Abstract

General formulas are derived for the caustic surface and irradiance over an arbitrary receiver surface for point source radiation on collimated rays that are reflected or refracted by a curved surface. Specific formulas are obtained for light from a point source that is deflected by an ellipsoid, an elliptic paiaboloid, and an elliptic cone. As a numerical example caustic surfaces are calculated for a concave spherical surface and a concave paraboloid.

© 1973 Optical Society of America

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References

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  1. D. G. Burkhard, D. L. Shealy, J. Opt. Soc. Am. 63, 299 (1973).
    [Crossref]
  2. D. L. Shealy, D. G. Burkhard, Opt. Acta 20, (1973).
    [Crossref]
  3. Erwin Kreyszig, Introduction to Differential Geometry and Riemannian Geometry (University of Toronto Press, Toronto, 1968).
  4. Von Z. Bartkowski, Optik 18, 22 (1961).
  5. Von Z. Bartkowski, Optik 19, 226 (1962).
  6. H. M. A. El-Sum, J. Opt. Soc. Am. 62, 1375A (1972).

1973 (2)

1972 (1)

H. M. A. El-Sum, J. Opt. Soc. Am. 62, 1375A (1972).

1962 (1)

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

1961 (1)

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

Bartkowski, Von Z.

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

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

Burkhard, D. G.

El-Sum, H. M. A.

H. M. A. El-Sum, J. Opt. Soc. Am. 62, 1375A (1972).

Kreyszig, Erwin

Erwin Kreyszig, Introduction to Differential Geometry and Riemannian Geometry (University of Toronto Press, Toronto, 1968).

Shealy, D. L.

J. Opt. Soc. Am. (2)

D. G. Burkhard, D. L. Shealy, J. Opt. Soc. Am. 63, 299 (1973).
[Crossref]

H. M. A. El-Sum, J. Opt. Soc. Am. 62, 1375A (1972).

Opt. Acta (1)

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

Optik (2)

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

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

Other (1)

Erwin Kreyszig, Introduction to Differential Geometry and Riemannian Geometry (University of Toronto Press, Toronto, 1968).

Cited By

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Figures (5)

Fig. 1
Fig. 1

Concave spherical mirror with origin of coordinate system at center of curvature. Light source is at (x0, O, z0).

Fig. 2
Fig. 2

Intersection with the xz plane of the caustic surface formed by reflection of light from a point source on the principal axis for a concave spherical mirror. One of the caustic surfaces is a straight line along the principal axis.

Fig. 3
Fig. 3

Same as Fig. 2 with light source displaced from principal axis by 0.2 radii.

Fig. 4
Fig. 4

Intersection of the caustic surface with the xz plane of a caustic surface formed by reflection of light from a point source on the principal axis of a concave paraboloid. For this case one of the caustic surfaces degenerates to a straight line (caustic spike) along the principal axis.

Fig. 5
Fig. 5

Same as Fig. 4 with light source displaced from principal axis by 0.2 radii.

Equations (40)

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F d S 1 - d S 2 = p σ cos ϕ i cos ψ a 0 + r 2 a 1 + ( r 2 ) 2 a 2 ,
a 0 = cos ϕ i
a 1 = - 2 ( 2 H cos 2 ϕ i + K N sin 2 ϕ i ) + 2 cos ϕ i / r 1
a 2 = 4 K cos ϕ i - 2 ( 2 H cos 2 ϕ i + K N sin 2 ϕ i ) r 1 + cos ϕ i / ( r 1 ) 2 ,
a 0 = cos ϕ s - [ 1 - ( n 0 / n 1 ) 2 sin 2 ϕ i ] 1 / 2 ,
a 1 = - [ ( n 0 cos ϕ i / n 1 cos ϕ s ) + 1 ] [ 2 H cos 2 ϕ s + ( n 0 / n 1 ) 2 K N sin 2 ϕ i ] + ( n 0 / n 1 ) ( cos 2 ϕ i + cos 2 ϕ s ) / ( r 1 cos ϕ s ) ,
a 2 = { [ ( n 0 / n 1 ) cos ϕ i + cos ϕ s ] 2 K - ( n 0 / n 1 ) [ ( n 0 / n 1 ) cos ϕ i + cos ϕ s ] ( 2 H cos 2 ϕ i + K N sin 2 ϕ i ) / r 1 + ( n 0 / n 1 ) 2 cos 2 ϕ i / ( r 1 ) 2 } / cos ϕ s ,
x ( u , v ) = x ( u , v ) I + y ( u , v ) J + z ( u , v ) K ,
K = b / g [ b u u b v v - ( b u v ) 2 ] / [ g u u g v v - ( g u v ) 2 ] ,
H = ( g u u b v v - 2 g u v b u v + g v v b u u ) / 2 g ,
K N sin 2 ϕ i = a u a u b u u + 2 a u a v b u v + a v a v b v v ,
g u v ( x / u ) · ( x / v ) ,
b u v = g - 1 / 2 [ ( x / u ) × ( x / v ) · 2 x / u v ] ,
a w = g w u a · ( x / u ) + g w v a · ( x / v ) ( w = u , v ) ,
g u u = g v v / g ; g v v = g u u / g ; g u v = - g u v / g .
a 0 + r 2 a 1 + ( r 2 ) 2 a 2 = 0.
X C = x ( u , v ) + r 2 ( u , v ) A ( u , v ) .
x ( u , v ) = A sin u cos v I + B sin u sin v J + C cos u K ,
N 1 = ( B C sin 2 u cos v I + A C sin 2 u sin v J + A B sin u cos u K ) / g 1 / 2 ,
g u u = A 2 cos 2 u cos 2 v + B 2 cos 2 u sin 2 v + C 2 sin 2 u , g v v = A 2 sin 2 u sin 2 v + B 2 sin 2 u cos 2 v , g u v = ( B 2 - A 2 ) sin u cos u sin v cos v , g = A 2 B 2 sin 2 u cos 2 u + A 2 C 2 sin 4 u sin 2 v + B 2 C 2 sin 4 u cos 2 v ,
b u u = - A B C sin u / g 1 / 2 , b v v = - A B C sin 3 u / g 1 / 2 , b u v = 0 , b = A 2 B 2 C 2 sin 4 u / g .
K = ( A B C ) sin 4 u / g 2 ,
H = - [ A 2 B C ( cos 2 u cos 2 v + sin 2 v ) + A B 3 C ( cos 2 u sin 2 v + cos 2 v ) + A B C 3 sin 2 u ] sin 3 u / 2 g 3 / 2 ,
K N sin 2 ϕ i = - A B C sin 5 u { [ a x A B 2 sin u cos v + a y A 2 B sin u sin v - a z C ( A 2 sin 2 v + B 2 cos 2 v ) ] 2 + [ - a x A sin v ( B 2 cos 2 u + C sin 2 u ) + a y B cos v ( A 2 cos 2 u + C 2 sin 2 u ) - a z C ( A 2 - B 2 ) sin u cos u sin v cos v ] 2 } / g 5 / 2 ,
K = 1 / R 2 H = - 1 / R , K N = - 1 / R .             ( sphere )
x ( u , v ) = A u cos v I + B u sin v J + u 2 K ,
N 1 = [ - 2 B u 2 cos v I - 2 A u 2 sin v J + A B u K ] / g 1 / 2 .
g u u = A 2 cos 2 v + B 2 sin 2 v + 4 u 2 , g v v = A 2 u 2 sin 2 v + B 2 u 2 cos 2 v , g u v = ( B 2 - A 2 ) u sin v cos v , g = ( A 2 B 2 u 2 + 4 u 4 ( A 2 sin 2 v + B 2 cos 2 v ) ,
b u u = 2 A B u / g 1 / 2 , b u v = 2 A B u 3 / g 1 / 2 , b u v = 0 , b = 4 A 2 B 2 u 4 / g .
K = 4 A 2 B 2 u 4 / g 2
H = A B u 3 ( A 2 + B 2 + 4 u 2 ) / g 3 / 2 ,
K N sin 2 ϕ = 2 A B u 5 { [ a x A B 2 cos v + a y A 2 B sin v + 2 a z u ( A 2 sin 2 v + B 2 cos 2 v ) ] 2 + [ - a x A sin v ( B 2 + 4 u 2 ) + a y B cos v ( A 2 + 4 u 2 ) + 2 a z u ( A 2 - B 2 ) sin v cos v ] 2 } / g 5 / 2 .
A = B = [ sin 30° / ( 1 - cos 30° ) 1 / 2 ] 13.6.
x ( u , v ) = A u cos v I + B u sin v J + C u K ,
N 1 = ( - B C u cos v I - A C u sin v J + A B u K ) / g 1 / 2 ,
g u u = A 2 cos 2 v + B 2 sin 2 v + C 2 , g v v = A 2 u 2 sin 2 v + B 2 u 2 cos 2 v , g u v = - A 2 u sin v cos v + B 2 u sin v cos v , g = A 2 B 2 u 2 + C 2 u 2 ( A 2 sin 2 v + B 2 cos 2 v ) ,
b u u = 0 , b v v = A B C u 2 / g 1 / 2 , b u v = 0 , b = 0 ,
K = 0 ,
H = A B C u 2 ( A 2 cos 2 v + B 2 sin 2 v + C 2 ) / 2 g 3 / 2 .
K N sin 2 ϕ i = A B C u 4 [ - a x A ( B 2 + C 2 ) sin v + a y B ( A 2 + C 2 ) cos v + a z C ( A 2 - B 2 ) sin v cos v ] 2 g 5 / 2 .

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