Abstract

Because of the distortions of geometrical optics, image curves and the outlines of the objects that generate them need not have the same topology. New image loops appear when the object curve touches the caustic of the family of (imaginary) rays emitted by the observing eye. Such disruption may be elliptic (loop born from an isolated point) or hyperbolic (loop pinched off from an already existing one). The number of images need not be odd (unlike the number of rays reaching the eye from each object point). Two examples are employed to illustrate caustic touching. The first is the Sun’s disk seen in rippled water (as the height of the eye increases, the boundary of all the images becomes a fractal curve with dimension 2). The second is sunset seen through the Earth’s atmosphere from near space (when there is an inversion layer) or from the Moon during a lunar eclipse (when there need not be one).

© 1987 Optical Society of America

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References

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  1. T. Gold, “Riddle of reflections in the water,” Nature 314, 12 (1985).
    [CrossRef]
  2. D. K. Lynch, “Reflections on closed loops,” Nature 316, 216–217 (1985).
    [CrossRef]
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    [CrossRef]
  6. W. Tape, “The topology of mirages,” Sci. Am. 252, 120–129 (1985).
    [CrossRef]
  7. R. Narayan, R. Blandford, R. Nityananda, “Multiple imaging of quasars by galaxies and clusters,” Nature 310, 112–115 (1984).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  19. M. V. Berry, “Singularities in waves and rays,” in Physics of Defects, Les Houches Lectures XXXIV, R. D. Balian, M. Kleman, J-P. Poirier, eds. (North-Holland, Amsterdam, 1981), pp. 453–543.
  20. M. V. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
    [CrossRef]
  21. J. L. Elliott, R. G. French, E. Dunham, P. J. Gierasch, J. Veverka, C. Church, C. Sagan, “Occultation of epsilon Geminorum by Mars. II. The structure and extinction of the Martian upper atmosphere,” Astrophys. J. 217, 661–679 (1977).
    [CrossRef]

1985 (3)

W. Tape, “The topology of mirages,” Sci. Am. 252, 120–129 (1985).
[CrossRef]

T. Gold, “Riddle of reflections in the water,” Nature 314, 12 (1985).
[CrossRef]

D. K. Lynch, “Reflections on closed loops,” Nature 316, 216–217 (1985).
[CrossRef]

1984 (2)

R. Narayan, R. Blandford, R. Nityananda, “Multiple imaging of quasars by galaxies and clusters,” Nature 310, 112–115 (1984).
[CrossRef]

C. Hogan, R. Narayan, “Gravitational lensing by cosmic strings,” Mon. Not. R. Astron. Soc. 211, 575–591 (1984).

1983 (1)

1977 (1)

J. L. Elliott, R. G. French, E. Dunham, P. J. Gierasch, J. Veverka, C. Church, C. Sagan, “Occultation of epsilon Geminorum by Mars. II. The structure and extinction of the Martian upper atmosphere,” Astrophys. J. 217, 661–679 (1977).
[CrossRef]

1976 (2)

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

A. B. Fraser, W. H. Mach, “Mirages,” Sci. Am. 234, 102–111 (1976).
[CrossRef]

1975 (1)

A. B. Fraser, “The green flash and clear air turbulence,” Atmosphere 13, 1–8 (1975).

1963 (1)

F. Link, “Eclipse phenomena,” Adv. Astron. Astrophys. 2, 87–198 (1963).

1960 (1)

1957 (1)

M. S. Longuet-Higgins, “The statistical analysis of a random, moving surface,” Philos. Trans. R. Soc. London Ser. A 249, 321–387 (1957).
[CrossRef]

Berry, M. V.

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

M. V. Berry, “Singularities in waves and rays,” in Physics of Defects, Les Houches Lectures XXXIV, R. D. Balian, M. Kleman, J-P. Poirier, eds. (North-Holland, Amsterdam, 1981), pp. 453–543.

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” in Progress in Optics XVIII, E. Wolf, ed. (North-Holland, Amsterdam, 1980), pp. 257–346.
[CrossRef]

Blandford, R.

R. Narayan, R. Blandford, R. Nityananda, “Multiple imaging of quasars by galaxies and clusters,” Nature 310, 112–115 (1984).
[CrossRef]

R. Blandford, R. Narayan, R. Nityananda, “Fermat’s principle, caustics, and the classification of gravitational lens images,” Preprint GRP-037 (California Institute of Technology, Pasadena, Calif., 1985).

Church, C.

J. L. Elliott, R. G. French, E. Dunham, P. J. Gierasch, J. Veverka, C. Church, C. Sagan, “Occultation of epsilon Geminorum by Mars. II. The structure and extinction of the Martian upper atmosphere,” Astrophys. J. 217, 661–679 (1977).
[CrossRef]

Dunham, E.

J. L. Elliott, R. G. French, E. Dunham, P. J. Gierasch, J. Veverka, C. Church, C. Sagan, “Occultation of epsilon Geminorum by Mars. II. The structure and extinction of the Martian upper atmosphere,” Astrophys. J. 217, 661–679 (1977).
[CrossRef]

Elliott, J. L.

J. L. Elliott, R. G. French, E. Dunham, P. J. Gierasch, J. Veverka, C. Church, C. Sagan, “Occultation of epsilon Geminorum by Mars. II. The structure and extinction of the Martian upper atmosphere,” Astrophys. J. 217, 661–679 (1977).
[CrossRef]

Fraser, A. B.

A. B. Fraser, W. H. Mach, “Mirages,” Sci. Am. 234, 102–111 (1976).
[CrossRef]

A. B. Fraser, “The green flash and clear air turbulence,” Atmosphere 13, 1–8 (1975).

French, R. G.

J. L. Elliott, R. G. French, E. Dunham, P. J. Gierasch, J. Veverka, C. Church, C. Sagan, “Occultation of epsilon Geminorum by Mars. II. The structure and extinction of the Martian upper atmosphere,” Astrophys. J. 217, 661–679 (1977).
[CrossRef]

Gierasch, P. J.

J. L. Elliott, R. G. French, E. Dunham, P. J. Gierasch, J. Veverka, C. Church, C. Sagan, “Occultation of epsilon Geminorum by Mars. II. The structure and extinction of the Martian upper atmosphere,” Astrophys. J. 217, 661–679 (1977).
[CrossRef]

Gold, T.

T. Gold, “Riddle of reflections in the water,” Nature 314, 12 (1985).
[CrossRef]

Greenler, R.

R. Greenler, Rainbows, Halos, and Glories (Cambridge U. Press, London, 1980).

Hogan, C.

C. Hogan, R. Narayan, “Gravitational lensing by cosmic strings,” Mon. Not. R. Astron. Soc. 211, 575–591 (1984).

Landau, L. D.

L. D. Landau, E. M. Lifshitz, L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, Oxford, 1984).

Lehn, W. H.

Lifshitz, E. M.

L. D. Landau, E. M. Lifshitz, L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, Oxford, 1984).

Link, F.

F. Link, “Eclipse phenomena,” Adv. Astron. Astrophys. 2, 87–198 (1963).

Longuet-Higgins, M. S.

M. S. Longuet-Higgins, “Reflection and refraction at a random moving surface. I. Pattern and paths of specular points,”J. Opt. Soc. Am. 50, 838–844 (1960).
[CrossRef]

M. S. Longuet-Higgins, “The statistical analysis of a random, moving surface,” Philos. Trans. R. Soc. London Ser. A 249, 321–387 (1957).
[CrossRef]

Lynch, D. K.

D. K. Lynch, “Reflections on closed loops,” Nature 316, 216–217 (1985).
[CrossRef]

Mach, W. H.

A. B. Fraser, W. H. Mach, “Mirages,” Sci. Am. 234, 102–111 (1976).
[CrossRef]

Mandelbrot, B. B.

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, Calif., 1982).

Narayan, R.

R. Narayan, R. Blandford, R. Nityananda, “Multiple imaging of quasars by galaxies and clusters,” Nature 310, 112–115 (1984).
[CrossRef]

C. Hogan, R. Narayan, “Gravitational lensing by cosmic strings,” Mon. Not. R. Astron. Soc. 211, 575–591 (1984).

R. Blandford, R. Narayan, R. Nityananda, “Fermat’s principle, caustics, and the classification of gravitational lens images,” Preprint GRP-037 (California Institute of Technology, Pasadena, Calif., 1985).

Nityananda, R.

R. Narayan, R. Blandford, R. Nityananda, “Multiple imaging of quasars by galaxies and clusters,” Nature 310, 112–115 (1984).
[CrossRef]

R. Blandford, R. Narayan, R. Nityananda, “Fermat’s principle, caustics, and the classification of gravitational lens images,” Preprint GRP-037 (California Institute of Technology, Pasadena, Calif., 1985).

Pitaevskii, L. P.

L. D. Landau, E. M. Lifshitz, L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, Oxford, 1984).

Poston, T.

T. Poston, I. N. Stewart, Catastrophe Theory and Its Applications (Pitman, London, 1978).

Sagan, C.

J. L. Elliott, R. G. French, E. Dunham, P. J. Gierasch, J. Veverka, C. Church, C. Sagan, “Occultation of epsilon Geminorum by Mars. II. The structure and extinction of the Martian upper atmosphere,” Astrophys. J. 217, 661–679 (1977).
[CrossRef]

Stewart, I. N.

T. Poston, I. N. Stewart, Catastrophe Theory and Its Applications (Pitman, London, 1978).

Tape, W.

W. Tape, “The topology of mirages,” Sci. Am. 252, 120–129 (1985).
[CrossRef]

Tricker, R. A. R.

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

Upstill, C.

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” in Progress in Optics XVIII, E. Wolf, ed. (North-Holland, Amsterdam, 1980), pp. 257–346.
[CrossRef]

Veverka, J.

J. L. Elliott, R. G. French, E. Dunham, P. J. Gierasch, J. Veverka, C. Church, C. Sagan, “Occultation of epsilon Geminorum by Mars. II. The structure and extinction of the Martian upper atmosphere,” Astrophys. J. 217, 661–679 (1977).
[CrossRef]

Adv. Astron. Astrophys. (1)

F. Link, “Eclipse phenomena,” Adv. Astron. Astrophys. 2, 87–198 (1963).

Adv. Phys. (1)

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

Astrophys. J. (1)

J. L. Elliott, R. G. French, E. Dunham, P. J. Gierasch, J. Veverka, C. Church, C. Sagan, “Occultation of epsilon Geminorum by Mars. II. The structure and extinction of the Martian upper atmosphere,” Astrophys. J. 217, 661–679 (1977).
[CrossRef]

Atmosphere (1)

A. B. Fraser, “The green flash and clear air turbulence,” Atmosphere 13, 1–8 (1975).

J. Opt. Soc. Am. (2)

Mon. Not. R. Astron. Soc. (1)

C. Hogan, R. Narayan, “Gravitational lensing by cosmic strings,” Mon. Not. R. Astron. Soc. 211, 575–591 (1984).

Nature (3)

T. Gold, “Riddle of reflections in the water,” Nature 314, 12 (1985).
[CrossRef]

D. K. Lynch, “Reflections on closed loops,” Nature 316, 216–217 (1985).
[CrossRef]

R. Narayan, R. Blandford, R. Nityananda, “Multiple imaging of quasars by galaxies and clusters,” Nature 310, 112–115 (1984).
[CrossRef]

Philos. Trans. R. Soc. London Ser. A (1)

M. S. Longuet-Higgins, “The statistical analysis of a random, moving surface,” Philos. Trans. R. Soc. London Ser. A 249, 321–387 (1957).
[CrossRef]

Sci. Am. (2)

A. B. Fraser, W. H. Mach, “Mirages,” Sci. Am. 234, 102–111 (1976).
[CrossRef]

W. Tape, “The topology of mirages,” Sci. Am. 252, 120–129 (1985).
[CrossRef]

Other (8)

R. Blandford, R. Narayan, R. Nityananda, “Fermat’s principle, caustics, and the classification of gravitational lens images,” Preprint GRP-037 (California Institute of Technology, Pasadena, Calif., 1985).

L. D. Landau, E. M. Lifshitz, L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, Oxford, 1984).

R. Greenler, Rainbows, Halos, and Glories (Cambridge U. Press, London, 1980).

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

M. V. Berry, C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” in Progress in Optics XVIII, E. Wolf, ed. (North-Holland, Amsterdam, 1980), pp. 257–346.
[CrossRef]

T. Poston, I. N. Stewart, Catastrophe Theory and Its Applications (Pitman, London, 1978).

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, Calif., 1982).

M. V. Berry, “Singularities in waves and rays,” in Physics of Defects, Les Houches Lectures XXXIV, R. D. Balian, M. Kleman, J-P. Poirier, eds. (North-Holland, Amsterdam, 1981), pp. 453–543.

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Figures (13)

Fig. 1
Fig. 1

Computer simulation of the reflection of a mast seen in rippled water whose deviation from a plane is a product of orthogonal sinusoids.

Fig. 2
Fig. 2

a, Object line L and eye caustic C in Ω space; b, topological structure of image R in ω space (magnifications8 not indicated).

Fig. 3
Fig. 3

Birth of loop in image R (sequence b) by elliptic process, as object line L touches eye caustic C from outside (sequence a).

Fig. 4
Fig. 4

Birth of loop in image R (sequence b) by hyperbolic process, as object line L touches eye caustic C from inside (sequence a).

Fig. 5
Fig. 5

Eye caustic in the far field of directions Ωx from a Gaussian water-wave pulse as a function of eye height h (h > 0 and h < 0 correspond to reflection from a crest and a trough, respectively). The sequences labeled a, b, and c represent three traversals of the Sun or pulse that, as described in the text, give the three sequences of images shown in Fig. 6.

Fig. 6
Fig. 6

Sequences of images of the Sun’s disk reflected from a traveling Gaussian water-wave trough, corresponding to the three traversals shown in Fig. 5. a, θs = 0.5, and t ranges from −4 to +4 in steps of 0.8; b, θs = 1, and t ranges from −3.8 to 3.8 in steps of 1.9; c, θs = 0.5, and t ranges from −2.5 to 2.5 in steps of 0.5.

Fig. 7
Fig. 7

Inner circuit: sequence of positions of the Sun’s disk (θs = 0.5) L relative to the cusped eye caustic C generated by the water surface [Eq. (18)]. Outer circuit: corresponding sequence of reflected images. Note the differences in the sizes of the images; all are drawn on the same scale.

Fig. 8
Fig. 8

Increasing disruption of the image of the Sun (radius θs = 1) reflected in the water surface [Eq. (27)], with N = 5, k = 1, A = 1/(10)1/2, ϕ = j2, for a, h = 2; b, h = 4;, c, h = 10. As h → ∞ the boundary of the images becomes a fractal curve with dimension 2.

Fig. 9
Fig. 9

Geometry and notation for rays from the setting Sun viewed from space after refraction by the Earth’s atmosphere.

Fig. 10
Fig. 10

Graph of true ray direction Ω versus apparent direction ω for rays [Eq. (34)] from the setting Sun viewed through the atmosphere [Eq. (33)]; the minimum corresponds to the eye caustic.

Fig. 11
Fig. 11

The black shapes are images of the setting Sun, and the open circles are the positions of the true Sun, at times t1 = θs = 15 arcmin, t2 = −Ωsh + 0.8θs = 46.7 arcmin, t3 = Ωsh + 110 = 51.2 arcmin, t4 = −Ωc − = 52.1 arcmin, t5 = −Ωc − 0.8θs = 55.1 arcmin, t6 = −Ωc + 0.8θs = 79.1 arcmin; vertical magnifications above the horizon are as indicated.

Fig. 12
Fig. 12

Geometry and coordinates in object space when the Sun is behind the Earth as seen by an observer on the Moon during a lunar eclipse.

Fig. 13
Fig. 13

The black shapes are images of the Sun seen from the Moon during a lunar eclipse, when the true Sun (small circles) is behind the Earth (large circles), at times t = −θe + 1.3θs = −37.48 arcmin, tb = −1.40 = −21 arcmin, tc = −1.01e = −15.15 arcmin, td = −0.99θs = −14.85 arcmin. Radial distances from the Earth’s limb are magnified 500 times.

Equations (42)

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L ( Ω ) = 0             ( object ) .
Ω = Ω ( ω )             ( rays ) .
L [ Ω ( ω ) ] R ( ω ) = 0             ( image ) .
R / ω 1 = L / Ω 1 Ω 1 / ω 1 + L / Ω 2 Ω 2 / ω 1 = 0 , R / ω 2 = L / Ω 1 Ω 1 / ω 2 + L / Ω 2 Ω 2 / ω 2 = 0 ,
Ω 1 / ω 1 Ω 2 / ω 1 = Ω 1 / ω 2 Ω 2 / ω 2 = - L / Ω 2 L / Ω 1 .
( Ω 1 , Ω 2 ) / ( ω 1 , ω 2 ) = 0             ( caustic ) .
d Ω 2 / d Ω 1 = - L / Ω 1 / L / Ω 2 .
d Ω 2 d Ω 1 = Ω 2 / ω 1 d ω 1 + Ω 2 / ω 2 d ω 2 Ω 1 / ω 1 d ω 1 + Ω 1 / ω 2 d ω 2 = Ω 2 / ω 1 Ω 1 / ω 1 ( 1 + d ω 2 / d ω 1 Ω 2 / ω 2 / Ω 2 / ω 1 1 + d ω 2 / d ω 1 Ω 1 / ω 2 / Ω 1 / ω 1 ) = Ω 2 / ω 1 Ω 1 / ω 1 ,
ω = ( ω x , ω y ) = h r .
Ω = ω - 2 f ( h ω ) .
L ( Ω ) = Ω x 2 + Ω y 2 - θ s 2 = 0 ,
1 - 2 h 2 f ( h ω ) + 4 h 2 [ 2 f ( h ω ) / x 2 2 f ( h ω ) / y 2 - ( 2 f ( h ω ) / x y ) 2 ] = 0.
d 2 f ( h ω x ) / d x 2 = 1 / 2 h .
F ( x ) = A exp [ - b ( x - v t ) 2 / 2 ]
h A b h ,             t t / v b 1 / 2 , ω x b 1 / 2 ( h ω x - v t ) ,             ω y b 1 / 2 h ω y
Ω x ( ω x ) = ω x + 2 h exp ( - ω x 2 / 2 )             ( rays ) , 2 h ( ω x 2 - 1 ) exp ( - ω x 2 / 2 ) = 1             ( caustics ) , ω y = ± { θ s 2 - [ Ω x ( ω x - t ) ] 2 } 1 / 2             ( image ) .
Ω x = 0 ,             h = - 1 / 2
Ω x = ± 3 3 / 2 / 2 = ± 2.5981 ,             h = exp ( 3 / 2 ) / 4 = 1.12042.
f ( r ) = A x 2 y
Ω x = ω x ( 1 - ω y ) ,             Ω y = ω y - ω x 2 / 2             ( rays ) , Ω x = ± [ 2 ( 1 - Ω y ) / 3 ] 3 / 2             ( caustic ) .
ω y = { 3 ω x 2 / 2 - X ω x + Y ± [ θ s 2 ( 1 + ω x 2 ) - ( ω x - ω x 3 / 2 - X - Y ω x ) 2 ] 1 / 2 } / ( 1 + ω x 2 ) .
ω max = θ s + 2 f ( r ) max .
A = d ω x d ω y θ { L [ Ω ( ω ) ] }
L = d ω x d ω y δ { L [ Ω ( ω ) ] } ω L [ Ω ( ω ) ]
ω L = L / Ω Ω / ω = L / ω ( 1 - 2 h 2 f / x 2 ) ( h ) 2 h L / ω 2 f / x 2 .
ω min = λ / h .
L ω min - 1 ,
f ( r ) = j = 1 N A j cos ( k j · r + ϕ j ) .
Ω ( ω ) = ω - D ( ω ) .
Ω / ω = 1 - D / ω = 0.
b = r e cos ω + l sin ω r e + l ω ,
D = 2 b r min d r / r [ n 2 ( r ) r 2 - b 2 ] 1 / 2 - π ,
D - 2 r b d r n ( r ) / ( r 2 - b 2 ) 1 / 2 = - 2 r e 0 d u n ( b cosh u )
n ( r ) = 1 + Δ { exp [ - ( r - r e ) / H ] - δ exp [ - ( r - r e ) / H ] } ,
Ω ( ω ) = ω - D 0 [ exp ( - l ω / H ) - ( δ / 1 / 2 ) exp ( - l ω / H ) ] ,
D 0 = Δ ( 2 π r e / H ) 1 / 2 = 0.02016 = 1.155° .
( δ / 3 / 2 - 1 ) l > H / D 0 .
T ( r e ) / r = - T ( r e ) δ / H = - 0.26 km - 1
Ω c = - 0.01951 = - 67.07 arcmin , ω c = 0.000178 = 0.612 arcmin ,
ω x = ± { θ s 2 - [ Ω ( ω y ) + t ] 2 } 1 / 2 .
Ω ( ω ) = ω - D 0 exp [ - ( ω - θ e ) l / H ] , ϕ Ω = ϕ ω ,
ϕ ω ( ω ) = cos - 1 { [ t 2 + Ω 2 ( ω ) - θ s 2 ] / 2 t Ω ( ω ) } .

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