for the Descartes rays for situations in which the glare points from Descartes rays were visible.

Fig. 5
Fig. 5

(a) Descartes glare point for the primary Airy caustic of an icicle is visible as the bright patch on the right-hand side of the icicle near its center. The weak glare point visible on the left-hand side is for a twice-reflected and twice-refracted ray. (b) The same icicle viewed from the other side of the meridional plane shows a line of primary Descartes glare points on its left-hand side with a line of specular glints to the right of an imaginary center line (shown as the cylinder’s bisector in Fig. 4). The lower of the two very bright glare points has a green hue. (c) Perspective as in (b) but with the camera’s focal plane conjugate to the far-field scattering. The glare points are now out of focus, and some of the Descartes glare points appear to be associated with colored regions.

Fig. 6
Fig. 6

Dashed curve shows the projected scattering angle 180° - θ D of the caustic for the region around the CMT from Eq. (A3) and is plotted as a function of tilt angle as in Fig. 3 for the case of n = 1.49. The solid curve is the cubic cusp fit from Eq. (A6). The caustics from both sides of the meridional plane are shown.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

n γ ,   n = n 2 - sin 2   γ 1 / 2 / cos   γ n .
n   sin   ϕ = sin   ϕ ,
θ = 180 °   m ± 2 ϕ - p ϕ ,   0 θ 180 ° ,
cos 2   ϕ D = n 2 - 1 / p - 1 p + 1 .
ρ f = a   cos 2   ϕ / cos   ϕ - n - 1 cos   ϕ .
h = 2 a   cos   ϕ .
γ c = arccos n 2 - 1 / 3 1 / 2 ,
d = 2 a cos   r = 2 na n 2 - sin 2   γ c 1 / 2 .
ρ p - 1 = cos   r - n - 1 cos   γ c / a .
180 ° - θ D 2 = D γ c - γ 3 ,

Metrics