Abstract

We analyze the optical caustic produced by light refracted at the curved meniscus surrounding a cylindrical rod standing partially out of a liquid-filled container. When the rod is tilted from the vertical or when light is diagonally incident, the caustic is a four-cusped astroid with two of its cusps obscured by the rod’s shadow. If a portion of the flat end of the rod is raised above the water level, the caustic evolves into a pattern of five interlocking cusps. The five cusps result from symmetry breaking of a three-cusped surface perturbation caustic.

© 2003 Optical Society of America

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References

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  1. C. Adler, “Shadow-sausage effect,” Am. J. Phys. 35, 774–776 (1967).
    [CrossRef]
  2. M. J. Smith, “Comment on: shadow-sausage effect,” Am. J. Phys. 36, 912–914 (1968).
    [CrossRef]
  3. J. Walker, The Flying Circus of Physics with Answers (Wiley, New York, 1977), topic 5.3, pp. 115, 266.
  4. J. Walker, “Shadows cast on the bottom of a pool are not like other shadows. Why?” Sci. Am. 259(1), 116–119 (1988).
    [CrossRef]
  5. M. V. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
    [CrossRef]
  6. W. P. Arnott, P. L. Marston, “Unfolding axial caustics of glory scattering with harmonic angular perturbations of toroidal wave fronts,” J. Acoust. Soc. Am. 85, 1427–1440 (1989).
    [CrossRef]
  7. P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” Phys. Acoust. 21, 1–234 (1992), pp. 220–221.
  8. C. Huh, L. E. Scriven, “Shapes of axisymmetric fluid interfaces of unbound extent,” J. Colloid. Interface Sci. 30, 323–337 (1969).
    [CrossRef]
  9. M. V. Berry, J. V. Hajnal, “The shadows of floating objects and dissipating vortices,” Opt. Acta 30, 23–40 (1983).
    [CrossRef]
  10. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), p. 618, Eq. (11.127).
  11. M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980), Sect. 3, pp. 277–297.
    [CrossRef]
  12. Ref. 11, Appen. 2, pp. 339–342.
  13. H. Hofer, D. R. Williams, “The eye’s mechanisms for autocalibration,” Opt. Photon. News (January2002), pp. 34–39.

2002

H. Hofer, D. R. Williams, “The eye’s mechanisms for autocalibration,” Opt. Photon. News (January2002), pp. 34–39.

1992

P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” Phys. Acoust. 21, 1–234 (1992), pp. 220–221.

1989

W. P. Arnott, P. L. Marston, “Unfolding axial caustics of glory scattering with harmonic angular perturbations of toroidal wave fronts,” J. Acoust. Soc. Am. 85, 1427–1440 (1989).
[CrossRef]

1988

J. Walker, “Shadows cast on the bottom of a pool are not like other shadows. Why?” Sci. Am. 259(1), 116–119 (1988).
[CrossRef]

1983

M. V. Berry, J. V. Hajnal, “The shadows of floating objects and dissipating vortices,” Opt. Acta 30, 23–40 (1983).
[CrossRef]

1980

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980), Sect. 3, pp. 277–297.
[CrossRef]

1976

M. V. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
[CrossRef]

1969

C. Huh, L. E. Scriven, “Shapes of axisymmetric fluid interfaces of unbound extent,” J. Colloid. Interface Sci. 30, 323–337 (1969).
[CrossRef]

1968

M. J. Smith, “Comment on: shadow-sausage effect,” Am. J. Phys. 36, 912–914 (1968).
[CrossRef]

1967

C. Adler, “Shadow-sausage effect,” Am. J. Phys. 35, 774–776 (1967).
[CrossRef]

Adler, C.

C. Adler, “Shadow-sausage effect,” Am. J. Phys. 35, 774–776 (1967).
[CrossRef]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), p. 618, Eq. (11.127).

Arnott, W. P.

W. P. Arnott, P. L. Marston, “Unfolding axial caustics of glory scattering with harmonic angular perturbations of toroidal wave fronts,” J. Acoust. Soc. Am. 85, 1427–1440 (1989).
[CrossRef]

Berry, M. V.

M. V. Berry, J. V. Hajnal, “The shadows of floating objects and dissipating vortices,” Opt. Acta 30, 23–40 (1983).
[CrossRef]

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980), Sect. 3, pp. 277–297.
[CrossRef]

M. V. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
[CrossRef]

Hajnal, J. V.

M. V. Berry, J. V. Hajnal, “The shadows of floating objects and dissipating vortices,” Opt. Acta 30, 23–40 (1983).
[CrossRef]

Hofer, H.

H. Hofer, D. R. Williams, “The eye’s mechanisms for autocalibration,” Opt. Photon. News (January2002), pp. 34–39.

Huh, C.

C. Huh, L. E. Scriven, “Shapes of axisymmetric fluid interfaces of unbound extent,” J. Colloid. Interface Sci. 30, 323–337 (1969).
[CrossRef]

Marston, P. L.

P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” Phys. Acoust. 21, 1–234 (1992), pp. 220–221.

W. P. Arnott, P. L. Marston, “Unfolding axial caustics of glory scattering with harmonic angular perturbations of toroidal wave fronts,” J. Acoust. Soc. Am. 85, 1427–1440 (1989).
[CrossRef]

Scriven, L. E.

C. Huh, L. E. Scriven, “Shapes of axisymmetric fluid interfaces of unbound extent,” J. Colloid. Interface Sci. 30, 323–337 (1969).
[CrossRef]

Smith, M. J.

M. J. Smith, “Comment on: shadow-sausage effect,” Am. J. Phys. 36, 912–914 (1968).
[CrossRef]

Upstill, C.

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980), Sect. 3, pp. 277–297.
[CrossRef]

Walker, J.

J. Walker, “Shadows cast on the bottom of a pool are not like other shadows. Why?” Sci. Am. 259(1), 116–119 (1988).
[CrossRef]

J. Walker, The Flying Circus of Physics with Answers (Wiley, New York, 1977), topic 5.3, pp. 115, 266.

Williams, D. R.

H. Hofer, D. R. Williams, “The eye’s mechanisms for autocalibration,” Opt. Photon. News (January2002), pp. 34–39.

Adv. Phys.

M. V. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
[CrossRef]

Am. J. Phys.

C. Adler, “Shadow-sausage effect,” Am. J. Phys. 35, 774–776 (1967).
[CrossRef]

M. J. Smith, “Comment on: shadow-sausage effect,” Am. J. Phys. 36, 912–914 (1968).
[CrossRef]

J. Acoust. Soc. Am.

W. P. Arnott, P. L. Marston, “Unfolding axial caustics of glory scattering with harmonic angular perturbations of toroidal wave fronts,” J. Acoust. Soc. Am. 85, 1427–1440 (1989).
[CrossRef]

J. Colloid. Interface Sci.

C. Huh, L. E. Scriven, “Shapes of axisymmetric fluid interfaces of unbound extent,” J. Colloid. Interface Sci. 30, 323–337 (1969).
[CrossRef]

Opt. Acta

M. V. Berry, J. V. Hajnal, “The shadows of floating objects and dissipating vortices,” Opt. Acta 30, 23–40 (1983).
[CrossRef]

Opt. Photon. News

H. Hofer, D. R. Williams, “The eye’s mechanisms for autocalibration,” Opt. Photon. News (January2002), pp. 34–39.

Phys. Acoust.

P. L. Marston, “Geometrical and catastrophe optics methods in scattering,” Phys. Acoust. 21, 1–234 (1992), pp. 220–221.

Prog. Opt.

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 18, 257–346 (1980), Sect. 3, pp. 277–297.
[CrossRef]

Sci. Am.

J. Walker, “Shadows cast on the bottom of a pool are not like other shadows. Why?” Sci. Am. 259(1), 116–119 (1988).
[CrossRef]

Other

Ref. 11, Appen. 2, pp. 339–342.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), p. 618, Eq. (11.127).

J. Walker, The Flying Circus of Physics with Answers (Wiley, New York, 1977), topic 5.3, pp. 115, 266.

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Figures (11)

Fig. 1
Fig. 1

Shadow-sausage effect for (a) a tree branch extending diagonally out of a shallow stream and (b) a 2.5-cm-diameter wooden dowel rod extending diagonally out of a 15-cm-deep plastic tub filled with water and illuminated by the Sun. The branch and the dowel rod lie in the plane formed by the Sun’s rays and the normal to the asymptotically flat water surface, and the portion above the surface is tilted away from the Sun.

Fig. 2
Fig. 2

Refraction caustic produced when a portion of the flat end of (a) a tree branch and (b)–(d) a 2.5-cm wooden dowel rod is raised above the water level and illuminated by the Sun. In (b)–(d) a progressively larger fraction of the flat end of the dowel rod lies above the water surface.

Fig. 3
Fig. 3

Geometry of the shadow-sausage caustic for a tilted rod and vertically incident light.

Fig. 4
Fig. 4

Caustic shape to first order for (a) p = 2 and (b) p = 3. The x 0 axis is horizontal, and the y 0 axis is vertical.

Fig. 5
Fig. 5

Caustic shape to second order for (a) p = 2, ∊2 = 0.5 and (b) p = 3, ∊3 = 0.3. The x 0 axis is horizontal, and the y 0 axis is vertical.

Fig. 6
Fig. 6

Separation of the pointed shadow ends Δx 0 of a tilted rod for vertically incident light as a function of the rod tilt angle γ. The data of Fig. 7 of Ref. 1 form the dashed curve, and the least-squares fit of Eq. (38) is the solid curve.

Fig. 7
Fig. 7

Geometry of the shadow-sausage caustic for a vertical rod and diagonally incident light.

Fig. 8
Fig. 8

Separation of the cusp points Δy 0 (filled circles) for a vertical rod and diagonally incident light as a function of the light’s angle of incidence Γ. The least-squares fit of Eq. (48) to the data is the solid curve, and the results of a ray-tracing computer program are the open circles.

Fig. 9
Fig. 9

Meniscus formation when a portion of the flat end of the rod is raised above the water surface.

Fig. 10
Fig. 10

p = 1, 2, 3, 4, 5 Fourier coefficients α p of a truncated ellipse as a function of the truncation fraction s for a rod tilt angle of γ = 30°.

Fig. 11
Fig. 11

Composite p = 2, 3 caustic of relation (54) for (a) ∊2:∊3 = 2:1, (b) ∊2:∊3 = 1:1, and (c) ∊2:∊3 = 1:2. The x 0 axis is horizontal, and the y 0 axis is vertical.

Equations (58)

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2Cmean-f/L2=0,
Lγ/ρg1/2
2f-f/L2=0,
fr, ϕ=A0K0r/L+p=1 pKpr/Lcospϕ+p=1 δpKpr/Lsinpϕ,
limrKpr/L=πL/2r1/2 exp-r/L.
fr, ϕ=A0K0r/L,
A0=h/K0a/L.
xrod2 cos2γ/a2+yrod2/a2=1,
rrodϕ=a/1-cos2ϕsin2γ1/2.
frrodϕ, ϕh1+q cosϕ,
A0=h/K0a/L+Oa/Lsin2γ,1=qK0a/L/K1a/L+Oqa/Lsin2γ,2=-a/4Lsin2γK0a/L/K2a/L+Oa/L2 sin4γ,3=-qa/8Lsin2γK0a/L×K1a/L/K3a/LK1a/L+Oqa/L2 sin4γ,4=Oa/L2 sin4γ,5=Oqa/L2 sin4γ,6=Oa/L3 sin6γ,
r0=x0ux+y0uy=r0 cosϕ0ux+r0 sinϕ0uy
Er0, z0=-dx -dy expikFr, r0,
Fr, r0=n-1fr, ϕ+nr2/2z0-nrr0 cosϕ-ϕ0/z0.
F=0,
x0=x+n-1z0/nf/rcosϕ-f/ϕsinϕ/r,
y0=y+n-1z0/nf/rsinϕ+f/ϕcosϕ/r.
n-12f/r2+n/z0n-1f/r/r+n/z0+n-12f/ϕ2/r2-n-122f/rϕ-f/ϕ/r2/r2=0.
n-1h/nRLK0R/L/K0a/L=-1/z0.
x0=y0=0.
0=n-1h/rLK0r/L/K0a/L+n/z0+n-1h/rL×p=1 pUpKpr/L/K0a/Lcospϕ,
Up=KpR/L-p2L/RKpR/L.
rϕ=R-L p=1 p cospϕUp/Wp,
Wp=K0R/L-L/RK0R/L.
x01=n-1hz0/nR×p=1 ppKpR/L/K0a/Lgpxϕ,y01=n-1hz0/nR×p=1 ppKpR/L/K0a/Lgpyϕ,
gpxϕ=p cospϕcosϕ+sinpϕsinϕ,
gpyϕ=p cospϕsinϕ-sinpϕcosϕ.
x01=n-1hz0/nR1K1R/L/K0a/L, y01=0.
x01=4n-1hz0/nR2×K2R/L/K0a/Lcos3ϕ,y01=-4n-1hz0/nR2×K2R/L/K0a/Lsin3ϕ.
x01=24n-1hz0/nR3K3R/L/K0a/L×cos4ϕ-cos2ϕ/2-1/8,y01=-24n-1hz0/nR3K3R/L/K0a/L×sin3ϕcosϕ,
x02=x01+pn-1hz0/nRp2Vp×pVp sin2pϕcosϕ-Up×cospϕgpxϕ/WpK0a/L,y02=y01+pn-1hz0/nRp2Vp×pVp sin2pϕsinϕ-Up×cospϕgpyϕ/WpK0a/L,
Vp=KpR/L-L/RKpr/L.
x02=n-1hz0/nR1K1R/L-12V12/W1cos3ϕ/K0a/L,y02=n-1hz0/nR12V12/W1×sin3ϕ/K0a/L,
q2<a/Lsin2γ.
x02pn-1hz0/nRa/R1/2 exp-R-a/L×pgpxϕ-p2g2pxϕ/2,y02pn-1hz0/nRa/R1/2 exp-R-a/L×pgpyϕ-p2g2pyϕ/2.
x0=x+xn-1hz0/nrLK0r/L+2K2r/L×x2-y2/r2+42K2r/LLy2/r3/K0a/L,
y0=y+yn-1hz0/nrLK0r/L+2K2r/L×x2-y2/r2-42K2r/LLx2/r3/K0a/L,
x0=xy0/y+x4n-1hz0/nr22×K2r/L/K0a/L.
x0±4n-1hz0/n2K2R/L/K0a/L×R2-a21/2/R2+O2.
x0±4n-1hz0/n2K2R/L/K0a/L×R2-a23/2/R4,
y0±4n-1hz0/n2K2R/L/K0a/La3/R4.
Δx0=C sin2γ
lr, ϕ=H1-Γ2/2+n-1h1+Γ2/2n×K0r/L/K0a/L-n-12h3/2nL2×K0r/LK02r/L/K03a/L+rΓ cosϕ+n-1NrΓ cosϕ+n-1MrΓ2 cos2ϕ+OΓ3,
Nr=-h2/nLK0r/LK0r/L/K02a/L,
Mr=h3/2n2L2K02r/LK0r/L/K03a/L+3+3n-2n2h3/4n3L2×K0r/LK02r/L/K03a/L,
x0=n-1z0/nRΓNR+TM-TNΓ2 cos3ϕ,y0=-n-1z0/nRTM-TNΓ2 sin3ϕ
TM=2MR,
TN=L2/hNR/R-NR2K0a/L/K0R/L-L/RK0R/L.
TMh3/2n3L23+5n-2n2a/R3/2×exp-3R-a/L,
TN4n-1h3/n2L2a/R3/2×exp-3R-a/L.
TM1.72TN,
Δy0=C sin2Γ,
rrodϕ=a/1-cos2ϕsin2γ1/2 if 0ϕϕ0,=-a1-s/cosϕcosγ if ϕ0ϕπ,
cosϕ0=-1-s/cos2γ+1-s2 sin2γ1/2.
rrodϕ=α0+p=1 αp cospϕ,
α0=1/π0πdϕrrodϕ,
αp=2/π0πdϕ cospϕrrodϕ.
x02g2xϕ+3g3xϕ, y02g2yϕ+3g3yϕ.

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