Abstract

The optical method of caustics, initially developed for recording abrupt plate slopes created by singularities in elastic stress fields, was extended to incorporate the study of the general case of any type of surface. A universal technique, based on the general theory of caustics developed in this paper, was formulated to study the topography of any surface from its corresponding caustics obtained by illuminating the surface by a parallel, convergent, or divergent light beam. The special case of an axisymmetric mirror with elliptical cross section, whose ellipticity varies from zero to infinity, was studied extensively to show the potentialities of the technique developed. It was shown that the caustics obtained are very sensitive to the particular form of the surface considered. From the procedure developed in this paper it was concluded that the method of caustics can be successfully used to record the topography of any surface with large or infinitesimal slopes.

© 1976 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1970), pp. 256–367.
  2. P. S. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, London, 1964), pp. 219–278.
  3. R. Platzeck, E. Gaviola, J. Opt. Soc. Am. 29, 484 (1939).
    [CrossRef]
  4. J. M. Simon, Opt. Acta 18, 369 (1971).
    [CrossRef]
  5. L. M. Foucault, C. R. Acad. Sci. Paris 47, 958 (1858).
  6. R. Platzeck, J. M. Simon, Opt. Acta 21, 267 (1974).
    [CrossRef]
  7. P. S. Theocaris, J. Appl. Mech., Trans. ASME E 37(2), 409 (1970).
    [CrossRef]
  8. P. S. Theocaris, J. Mech. Phys. Solids 20, 215 (1972).
    [CrossRef]
  9. P. S. Theocaris, Exp. Mech. 13, 511 (1973).
    [CrossRef]

1974 (1)

R. Platzeck, J. M. Simon, Opt. Acta 21, 267 (1974).
[CrossRef]

1973 (1)

P. S. Theocaris, Exp. Mech. 13, 511 (1973).
[CrossRef]

1972 (1)

P. S. Theocaris, J. Mech. Phys. Solids 20, 215 (1972).
[CrossRef]

1971 (1)

J. M. Simon, Opt. Acta 18, 369 (1971).
[CrossRef]

1970 (1)

P. S. Theocaris, J. Appl. Mech., Trans. ASME E 37(2), 409 (1970).
[CrossRef]

1939 (1)

1858 (1)

L. M. Foucault, C. R. Acad. Sci. Paris 47, 958 (1858).

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1970), pp. 256–367.

Foucault, L. M.

L. M. Foucault, C. R. Acad. Sci. Paris 47, 958 (1858).

Gaviola, E.

Platzeck, R.

R. Platzeck, J. M. Simon, Opt. Acta 21, 267 (1974).
[CrossRef]

R. Platzeck, E. Gaviola, J. Opt. Soc. Am. 29, 484 (1939).
[CrossRef]

Simon, J. M.

R. Platzeck, J. M. Simon, Opt. Acta 21, 267 (1974).
[CrossRef]

J. M. Simon, Opt. Acta 18, 369 (1971).
[CrossRef]

Theocaris, P. S.

P. S. Theocaris, Exp. Mech. 13, 511 (1973).
[CrossRef]

P. S. Theocaris, J. Mech. Phys. Solids 20, 215 (1972).
[CrossRef]

P. S. Theocaris, J. Appl. Mech., Trans. ASME E 37(2), 409 (1970).
[CrossRef]

P. S. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, London, 1964), pp. 219–278.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1970), pp. 256–367.

C. R. Acad. Sci. Paris (1)

L. M. Foucault, C. R. Acad. Sci. Paris 47, 958 (1858).

Exp. Mech. (1)

P. S. Theocaris, Exp. Mech. 13, 511 (1973).
[CrossRef]

J. Appl. Mech., Trans. ASME E (1)

P. S. Theocaris, J. Appl. Mech., Trans. ASME E 37(2), 409 (1970).
[CrossRef]

J. Mech. Phys. Solids (1)

P. S. Theocaris, J. Mech. Phys. Solids 20, 215 (1972).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Acta (2)

J. M. Simon, Opt. Acta 18, 369 (1971).
[CrossRef]

R. Platzeck, J. M. Simon, Opt. Acta 21, 267 (1974).
[CrossRef]

Other (2)

M. Born, E. Wolf, Principles of Optics (Pergamon, London, 1970), pp. 256–367.

P. S. Theocaris, Moiré Fringes in Strain Analysis (Pergamon, London, 1964), pp. 219–278.

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Figures (10)

Fig. 1
Fig. 1

Geometry of formation of a caustic by illuminating a surface by a parallel or divergent light beam.

Fig. 2
Fig. 2

Geometry of the elliptical cross section of an axisymmetric mirror illuminated by a point-light source S.

Fig. 3
Fig. 3

Caustics formed by a spherical mirror of radius R illuminated by a point-light source placed at the following distances A from the face of the mirror: (A/R) = 0, 0.1, 0.2, 0.4, 0.6,1.0, 2.0, 4.0, ∞, −0.1, −0.2, −0.3, −0.5, −0.4, −0.6, −0.8, −1.0, −2.0, −4.0. {Positive values of A correspond to the region where the normal to the front face of the mirror is directed outside the mirror. The locus of the extremities of the caustics forms an ellipse with semiaxes (R/2) and [R/(2√2)] and the origin lying at a point with r/R = 0.5 and z = 0.}

Fig. 4
Fig. 4

Caustics formed by an axisymmetric mirror with an elliptical cross section, whose ratio of its semiaxes is equal to (a/b) = 2. The mirror is illuminated by a point-light source placed at the following distances A from the front face of the mirror: (A/b) = 0, 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 4.0, 6.0, ∞, −0.2, −0.4, −0.6, −0.8, −1.0, −1.2, −1.4, −1.6, −1.8, −2.0, −4.0. Positive values of A are defined as in Fig. 3.

Fig. 5
Fig. 5

Position of the elliptical cross section of the axisymmetric mirror of Fig. 4 with the locus of the terminal points of the caustics formed by reflections of one-half of the mirror. This locus constitutes a hyperbola with the center at the point (2.0) and asymptotes inclined by 45° to the axis of symmetry of the mirror.

Fig. 6
Fig. 6

Caustics formed by a point-light source placed at a position with (A/b) = 0 and illuminating axisymmetric mirrors with elliptical cross sections with the following values of the ratio (a/b) of their semiaxes: (a/b) = 0.80, 0.85, 0.90, 0.95, 1.00, 1.05, 1.10, 1.15, 1.20.

Fig. 7
Fig. 7

Caustics formed by a point-light source placed at a position with (A/b) = 0.2 and illuminating the axisymmetric mirrors of Fig. 6.

Fig. 8
Fig. 8

Caustics formed by a point-light source placed at positions with (A/b) = 0.6 and 1.0, respectively, and illuminating the axisymmetric mirrors of Fig. 6.

Fig. 9
Fig. 9

Caustics formed by illuminating the axisymmetric mirrors of Fig. 6 with a parallel light beam.

Fig. 10
Fig. 10

Variation of the radii r1.00 and r1.01, r1.02, r1.03, r1.04, r1.05 of the caustics obtained by illuminating a sphere and the ellipsoids with the ratio of their semiaxes equal to (a/b) = 1.01, 1.02,1.03, 1.04, 1.05, respectively, by a light source vs the distance (A/b) of the source from the mirror. The caustics obtained by placing the reference screen at the positions indicated by the curve (zc/b) = f(A/b), which correspond to planes in contact with the caustics obtained from the sphere.

Equations (32)

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z = f ( x , y )
w = w x i + w y j ,
w x = ( z - z 0 ) tan 2 α w y = ( z - z 0 ) tan 2 β ,
tan α = f ( x , y ) x             tan β = f ( x , y ) y ,
W = W x i + W y j ,
W x = x + [ f ( x , y ) - z 0 ] 2 f ( x , y ) / x 1 - [ f ( x , y ) / x ] 2 , W y = y + [ f ( x , y ) - z 0 ] 2 f ( x , y ) / x 1 - [ f ( x , y ) / y ] 2 .
J = ( W x , W y ) ( x , y ) = | W x x W x y W y x W y y | = 0.
W x = x - 2 z 0 f ( x , y ) x , W y = y - 2 z 0 f ( x , y ) y
W x , y = x , y + [ f ( x , y ) - z 0 ] tan ( 2 α , β + φ )
W x , y x , y - z 0 ( 2 tan α , β + tan φ ) ,
W x , y = λ m x , y + w x , y ,
λ m = ( z 0 + z i ) / z i ,
W ( r ) = r - 2 z 0 d f ( r ) d r
[ d W ( r ) ] / ( d r ) = 0.
[ d 2 f ( r ) ] / ( d r 2 ) = 1 / ( 2 z 0 ) ,
r = W ( r ) = r + ( z - z 0 ) tan ( 2 α + φ ) ,
z = a b ( b 2 - r 2 ) 1 / 2 , tan α = d z d r , tan φ = r A + z .
z 0 b = B 1 [ 1 - ( r / b ) 2 ] 1 / 2 + B 2 Δ 1 [ 1 - ( r / b ) 2 ] 1 / 2 + Δ 2 ,
B 1 = 2 ( a b ) 2 { 1 + [ ( a b ) 2 - 1 ] ( r b ) 2 - ( a b ) 2 } + { 1 - [ ( a b ) 2 + 1 ] ( r b ) 2 - 2 ( a b ) 2 } ( A b ) 2 , B 2 = ( A b ) ( a b ) ( 3 { 1 + [ ( a b ) 2 - 1 ] ( r b ) 2 } - 4 ( a b ) 2 ) , Δ 1 = ( A b ) { 1 + [ ( a b ) 2 - 1 ] ( r b ) 2 - 4 ( a b ) 2 } , Δ 2 = ( a b ) ( { 1 + [ ( a b ) 2 - 1 ] ( r b ) 2 } - 2 { ( a b ) 2 - [ ( a b ) 2 - 1 ] ( r b ) 2 } - 2 ( A b ) 2 ) .
r b = 2 ( a / b ) { [ ( a / b ) 2 - 1 ] - ( A / b ) 2 } ( r / b ) 3 Δ 1 [ 1 - ( r / b ) 2 ] 1 / 2 + Δ 2 ,
z 0 R = ( A / R ) 2 [ 1 + 2 ( r / R ) 2 ] [ 1 - ( r / R ) 2 ] 1 / 2 + ( A / R ) 3 ( A / R ) [ 1 - ( r / R ) 2 ] 1 / 2 + 2 ( A / R ) 2 + 1 , r R = 2 ( A / R ) 2 ( r / R ) 3 3 ( A / R ) [ 1 - ( r / R ) 2 ] 1 / 2 + 2 ( A / R ) 2 + 1 .
z 0 b = - { 1 - [ ( a / b ) 2 + 1 ] ( r / b ) 2 - 2 ( a / b ) 2 } 2 ( a / b ) × [ 1 - ( r b ) 2 ] 1 / 2 r / b = ( r / b ) 3 .
z 0 R = [ 1 2 + ( r R ) 2 ] [ 1 - ( r R ) 2 ] 1 / 2 , r / R = ( r / R ) 3 .
2 b 2 z 0 2 + ( 2 b 2 - a 2 ) r 2 - ( 4 b 2 - 3 a 2 ) r 2 + 2 b 2 ( b 2 - a 2 ) = 0.
r 0 b = 4 b 2 - 3 a 2 2 ( 2 b 2 - a 2 )
± z 0 2 { a 2 2 [ 2 b 2 ( ± 2 b 2 a 2 ) ] 1 / 2 } 2 + r 2 [ a 2 2 ( ± 2 b 2 a 2 ) ] 2 = b 2 ,
z = Σ A k r k .
r = λ m r - 2 z 0 Σ k A k r k - 1 , d r d r = λ m - 2 z 0 Σ k ( k - 1 ) A k r k - 2 = 0 ,
r = f ( A k , z 0 , λ m ) .
r = λ m f ( A k , z 0 , λ m ) - 2 z 0 Σ k A k [ f ( A k , z 0 , λ m ) ] k - 1 .
S = 1 2 [ | ( D 2 - D 0 D 2 + D 0 ) | + | ( D 0 - D 1 D 0 + D 1 ) | ] C 2 - C 0 + C 0 - C 1 .
S = 1 2 [ | ( 1.45 - 0.22 1.45 + 0.22 ) | + | ( 0.22 - 0 0.22 + 0 ) | ] 0.80 - 0.85 + 0.85 - 0.90 = 8.67 ,

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