Abstract

The flux distribution resulting from the light from an emitting surface being scattered by reflection from an irregular mirror surface is formulated. The particular case of the limb-darkened sun, a parabolic mirror, and a plane focal surface are analyzed using an example of geometrical scattering that occurs for thin metal optical surfaces possessing a show-through due to honeycomb backing. A matrix algorithm for computing the flux distribution is developed.

© 1966 Optical Society of America

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References

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  1. W. Graustein, Differential Geometry (The Macmillan Company, Inc., New York, 1931), Chap. 4.
  2. L. Mirsky, Introduction to Linear Algebra (Oxford Press, London, 1955), Chap. 3.
  3. G. Kuiper, Editor, The Sun (University of Chicago Press, Chicago, 1953), p. 99.

Graustein, W.

W. Graustein, Differential Geometry (The Macmillan Company, Inc., New York, 1931), Chap. 4.

Mirsky, L.

L. Mirsky, Introduction to Linear Algebra (Oxford Press, London, 1955), Chap. 3.

Other (3)

W. Graustein, Differential Geometry (The Macmillan Company, Inc., New York, 1931), Chap. 4.

L. Mirsky, Introduction to Linear Algebra (Oxford Press, London, 1955), Chap. 3.

G. Kuiper, Editor, The Sun (University of Chicago Press, Chicago, 1953), p. 99.

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Figures (5)

F. 1
F. 1

Radiance (solar) matrix.

F. 2
F. 2

Scattering matrix.

F. 3
F. 3

Image matrix.

F. 4
F. 4

Pseudosun for various tm/R.

F. 5
F. 5

Focal-plane flux distribution measured in relative units.

Equations (37)

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ϕ 1 ( u 1 , v 1 , α 1 , β 1 ) d A 1 ,
ϕ 1 ( u 1 , v 1 , α 1 , β 1 ) d A 1 ( d A 2 / r 2 ) cos α 2 ,
ϕ 2 ( u 1 , v 1 , u 2 , v 2 , u 3 , v 3 ) ϕ 1 ( u 1 , v 1 , α 1 , β 1 ) d A 1 ( d A 2 / r 2 ) cos α 2
d E ( u 1 , v 1 , u 2 , v 2 , u 3 , v 3 ) = ϕ 1 ϕ 2 d A 1 ( d A 2 / r 2 ) × cos α 2 ( d A 3 / r 2 ) cos α 3 ,
( 2 / A 1 A 2 ) ( E / A 3 ) = ϕ 1 ϕ 2 cos α 2 cos α 3 ( r 1 r 2 ) 2 .
F ( u 3 , v 3 ) = u 1 v 2 u 2 v 2 ϕ 1 ϕ 2 r 1 2 × D 1 d u 1 r 2 2 D 2 d u 2 d v 2 cos α 2 cos α 3 ,
d A 1 = D 1 ( u 1 , v 1 ) d u 1 d v 1 d A 2 = D 2 ( u 2 , v 2 ) d u 2 d v 2 .
lim Δ F Δ A 2 0 = lim Δ A 2 0 Δ A 2 S 1 ϕ 1 ϕ 2 cos α 2 × cos α 3 ( d A 1 / r 1 2 ) ( d A 2 / r 2 2 ) .
lim Δ A 2 0 Δ F r 2 2 cos α 2 Δ A 2 = S 1 ϕ 1 ϕ 2 d ω 1 ,
ϕ 1 = ϕ 1 cos α 1 , d ω 1 = r 1 2 cos α 1 d A 1 .
ψ = F ω = S 1 ϕ 1 ϕ 2 d ω 1 .
ψ ( ρ ) = r = 0 R θ = 0 2 π ϕ 1 ( r ) ϕ 2 ( r 2 + ρ 2 2 r ρ cos θ ) 1 2 rdrd θ ,
A = ( Φ i j ) , B = ( Φ i j ) , C = ( ψ i j ) ,
Φ i j = ϕ i j Δ ω .
ψ i 3 = α { 1 , i 3 i 2 , i 3 α = { i 3 [ ϕ ( i + 1 α ) 3 ϕ ( i + 1 α ) 2 ϕ ( i + 1 α ) 1 ] T [ Φ 1 Φ 2 Φ 3 ] ; i = 1 , , 5 .
ψ i 3 = α = { 1 , i 3 i 2 , i 3 α = { i 3 B ( i + 1 α ) [ A α * ] T ,
a = ( A 3 3 O 3 2 O 2 3 O 2 2 ) , B = ( B 3 3 O 3 2 O 2 3 O 2 2 ) ,
ψ i 3 = α = { 1 , i 3 i 2 , i 3 α = { i 3 B ( i + 1 α ) * [ a α * ] T .
ψ i n = α = 1 i B ( i + 1 α ) * [ a α * ] T ; i = 1 , , n .
B ( i + 1 α ) * [ a α * ] T = B ( i + 1 α ) [ a T ] * α = ( Ba T ) ( i + 1 α ) α = { [ ( Ba ) T ] T } ( i + 1 α ) α = { [ aB T ] T } ( i + 1 α ) α = { [ aB ] T } ( i + 1 α ) α = ( aB ) α ( i + 1 α ) ,
ψ i n = α = 1 n ( aB ) α ( i + 1 α ) ; i = 1 , , u .
aB = ( A n n O n n 1 O n 1 n O n 1 n 1 ) ( B n n O n n 1 O n 1 n O n 1 n 1 ) = ( A B O n n 1 O n 1 n O n 1 n 1 ) .
ψ i n = α = 1 n ( A B ) α ( i + 1 α ) ; i = 1 , , n ,
ψ i n = T r [ C ¯ i ] ,
ρ = ( t m + R ) i / ( n 1 2 ) ,
ϕ 2 Δ ω = F ( u 3 , v 3 ) / 2 π I r d ω ,
z = ( b / 2 ) cos ( 2 π x / λ ) , ( λ / 4 ) x ( λ / 4 ) ,
t = 2 r 2 ( d z / d x ) = b r 2 [ sin ( 2 π x / λ ) ] .
P ( x ) = [ x 2 / ( λ / 4 ) 2 ] P 0 .
d P / d t = ( d P / d x ) / ( d t / d x )
d P d t = 8 P 0 π 2 b r 2 · x cos ( 2 π x / λ ) = 8 π 2 sin 1 ( t / t m ) t m [ 1 ( t / t m ) 2 ] 1 2 ,
t m = 2 π r 2 ( b / λ ) .
d P d A 3 = d P 2 π tdt = 4 π 3 sin 1 ( t / t m ) t t m [ 1 ( t / t m ) 2 ] 1 2 P 0 .
ϕ ( t ) = 4 π 3 sin 1 ( t / t m ) t t m [ 1 ( t / t m ) 2 ] 1 2 .
ρ = 0 t m + R θ = 0 2 π ψ ( ρ ) ρ d ρ d θ
A 3 F d A 3
ξ max f ( 1 1 cos γ 1 + cos γ ) sin ρ sec γ cos ( γ + ρ ) .