Abstract

The intensity of high-order rainbows for normally incident light and certain rotation angles of a cylinder with an elliptical cross section is greatly amplified with respect to the intensity for a circular cross-sectional cylinder. The amplification is due to a number of the internal reflections occurring past the critical angle for total internal reflection, and the effect is especially strong for odd-order rainbows, beginning with the third order. Experimentally, the fourth- and the fifth-order rainbows of a nearly elliptical cross-sectional glass rod were observed and analyzed.

© 1998 Optical Society of America

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References

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  1. J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
    [CrossRef]
  2. J. A. Lock, “Theory of the observations made of high-order rainbows from a single water droplet,” Appl. Opt. 26, 5291–5298 (1987).
    [CrossRef] [PubMed]
  3. C. W. Chan, W. K. Lee, “Measurement of a liquid refractive index by using high-order rainbows,” J. Opt. Soc. Am. B 13, 532–535 (1996).
    [CrossRef]
  4. J. Walker, “How to create and observe a dozen rainbows in a single drop of water,” Sci. Am. 237(1), 138–144 (1977).
    [CrossRef]
  5. J. A. Lock, “Contribution of high-order rainbows to the scattering of a Gaussian laser beam by a spherical particle,” J. Opt. Soc. Am. A 10, 693–706 (1993).
    [CrossRef]
  6. C. L. Adler, J. A. Lock, B. R. Stone, “Rainbow scattering by a cylinder with a nearly elliptical cross section,” Appl. Opt. 37, 1540–1550 (1998).
    [CrossRef]
  7. 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]
  8. C. L. Adler, J. A. Lock, B. R. Stone, C. J. Garcia, “High-order interior caustics produced in scattering of a diagonally incident plane wave by a circular cylinder,” J. Opt. Soc. Am. A 14, 1305–1315 (1997).
    [CrossRef]
  9. R. T. Wang, H. C. van de Hulst, “Rainbows: Mie computations and the Airy theory,” Appl. Opt. 30, 106–117 (1991).
    [CrossRef] [PubMed]
  10. E. A. Hovenac, J. A. Lock, “Assessing the contributions of surface waves and complex rays to far-field Mie scattering by use of the Debye series,” J. Opt. Soc. Am. A 9, 781–795 (1992).
    [CrossRef]
  11. P. L. Marston, “Geometrical and catastrophe methods in scattering,” Phys. Acoust. 21, 1–234 (1992).
  12. J. A. Lock, “Ray scattering by an arbitrarily oriented spheroid. I. Diffraction and specular reflection,” Appl. Opt. 35, 500–514 (1996).
    [CrossRef] [PubMed]
  13. H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
    [CrossRef]
  14. J. J. D’Amico, M. D. Knudson, D. S. Langley, “Rainbow-enhanced forward glory from fused-silica spheres,” Appl. Opt. 33, 4672–4676 (1994).
    [CrossRef] [PubMed]
  15. J. U. Nöckel, A. D. Stone, R. K. Chang, “Q spoiling and directionality in deformed ring cavities,” Opt. Lett. 19, 1693–1695 (1994).
    [CrossRef] [PubMed]
  16. J. U. Nöckel, A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature (London) 385, 45–47 (1997).
    [CrossRef]
  17. J. U. Nöckel, A. D. Stone, “Chaotic light: a theory of asymmetric resonant cavities,” in Optical Processes in Microcavities, R. K. Chang, A. J. Campillo, eds. (World Scientific, Singapore, 1996), pp. 389–426.
    [CrossRef]
  18. H. R. Pruppacher, R. L. Pitter, “A semi-empirical determination of the shape of cloud and rain drops,” J. Atmos. Sci. 28, 86–94 (1971).
    [CrossRef]
  19. D. E. Pedgley, “A tertiary rainbow,” Weather 41, 401 (1986).

1998

1997

1996

1994

1993

1992

1991

1987

1986

D. E. Pedgley, “A tertiary rainbow,” Weather 41, 401 (1986).

1977

J. Walker, “How to create and observe a dozen rainbows in a single drop of water,” Sci. Am. 237(1), 138–144 (1977).
[CrossRef]

1976

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

1971

H. R. Pruppacher, R. L. Pitter, “A semi-empirical determination of the shape of cloud and rain drops,” J. Atmos. Sci. 28, 86–94 (1971).
[CrossRef]

1969

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

Adler, C. L.

Chan, C. W.

Chang, R. K.

D’Amico, J. J.

Garcia, C. J.

Hovenac, E. A.

Knudson, M. D.

Langley, D. S.

Lee, W. K.

Lock, J. A.

Marston, P. L.

P. L. Marston, “Geometrical and catastrophe methods in scattering,” Phys. Acoust. 21, 1–234 (1992).

Nöckel, J. U.

J. U. Nöckel, A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature (London) 385, 45–47 (1997).
[CrossRef]

J. U. Nöckel, A. D. Stone, R. K. Chang, “Q spoiling and directionality in deformed ring cavities,” Opt. Lett. 19, 1693–1695 (1994).
[CrossRef] [PubMed]

J. U. Nöckel, A. D. Stone, “Chaotic light: a theory of asymmetric resonant cavities,” in Optical Processes in Microcavities, R. K. Chang, A. J. Campillo, eds. (World Scientific, Singapore, 1996), pp. 389–426.
[CrossRef]

Nussenzveig, H. M.

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

Pedgley, D. E.

D. E. Pedgley, “A tertiary rainbow,” Weather 41, 401 (1986).

Pitter, R. L.

H. R. Pruppacher, R. L. Pitter, “A semi-empirical determination of the shape of cloud and rain drops,” J. Atmos. Sci. 28, 86–94 (1971).
[CrossRef]

Pruppacher, H. R.

H. R. Pruppacher, R. L. Pitter, “A semi-empirical determination of the shape of cloud and rain drops,” J. Atmos. Sci. 28, 86–94 (1971).
[CrossRef]

Stone, A. D.

J. U. Nöckel, A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature (London) 385, 45–47 (1997).
[CrossRef]

J. U. Nöckel, A. D. Stone, R. K. Chang, “Q spoiling and directionality in deformed ring cavities,” Opt. Lett. 19, 1693–1695 (1994).
[CrossRef] [PubMed]

J. U. Nöckel, A. D. Stone, “Chaotic light: a theory of asymmetric resonant cavities,” in Optical Processes in Microcavities, R. K. Chang, A. J. Campillo, eds. (World Scientific, Singapore, 1996), pp. 389–426.
[CrossRef]

Stone, B. R.

van de Hulst, H. C.

Walker, J.

J. Walker, “How to create and observe a dozen rainbows in a single drop of water,” Sci. Am. 237(1), 138–144 (1977).
[CrossRef]

Walker, J. D.

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

Wang, R. T.

Am. J. Phys.

J. D. Walker, “Multiple rainbows from single drops of water and other liquids,” Am. J. Phys. 44, 421–433 (1976).
[CrossRef]

Appl. Opt.

J. Atmos. Sci.

H. R. Pruppacher, R. L. Pitter, “A semi-empirical determination of the shape of cloud and rain drops,” J. Atmos. Sci. 28, 86–94 (1971).
[CrossRef]

J. Math. Phys.

H. M. Nussenzveig, “High-frequency scattering by a transparent sphere. I. Direct reflection and transmission,” J. Math. Phys. 10, 82–124 (1969).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Nature (London)

J. U. Nöckel, A. D. Stone, “Ray and wave chaos in asymmetric resonant optical cavities,” Nature (London) 385, 45–47 (1997).
[CrossRef]

Opt. Lett.

Phys. Acoust.

P. L. Marston, “Geometrical and catastrophe methods in scattering,” Phys. Acoust. 21, 1–234 (1992).

Sci. Am.

J. Walker, “How to create and observe a dozen rainbows in a single drop of water,” Sci. Am. 237(1), 138–144 (1977).
[CrossRef]

Weather

D. E. Pedgley, “A tertiary rainbow,” Weather 41, 401 (1986).

Other

J. U. Nöckel, A. D. Stone, “Chaotic light: a theory of asymmetric resonant cavities,” in Optical Processes in Microcavities, R. K. Chang, A. J. Campillo, eds. (World Scientific, Singapore, 1996), pp. 389–426.
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Rainbow angle θ6 R and the rainbow Fresnel coefficient factor I 6 as defined in Eq. (10) for the P = 6 rainbow of an elliptical-cross-section cylinder as a function of the cylinder rotation angle ξ for (a) b/ a = 1.0, (b) b/ a = 0.987, (c) b/ a = 0.975, (d) b/ a = 0.963, (e) b/ a = 0.950.

Fig. 2
Fig. 2

Rainbow Fresnel coefficient factor as defined in Eq. (10) and the values of the individual TE Fresnel coefficients for the P = 6 rainbow of an elliptical-cross-section cylinder for the 0 ≤ p ≤ 6 interactions of the rainbow ray with the cylinder surface. The p = 0 curve is the initial transmission coefficient, the 1 ≤ p ≤ 5 curves are the internal reflection coefficients, and the p = 6 curve is the final transmission coefficient minus one.

Fig. 3
Fig. 3

Ray path of the rainbow ray for 2 ≤ P ≤ 8 and n = 1.474 at the cylinder rotation angle corresponding to the maximum rainbow Fresnel coefficient amplification given in Table 2. The large filled circles represent total internal reflection.

Fig. 4
Fig. 4

Rainbow angle θ6 R and the rainbow Fresnel coefficient factor I 6 as defined in Eq. (10) for the P = 6 rainbow as a function of the cylinder rotation angle ξ for an elliptical cross section with b/ a = 0.963 (solid curve) and a two-half-ellipse cross section with b 1/a = 0.950 and b 2/a = 0.976 (dashed curve).

Fig. 5
Fig. 5

Rainbow angle θ P R and the rainbow Fresnel coefficient factor I P as defined in Eq. (10) for the P = 3 (solid curve) and P = 5 (dashed curve) rainbows as a function of the cylinder rotation angle ξ for a two-half-ellipse cross section with b 1/a = 0.950 and b 2/a = 0.976. The two rainbows are produced by rays incident at opposite sides of the cylinder.

Fig. 6
Fig. 6

Experimental light intensity in the vicinity of the P = 3 rainbow at θ3 R ≈ 270°. The distance from the glass rod to the detector was 49.5 cm.

Fig. 7
Fig. 7

Experimental light intensity in the vicinity of the P = 5 rainbow at θ5 R ≈ 275°. The distance from the glass rod to the detector was 49.5 cm.

Fig. 8
Fig. 8

Experimental light intensity in the vicinity of the P = 6 rainbow at θ6 R ≈ 175°. The distance from the glass rod to the detector was 73.5 cm.

Tables (2)

Tables Icon

Table 1 Maximum Fresnel Coefficient Contribution to the Rainbow Intensity for an Elliptical Cross-Sectional Cylinder, the Cylinder Rotation Angle ξ at which it occurs, and the Fresnel Coefficient Amplification Ratio with respect to a Circular Cross-Sectional Cylinder for the 2 ≤ P ≤ 8 Rainbows and b/a = 0.975

Tables Icon

Table 2 Maximum Fresnel Coefficient Contribution to the Rainbow Intensity for an Elliptical Cross-Sectional Cylinder, the Cylinder Rotation Angle ξ at which it occurs, and the Fresnel Coefficient Amplification Ratio with respect to a Circular Cross-Sectional Cylinder for the 2 ≤ P ≤ 8 Rainbows and b/a = 0.950

Equations (10)

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x 2 a 2 + y 2 b 1 2 = 1     for   y 0 , x 2 a 2 + y 2 b 2 2 = 1     for   y < 0 ,
tan   γ p = b j 2 / a 2 sin 2   ξ + cos 2   ξ - x p 2 b j 2 / a 2 - 1 sin   ξ   cos   ξ + x p y p
θ 0 i = π / 2 - γ 0 ,
θ 0 t = arcsin sin   θ 0 i / n .
δ 0 = γ 0 + θ 0 t , δ p = 2 γ p - δ p - 1   if   p 1 ,   p = odd , = 2 γ p - δ p - 1 - π   if   p 1 ,   p = even ,
θ p i = γ p - δ p - 1 .
θ P ξ = P π / 2 - γ P + θ P t P + 1 π / 2 - γ P + θ P t if   P = odd if   P = even ,
θ P t = arcsin n   sin   θ P i .
tan   δ p = y p + 1 - y p / x p + 1 - x p .
I P = 1 2 t 0 TE p = 1 P - 1   r p TE t P TE 2 + 1 2 t 0 TM p = 1 P - 1   r p TM t P TM 2 ,

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