Abstract

General relations for the caustic surfaces obtained by illuminating an ellipsoid, paraboloid, or hyperboloid reflector by a point-light source, lying along the principal axis of the reflector, were derived. A thorough study of the evolution of caustics with the particular type of reflector, as well as with the relative position of the point-light source and the reflector, was undertaken. Interesting laws for the shape, position, and properties of the caustics were derived, depending on the shape of the particular reflector used, its aperture, and the relative position of the light source and the reflector. These properties may be useful for the creation of wide angle all-reflective multimirror systems designed for definite limits of distances and which, by adjacent pairs of reflectors, present only a limited and reduced amount of third-order aberrations.

© 1977 Optical Society of America

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References

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  1. Z. Bartkowski, “O pewnych r⊙wnaniach r⊙zniczkowych katoptryki (Über einige Differentialgleichungen der Katoptrik),” Candidate’s Dissertation in Manuscript (1957).
  2. Z. Bartkowski, Optik 18, 22 (1961).
  3. Z. Bartkowski, Optik 19, 226 (1962).
  4. D. G. Burkhard, D. L. Shealy, Opt. Acta 20, 287 (1973).
    [CrossRef]
  5. D. G. Burkhard, D. L. Shealy, J. Opt. Soc. Am. 63, 299 (1973).
    [CrossRef]
  6. D. L. Shealy, D. G. Burkhard, Appl. Opt. 12, 2955 (1973).
    [CrossRef] [PubMed]
  7. P. S. Theocaris, Appl. Opt. 10, 2240 (1971).
    [CrossRef] [PubMed]
  8. P. S. Theocaris, E. E. Gdoutos, Appl. Opt. 15, 1629 (1976).
    [CrossRef] [PubMed]
  9. P. S. Theocaris, E. E. Gdoutos, Appl. Opt. 16, 722 (1977).
    [CrossRef] [PubMed]

1977 (1)

1976 (1)

1973 (3)

1971 (1)

1962 (1)

Z. Bartkowski, Optik 19, 226 (1962).

1961 (1)

Z. Bartkowski, Optik 18, 22 (1961).

Appl. Opt. (4)

J. Opt. Soc. Am. (1)

Opt. Acta (1)

D. G. Burkhard, D. L. Shealy, Opt. Acta 20, 287 (1973).
[CrossRef]

Optik (2)

Z. Bartkowski, Optik 18, 22 (1961).

Z. Bartkowski, Optik 19, 226 (1962).

Other (1)

Z. Bartkowski, “O pewnych r⊙wnaniach r⊙zniczkowych katoptryki (Über einige Differentialgleichungen der Katoptrik),” Candidate’s Dissertation in Manuscript (1957).

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Figures (18)

Fig. 1
Fig. 1

Caustics obtained from a shallow ellipsoid reflector with (a/b) = 0.50 illuminated by a point-light source placed at various distances (A/b) from the equator of the reflector and along its principal axis. Positive values of A correspond to positions of the light source lying on the positive z semiaxis.

Fig. 2
Fig. 2

As in Fig. 1 with (a/b) = 2.00.

Fig. 3
Fig. 3

Variation of the projections along the z axis of the extremities of the caustics formed by ellipsoid reflectors with (a/b) = 0.25 (a) and 0.50 (b) vs the relative position of the light source. The position of the cusp point is also indicated.

Fig. 4
Fig. 4

As in Fig. 3 with (a/b) = 1.00 (a) and 2.00 (b).

Fig. 5
Fig. 5

Loci of the terminal points of the caustics corresponding to ellipsoid reflectors with (a/b) = 0.25 and 0.50 and various apertures of angles 20 of the reflectors.

Fig. 6
Fig. 6

As in Fig. 5 with (a/b) = 1.00 and 2.00.

Fig. 7
Fig. 7

Variation of the ratio (a′/b′) of the principal semiaxes (a) as well as the angle of inclination φ of the major semiaxis with the r axis (b) of the elliptical type of the loci of terminal points of caustics corresponding to various apertures (r/b) of the ellipsoid reflectors with (a/b) = 0.25, 0.50, and 1.00.

Fig. 8
Fig. 8

Caustics obtained from a paraboloid reflector with aperture equal to (r/b) = 1.00 illuminated by a point-light source placed at various distances (A/b) from the tangential plane of the reflector and along its principal axis. Positive values of A correspond to positions of the light source lying on the positive z semiaxis.

Fig. 9
Fig. 9

Variation of the projections along the z axis of the extremities of the caustics formed by a paraboloid (a) and a hyperboloid (b) reflector with (a/b) = 0.50 vs the relative position of the light source. The position of the cusp point is also indicated.

Fig. 10
Fig. 10

Loci of the terminal points of the caustics corresponding to a paraboloid (a) and to a hyperboloid (b) reflector with (a/b) = 0.50 (b) and apertures equal to (r/b) = 0.25, 0.50, 0.75, and 1.00.

Fig. 11
Fig. 11

Caustics obtained from a hyperboloid reflector with (a/b) = 0.50 and aperture equal to (r/b) = 1.00 illuminated by a point-light source placed at various distances (A/b) from the reference plane Orr of the reflector and along its principal axis. Positive values of A correspond to positions of the light source lying on the positive z semiaxis.

Fig. 12
Fig. 12

As in Fig. 9 with (a/b) = 2.00.

Fig. 13
Fig. 13

As in Fig. 9 for hyperboloid reflectors with (a/b) = 1.00 (a) and 2.00 (b).

Fig. 14
Fig. 14

As in Fig. 10 for hyperboloid reflectors with (a/b) = 1.00 (a) and 2.00 (b).

Fig. 15
Fig. 15

Caustics obtained by illuminating a series of ellipsoid reflectors defined by the ratio (a/b) by a point-light source placed at point (A/b) = 1.00. The terminal points of all these caustics lie on two straight lines defined by point (r/b) = 1, z = 0, and the symmetric of the point source with respect to the equatorial plane of the ellipsoid.

Fig. 16
Fig. 16

Caustics obtained by illuminating a series of ellipsoid reflectors by a parallel light beam. The terminal points of all these caustics lie on the circumference of the common equator of all the above ellipsoids.

Fig. 17
Fig. 17

Caustics obtained by illuminating a series of hyperboloid reflectors with (r/b) = 1.00 by a point light source placed at the plane Orr to whom all these hyperboloids are referred.

Fig. 18
Fig. 18

Caustics obtained by illuminating a series of hyperboloid reflectors with (r/b) = 1.00 by a parallel light beam. The terminal points of all these caustics lie on a cylinder with radius (r/b) = 1.0.

Equations (28)

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z = f ( r )
w = ( z - z o ) tan ( 2 α + φ )
tan α = d z / d r             and             tan φ = r / ( A + z ) .
r = r + ( z - z o ) tan ( 2 α + φ ) .
( d r ) / ( d r ) = 0.
z 2 a 2 + r 2 b 2 = 1 ,
z o 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 o 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 o 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 o R = [ 1 2 + ( r R ) 2 ] [ 1 - ( r R ) 2 ] 1 / 2 r R = ( r R ) 3 .
2 b 2 z o 2 + ( 2 b 2 - a 2 ) r 2 - ( 4 b 2 - 3 a 2 ) r + 2 b 2 ( b 2 - a 2 ) = 0 ,
z o = 0             and             r o b = 4 b 2 - 3 a 2 2 ( 2 b 2 - a 2 ) .
± z o 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 ,
( A b ) = ± [ ( a b ) 2 - 1 ] 1 / 2 .
z o = - A .
z = ( r 2 ) / ( 4 b ) ,
z o b = { [ 4 - ( r b ) 2 ] [ 4 ( A b ) - ( r b ) 2 ] + 16 ( r b ) 2 } [ 8 ( A b ) ( r b ) 2 + 16 ( A b ) + ( r b ) 4 ] + 16 ( r b ) 4 [ ( A b ) - 1 ] [ 2 ( A b ) - ( r b ) 2 - 6 ] 16 [ ( A b ) - 1 ] [ 4 + ( r b ) 2 ] [ 4 ( A b ) - 3 ( r b ) 2 ]
r b = - 4 ( r b ) 3 4 ( A b ) - 3 ( r b ) 2
z o = b             and             r = 0 ,
z 2 a 2 - r 2 b 2 = 1 ,
z o 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 + [ 1 + ( a b ) 2 ] ( r b ) 2 + ( a b ) 2 } + { 1 + [ 1 - ( a b ) 2 ] ( r b ) 2 + 2 ( a b ) 2 } ( A b ) 2 , B 2 = ( A b ) ( a b ) ( 3 { 1 + [ 1 + ( a b ) 2 ] ( r b ) 2 } + 4 ( a b ) 2 ) , Δ 1 = ( A b ) { 1 + [ 1 + ( a b ) 2 ] ( r b ) 2 + 4 ( a b ) 2 } , Δ 2 = ( a b ) ( 1 + [ 1 + ( a b ) 2 ] ( r b ) 2 + 2 { ( a b ) 2 + [ 1 + ( a b ) 2 ] ( r b ) 2 } + 2 ( A b ) 2 ) ,
r b = 2 ( a b ) { ( A b ) 2 - [ 1 + ( a b ) 2 ] } Δ 1 [ 1 + ( r b ) 2 ] 1 / 2 + Δ 2 .
( A / b ) = ± [ 1 + ( a / b ) 2 ] 1 / 2 ,
r b = [ 1 - ( z o / b ) ( A / b ) ] .
( r / b ) = ( r / b ) 3 ,

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