Abstract

The geometrical flux density (irradiance) is singular over caustic surfaces and, therefore, cannot be used effectively as a measure of the concentration of rays at or near the caustic surfaces. A finite substitute measure, the density of rays tangent to the caustic, may be obtained by dividing an element of incident flux by the area of the caustic formed by the associated rays. This gives a measure of the energy density over different regions of the caustic. As an example, the ray density over the caustic is evaluated for collimated light reflected from a spherical mirror. A similar calculation is performed for collimated light refracted by a plano-convex singlet lens. General formulas are presented for computing the ray density over the caustic for reflection of meridional rays by an aspheric surface. Also analytical and numerical algorithms are given for evaluating the ray density over the caustic in a multiinterface optical system.

© 1982 Optical Society of America

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References

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  1. D. G. Burkhard, D. L. Shealy, Appl. Opt. 20, 897 (1981).
    [CrossRef] [PubMed]
  2. M. Herzberger, Modern Geometrical Optics (Wiley-Interscience, New York, 1958).
  3. O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972).
  4. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 127.
  5. E. Buchove, J. Opt. Soc. Am. 69, 891 (1979).
    [CrossRef]
  6. M. V. Berry, Adv. Phys. 25, 1 (1976).
    [CrossRef]
  7. D. L. Shealy, Appl. Opt. 15, 2588 (1976).
    [CrossRef] [PubMed]
  8. Ref. 1, p. 905.
  9. D. L. Shealy, D. G. Burkhard, Opt. Acta 22, 485 (1975).
    [CrossRef]
  10. D. G. Burkhard, Appl. Opt. 19, 3682 (1980).
    [CrossRef] [PubMed]
  11. G. Dahlquist, A. Björch, Numerical Methods (Prentice-Hall, Englewood Cliffs, N.J., 1974), p. 342.
  12. W. H. Beyer, CRC Handbook of Mathematical Series (CRC Press, West Palm Beach, Fla.1978), pp. 647–649.
  13. O. N. Stavroudis, L. E. Sutton, “Spot Diagrams for the Prediction of Lens Performance from Design Data,” Natl. Bur. Stand. U.S. Monogr. 93 (1965).
  14. S. C. Parker, “Properties and Applications of Generalized Ray Tracing,” Technical Report 71, Optical Sciences Center, U. Arizona, Tucson (1971).

1981 (1)

1980 (1)

1979 (1)

1976 (2)

1975 (1)

D. L. Shealy, D. G. Burkhard, Opt. Acta 22, 485 (1975).
[CrossRef]

1965 (1)

O. N. Stavroudis, L. E. Sutton, “Spot Diagrams for the Prediction of Lens Performance from Design Data,” Natl. Bur. Stand. U.S. Monogr. 93 (1965).

Berry, M. V.

M. V. Berry, Adv. Phys. 25, 1 (1976).
[CrossRef]

Beyer, W. H.

W. H. Beyer, CRC Handbook of Mathematical Series (CRC Press, West Palm Beach, Fla.1978), pp. 647–649.

Björch, A.

G. Dahlquist, A. Björch, Numerical Methods (Prentice-Hall, Englewood Cliffs, N.J., 1974), p. 342.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 127.

Buchove, E.

Burkhard, D. G.

Dahlquist, G.

G. Dahlquist, A. Björch, Numerical Methods (Prentice-Hall, Englewood Cliffs, N.J., 1974), p. 342.

Herzberger, M.

M. Herzberger, Modern Geometrical Optics (Wiley-Interscience, New York, 1958).

Parker, S. C.

S. C. Parker, “Properties and Applications of Generalized Ray Tracing,” Technical Report 71, Optical Sciences Center, U. Arizona, Tucson (1971).

Shealy, D. L.

Stavroudis, O. N.

O. N. Stavroudis, L. E. Sutton, “Spot Diagrams for the Prediction of Lens Performance from Design Data,” Natl. Bur. Stand. U.S. Monogr. 93 (1965).

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972).

Sutton, L. E.

O. N. Stavroudis, L. E. Sutton, “Spot Diagrams for the Prediction of Lens Performance from Design Data,” Natl. Bur. Stand. U.S. Monogr. 93 (1965).

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 127.

Adv. Phys. (1)

M. V. Berry, Adv. Phys. 25, 1 (1976).
[CrossRef]

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

Natl. Bur. Stand. U.S. Monogr. 93 (1)

O. N. Stavroudis, L. E. Sutton, “Spot Diagrams for the Prediction of Lens Performance from Design Data,” Natl. Bur. Stand. U.S. Monogr. 93 (1965).

Opt. Acta (1)

D. L. Shealy, D. G. Burkhard, Opt. Acta 22, 485 (1975).
[CrossRef]

Other (7)

Ref. 1, p. 905.

M. Herzberger, Modern Geometrical Optics (Wiley-Interscience, New York, 1958).

O. N. Stavroudis, The Optics of Rays, Wavefronts, and Caustics (Academic, New York, 1972).

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1965), p. 127.

S. C. Parker, “Properties and Applications of Generalized Ray Tracing,” Technical Report 71, Optical Sciences Center, U. Arizona, Tucson (1971).

G. Dahlquist, A. Björch, Numerical Methods (Prentice-Hall, Englewood Cliffs, N.J., 1974), p. 342.

W. H. Beyer, CRC Handbook of Mathematical Series (CRC Press, West Palm Beach, Fla.1978), pp. 647–649.

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Figures (6)

Fig. 1
Fig. 1

Caustic surface formed when collimated light is reflected from a concave spherical mirror.

Fig. 2
Fig. 2

Caustic surfaces and ray density over caustic surfaces for collimated light reflected by spherical mirror with unit reflectivity ρ. σ is the illuminance of the incident beam.

Fig. 3
Fig. 3

Spherical mirror; collimated incident light. Coordinates shown are employed in the general formula for density of tangent rays over the caustic.

Fig. 4
Fig. 4

Refraction of collimated beam by plano-convex singlet lens. Top figure shows density of tangent rays over the tangential caustic. The lower figure shows ray density over the sagittal caustic.

Fig. 5
Fig. 5

(a) Density of tangent rays over the tangential caustic for the ME-10 aerial camera to lens12; (b) ray density over the sagittal caustic for the ME-10.

Fig. 6
Fig. 6

Caustics formed when light from a point source is reflected from a spherical mirror.

Equations (120)

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E out = σ ρ cos φ i d S .
x c ( u , v ) = x ( u , v ) + r c ( u , v ) A ^ ( u , v ) .
F c = E out / d S c = ρ cos φ i d S / d S c = ρ cos φ i g 1 / 2 ( u , v ) g c 1 / 2 ( u , v ) ,
d S = | x u × x v | d u d v = g 1 / 2 d u d v , g = g u u g v v - g u v 2 ,             g u v = x u · x v
g u v ( c ) = ( x u + A ^ r c u + r c A ^ u ) · ( x v + A ^ r c v + r c A ^ v ) .
1 r c = γ - r 0 + Ω R ,
1 r c = γ cos 2 φ i - r 0 cos 2 φ s + Ω R cos 2 φ s ,
1 r c = 1 ( - r 0 ) - 2 cos φ i R ,
1 r c = 1 ( - r 0 ) - 2 R cos φ i ,
d F = σ d S = σ 2 π R 2 sin θ cos θ d θ ,
x c = R ( - sin θ + cos θ 2 sin 2 θ ) = - R sin 3 θ ,
z c = ( - cos θ + cos θ 2 cos 2 θ ) = R cos θ 2 ( - 3 + 2 cos 2 θ ) ,
d l c = ± [ ( d x c d θ ) 2 + ( d z c d θ ) 2 ] 1 / 2 d θ
d l c = - 3 2 R sin θ d θ ,
d S c = 2 π x c d l c = 3 π R 2 sin 4 θ d θ
F c = d F d S c = 2 σ ρ cos θ 3 sin 3 θ .
F c r 2 π x c F c = - 4 π 3 σ ρ R cos θ .
0 φ F c r d l c = 2 π σ ρ R 2 0 θ cos θ sin θ d θ = π σ ρ R 2 sin 2 θ
x c = 0             z c = - R / 2 cos .
d S c = d l c = d z c d θ = - R 2 sin θ cos 2 θ d θ , F c = d F d S = - 4 π σ ρ R cos 3 θ .
x = - R sin θ cos φ I ^ - R sin θ sin φ J ^ - R cos θ K ^ ;
x θ = x θ = - R cos θ cos φ I ^ - R cos θ sin φ J ^ + R sin θ K ^ ,
x φ = x φ = + R sin θ sin φ I ^ - R sin θ cos φ J ^ .
A ^ = - sin 2 θ cos φ I ^ - sin 2 θ sin φ J ^ - cos 2 θ K ^
A ^ θ A ^ θ = - 2 cos 2 θ cos φ I ^ - 2 cos 2 θ sin φ J ^ + 2 sin 2 θ K ^ ,
A ^ ϕ A ^ φ = sin 2 θ sin φ I ^ - sin 2 θ cos φ J ^ .
r c = - R cos θ 2 .
r c θ = R sin θ 2 and r c φ = 0.
g φ φ ( c ) = R 2 sin 6 θ ,
g θ θ ( c ) = 9 R 2 cos 2 θ sin 4 θ + R 2 sin 2 θ 4 ( 3 - 6 cos 2 θ ) 2 ,
g θ φ ( c ) = 0 ,
g ( c ) = 9 4 R 4 sin 8 θ .
g θ θ = R 2 ;             g φ φ = R 2 sin 2 θ ;             g θ φ = 0 ,
g = R 4 sin 2 θ .
F c = 2 σ ρ cos θ 3 sin 3 θ ,
r c = - R 2 cos θ ,
r c θ = - R 2 sin θ cos 2 θ ,             r c φ = 0.
g θ θ ( c ) = R 2 sin 2 θ cos 2 2 θ 4 cos 2 θ ,
g φ φ ( c ) = 0 ,             g θ φ ( c ) = 0 ,
g ( c ) = 0.
F c = σ ρ R 2 sin θ cos θ d θ x c d l c ,
x c = - R sin θ - r c sin ( φ s - θ ) ,
z c = - R cos θ + r c cos ( φ s - θ ) ,
r c = R cos 2 φ s - n cos θ + cos φ s ,
sin φ s = n sin θ .
d x c d θ = - R cos θ - sin ( φ s - θ ) d r c d θ - r c cos ( φ s - θ ) ( d φ s d θ - 1 ) ,
d z c d θ = R sin θ + cos ( φ s - θ ) d r c d θ - r c sin ( φ s - θ ) ( d φ s d θ - 1 ) ,
d r c d θ = R n sin θ [ n 2 ( 1 + cos 2 θ ) - n cos θ cos φ s - 1 ] ( - n cos θ + cos φ s ) 2 ,
d φ s d θ = n cos θ cos φ s .
F c r = 2 π x c F c ,
F c = 2 π σ ρ R 2 sin θ cos θ ( d z c / d θ ) ,
x c = 0
z c = - R cos θ + r c cos ( φ s - θ ) ,
r c = R - n cos θ + cos φ s ,
z c d θ = R sin θ + d r c d θ cos ( φ s - θ ) - r c sin ( φ s - θ ) ( d φ s d θ - 1 ) ,
d r c d θ = R n sin θ ( d φ s d θ - 1 ) ( - n cos θ + cos φ s ) 2 .
z = z ( R ) .
a = - I ^ sin α - K ^ cos α .
x ( R , φ ) = I ^ R cos φ + J ^ R sin φ + K ^ z ( R ) .
x R x / R = I ^ cos φ + J ^ sin φ + K ^ z ,
x φ x / φ = - I ^ R sin φ + J ^ R cos φ ,
x R R 2 x / R 2 = K ^ z ,
x φ φ 2 x / φ 2 = - I ^ R cos φ - J ^ R sin φ ,
x R φ 2 x / R φ = - I ^ sin φ + J ^ cos φ ,
N ^ = ( - I ^ z cos φ - J ^ z sin φ + K ^ ) / ( 1 + z 2 ) 1 / 2 .
A ^ = a - 2 N ^ ( a · N ^ ) , A ^ = { I ^ [ - ( 1 + z 2 cos 2 φ ) sin α - 2 z cos φ cos α ] + J ^ ( z 2 sin 2 φ sin α - 2 z sin φ cos α ) + K ^ [ ( 1 - z 2 ) cos α - 2 z cos φ sin α ] } / ( 1 + z 2 ) .
g R R = x R · x R = 1 + z 2 ; g φ φ = R 2 ; g R φ = 0 ; b R R = x R R · N ^ = z / ( 1 + z 2 ) 1 / 2 ;
b φ φ = x φ φ · N ^ = R z / ( 1 + z 2 ) 1 / 2 ; b R φ = x R φ · N ^ = 0.
κ R = b R R g R R = z ( 1 + z 2 ) 1 / 2 ,
κ φ = b φ φ g φ φ = z R ( 1 + z 2 ) 1 / 2 .
1 r c = 2 z cos φ i R ( 1 + z 2 ) 1 / 2             1 r c = 2 z ( 1 + z 2 ) 3 / 2 cos φ i ,
cos φ i = - a · N ^ = - z cos φ sin α + cos α ( 1 + z 2 ) 1 / 2 ,
r c = R ( 1 + z 2 ) 2 z ( - z cos φ sin α + cos α ) ,
r c = ( 1 + z 2 ) ( - z cos φ sin α + cos α ) 2 z ,
x c = x ( R , φ ) + r c ( R , φ ) A ( r , φ ) ,             φ = 0 , 180 ° ,
x c R = x R + ( r c A ^ ) R ,
x c φ = x φ + ( r c A ^ ) φ .
r c A ^ = R A ¯ / [ 2 z ( - z cos φ sin α + cos α ) ] ,
r c A ^ = ( - z cos φ sin α + cos α ) A ¯ / 2 z ,
g u v ( c ) = x u · x v + x u · ( r c A ^ ) / v + x v · ( r c A ^ ) / u + [ ( r c A ^ ) / u ] · [ ( r c A ^ ) / v ] ,
( r c A ^ ) φ = A ¯ ( z sin φ sin α ) 2 z + w 2 z A ¯ φ ,
( r c A ^ ) R = A ¯ ( [ - ( z ) 2 cos φ sin α - z w ] 2 ( z ) 2 + w 2 z A ¯ R ,
( r c A ^ ) φ = - R z sin φ sin α 2 z w 2 A ¯ + R 2 z w A ¯ φ ,
( r c A ^ ) R = { z w - R [ z w - z z cos φ sin α ] } A ¯ 2 ( z ) 2 w 2 + R 2 z w A ^ R ,
A ¯ φ = I ^ [ 2 ( z ) 2 sin α sin 2 α + 2 z cos α sin φ ] + J ^ [ 2 ( z ) 2 sin α cos 2 φ - 2 z cos α cos φ ] + K ^ ( 2 z sin α sin φ ) ,
A ¯ φ = I ^ ( - 2 z z sin α cos 2 φ - 2 z cos α cos φ ) + J ^ ( 2 z z sin α sin 2 φ - 2 z cos α sin φ ) + K ^ ( - 2 z z cos α - 2 z sin α cos φ ) .
z = c R 2 / [ 1 + ( 1 - c 2 R 2 ) 1 / 2 ] , c = curvature of sphere , 1 / radius , z = c R / ( 1 - c 2 R 2 ) 1 / 2 , z = c / ( 1 - c 2 R 2 ) 3 / 2 , z = 3 c 3 R / ( 1 - c 2 R 2 ) 5 / 2 ;
z = R 2 2 p - p 2 , z = R / p , z = 1 / p , z = 0 ;
z = ± a ( 1 - R 2 / b 2 ) 1 / 2 , z = ± a R / b ( b 2 - R 2 ) 1 / 2 , z = ± a b / ( b 2 - R 2 ) 3 / 2 , z = ± 3 a b R / ( b 2 - R 2 ) 5 / 2 ;
z 2 a 2 - R 2 b 2 = 1 , z = a ( 1 + R 2 / b 2 ) 1 / 2 , z = ± a R / b ( b 2 + R 2 ) 1 / 2 , z = ± a b / ( b 2 - R 2 ) 3 / 2 , z = ± 3 a b R / ( b 2 - R 2 ) 5 / 2 .
E out = σ ( i ) ρ ( i ) cos φ i ( i ) d S ( i ) .
F c = σ ( i ) ρ ( i ) cos φ i ( i ) g 1 / 2 ( i ) g 1 / 2 ( c ) .
I 0 ( u , v ) + r c I 1 ( u , v ) + r c 2 I 2 ( u , v ) = 0 ,
I 0 = A ^ · x u × x v = L 0 g 1 / 2 ,
I 1 = A ^ · ( x u × A ^ u + A ^ u × x v ) = L 1 g 1 / 2 ,
I 2 = A ^ · A ^ u × A ^ v = L 2 g 1 / 2 .
d S c = g 1 / 2 ( c ) d u d v = [ g u u ( c ) g v v ( c ) - g v v 2 ( c ) ] 1 / 2 d u d v ,
g u v ( c ) = ( x u + A ^ r c u + r c A ^ u ) · ( x u + A ^ r c v + r c A ^ v ) .
r c ω = ( I 0 ω + r c I 1 ω + r c 2 I 2 ω ) I 1 + 2 r c I 2 ,
I 0 ω = A ^ ω · x u × x v + A ^ · x u ω × x v + A ^ · x u × x v ω ,
I 1 ω = A ^ ω · [ x u × A ^ v + A ^ u × x v ) + A ^ · ( x u ω × A ^ v + x u × A ^ v ω + A ^ u ω × x v + A ^ u × x v ω ) ,
I 2 ω = A ^ · A ^ u ω × A ^ v + A ^ · A ^ u × A ^ v ω .
x c ( u , v ) u = 1 2 Δ u [ - x c ( u + 2 Δ u , v ) + 4 x c ( u + Δ u , v ) - 3 x c ( u , v ) ] ,
x c ( u , v ) v = 1 2 Δ v [ - x c ( u , v + 2 Δ v ) + 4 x c ( u , v + Δ v ) - 3 x c ( u , v ) ] ,
r c = - R ( R / r 0 ) + 2 cos φ
r c = - R cos φ 2 + ( R / r 0 ) cos φ ,
x c = 0 ,
z c = - R [ cos ( α + φ ) - cos ( 2 φ + α ) A ] ,
x c = - R [ sin ( α + φ ) - cos φ sin ( 2 φ + α ) A ] ,
z c = R [ cos ( α + φ ) - cos φ cos ( 2 φ + α ) A ] ,
- R r 0 = sin α sin ( α + φ ) ,             d - R = sin φ sin α ,
d α d φ = - R cos φ d cos α ,
d d φ ( R r 0 ) = - [ sin φ d α d φ - sin α cos ( α + φ ) ] / sin 2 ( α + φ ) ,
d A d φ = + [ - 2 sin φ + d d φ ( R r 0 ) ] ,
d A d φ = - R r 0 sin φ + cos φ d d φ ( R r 0 ) .
d l c = z c φ d φ = ( - R ) [ - sin ( α + φ ) ( 1 + d α d φ ) + sin ( 2 φ + α ) A ( 2 + d α d φ ) + cos ( 2 φ + α ) A 2 d A d φ ] d φ ,
d α d φ = 0 ,             A = 2 cos φ ,             d d φ R r 0 = 0 ,             d A d φ = - 2 sin φ ,
d l c = - R sin φ d φ cos 2 φ ,
( d l c ) 2 = [ ( d x c d φ ) 2 + ( d z c d φ ) 2 ] d 2 φ = { ( 1 + d α d φ ) 2 + 1 A 2 [ sin 2 φ + cos 2 φ ( 2 + d α d φ ) 2 ] + 1 A 4 ( d A d φ ) 2 cos 2 φ + 2 A ( 1 + d α d φ ) [ sin 2 φ - ( 2 + d α d φ ) cos 2 φ ] + 2 A 2 d A d φ sin φ cos φ ( 1 + d α d φ ) + 2 A 3 d A d φ sin φ cos φ } ( - R d φ ) 2 .
d l c = - 3 R 2 sin φ d φ ,

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