Abstract

A general formula is derived that specifies the illumination (flux density) over an arbitrary receiver surface when light rays are reflected by or refracted through a curved surface. The direction of the deflected ray and its intersection with the receiving surface, used with the equation for the surfaces, lead to a transformation that maps an element of deflecting area onto the receiving area, by means of the jacobian determinant. A formula for the flux density along a ray path follows as a special case. An equation for the caustic surface is obtained from the latter. As an example radiation flux-density contours are calculated for a plane wave reflected from a sphere. Flux density and the caustic surface are calculated for a plane wave reflected onto a plane from a concave spherical lens and also for a plane wave refracted onto a plane through a hemisphere.

© 1973 Optical Society of America

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References

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  1. Max Herzberger, Modern Geometrical Optics (Wiley–Interscience, New York, 1958).
  2. E. R. G. Eckert and E. M. Sparrow, Int. J. Heat Mass Transfer 3.1, 42 (1961).
    [Crossref]
  3. S. H. Lin and E. M. Sparrow, J. Heat Transfer 87C, 299 (1965).
    [Crossref]
  4. J. A. Plamondon and T. E. Horton, Int. J. Heat Mass Transfer 10, 665 (1967).
    [Crossref]
  5. Joseph B. Keller and Herbert B. Keller, J. Opt. Soc. Am. 40, 48 (1950).
    [Crossref]
  6. Joseph B. Keller and William Streifer, J. Opt. Soc. Am. 61, 40 (1971).
    [Crossref]
  7. V. A. Fock, Electromagnetic Diffraction and Propagation Problems (Pergamon, New York, 1965).
  8. Hilbert Schenck, J. Opt. Soc. Am. 47, 653 (1957).
    [Crossref]
  9. Van Z. Barkowski, Optik 18, 22 (1961).
  10. Van Z. Barkowski, Optik 19, 226 (1962).
  11. M. M. Lipschutz, Differential Geometry (Schaum’s Outline Series)(McGraw–Hill, New York, 1969).

1971 (1)

1967 (1)

J. A. Plamondon and T. E. Horton, Int. J. Heat Mass Transfer 10, 665 (1967).
[Crossref]

1965 (1)

S. H. Lin and E. M. Sparrow, J. Heat Transfer 87C, 299 (1965).
[Crossref]

1962 (1)

Van Z. Barkowski, Optik 19, 226 (1962).

1961 (2)

Van Z. Barkowski, Optik 18, 22 (1961).

E. R. G. Eckert and E. M. Sparrow, Int. J. Heat Mass Transfer 3.1, 42 (1961).
[Crossref]

1957 (1)

1950 (1)

Barkowski, Van Z.

Van Z. Barkowski, Optik 19, 226 (1962).

Van Z. Barkowski, Optik 18, 22 (1961).

Eckert, E. R. G.

E. R. G. Eckert and E. M. Sparrow, Int. J. Heat Mass Transfer 3.1, 42 (1961).
[Crossref]

Fock, V. A.

V. A. Fock, Electromagnetic Diffraction and Propagation Problems (Pergamon, New York, 1965).

Herzberger, Max

Max Herzberger, Modern Geometrical Optics (Wiley–Interscience, New York, 1958).

Horton, T. E.

J. A. Plamondon and T. E. Horton, Int. J. Heat Mass Transfer 10, 665 (1967).
[Crossref]

Keller, Herbert B.

Keller, Joseph B.

Lin, S. H.

S. H. Lin and E. M. Sparrow, J. Heat Transfer 87C, 299 (1965).
[Crossref]

Lipschutz, M. M.

M. M. Lipschutz, Differential Geometry (Schaum’s Outline Series)(McGraw–Hill, New York, 1969).

Plamondon, J. A.

J. A. Plamondon and T. E. Horton, Int. J. Heat Mass Transfer 10, 665 (1967).
[Crossref]

Schenck, Hilbert

Sparrow, E. M.

S. H. Lin and E. M. Sparrow, J. Heat Transfer 87C, 299 (1965).
[Crossref]

E. R. G. Eckert and E. M. Sparrow, Int. J. Heat Mass Transfer 3.1, 42 (1961).
[Crossref]

Streifer, William

Int. J. Heat Mass Transfer (2)

J. A. Plamondon and T. E. Horton, Int. J. Heat Mass Transfer 10, 665 (1967).
[Crossref]

E. R. G. Eckert and E. M. Sparrow, Int. J. Heat Mass Transfer 3.1, 42 (1961).
[Crossref]

J. Heat Transfer (1)

S. H. Lin and E. M. Sparrow, J. Heat Transfer 87C, 299 (1965).
[Crossref]

J. Opt. Soc. Am. (3)

Optik (2)

Van Z. Barkowski, Optik 18, 22 (1961).

Van Z. Barkowski, Optik 19, 226 (1962).

Other (3)

M. M. Lipschutz, Differential Geometry (Schaum’s Outline Series)(McGraw–Hill, New York, 1969).

V. A. Fock, Electromagnetic Diffraction and Propagation Problems (Pergamon, New York, 1965).

Max Herzberger, Modern Geometrical Optics (Wiley–Interscience, New York, 1958).

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Figures (6)

F. 1
F. 1

Geometrical configuration for source, deflector, and receiver.

F. 2
F. 2

Contours of equal flux density (irradiance) for plane wave reflected from sphere to plane Z = −0.5, in units of the radius of the sphere. Flux-density values are expressed in terms of percent of incident flux density. Reflectance is taken as 1 for all angles of incidence.

F. 3
F. 3

Intersection of caustic surface with ZX plane for reflected plane wave propagating from negative-Z direction and incident upon concave spherical mirror with center at Z = 0 and principal axis in the Z direction. Caustic surface has rotational symmetry about the Z axis.

F. 4
F. 4

Flux contours plotted in one quadrant for a plane wave incident upon a concave mirror. Receiver plane is located at paraxial point Z = −0.5 in units of the radius of the sphere.

F. 5
F. 5

Intersection of caustic surface with ZX plane for refracted plane wave incident upon quartz hemisphere. (n1/n0 = 1.544). Center of sphere at Z = 0. The light is incident from the positive-z direction. Caustic surface has rotational symmetry about the Z axis, which is the principal axis of the lens. Caustic surface lies between z = −0.5 and the paraxial focal point Z = −1.838. The caustic spike extends from Z = −0.85 to Z = −1.838.

F. 6
F. 6

Contours of equal flux density for collimated light refracted by quartz hemisphere of index of refraction 1.544 to plane Z = −1.838 at the paraxial focal point in units of radius of sphere. Flux-density values are expressed in terms of percent of incident flux density. Transmittance is taken as 1 for all angles of incidence.

Equations (39)

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F ζ ( u , υ ) [ X ( U , V ) x ( u , υ ) ] ξ ( u , υ ) [ Z ( U , V ) z ( u , υ ) ] = 0 ,
G ζ ( u , υ ) [ Y ( U , V ) y ( u , υ ) ] η ( u , υ ) [ Z ( U , V ) z ( u , υ ) ] = 0 ,
A = a 2 N 1 ( a N 1 ) ,
A = n 0 n 1 a + [ n 0 n 1 cos ϕ i cos ϕ s ] N 1 ,
cos ϕ s = [ 1 ( n 0 / n 1 ) 2 sin 2 ϕ i ] 1 2 .
F inc = σ cos ϕ i d S 1 ,
F d S 1 d S 2 = ρ σ cos ϕ i d S 1 / d S 2 ;
d S 1 = | ( x / u ) × ( x / υ ) | d u d υ , d S 2 = | ( X / U ) × ( X / V ) | d U d V .
d U d V d u d υ = U u V υ U υ V u = ( U , V ) ( u , υ ) ,
d S 2 d S 1 = | ( X / U ) × ( X / V ) | | ( x / u ) × ( x / υ ) | d ( U , V ) d ( u , υ ) .
F ( u , υ ; U , V ) = 0 , G ( u , υ ; U , V ) = 0 ,
( U , V ) ( u , υ ) = ( F , G ) ( u , υ ) / ( F , G ) ( U , V ) .
d F = F u d u + F υ d υ + F U d U + F V d V = 0 , d G = G u d u + G υ d υ + G U d U + G V d V = 0 .
F = F ( x , z , ξ , ζ ; X , Z ) = 0 ,
G = G ( y , z , η , ζ ; Y , Z ) = 0
F ω = ζ x ω + ξ z ω ( Z z ) ξ ω + ( X x ) ζ ω ( ω = u , υ ) , G ω = ζ y ω + η z ω ( Z z ) η ω + ( Y y ) ζ ω ( ω = u , υ ) ,
F Ω = ζ Y Ω ξ Z Ω ( Ω = U , V ) , G Ω = ζ Y Ω η Z Ω ( Ω = U , V ) .
( F , G ) ( u , υ ) = ζ [ I 0 + r 2 I 1 + ( r 2 ) 2 I 2 ] ,
r 2 = ( Z z ) / ζ , I 0 = A ( x / u ) × ( x / υ ) ,
I 1 = A [ ( x / u ) × ( A / υ ) + ( A / u ) × ( x / υ ) ] ,
I 2 = A ( A / u ) × ( A / υ ) ,
( F , G ) ( U , V ) = ζ A X U × X V .
F d S 1 d S 2 = σ ρ | ( x / u ) × ( x / υ ) | cos ϕ i cos ψ | I 0 + r 2 I 1 + ( r 2 ) 2 I 2 | ,
cos ψ = A ( X / U ) × ( X / V ) | ( X / U ) × ( X / V ) |
I 0 + r 2 I 1 + ( r 2 ) 2 I 2 = 0 .
X c = x ( u , υ ) + r 2 ( u , υ ) A ( u , υ ) .
x = R sin θ cos ϕ I + R sin θ sin θ J + R cos θ K , X = X J + Y J + ( p a X b Y ) K ,
N 1 = sin θ cos ϕ I + sin θ sin ϕ J + cos θ K , N 2 = ( a I + b J + K ) / ( 1 + a 2 + b 2 ) 1 2 , a = α I β J γ K .
ξ = α ( 2 sin 2 θ cos 2 ϕ 1 ) + 2 β sin 2 θ sin ϕ cos ϕ + 2 γ sin θ cos θ cos ϕ , η = 2 α sin 2 θ sin ϕ cos ϕ + β ( 2 sin 2 θ sin 2 ϕ 1 ) + 2 γ sin θ cos θ sin ϕ , ζ = 2 α sin θ cos θ cos ϕ + 2 β sin θ cos θ sin ϕ + γ ( 2 cos 2 θ 1 ) .
I 0 = R 2 sin θ [ α sin θ cos ϕ + β sin θ sin ϕ + γ cos θ ] , I 1 = 2 R sin θ { α 2 ( 1 + sin 2 θ cos 2 ϕ ) + 2 α β sin 2 θ sin ϕ cos ϕ + 2 α γ sin θ cos θ cos ϕ + β 2 ( 1 + sin 2 θ sin 2 ϕ ) + 2 β γ sin θ cos θ sin ϕ + γ 2 ( 1 + cos 2 θ ) } , I 2 = 4 sin θ [ α sin θ cos ϕ + β sin θ sin ϕ + γ cos θ ] .
F d S 1 d S 2 = σ ρ R 2 sin θ | ( α sin θ cos ϕ + β sin θ sin ϕ + γ cos θ ) ( a ξ + b η + ζ ) | ( 1 + a 2 + b 2 ) 1 2 | I 0 + r 2 I 1 + ( r 2 ) 2 I 2 | .
X ( θ , ϕ ) = { R sin θ cos ϕ [ 1 + b ( η / ζ ) ] + ( ξ / ζ ) [ p R cos θ b R sin θ sin ϕ ] } [ 1 + a ( ξ / ζ ) + b ( η / ζ ) ] 1 ,
Y ( θ , ϕ ) = { R sin θ sin ϕ [ 1 + a ( ξ / ζ ) ] + ( η / ζ ) [ p R cos θ a R sin θ sin ϕ ] } [ 1 + a ( ξ ζ ) + b ( η / ζ ) ] 1 ,
Z ( θ , ϕ ) = p a X ( θ , ϕ ) b Y ( θ , ϕ ) .
F d S 1 d S 2 = 2 ρ σ 0 | sin 3 θ cos 3 θ sin 3 ϕ | | sin 2 θ cos 2 θ sin 2 ϕ + cos θ [ ( p / R ) cos θ ] ( 1 + sin 2 θ sin 2 ϕ ) + [ ( p / R ) cos θ ] 2 | .
X ( θ , ϕ ( θ ) ) = p sin θ cos ϕ ( θ ) / cos θ , Y ( θ , ϕ ( θ ) ) = [ p ( 2 sin 2 θ sin 2 ϕ ( θ ) 1 ) + R cos θ ] 2 sin θ cos θ sin ϕ ( θ ) , Z = p ,
A = { γ sin θ cos θ cos ϕ sin θ cos ϕ [ 1 γ 2 sin 2 θ ] 1 2 } I + { γ sin θ cos θ sin ϕ sin θ sin ϕ [ 1 γ 2 sin 2 θ ] 1 2 } J + { γ sin 2 θ cos θ [ 1 γ 2 sin 2 θ ] 1 2 } K ,
I 0 = R 2 sin θ cos ϕ s , I 1 = R sin θ ( 1 + cos 2 ϕ s ) ( 1 + γ cos θ / cos ϕ s ) , I 2 = sin θ ( γ cos θ + cos ϕ s ) 2 / cos ϕ s , cos ψ = cos θ cos ϕ s γ sin 2 θ , cos ϕ s = ( 1 γ 2 sin 2 θ ) 1 2 .
F d S 1 d S 2 = ρ σ R 2 sin θ cos θ | cos θ cos ϕ s γ sin 2 θ | | I 0 + ( p R cos θ ) I 1 / ζ + ( p R cos θ ) 2 I 2 / ζ 2 | .